10169:"It is obvious that this middle part, dotted in our curves , can have no physical reality. In fact, let us imagine the fluid in a state corresponding to this part of the curve contained in a heat conducting vertical cylinder whose top is formed by a piston. The piston can slide up and down in the cylinder, and we put on it a load exactly balancing the pressure of the gas. If we take a little weight off the piston, there will no longer be equilibrium and it will begin to move upward. However, as it moves the volume of the gas increases and with it its pressure. The resultant force on the piston gets larger, retaining its upward direction. The piston will, therefore, continue to move and the gas to expand until it reaches the state represented by the maximum of the isotherm. Vice versa, if we add ever so little to the load of the balanced piston, the gas will collapse to the state corresponding to the minimum of the isotherm"
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of the particles do not feel that the container is of finite size. Given the homogeneity of the fluid, the bulk of the particles do not experience a net force pulling them to the right or to the left. This is different for the particles in surface layers directly adjacent to the walls. They feel a net force from the bulk particles pulling them into the container, because this force is not compensated by particles on the side where the wall is (another assumption here is that there is no interaction between walls and particles, which is not true, as can be seen from the phenomenon of droplet formation; most types of liquid show adhesion). This net force decreases the force exerted onto the wall by the particles in the surface layer. The net force on a surface particle, pulling it into the container, is proportional to the number density. On considering one mole of gas, the number of particles will be
3541:. Andrews called the isotherm at which the jump just disappeared the critical point. Given the similarity of the titles of this paper and van der Waals subsequent thesis one might think that van der Waals set out to develop a theoretical explanation of Andrews' experiments. However the opposite is true, van der Waals began work by trying to determine a mollecular attraction that appeared in Laplace's theory of capillarity, and only after establishing his equation tested it using Andrews results.
3470:. This led, in 1873, to a dissertation that provided a simple, particle based, equation that described the gas–liquid change of state, the origin of a critical temperature, and the concept of corresponding states. The equation is based on two premises, first that fluids are composed of particles with non-zero volumes, and second that at a large enough distance each particle exerts an attractive force on all other particles in its vicinity. These forces were called by Boltzmann
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13235:. The values of all other property discontinuities across the saturation curve also follow from this solution. These functions define the coexistence curve which is the locus of the saturated liquid and saturated vapor states of the vdW fluid. The curve is plotted in Fig. 1 and Fig. 2, two projections of the state surface. These curves agree exactly with the numerical results referenced earlier.
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13332:. Moreover, it has now become clear that these metastable states occur regularly in the phase transition process. In particular processes that involve very high heat fluxes create large numbers of these states, and transition to their stable alternative with a corresponding release of energy can be dangerous. Consequently there is a pressing need to study their thermal properties.
26217:. In order to derive the Van der Waals equation we assume now that each particle moves independently in an average potential field offered by the other particles. The averaging over the particles is easy because we will assume that the particle density of the Van der Waals fluid is homogeneous. The interaction between a pair of particles, which are hard spheres, is taken to be:
22202:
2117:. The orange isobar is the critical one on which the minimum and maximum are equal. The black isobar is the limit of positive pressures, although drawn solid none of its points represent stable solutions, they are either metastable (positive or zero slope) or unstable (negative slope. All this is a good explanation of the observed behavior of fluids.
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21750:. The double tangent line is rendered solid between its saturated liquid and vapor values to indicate that states on it are stable, as opposed to the metastable and unstable states, above it (with larger Helmholtz free energy), but black, not green, to indicate that these states are heterogeneous, not homogeneous solutions of the vdW equation.
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16375:. Note from Fig. 5 that there is an overlap between the saturation curve and the inversion curve plotted there. This region is shown enlarged in the right hand graph of the figure. Therefore a van der Waals gas can be liquified by passing it through a throttle under the proper conditions; real gases are liquified in this way.
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about the various types of phase equilibria that are possible in fluid mixtures, complete solutions to the problem were only obtained after 1967, when the availability of modern computers made calculations of mathematical problems of this complexity feasible for the first time. The results obtained were, in
Rowlinson's words,
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1932:, superheated liquid, and those below it, subcooled vapor, are metastable; a sufficiently strong disturbance causes them to transform to the stable alternative. Consequently they are shown dashed. Finally the points in the region of negative slope are unstable. All this describes a fluid as a stable gas for
17562:. Also shown are the spinodal and coexistence curves described previously. The subcritical isotherm consists of stable, metastable, and unstable segments, and are identified the same as they were in Fig. 1. Also included are the zero initial slope isotherm and the one corresponding to infinite temperature.
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of the title equation. One is the conventional derivation that goes back to Van der Waals, a mechanical equation of state that cannot be used to specify all thermodynamic functions; the other is a statistical mechanics derivation that makes explicit the intermolecular potential neglected in the first
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direction. Therefore, there can be two liquid phases that can be either miscible, or wholly or partially immiscible, as well as a vapor phase. Despite a great deal of both theoretical and experimental work on this problem by van der Waals and his successors, work which produced much useful knowledge
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is the mole fraction of the vapor. This equation is called the lever rule and applies to other properties as well. The states it represents form a horizontal line connecting the same colored dots on an isotherm, but not shown in Fig. 1 as noted already since it is a distinct equation of state for the
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are vapor; due to gravity the denser liquid lies below the vapor. The transition points, states with zero slope, are called spinodal points. Their locus is the spinodal curve that separates the regions of the plane for which liquid, vapor, and gas exist from a region where no observable states exist.
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All this labor required considerable faith in the belief that gas–liquid systems were all basically the same, even if no one had ever seen the liquid phase. This faith arose out of the repeated success of the van der Waals theory, which is essentially a universal equation of state, independent of the
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particles. This is a mean field approximation; the position of the particles is averaged. In reality the density close to the particle is different than far away as can be described by a pair correlation function. Furthermore, it is neglected that the fluid is enclosed between walls. Performing the
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Next, we introduce a (not necessarily pairwise) attractive force between the particles. Van der Waals assumed that, notwithstanding the existence of this force, the density of the fluid is homogeneous; furthermore, he assumed that the range of the attractive force is so small that the great majority
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Figure 5: Curves of constant enthalpy in this plane have negative slope above this (green) inversion curve, positive slope below it and zero slope on it; they are S-shaped. A gas entering a throttle at a state corresponding to a point on this curve to the right of its maximum will cool if the final
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as an example, it consists of the two separate green segments. The solid segment above the green circle on the left, and below the one on the right correspond to stable states, the dots represent the saturated liquid and vapor states that comprise the phase change, and the two green dotted segments
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shown in Fig. 1) is the origin of the phase change, the spinodal points do not represent its full extent, because both states, saturated liquid and saturated vapor coexist in equlilbrium; they both must have the same pressure as well as the same temperature. Thus the phase change is characterized,
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in his classic thermodynamics textbook began his discussion of the van der Waals equation by writing, "In spite of its simplicity, it comprehends both the gaseous and the liquid state and brings out, in a most remarkable way, all the phenomena pertaining to the continuity of these two states". Also
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All the constants characterizing the gas have dropped out of this equation. If one bases measurements on the van der Waals units , then he obtains the same equation of state for all gases. ... Only the values of the critical volume, pressure, and temperature depend on the nature of the particular
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As noted previously, modern day studies of first order phase changes make use of the van der Waals equation together with the Gibbs criterion, equal chemical potential of each phase, as a model of the phenomenon. This model has an analytic coexistence (saturation) curve expressed parametrically,
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a constant of proportionality. Hence temperature was proportional to the average kinetic energy of the particles. This article inspired further work based on the twin ideas that substances are composed of indivisible particles, and that heat is a consequence of the particle motion as governed by
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a spectacular vindication of the essential physical correctness of the ideas behind the van der Waals equation, for almost every kind of critical behavior found in practice can be reproduced by the calculations, and the range of parameters that correlate with the different kinds of behavior are
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The equation also plays an important role in the modern theory of phase transitions. It depicts the liquid metals, Mercury and Cesium, quantitatively, and describes most real fluids qualitatively. Consequently it can be regarded as one member of a family of equations of state, that depend on a
5708:. Recent research has suggested that there is a family of equations of state that depend on an additional dimensionless group, and this provides a more exact correlation of properties. Nevertheless, as Boltzmann observed, the van der Waals equation provides an essentially correct description.
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18858:, which is approximately the close packing distance for hard spheres. This was one of the first of many equations of state proposed over the years that attempted to make quantitative improvements to the remarkably accurate explanations of real gas behavior produced by the vdW equation.
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substance; the numbers that express the actual volume, pressure, and temperature as multiples of the critical values satisfy the same equation for all substances. In other words, the same equation relates the reduced volume, reduced pressure, and reduced temperature for all substances.
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The number of particles in the surface layers is, again by assuming homogeneity, also proportional to the density. In total, the force on the walls is decreased by a factor proportional to the square of the density, and the pressure (force per unit surface) is decreased by:
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Using these expressions in the vdW equation is apparently helpful for divers, as well as being important for physicists, physical chemists, and chemical engineers in their study and management of the various phase equilibria and critical behavior observed in fluid mixtures.
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as a parameter, one obtains the generalized compressibility chart for a vdW gas, which is shown in Fig. 7. Like all other vdW properties, this is not quantitatively correct for most gases but it has the correct qualitative features as can be seen by comparison with this
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22416:{\displaystyle {\frac {\partial ^{2}f}{\partial v^{2}}}>0\qquad {\frac {\partial ^{2}f}{\partial x^{2}}}>0\qquad {\frac {\partial ^{2}f}{\partial v^{2}}}{\frac {\partial ^{2}f}{\partial x^{2}}}-\left({\frac {\partial ^{2}f}{\partial x\partial v}}\right)^{2}>0}
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In 1890 van der Waals published an article that initiated the study of fluid mixtures. It was subsequently included as Part III of a later published version of his thesis. His essential idea was that in a binary mixture of vdw fluids described by the equations
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individually, which are just the Gibbs conditions for material equilibrium in this situation. Although this case is similar to the previous one of a single component, here the geometry can be much more complex. The surface can develop a wave (called a
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are usually lower. Of course, molecules are not infinitely hard, as Van der Waals thought, and are often fairly soft. To obtain the excluded volume per mole we just need to multiply by the number of molecules in a mole, i.e. by the
Avogadro number:
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is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. To see this, we must realize that a particle is surrounded by a sphere of radius
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This is the energetic equation of state for a perfect vdW fluid. By making a dimensional analysis (what might be called extending the principle of corresponding states to other thermodynamic properties) it can be written simply in reduced form as
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state is below the curve. The other (dashed purple) curve in the graph is the saturation curve. The graph on the right is the square (0,0),(1.1,1.1) of the left graph expanded to display the overlap between the inversion and saturation curves.
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5910:, defined by the first and second laws of thermodynamics, hence all thermodynamic properties of a simple compressible substance, can be specified, up to a constant of integration, by two measurable functions, a mechanical equation of state,
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5838:. Thus most real fluids do not satisfy this condition, and consequently their behavior is only described qualitatively by the vdW equation. However, the vdW equation of state is a member of a family of state equations based on the Pitzer (
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23040:. As in the case of a single substance, here the plane and the surface can have a double tangent and the locus of the coexisting phase points forms a curve on each surface. The coexistence conditions are that the two phases have the same
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11339:, so another equation is required in order to specify the values of 3 of these variables uniquely in terms of a fourth. Such an equation is provided here by the equality of the Gibbs free energy in the saturated liquid and vapor states,
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derivation. A particular advantage of the statistical mechanical derivation is that it yields the partition function for the system, and allows all thermodynamic functions to be specified (including the mechanical equation of state).
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details of any particular substance once it has been properly scaled. ... As a result, not only was it possible to believe that hydrogen could be liquefied. but it was even possible to predict the necessary temperature and pressure.
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3991:, that was first obtained by Plank, known to Gibbs, and later derived in a beautifully simple and elegant manner by Lekner. A summary of Lekner's solution is presented in a subsequent section, and a more complete discussion in the
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the curvature decreases to a minimum then increases to another spinodal point; between these two spinodal points is the unstable region in which the fluid cannot exist in a homogeneous equilibrium state. With a further increase in
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Van der Waals wrote this relation, but did not make use of it initially. However, it has been used frequently in subsequent studies, and its use is said to produce good agreement with experimental results at high pressure.
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Van der Waals introduced the
Helmholtz function because its properties could be easily extended to the binary fluid situation. In a binary mixture of vdW fluids the Helmholtz potential is a function of 2 variables,
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Figure 3: The family of saturation curves showing the vdw curve as a member. The blue dots are calculated from Lekner's solution. The orange dots are calculated from data in the ASME Steam Tables
Compact Edition,
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25028:, namely for molecules of not too different diameters. In fact Rowlinson said of this approximation, "It was, and indeed still is, hard to improve on the original van der Waals recipe when expressed in form".
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many years later who wrote "The van der Waals equation predicts that at low temperatures liquids sustain enormous tension...In recent years measurements have been made that reveal this to be entirely correct."
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the first of which follows from a simple interpretation of the dispersion forces in terms of polarizabilities of the individual molecules while the second is exact for rigid molecules. Then, generalizing for
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no matter how high the pressure), and other attributes. These predictions are accurate for only a few substances. For most simple fluids they are only a valuable approximation. The equation also explains why
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which is the ideal gas law. This is not surprising since the van der Waals equation was constructed from the ideal gas equation in order to obtain an equation valid beyond the limit of ideal gas behavior.
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which is the van der Waals approximation expressed in terms of the intermolecular constants. This approximation, when compared with computer simulations for mixtures, are in good agreement over the range
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as given by this equation. The ordinate is logarithmic in order to show the behavior at pressures far below the critical where differences among the various substances (indicated by varying values of
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In his treatise of 1898 in which he described the van der Waals equation in great detail
Boltzmann discussed these states in a section titled "Undercooling, Delayed evaporation"; they are now denoted
12196:. Using the equation for the Gibbs free energy obtained previously for the vdW equation applied to the saturated vapor state and subtracting the result applied to the saturated liquid state produces,
4010:
Figure 1: Four isotherms of the van der Waals equation along with the black dash dot spinodal curve and the red dash dot coexistence (saturation) curve plotted using reduced (dimensionless) variables.
2496:", also wrote "It is very remarkable that the theory due to van der Waals is in a position to predict, at least qualitatively, the unstable states" that are associated with the phase change process.
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3556:, had been opened. The van der Waals equation played a part in all this especially with respect to the liquefaction of hydrogen and helium which was finally achieved in 1908. From measurements of
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is the one shown in Fig. 8 along with the line tangent to it at its two coexisting saturation points. The data illustrated in Fig. 8 is exactly the same as that shown in Fig.1 for this isotherm.
10825:{\displaystyle T_{\rm {sp}}=2a{\frac {(v-b)^{2}}{Rv^{3}}}=T_{\text{c}}{\frac {(3v_{r}-1)^{2}}{4v_{r}^{3}}}\qquad p_{\rm {sp}}={\frac {a(v-2b)}{v^{3}}}=p_{\text{c}}{\frac {(3v_{r}-2)}{v_{r}^{3}}}}
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Obviously such a broad general relation is unlikely to be correct; nevertheless, the fact that one can obtain from it an essentially correct description of actual phenomena is very remarkable.
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12476:, and tabular results presented; however, the equations also admit an analytic parametric solution obtained most simply and elegantly, by Lekner. Details of this solution may be found in the
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are close to the vdW value. However, it describes the behavior of other fluids only qualitatively, because specific numerical values are modified by differing values of their Pitzer factor,
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27346:{\displaystyle p={\frac {NkT}{V-Nb'}}-{\frac {N^{2}a'}{V^{2}}}\Rightarrow \left(p+{\frac {N^{2}a'}{V^{2}}}\right)(V-Nb')=NkT\Rightarrow \left(p+{\frac {n^{2}a}{V^{2}}}\right)(V-nb)=nRT.}
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All this makes it a worthwhile pedagogical tool for physics, chemistry, and engineering lecturers, in addition to being a useful mathematical model which can aid student understanding.
1804:, shown as green circles that designate the coexisting boiling liquid and condensing gas respectively. Heating the fluid in this state increases the fraction of gas in the mixture; its
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23934:{\displaystyle a_{x}=\sum _{i=1}^{n}\sum _{j=1}^{n}(a_{ii}a_{jj})^{1/2}x_{i}x_{j}\quad {\mbox{which can be written as}}\quad a_{x}={\Big (}\sum _{i=1}^{n}a_{ii}^{1/2}x_{i}{\Big )}^{2}}
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remains the same. This is shown as the dotted gray line, because it does not represent a solution of the equation; however, it does describe the observed behavior. The points above
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This environment influenced
Johannes Diderik van der Waals. After initially pursuing a teaching credential, he was accepted for doctoral studies at the University of Leiden under
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Another method of specifying the vdW constants pioneered by W.B. Kay, and known as Kay's rule. specifies the effective critical temperature and pressure of the fluid mixture by
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13400:, there is no physical discontinuity. As the liquid begins to vaporize the fluid becomes a heterogeneous mixture of liquid and vapor whose molar volume varies continuously from
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12912:{\displaystyle v_{rf}=\left({\frac {1}{3}}\right){\frac {1+f(y)e^{y}}{f(y)e^{y}}}\qquad \qquad \qquad \quad v_{rg}=\left({\frac {1}{3}}\right){\frac {1+f(y)e^{-y}}{f(y)e^{-y}}}}
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13744:, namely all substances have the same dimensionless coexistence curve. In order to avoid this paradox an extended principle of corresponding states has been suggested in which
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which was produced from data using real gases. The two graphs are similar, including the caustic generated by the crossing isotherms; they are qualitatively very much alike.
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represent collisions between one molecule of each of the two different component fluids. This idea of van der Waals was later called a one fluid model of mixture behavior.
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997:, is used by physicists, but there is no essential difference between equations written with any of these properties. Equations of state written using molar volume contain
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25858:{\displaystyle p={\frac {RT}{V_{\mathrm {m} }-b}}-{\frac {a}{V_{\mathrm {m} }^{2}}}\Rightarrow \left(p+{\frac {a}{V_{\mathrm {m} }^{2}}}\right)(V_{\mathrm {m} }-b)=RT.}
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16579:, called the compressibility factor, is usually expressed either as a function of pressure and temperature, or density and temperature, and in each case in the limit
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8456:{\displaystyle {\frac {\partial p}{\partial T}}={\frac {R}{v-b}}\quad {\mbox{and}}\quad {\frac {\partial p}{\partial v}}=-{\frac {RT}{(v-b)^{2}}}+{\frac {2a}{v^{3}}}}
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Figure 1 shows four isotherms of the van der Waals equation (abbreviated as vdW) on a pressure, molar volume plane. The essential character of these curves is that:
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exp to take care of the average intermolecular potential. We divide here the potential by two because this interaction energy is shared between two particles. Thus:
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Figure 2: The dashed dot black curve is the stability limit (spinodal curve) and the dashed dot blue curve is the coexistence, or saturation curve, plotted in the
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In this article van der Waals used the
Helmholtz Potential Minimum Principle to establish the conditions of stability. This principle states that in a system in
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26342:{\displaystyle u(r)={\begin{cases}\infty &{\hbox{when}}\quad r<d,\\-\epsilon \left({\frac {d}{r}}\right)^{6}&{\hbox{when}}\quad r\geq d,\end{cases}}}
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The Gibbs free energy is one of the 4 thermodynamic potentials whose partial derivatives produce all other thermodynamics state properties; its differential is
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2372:
2253:
2143:
2115:
1822:
1419:
1331:
1311:
1148:
1128:
1105:
1015:
853:
613:
589:
500:
480:
453:
433:
409:
306:
282:
179:
155:
26022:
23548:
the geometric and algebraic means respectively can be used. These relations correspond to the empirical combining laws for the intermolecular force constants,
13685:. However, as Boltzmann noted, such a simple representation could not correctly describe all substances. Indeed, the saturation analysis of this form produces
9628:
22498:
22495:
are both positive for stable states while the third condition indicates that stable states correspond to elliptic points on this surface. Moreover its limit,
25455:
is four times the proper volume of the particle. It was a point of concern to Van der Waals that the factor four yields an upper bound; empirical values for
17733:
16793:
10832:
A projection of this space curve is plotted in Fig. 1 as the black dash dot curve. It passes through the critical point which is also a spinodal point.
5112:
3903:
Van der Waals was awarded the Nobel Prize in 1910, in recognition of the contribution of his formulation of this "equation of state for gases and liquids".
71:. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One way to write this equation is:
3537:, that showed at low temperatures a jump in density at a certain pressure, while at higher temperatures there was no abrupt change; the figure can be seen
25049:
Consider one mole of gas composed of non-interacting point particles that satisfy the ideal gas law: (see any standard
Physical Chemistry text, op. cit.)
20231:
8662:, the coefficient of thermal expansion, is a measure of the relative increase of volume from an increase of temperature, at constant pressure. Therefore
17699:
13457:
26382:
Because the particles are not coupled under the mean field
Hamiltonian, the mean field approximation of the total partition function still factorizes:
19336:{\displaystyle a_{x}=a_{11}x_{1}^{2}+2a_{12}x_{1}x_{2}+a_{22}x_{2}^{2}\quad {\mbox{and}}\quad b_{x}=b_{11}x_{1}^{2}+2b_{12}x_{1}x_{2}+b_{22}x_{2}^{2}}
14875:
is positive the gas temperature decreases when it passes through a throttle, and if it is negative the temperature increases. Therefore the condition
25113:). The effect of the finite volume of the particles is to decrease the available void space in which the particles are free to move. We must replace
13325:
13318:
13127:
4155:
23376:, must also be specified. In he absence of experimental data, or computer modelling results to estimate their value, the empirical combining laws,
19036:
11266:{\displaystyle p_{s}={\frac {RT_{s}}{v_{f}-b}}-{\frac {a}{v_{f}^{2}}}\quad {\mbox{and}}\quad p_{s}={\frac {RT_{s}}{v_{g}-b}}-{\frac {a}{v_{g}^{2}}}}
9838:
4996:
13832:
is a substance dependent dimensionless parameter related to the only physical feature associated with an individual substance, its critical point.
8870:{\displaystyle \kappa _{T}={\frac {v^{2}(v-b)^{2}}{RTv^{3}-2a(v-b)^{2}}}\quad {\mbox{and}}\quad \alpha ={\frac {Rv^{2}(v-b)}{RTv^{3}-2a(v-b)^{2}}}}
4521:
25346:
24945:{\displaystyle \sigma _{x}^{3}=\sum _{i=i}^{n}\sum _{j=1}^{n}\,\sigma _{ij}^{3}x_{i}x_{j}\qquad {\mbox{and}}\qquad \epsilon _{x}=\left\left^{-1}}
24416:
and Kay used these specifications of the mixture critical constants as the basis for calculations of the thermodynamic properties of mixtures.
19026:{\displaystyle p_{1}={\frac {RT}{v-b_{11}}}-{\frac {a_{11}}{v^{2}}}\quad {\mbox{and}}\quad p_{2}={\frac {RT}{v-b_{22}}}-{\frac {a_{22}}{v^{2}}}}
9064:. Both of these are the ideal gas values, which is consistent because, as noted earlier, the vdW fluid behaves like an ideal gas in this limit.
8606:, the isothermal compressibility, is a measure of the relative increase of volume from an increase of pressure, at constant temperature, while
5714:
28569:
26623:
19697:
are a consequence of the forces between molecules. This was first shown by
Lorentz, and was credited to him by van der Waals. The quantities
14784:
14778:, do not change temperature in such a process, real gases do, and it is important in applications to know whether they heat up or cool down.
27544:
Lectures on Theoretical Physics Volume V, Edited by Bopp, F. and Meixner, J., Translated by Kestin, J., Academic Press, NY (1956), pp 55, 66
12422:
This is a third equation that along with the two vdW equations above can be solved numerically. This has been done given a value for either
7927:
6874:
1712:
The thermodynamic requirements of mechanical, thermal, and material equilibrium together with the equation specify two points on the curve,
74:
13858:
13335:
In the same section Boltzmann also addressed and explained the negative pressures which some liquid metastable states exhibit (for example
13122:
plane. At every point in the region between the two curves there are two states, one stable and another metastable. The metastable states,
2938:
28513:
26518:
11555:
18402:
14078:. Several investigators have produced correlations of saturation data for a number of substances, the best is that of Dong and Lienhard,
615:
denotes an excluded molar volume due to a finite molecule size, since the centers of two molecules cannot be closer than their diameter,
23674:{\displaystyle \epsilon _{12}=(\epsilon _{11}\epsilon _{22})^{1/2}\qquad {\mbox{and}}\qquad \sigma _{12}=(\sigma _{11}+\sigma _{22})/2,}
15624:
25999:
25891:
21077:
18493:, which illustrates the effect of the excluded volume alone. It was recognized early on that this was in error beginning with the term
13321:, that are created in the process of phase transition, have a short lifetime, then devolve into their lower energy stable alternative.
4930:{\displaystyle b+RT_{\text{c}}/p_{\text{c}}=3v_{\text{c}}\quad a/p_{\text{c}}=3v_{\text{c}}^{2}\quad ab/p_{\text{c}}=v_{\text{c}}^{3}}
3119:
17141:
16524:
Real gases can be characterized by their difference from ideal. This is done by writing the mechanical equation of state in the form
12477:
3992:
3737:
41:
and the interactions between them (both of which depend on the specific substance). As a result the equation is able to model the
28720:
28700:
26359:
is the distance where the hard spheres touch each other (twice the Van der Waals radius). The depth of the Van der Waals well is
24956:
2930:
and associated inversion curve, which were instrumental in the commercial liquefaction of gases have simple analytic expressions.
28411:
26129:
25243:
24409:{\displaystyle a_{x}={\frac {27}{64}}{\frac {(RT_{cx})^{2}}{p_{cx}}}\qquad \qquad b_{x}={\frac {1}{8}}{\frac {RT_{cx}}{p_{cx}}}}
11931:{\displaystyle g_{g}-g_{f}=\sum _{j=1}^{3}\int _{p_{j}}^{p_{j}+1}\,v_{j}\,dp=-\int _{v_{f}}^{v_{g}}\,p\,dv+p_{s}(v_{g}-v_{f})=0}
20059:
27011:
21538:
15370:
7782:
27614:
The Volumetric and Thermodynamic Properties of Fluids. II. Compressibility Factor, Vapor Pressure and Entropy of Vaporization
10935:
10363:(shown dashed gray in Fig. 1), are unstable and thus not observed. This is the genesis of the phase change; there is a range
6354:
21477:
18585:
4940:
4369:
11276:
10366:
1650:
12702:{\displaystyle T_{rs}(y)=\left({\frac {27}{8}}\right){\frac {2f(y)}{g(y)^{2}}}\quad p_{rs}=27{\frac {f(y)^{2}}{g(y)^{2}}}}
2573:
28715:
27402:
27397:
25690:{\displaystyle a'C^{2}=a'\left({\frac {N_{\mathrm {A} }}{V_{\mathrm {m} }}}\right)^{2}={\frac {a}{V_{\mathrm {m} }^{2}}}}
25524:
25139:
14542:) is a member of the class of real fluids; indeed it quantitatively describes the behavior of the liquid metals cesium (
10469:
635:. A theoretical calculation of these constants for spherical molecules with an interparticle potential characterized by,
28563:
25055:
8550:
6459:
3639:
15729:
8609:
6588:
4790:, and noting that these two cubic equations are the same when all their coefficients are equal gives three equations
28662:
28434:
21697:; the region between this point and the second spinodal point is the metastable subcooled vapor. Finally, the region
10314:
10105:
7206:
6814:
4051:
2511:
21602:, is a spinodal point, and between these two points is the metastable superheated liquid. For further increases in
14222:{\displaystyle \ln p_{rs}=5.37270(1-1/T_{r})+\omega (7.49408-11.181777T_{r}^{3}+3.68769T_{r}^{6}+17.92998\ln T_{r})}
13380:
Even though the phase change produces a mathematical discontinuity in the homogeneous fluid properties, for example
2031:, that also supports metastable states of subcooled gas and superheated liquid. It is characteristic of all isobars
24514:
18322:
17138:); the fluid behaves like an ideal gas, a point noted several times earlier. Note additionally that the derivative
16878:
16092:
4103:
3910:
25226:, it would mean that the two particles penetrate each other, which, by definition, hard spheres are unable to do.
16387:
Figure 6: The isotherms, spinodal and coexistence curves here are the same as in Fig. 1. In addition the isotherm
15981:
15208:{\displaystyle {\frac {2a(v-b)^{2}-RTv^{2}b}{RTv^{3}-2a(v-b)^{2}}}=0\quad {\mbox{or}}\quad 2a(v-b)^{2}-RTv^{2}b=0}
14651:
13932:
11386:
7706:
5802:
796:
27392:
26841:{\displaystyle \phi =-2a'{\frac {N}{V}}\quad {\hbox{with}}\quad a'=\epsilon {\frac {2\pi d^{3}}{3}}=\epsilon b'.}
23259:
22753:
14016:
13929:, but because that quantity is difficult to measure accurately, the acentric factor developed by Kenneth Pitzer,
2660:
2646:
1553:
1497:
1155:
23541:{\displaystyle a_{12}=(a_{11}a_{22})^{1/2}\qquad {\mbox{and}}\qquad b_{12}^{1/3}=(b_{11}^{1/3}+b_{22}^{1/3})/2,}
17254:
16984:
16718:
14013:, is more useful. The saturation pressure in this situation is represented by a one parameter family of curves,
13747:
28681:
28604:
22609:
11456:
9097:
3367:
3219:
26:
25986:
It is of some historical interest to point out that Van der Waals, in his Nobel prize lecture, gave credit to
24261:{\displaystyle T_{cx}=\sum _{i=1}^{n}T_{ci}x_{i}\qquad {\mbox{and}}\qquad p_{cx}=\sum _{i=1}^{n}\,p_{ci}x_{i}}
19946:
9722:
26120:
23322:
must be known. In addition, the effect of collisions between molecules of the different components, given by
6964:
5435:
17617:
17062:
16602:
16464:
15815:
14481:
13623:
The idea of corresponding states originated when van der Waals cast his equation in the dimensionless form,
11052:
11004:
7650:
6755:
for a vdW fluid exactly as it is for an ideal gas. To keep things simple it is regarded as a constant here,
6351:
be continuous with continuous partial derivatives, its second mixed partial derivatives must also be equal,
4447:
3538:
2034:
28627:, Pasadena, California: Division of Physics, Mathematics, and Astronomy, California Institute of Technology
23702:
fluid components, and using these empirical combinig laws, the expressions for the material constants are:
15874:
15290:
13069:{\displaystyle f(y)={\frac {y\cosh y-\sinh y}{\sinh y\cosh y-y}}\qquad \qquad g(y)=1+2f(y)\cosh y+f(y)^{2}}
12039:); the result is a special case of material equilibrium. The last equality, which follows from integrating
9589:
5640:
1150:
are experimentally determined for a given substance, the van der Waals equation can be used to predict the
531:
24550:
23256:
In order to obtain these numerical results the values of the constants of the individual component fluids
20872:; there will be a double tangent. Furthermore, each of these points is characterized by the same value of
19600:
14967:
from the region where it is less than zero. This curve is called the inversion curve, and its equation is
13626:
13169:
9488:
8880:
8501:
6666:
6416:
5569:
28705:
27644:
The Collected Works of J. Willard Gibbs Volume II Part One Elementary Principles in Statistical Mechanics
19531:) are the mole fractions of the two fluid substances. Adding the equations for the two fluids shows that
16668:
14255:
13688:
11978:
integral must be divided into 3 parts corresponding to the 3 real roots of the vdW equation in the form,
11379:. This condition of material equilibrium can be obtained from a simple physical argument as follows: the
10532:
8466:
3333:
2852:
1609:
1460:
28551:
20956:
18240:
18001:
17919:
17568:
16332:
16286:
13137:
10428:
10175:
9991:
8939:
4498:
This calculation can be done algebraically by noting that the vdW equation can be written as a cubic in
4366:
Evaluating the two partial derivatives in 1) using the vdW equation and equating them to zero produces,
1424:
1372:
24422:
23083:
22956:
20336:
19534:
18532:
18237:
This is the virial expansion for the van der Waals fluid. The first virial coefficient is the slope of
16938:
12042:
10211:
7484:{\displaystyle h_{r}-{\mbox{C}}_{u}=T_{r}-9/(4v_{r})\quad {\mbox{where}}\quad h_{r}=h/(RT_{\text{c}}c)}
5512:
5248:
3478:
1968:
1935:
1248:
1215:
188:
25469:
21960:
21730:
is the vapor. In this region the curvature continually decreases until it is zero at infinitely large
15567:
7117:
2261:
51:
26455:
20611:
19779:
in these expressions characterize collisions between two molecules of the same fluid component while
19741:
19700:
11001:
that lies between that of the minimum and maximum spinodal points, and with molar volumes of liquid,
9026:
6241:{\displaystyle u-C_{u}=\int \,c_{v}(v,T)\,dT+\int \,\left\,dv=\int \,c_{v}(v,T)\,dT+\int \,\left\,dv}
3020:
1336:
926:
681:
354:
26247:
20678:
17968:
17496:
14545:
14516:
10529:
This spinodal curve is obtained here from the vdW equation by differentiation (or equivalently from
9560:
7604:{\displaystyle s-C_{s}=\int \,c_{v}(T)\,{\frac {dT}{T}}+\int \,{\frac {\partial p}{\partial T}}\,dv}
6709:
4018:
2001:
761:
311:
25340:
which, divided by two (the number of colliding particles), gives the excluded volume per particle:
22426:
21207:
20013:
19822:
19782:
19488:
19442:
19394:
19346:
16431:
15930:
14937:
14574:
13576:
10279:
10244:
9778:
9514:
5337:
4756:{\displaystyle (v-v_{\text{c}})^{3}=v^{3}-3v_{\text{c}}v^{2}+3v_{\text{c}}^{2}v-v_{\text{c}}^{3}=0}
2927:
1761:
1715:
1283:
can exist above their boiling point and subcooled vapors can exist below their condensation point.
658:
24110:{\displaystyle b_{x}=(1/8)\sum _{i=1}^{n}\sum _{j=1}^{n}(b_{ii}^{1/3}+b_{jj}^{1/3})^{3}x_{i}x_{j}}
22062:
21005:
20832:
20789:
18742:
18496:
14970:
12104:
5385:
4766:
2073:
1188:
1161:
1020:
793:, gives the excluded volume as 4 times the volume of all the molecules. This theory also produces
595:
pressure times molar volume squared, which is also molar energy times molar volume. The constant
505:
236:
24474:
21700:
21350:
21041:
20195:
18289:
18230:{\displaystyle Z(\rho _{r},T_{r})=1+^{-1}](\rho _{r}/3)+\sum _{k=2}^{\infty }\,(\rho _{r}/3)^{k}}
17324:
16177:
15527:
14706:
13085:
11690:
11650:
11342:
10069:
9811:
8905:
5254:
3890:, and from these values calculate the expected critical pressure, temperature, and molar volume.
3599:
3559:
3440:
858:
28505:
26388:
21308:
20747:
20569:
20484:
20406:
For a single substance the definition of the molar Gibbs free energy can be written in the form
17357:
16390:
14878:
13241:
5954:
1057:
26362:
22163:
22017:
21243:
20648:, the curve and the straight line will be have a single tangent. However, for a subcritical,
18792:
17673:
17538:
16217:
15255:
13338:
13274:
12016:
11727:, noting that the pressure is the same at each endpoint, and setting the result to zero yields
10866:
8278:{\displaystyle g_{r}-{\mbox{C}}_{u}=\{c+3v_{r}/(3v_{r}-1)-{\mbox{C}}_{s}-\ln\}T_{r}-9/(4v_{r})}
5301:, and 3 dimensions, , , , must be expressible in terms of 6 − 3 = 3 dimensionless groups. Here
5204:
This dimensionless form is a similarity relation; it indicates that all vdW fluids at the same
4302:
at lower temperatures there are two points on each isotherm where the slope is zero (values of
2316:
2148:
388:
28257:"On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions"
26102:
23177:
23150:
23123:
23016:
22886:
22859:
21950:{\displaystyle f_{r}=u_{r}-T_{r}s_{r}={\mbox{C}}_{u}+T_{r}\{c-{\mbox{C}}_{s}-\ln\}-9/(8v_{r})}
20228:, namely at equilibrium the Helmholtz potential is a minimimum. This leads to the requirement
17834:
14851:
14232:
12158:
11380:
7614:
6758:
5913:
1494:, has a physically unreal negative slope, hence shown dotted gray, between its local minimum,
28256:
27372:
22130:
20133:
15005:
14623:
14603:
14431:
14381:
14361:
10208:
this requirement is satisfied everywhere so all states are gas, those states on an isotherm,
9755:
6281:
5845:
5304:
3452:
638:
618:
23352:
23325:
21184:
20915:
20409:
19919:
19892:
19865:
16527:
15229:
14746:
14451:
14304:
11981:
11517:
8989:
7892:
7059:
6319:
2817:
2782:
2186:
28642:
27382:
25110:
22807:
21673:
21646:
21430:
21383:
20651:
19673:
19646:
17398:
16038:
15495:
14404:
14334:
13838:
13815:
13430:
13403:
12452:
12425:
12131:
12098:
10908:
9310:
9300:{\displaystyle c_{p}-c_{v}=-T(\partial _{T}p)^{2}/\partial _{v}p=Tv\alpha ^{2}/\kappa _{T}}
9152:
9070:
6251:
5207:
3448:
2755:
2668:
1908:
1881:
1854:
1827:
962:
592:
28615:
26995:{\displaystyle \ln Z=N\ln {(V-Nb')}+{\frac {N^{2}a'}{VkT}}-N\ln {(\Lambda ^{3})}-\ln {N!}}
23203:
20166:
17425:
16258:
10040:
7021:
3514:
3488:
1401:(although the plot only shows a finite quadrant). This describes a fluid as a gas for all
8:
26089:{\displaystyle Z={\frac {z^{N}}{N!}}\quad {\hbox{with}}\quad z={\frac {V}{\Lambda ^{3}}}}
24587:
24448:
18833:
18376:
17890:
17095:
16642:
15342:
14911:
12128:
be equal to the lower one. This form means that the thermodynamic restriction that fixes
9712:{\displaystyle \partial _{T}p=-\partial _{T}v/\partial _{p}v=\alpha /\kappa _{T}=\infty }
6560:
4630:. Moreover, at the critical point all three roots coalesce so it can also be written as
4256:
3471:
3444:
2480:
2212:
968:
894:
348:
28646:
28622:
22913:
22598:{\displaystyle \partial _{x^{2}}^{2}f\partial _{v^{2}}^{2}f-(\partial _{xv}^{2}f)^{2}=0}
22455:). The first two stability conditions show that the curvature in each of the directions
20526:
13609:
10102:, namely that at equilibrium the entropy is a maximum. This leads to a requirement that
7094:
27070:
26202:
25200:
23685:
23229:
23209:
23063:
23043:
22996:
22936:
22839:
22801:
22478:
22458:
22110:
21733:
21626:
21605:
21457:
21410:
21284:
20895:
20875:
20727:
20549:
20464:
20444:
20386:
19580:
17870:
17713:
17473:
17453:
17121:
16582:
16562:
16157:
16072:
13383:
13368:
of Fig. 1). He concluded that such liquid states of tensile stresses were real, as did
13329:
13314:
13123:
11961:
11941:
10843:
7186:
6794:
5893:
5873:
4501:
4345:
4325:
4305:
4282:
3873:
3853:
3421:
3313:
3293:
3273:
2735:
2715:
2695:
2570:
has a simple analytic expression [this is also true of its isothermal compressibility,
2381:
2357:
2238:
2128:
2100:
1807:
1404:
1316:
1296:
1133:
1113:
1090:
1000:
838:
598:
574:
485:
465:
438:
418:
412:
394:
291:
267:
164:
140:
28654:
28710:
28677:
28658:
28633:
Dalgarno, A.; Davison, W.D. (1966). "The Calculation of Van Der Waals Interactions".
28600:
28559:
28472:
Leland, T. W., Rowlinson, J.S., Sather, G.A., and Watson, I.D., Trans. Faraday Soc.,
28403:
25990:
for the argument that pressure is reduced proportional to the square of the density.
7886:
2485:
344:
261:
30:
24419:
Kay's idea was adopted by T. W. Leland, who applied it to the molecular parameters,
12101:
which requires that the upper area between the vdW curve and the horizontal through
9478:{\displaystyle c_{p}(v,T)-c_{v}(T)={\frac {R^{2}Tv^{3}}{RTv^{3}-2a(v-b)^{2}}}\geq R}
4253:
at higher temperatures the slope of the isotherms is everywhere negative (values of
3636:
in two states with the same density, the van der Waals equation produces the values
28650:
28271:
26509:
22743:{\displaystyle f=-pv+\mu _{1}x_{1}+\mu _{2}x_{2}=-pv+(\mu _{2}-\mu _{1})x+\mu _{1}}
20322:{\displaystyle \partial ^{2}f/\partial v^{2}|_{T}=-\partial p/\partial v|_{T}>0}
14781:
This coefficient can be found in terms of the previously described derivatives as,
13373:
9177:
7779:
This is the entropic equation of state for a perfect vdW fluid, or in reduced form
7297:
This is the enthalpic equation of state for a perfect vdW fluid, or in reduced form
3364:. He then noted that using the classical laws of Boyle and Charles one could write
2650:
888:
42:
27688:
The Bakerian Lecture: On the Continuity of the Gaseous and Liquid States of Matter
12415:{\displaystyle RT_{s}\left-2a\left({\frac {1}{v_{g}}}-{\frac {1}{v_{f}}}\right)=0}
3850:. Thus from two such measurements of pressure and temperature one could determine
1333:
calculated from the equation at four constant pressure values. On the red isobar,
458:
26190:
20380:
20127:
14703:, is of practical importance because the two end states of a throttling process (
13614:
5839:
3891:
3549:
3014:
2479:
What is truly remarkable is the extent to which van der Waals succeeded. Indeed,
920:
11097:. Then from the vdW equation applied to these saturated liquid and vapor states
2468:{\displaystyle p=RT/v\quad {\mbox{or in terms of }}V{\mbox{ and }}N\quad pV=NkT}
13369:
4243:{\displaystyle \partial p/\partial v|_{T}=\partial ^{2}p/\partial v^{2}|_{T}=0}
3010:
994:
9977:{\displaystyle T=2a(v-b)^{2}/(Rv^{3})=T_{\text{c}}(3v_{r}-1)^{2}/(4v_{r}^{3})}
5102:{\displaystyle p_{r}=p/p_{\text{c}},T_{r}=T/T_{\text{c}},v_{r}=v/v_{\text{c}}}
28694:
28046:
Generalized Saturation Properties of Pure Fluids Via Cubic Equations of State
27597:
The Assimilation of van der Waals Equation in the Corresponding States Family
22423:
and the Helmholtz potential is a surface (of physical interest in the region
15333:
Air can be cooled at will by repeated expansion and can finally be liquified.
15331:. As Sommerfeld noted, "This is the case with air and with most other gases.
14478:. It includes data from 51 substances including the vdW fluid over the range
4623:{\displaystyle p_{\text{c}}cv^{3}-(p_{\text{c}}b+RT_{\text{c}})v^{2}+av-ab=0}
3545:
2654:
1151:
34:
25437:{\displaystyle b'=b'_{2}/2\quad \rightarrow \quad b'=4\times (4\pi r^{3}/3)}
25105:
Next, assume that all particles are hard spheres of the same finite radius
25037:
14297:
Figure 4: A plot of the correlation including data from various substances.
13078:
10155:
3485:, displayed an experimentally obtained set of isotherms of carbonic acid, H
3467:
2489:
1280:
230:
21301:, but its numerical value is arbitrary due to a constant of integration.
8071:
This is the Gibbs free energy for a perfect vdW fluid, or in reduced form
5862:, and the liquid metals, Mercury and Cesium, are well approximated by it.
3201:
is a dimensionless saturation pressure, and log is the logarithm base 10.
28404:"Who was Van der Waals anyway and what has he to do with my Nitrox fill?"
20953:
Figure 8: The straight line (dotted-solid black) is tangent to the curve
18582:, and used the result to propose an enhanced version of the vdW equation
18319:. Notice that it can be positive or negative depending on whether or not
9583:
gives the same result, which does not agree with experiments on liquids.
5792:{\displaystyle Z_{\text{c}}=p_{\text{c}}v_{\text{c}}/(RT_{\text{c}})=3/8}
182:
28275:
27421:
van der Waals, J.D., Edited and with an Introduction by J.S. Rowlinson,
26698:{\displaystyle \phi =\int _{d}^{\infty }u(r){\frac {N}{V}}4\pi r^{2}dr,}
26213:
is the partition function of a single particle in a container of volume
6811:
a number. Then both integrals can be easily evaluated and the result is
5109:
renders the equation in the dimensionless form used to construct Fig. 1
28112:
Corresponding States Correlation of Saturated and Metastable Properties
27852:
Corresponding States Correlation of Saturated and Metastable Properties
27770:
Parametric solution of the van der Waals liquid–vapor coexistence curve
27646:, Yale University Press, New Haven (1901 reprinted in 1948), pp vii–xii
27612:
Pitzer, K.S., Lippman D.Z., Curl R.F., Huggins C.M. and Peterson D.E.,
20942:
These are the same three specifications for coexistence that were used
17824:{\displaystyle Z(\rho ,T)=1+\sum _{k=1}^{\infty }\,V_{k}(T)(\rho )^{k}}
16868:{\displaystyle Z={\frac {3}{3-\rho _{r}}}-{\frac {9\rho _{r}}{8T_{r}}}}
10161:
5996:
5197:{\displaystyle p_{r}={\frac {8T_{r}}{3v_{r}-1}}-{\frac {3}{v_{r}^{2}}}}
4006:
3894:
summarized this contribution of the van der Waals equation as follows:
3553:
2488:, in addition to noting that "Boltzmann described van der Waals as the
1084:
19643:
with equality holding in the ideal gas limit. The quadratic forms for
14293:
2378:
Putting these two approximations into the van der Waals equation when
2145:
is sufficiently large (or correspondingly whenever the molar density,
27377:
27367:
13922:{\displaystyle Z_{\text{c}}=p_{\text{c}}v_{\text{c}}/(RT_{\text{c}})}
13610:
Extended corresponding states behavior and the van der Waals Equation
3002:{\displaystyle Z_{\text{c}}=p_{\text{c}}v_{\text{c}}/(RT_{\text{c}})}
26607:{\displaystyle z={\frac {(V-Nb')\,e^{-\phi /(2kT)}}{\Lambda ^{3}}}.}
18370:, which agrees with the result found previously by differentiation.
16520:
Figure 7: Generalized compressibility chart for a van der Waals gas.
14743:) lie on a constant enthalpy curve. Although ideal gases, for which
14643:
11640:{\displaystyle dg=\partial _{p}g\,dp+\partial _{T}g\,dT=v\,dp-s\,dT}
9557:, which is also the ideal gas result as expected; however the limit
3458:
2398:
is large enough that both inequalities are satisfied reduces it to
27387:
27362:
26433:
but the intermolecular potential necessitates two modifications to
18486:{\displaystyle Z=(1-b\rho )^{-1}=\sum _{i=0}^{\infty }(b\rho )^{i}}
10154:. This mathematical criterion expresses a physical condition which
9985:
9307:. Then the two partials of the vdW equation can be used to express
7053:
2493:
2125:
The ideal gas law follows from the van der Waals equation whenever
1606:. This describes the fluid as two disconnected branches; a gas for
158:
38:
24268:
In terms of these quantities the vdW mixture constants are then,
13566:{\displaystyle v=v_{f}+x(v_{g}-v_{f})\qquad x=N_{g}/(N_{f}+N_{g})}
12013:(this can be visualized most easily by imagining Fig. 1 rotated 90
28346:
Van der Waals in his time and the present revival opening address
28012:
On the Dynamical Evidence of the Molecular Constitution of Bodies
26437:. First, because of the finite size of the particles, not all of
25987:
25976:{\displaystyle \left(p+{\frac {n^{2}a}{V^{2}}}\right)(V-nb)=nRT.}
23252:
intelligible in terms of the expected effects of size and energy.
21174:{\displaystyle \partial _{v_{r}}f_{r}=-p_{c}v_{c}/(RT_{c})p_{rs}}
20949:
16428:, which has zero slope at the origin is plotted and the isotherm
15217:
7499:
6001:
The internal energy is given by the energetic equation of state,
5432:
a characteristic temperature, and the 3 dimensionless groups are
2935:
molecular parameter such as the critical compressibility factor,
855:
is a number that depends on the shape of the potential function,
28392:
Hirschfelder, J. O., Curtis, C. F., and Bird, R. B., pp. 168-169
20786:
at the molar volume of each coexisting phase, saturated liquid,
18047:
can be expanded in an absolutely convergent series; this yields
9625:, namely that the vdW liquid is incompressible. Moreover, since
3194:{\displaystyle p_{s}/p_{\text{c}}=p_{s}(T/T_{\text{c}},\omega )}
2779:
have simple analytic expressions [this is also true of enthalpy
455:
are experimentally determinable, substance-specific constants.
28029:
Saturation and Metastable Properties of the van der Waals Fluid
25229:
The excluded volume for the two particles (of average diameter
17244:{\displaystyle D_{\rho _{r}}Z=3/(3-\rho _{r})^{2}-9/(8T_{r})=0}
16715:. Thus for a van der Waals fluid the compressibility factor is
285:
46:
16:
Gas equation of state which accounts for non-ideal gas behavior
19106:{\displaystyle p={\frac {RT}{v-b_{x}}}-{\frac {a_{x}}{v^{2}}}}
17730:
can be expressed by a power series called a virial expansion,
16516:
1286:
27869:
Thermodynamics: Basic Principles and Engineering Applications
27690:, Philosophical Transactions of the Royal Society of London,
15339:
The inversion curve can be found by solving its equation for
591:
expresses the strength of the molecular interactions. It has
68:
16383:
3843:{\displaystyle a=v^{2}(p_{2}T_{1}-p_{1}T_{2})/(T_{2}-T_{1})}
3483:
On the Continuity of the Gaseous and Liquid States of Matter
28676:(Revised ed.). New York: Macmillan. pp. 287–295.
27633:, Cambridge University Press, Cambridge, UK (2021), pp. 4–5
27067:
so that we only have to differentiate the terms containing
26335:
25021:{\displaystyle 1/2<(\sigma _{11}/\sigma _{22})^{3}<2}
22910:, all constant this is the equation of a plane with slopes
21240:. All this is consistent with the data of the green curve,
4100:, everywhere except at a single point, the critical point,
28317:, University of Toronto Press, Toronto (1959), pp. 124-128
27531:
Boltzmann, L. Enzykl. der Mathem. Wiss., Vol. V. 1, p. 550
27473:
Revised Printing, Hemisphere Publishing, NY (1979), p. 250
26175:{\displaystyle \Lambda ={\sqrt {\frac {h^{2}}{2\pi mkT}}}}
25329:{\displaystyle b'_{2}=4\pi d^{3}/3=8\times (4\pi r^{3}/3)}
25133:(per mole) or "co-volume". The corrected equation becomes
21407:
is designated by the left green circle) is the liquid. As
15022:
derived previously for the van der Waals equation this is
13606:
heterogeneous combination of liquid and vapor components.
8309:
The two first partial derivatives of the vdW equation are
8287:
5251:
in which an equation containing 6 dimensional quantities,
3998:
26449:′, where (just as in the conventional derivation above):
17395:. The slope is positive or negative depending on whether
15367:, and substituting into the vdW equation. This produces
4993:
Using these critical values to define reduced properties
462:
The force exerted by a molecule on another at a distance
27056:{\displaystyle p=kT{\frac {\partial \ln Z}{\partial V}}}
25993:
22605:, is in this case the specification of spinodal points.
21753:
For a vdW fluid the molar Helmholtz potential is simply
21595:{\displaystyle \partial _{v^{2}}^{2}f=-\partial _{v}p=0}
20010:, substituting into the quadratic form, and noting that
15485:{\displaystyle p/p^{*}=-1+4(T/2T^{*})^{1/2}-3(T/2T^{*})}
3451:
also contributed, and advanced it by converting it into
2712:
only, and its thermodynamic properties, internal energy
2504:
The equation has been, and remains very useful because:
1212:
such that the substance cannot be liquefied either when
26355:
is the distance between the centers of the spheres and
22014:. A plot of this function for the suncritical isotherm
14841:{\displaystyle \mu _{J}={\frac {v(\alpha T-1)}{c_{p}}}}
10994:{\displaystyle p_{\rm {min}}<p_{s}<p_{\rm {max}}}
10425:, for which no observable states exist. The states for
6406:{\displaystyle \partial _{vT}^{2}u=\partial _{Tv}^{2}u}
28048:, ASEE Chemical Engineering Education, 168–175, (1989)
27997:
27995:
27993:
27833:
27831:
27574:
27572:
26775:
26313:
26256:
26056:
24718:
24190:
23833:
23611:
23439:
21847:
21811:
21528:{\displaystyle \partial _{v^{2}}^{2}f=-\partial _{v}p}
20354:
19224:
18946:
18340:
15142:
14401:
Figure 4 is another plot of the same equation showing
13947:
11514:. Equating these two, rearranging, and recalling that
11181:
10037:
According to the extremum principle of thermodynamics
8771:
8359:
8165:
8093:
8064:{\displaystyle g-C_{u}=\{c+v/(v-b)-C_{s}-\ln\}RT-2a/v}
7801:
7431:
7319:
6954:{\displaystyle u_{r}-{\mbox{C}}_{u}=cT_{r}-9/(8v_{r})}
6893:
4983:{\displaystyle p_{\text{c}},v_{\text{c}},T_{\text{c}}}
4437:{\displaystyle v_{\text{c}}=3b,T_{\text{c}}=8a/(27Rb)}
2437:
2427:
965:
did as well, so do most engineers. Also the property,
130:{\displaystyle p={\frac {RT}{v-b}}-{\frac {a}{v^{2}}}}
28599:. Oxford: Oxford University Press. pp. 287–295.
28379:
Hirschfelder, J. O., Curtis, C. F., and Bird, R. B.,
27657:
Over de Continuïteit van den Gas en Vloeistoftoestand
27096:
27073:
27014:
26860:
26742:
26626:
26521:
26458:
26391:
26365:
26226:
26132:
26105:
26025:
25894:
25710:
25587:
25527:
25472:
25349:
25246:
25203:
25142:
25058:
24959:
24616:
24590:
24553:
24517:
24477:
24451:
24425:
24274:
24127:
23946:
23708:
23688:
23554:
23382:
23355:
23328:
23262:
23232:
23212:
23180:
23153:
23126:
23086:
23066:
23046:
23019:
22999:
22959:
22939:
22916:
22889:
22862:
22842:
22810:
22756:
22618:
22501:
22481:
22461:
22429:
22205:
22166:
22133:
22113:
22065:
22020:
21963:
21759:
21736:
21703:
21676:
21649:
21629:
21608:
21541:
21480:
21460:
21433:
21413:
21386:
21353:
21311:
21305:
As depicted in Fig. 8, the region on the green curve
21287:
21246:
21210:
21187:
21080:
21044:
21008:
20959:
20918:
20898:
20878:
20835:
20792:
20750:
20730:
20681:
20654:
20614:
20572:
20552:
20529:
20487:
20467:
20447:
20412:
20389:
20339:
20234:
20198:
20169:
20136:
20062:
20016:
19949:
19922:
19895:
19868:
19825:
19785:
19744:
19703:
19676:
19649:
19603:
19583:
19537:
19491:
19445:
19397:
19349:
19119:
19039:
18873:
18836:
18795:
18745:
18588:
18535:
18499:
18405:
18379:
18325:
18292:
18243:
18053:
18004:
17971:
17922:
17893:
17873:
17837:
17736:
17716:
17676:
17620:
17571:
17541:
17499:
17476:
17456:
17428:
17401:
17360:
17327:
17257:
17144:
17124:
17098:
17065:
16987:
16941:
16881:
16796:
16721:
16671:
16645:
16605:
16585:
16565:
16530:
16467:
16434:
16393:
16335:
16289:
16261:
16220:
16180:
16160:
16095:
16075:
16041:
15984:
15933:
15877:
15818:
15732:
15627:
15570:
15564:. The maximum of this, quadratic, curve occurs, with
15530:
15498:
15373:
15345:
15293:
15258:
15232:
15028:
15008:
14973:
14940:
14914:
14881:
14854:
14787:
14749:
14709:
14654:
14626:
14606:
14577:
14548:
14519:
14484:
14454:
14434:
14407:
14384:
14364:
14337:
14307:
14258:
14235:
14084:
14019:
13935:
13861:
13841:
13818:
13750:
13691:
13629:
13579:
13460:
13433:
13406:
13386:
13341:
13277:
13244:
13172:
13140:
13088:
12925:
12714:
12486:
12455:
12428:
12202:
12161:
12134:
12107:
12045:
12019:
11984:
11964:
11944:
11733:
11693:
11653:
11558:
11520:
11459:
11389:
11345:
11332:{\displaystyle p_{\text{s}},T_{\text{s}},v_{f},v_{g}}
11279:
11103:
11055:
11007:
10938:
10911:
10869:
10860:
delimited by the two spinodal points on an isotherm (
10846:
10568:
10535:
10472:
10431:
10418:{\displaystyle v_{\rm {min}}\leq v\leq v_{\rm {max}}}
10369:
10317:
10282:
10247:
10214:
10178:
10108:
10072:
10043:
9994:
9841:
9814:
9781:
9758:
9725:
9631:
9592:
9563:
9517:
9491:
9340:
9313:
9186:
9155:
9100:
9073:
9029:
8992:
8942:
8908:
8883:
8668:
8612:
8553:
8504:
8469:
8315:
8077:
7930:
7895:
7785:
7709:
7653:
7617:
7508:
7303:
7209:
7189:
7120:
7097:
7062:
7024:
6967:
6877:
6817:
6797:
6761:
6712:
6669:
6591:
6563:
6462:
6419:
6357:
6322:
6284:
6254:
6007:
5957:
5916:
5896:
5876:
5848:
5805:
5717:
5643:
5572:
5515:
5438:
5388:
5340:
5307:
5257:
5210:
5115:
4999:
4943:
4796:
4769:
4636:
4524:
4504:
4450:
4372:
4348:
4328:
4308:
4285:
4259:
4158:
4106:
4054:
4021:
3913:
3876:
3856:
3740:
3642:
3602:
3562:
3517:
3491:
3424:
3370:
3336:
3316:
3296:
3276:
3222:
3122:
3023:
2941:
2855:
2820:
2785:
2758:
2738:
2718:
2698:
2671:
2576:
2514:
2404:
2384:
2360:
2319:
2264:
2241:
2215:
2189:
2151:
2131:
2103:
2076:
2037:
2004:
1971:
1938:
1911:
1884:
1857:
1830:
1810:
1764:
1718:
1702:{\displaystyle v\leq v_{\rm {max}}\leq v_{\rm {min}}}
1653:
1612:
1556:
1500:
1463:
1427:
1407:
1375:
1339:
1319:
1299:
1251:
1218:
1191:
1164:
1136:
1116:
1093:
1060:
1023:
1003:
971:
929:
897:
861:
841:
799:
764:
684:
661:
641:
621:
601:
577:
534:
508:
488:
468:
441:
421:
397:
357:
314:
294:
270:
239:
191:
167:
143:
77:
54:
27705:
The Historical Origins of the Van der Waals Equation
27484:
Classical Thermodynamics as a Theory of Heat Engines
25885:, the equation obtains the second form given above:
22199:. In this case there are three stability conditions
21535:) continually decreases. The point characterized by
21002:(solid-dashed green, dotted gray) at the two points
20329:, which is the previous stability condition for the
18373:
For molecules that are non attracting hard spheres,
5997:
Internal energy and specific heat at constant volume
2752:, as well as the specific heat at constant pressure
2635:{\displaystyle \kappa _{T}=-(\partial _{p}v|_{T})/v}
2120:
28169:
Modified Law of Corresponding States for Real Gases
27990:
27839:
Thermodynamic Properties of the van der Waals Fluid
27828:
27569:
26617:The total attraction felt by a single particle is:
25566:{\displaystyle C=N_{\mathrm {A} }/V_{\mathrm {m} }}
25187:{\displaystyle p={\frac {RT}{V_{\mathrm {m} }-b}}.}
25036:Textbooks in physical chemistry generally give two
24584:). Using these together with the quadratic form of
10521:{\displaystyle v>v_{\rm {max}}>v_{\rm {min}}}
2484:in Volume 5 of his Lectures on Theoretical Physics
28255:Redlich, Otto; Kwong, J. N. S. (1 February 1949).
27580:The Properties and Behavior of Superheated Liquids
27423:On the Continuity of the Gaseous and Liquid States
27345:
27079:
27055:
26994:
26840:
26697:
26606:
26496:
26421:
26371:
26341:
26174:
26111:
26088:
25975:
25857:
25689:
25565:
25499:
25436:
25328:
25209:
25186:
25094:
25020:
24944:
24602:
24572:
24539:
24503:
24463:
24437:
24408:
24260:
24109:
23933:
23694:
23673:
23540:
23368:
23341:
23314:
23238:
23218:
23193:
23166:
23139:
23112:
23072:
23052:
23032:
23005:
22985:
22945:
22925:
22902:
22875:
22848:
22828:
22792:
22742:
22597:
22487:
22467:
22447:
22415:
22191:
22152:
22119:
22099:
22047:
22006:
21949:
21742:
21722:
21689:
21662:
21635:
21614:
21594:
21527:
21466:
21446:
21419:
21399:
21372:
21339:
21293:
21273:
21232:
21196:
21173:
21066:
21030:
20994:
20934:
20904:
20884:
20864:
20821:
20778:
20736:
20716:
20667:
20640:
20600:
20558:
20538:
20515:
20473:
20453:
20433:
20395:
20371:
20321:
20220:
20184:
20155:
20114:
20048:
20002:
19935:
19908:
19881:
19851:
19811:
19771:
19730:
19689:
19662:
19635:
19589:
19569:
19523:
19477:
19431:
19383:
19335:
19105:
19025:
18850:
18822:
18781:
18731:
18574:
18521:
18485:
18391:
18362:
18311:
18278:
18229:
18039:
17990:
17957:
17905:
17879:
17859:
17823:
17722:
17689:
17662:
17606:
17554:
17527:
17482:
17462:
17442:
17414:
17387:
17346:
17313:
17243:
17130:
17110:
17084:
17048:
16973:
16927:
16867:
16782:
16707:
16657:
16631:
16591:
16571:
16551:
16504:
16453:
16420:
16367:
16321:
16275:
16247:
16206:
16166:
16146:
16081:
16061:
16027:
15970:
15919:
15863:
15804:
15719:{\displaystyle D_{z}p/p^{*}=4-6(T/2T^{*})^{1/2}=0}
15718:
15613:
15556:
15516:
15484:
15359:
15323:
15279:
15244:
15207:
15014:
14994:
14959:
14926:
14900:
14867:
14840:
14770:
14735:
14695:
14632:
14612:
14592:
14563:
14534:
14505:
14470:
14440:
14420:
14390:
14370:
14350:
14323:
14283:
14244:
14221:
14070:
14005:
13921:
13847:
13824:
13804:
13736:
13677:
13597:
13565:
13446:
13419:
13392:
13360:
13304:
13263:
13227:
13158:
13114:
13068:
12911:
12701:
12468:
12441:
12414:
12188:
12147:
12120:
12089:
12031:
12005:
11970:
11950:
11930:
11719:
11679:
11639:
11541:
11506:
11445:
11371:
11331:
11265:
11089:
11041:
10993:
10924:
10896:
10852:
10824:
10554:
10520:
10458:
10417:
10355:
10303:
10268:
10233:
10200:
10146:
10094:
10058:
10029:, and is discussed at length in the next section.
10021:
9976:
9827:
9800:
9764:
9744:
9719:, it is also mechanically incompressible, that is
9711:
9617:
9575:
9549:
9503:
9477:
9326:
9299:
9168:
9141:
9086:
9056:
9015:
8978:
8928:
8895:
8869:
8654:
8598:
8539:
8490:
8455:
8277:
8063:
7916:
7870:
7771:
7695:
7639:
7603:
7483:
7289:
7195:
7175:
7106:
7083:
7036:
7010:
6953:
6862:
6803:
6783:
6747:
6698:
6655:
6577:
6549:
6448:
6405:
6343:
6308:
6267:
6240:
5985:
5943:
5902:
5882:
5854:
5830:
5791:
5700:
5629:
5558:
5501:
5424:
5374:
5326:
5293:
5223:
5196:
5101:
4982:
4929:
4782:
4755:
4622:
4510:
4487:
4436:
4354:
4334:
4314:
4291:
4271:
4242:
4144:
4092:
4040:
3983:
3882:
3862:
3842:
3726:
3628:
3588:
3529:
3503:
3481:at Queen's University Belfast in a paper entitled
3430:
3410:
3356:
3322:
3302:
3282:
3262:
3193:
3108:
3001:
2915:
2841:
2806:
2771:
2744:
2724:
2704:
2684:
2634:
2562:
2467:
2390:
2366:
2346:
2305:
2247:
2227:
2201:
2171:
2137:
2109:
2089:
2062:
2023:
1990:
1957:
1924:
1897:
1870:
1843:
1816:
1796:
1750:
1701:
1639:
1598:
1542:
1486:
1449:
1413:
1393:
1361:
1325:
1305:
1270:
1237:
1204:
1177:
1142:
1122:
1099:
1075:
1046:
1009:
985:
953:
911:
876:
847:
827:
785:
750:
670:
647:
627:
607:
583:
559:
520:
494:
474:
447:
427:
403:
379:
335:
300:
276:
252:
221:
173:
149:
129:
60:
28066:Lienhard. J.H., Shamsundar, N., and Biney, P.O.,
25095:{\displaystyle p={\frac {RT}{V_{\mathrm {m} }}}.}
23920:
23855:
23147:; the last two are equivalent to having the same
20481:are constant along with temperature the function
14908:defines a curve that separates the region of the
14644:Joule–Thomson coefficient and the inversion curve
8599:{\displaystyle \kappa _{T}=-v^{-1}\partial _{p}v}
7647:as before, and integrating the second term using
6550:{\displaystyle \partial _{v}c(v,T)=\partial _{T}}
4937:whose solution produces the previous results for
3727:{\displaystyle b=v-R(T_{2}-T_{1})/(p_{2}-p_{1})]}
3310:particles per unit volume (number density), mass
1087:), and those written with number density contain
28692:
15805:{\displaystyle (T/2T^{*})_{\rm {max}}^{1/2}=2/3}
13313:below and above the dots are metastable states,
8655:{\displaystyle \alpha =v^{-1}\partial _{T}v_{p}}
6656:{\displaystyle T^{2}\partial _{T}(p/T)]=a/v^{2}}
5509:. The reduced properties defined previously are
3552:had been produced, and a new field of research,
25:, named for its originator, the Dutch physicist
28632:
28558:. CreateSpace Independent Publishing Platform.
28335:van der Waals, Rowlinson ED, pp. 23-27, 253-258
28068:Spinodal Lines and Equations of State: A Review
27951:, John Wiley \& Sons, NY (1960), pp 131–135
27825:, Cambridge University Press, pp. 22–26, (2003)
26708:where we assumed that in a shell of thickness d
11383:is from the second law at constant temperature
10356:{\displaystyle \partial p/\partial v|_{T}>0}
10147:{\displaystyle \partial p/\partial v|_{T}<0}
8986:. Since the vdW equation in this limit becomes
4152:, where both the slope and curvature are zero,
4093:{\displaystyle \partial p/\partial v|_{T}<0}
3447:, and Ludwig Boltzmann. At about the same time
2563:{\displaystyle \alpha =(\partial _{T}v|_{p})/v}
1369:, the slope is positive over the entire range,
27005:From statistical thermodynamics we know that:
20333:, but in addition requires that the function,
18789:, this produced the correct coefficients thru
17493:Figure 6 shows a plot of various isotherms of
14513:. This plot shows clearly that the vdW fluid (
12155:is specified by the equation of state itself,
11453:, and from the first law at constant pressure
5231:will plot on the same curve. It expresses the
482:is the negative of the slope of this curve at
28671:
28624:Physics 127: Statistical Physics, Second Term
28552:"Chapter 26.9, Derivation: Particle in a Box"
28537:An Introduction to Statistical Thermodynamics
28383:, John Wiley and Sons, NY (1964), pp. 252-253
28361:van der Waals, Rowlinson ED, pp. 23-27, 64-66
24540:{\displaystyle a\propto \epsilon \sigma ^{3}}
18363:{\displaystyle T_{r}>{\mbox{ or}}<27/8}
16928:{\displaystyle 0\leq \rho _{r}=1/v_{r}\leq 3}
16147:{\displaystyle T/T^{*}=2/9=0.{\overline {2}}}
11273:These two vdW equations contain 4 variables,
4145:{\displaystyle p=p_{\text{c}},v=v_{\text{c}}}
3984:{\displaystyle p_{s}=f_{p}(y),T_{s}=f_{T}(y)}
1158:(defined by pressure and temperature values,
678:, as shown in the accompanying plot produces
28597:Introduction to Modern Statistical Mechanics
28254:
21914:
21836:
18529:. Boltzmann calculated its correct value as
16174:). In terms of the dimensionless variables,
16028:{\displaystyle z=(4\pm {\sqrt {16-12}}\,)/6}
15978:, are, making use of the quadratic formula,
14696:{\displaystyle \mu _{J}=\partial _{p}T|_{h}}
14006:{\displaystyle \omega =-{\mbox{log}}_{10}-1}
11446:{\displaystyle q_{\rm {vap}}=T(s_{g}-s_{f})}
8232:
8108:
8035:
7950:
5831:{\displaystyle 0.23<Z_{\text{c}}<0.31}
4342:for which the equation has 3 real roots for
828:{\displaystyle a=I\varepsilon N_{\text{A}}b}
23315:{\displaystyle a_{11},a_{22},b_{11},b_{22}}
22793:{\displaystyle \mu _{j}=\partial _{x_{j}}f}
21074:. The slope of the straight line, given by
17450:, and becomes infinitely large negative as
14071:{\displaystyle p_{rs}=p_{s}(T_{r},\omega )}
13238:Referring back to Fig. 1 the isotherms for
7502:is given by the entropic equation of state:
5865:
4279:for which the equation has 1 real root for
3214:The Nature of the Motion which We Call Heat
1878:weighted by this fraction, increases while
1599:{\displaystyle T_{\rm {max}},v_{\rm {max}}}
1543:{\displaystyle T_{\rm {min}},v_{\rm {min}}}
1245:no matter how low the temperature, or when
758:. Multiplying this by the number of moles,
28672:Kittel, Charles; Kroemer, Herbert (1980).
28616:"Lecture 3: First Order Phase Transitions"
28158:Second Edition, Wiley, N.Y., p. 49, (1973)
27882:Fundamentals of Engineering Thermodynamics
26209:the absolute temperature. In an ideal gas
25044:
21281:, of Fig. 1. The intercept on the line is
18399:, the vdW virial expansion becomes simply
17314:{\displaystyle T_{r}=3(3-\rho _{r})^{2}/8}
17049:{\displaystyle Z=Z_{\text{c}}=3/2-9/8=3/8}
16783:{\displaystyle Z=1/(1-b\rho )-a\rho /(RT)}
16665:, the ideal gas value. In the second case
13805:{\displaystyle p_{r}=p(v_{r},T_{r},\phi )}
3462:Van der Waals equation on a wall in Leiden
2499:
28503:
27871:, Springer, N.Y., pp 155, 203, 204 (2023)
27764:
27762:
27447:Boltzmann, L., Translated by S.G. Brush,
26554:
24889:
24677:
24234:
20608:, has positive curvature everywhere when
20115:{\displaystyle b=b_{11}x_{1}+b_{22}x_{2}}
19758:
19717:
19033:the mixture is also a vdW fluid given by
18195:
17785:
16378:
16013:
12080:
12049:
11873:
11869:
11827:
11816:
11647:. Integrating this over an isotherm from
11630:
11617:
11604:
11581:
11507:{\displaystyle q_{\rm {vap}}=h_{g}-h_{f}}
9142:{\displaystyle c_{p}=\partial _{T}h|_{p}}
7594:
7573:
7551:
7531:
6316:to be an exact differential, namely that
6275:is an arbitrary constant of integration.
6231:
6176:
6163:
6137:
6124:
6069:
6056:
6030:
3411:{\displaystyle m{\overline {v^{2}}}/3=kT}
3263:{\displaystyle p=nm{\overline {v^{2}}}/3}
1473:
502:. The force is repulsive, and large, for
28635:Advances in Atomic and Molecular Physics
28594:
28304:van der Waals, Rowlinson ED, pp. 245-247
28220:van der Waals, Rowlinson ED, pp. 243-282
28156:Fundamentals of Classical Thermodynamics
27668:van der Waals, Rowlinson ED, pp.121–282
27542:Thermodynamics and Statistical Mechanics
21643:the curvature increases to a maximum at
20948:
20003:{\displaystyle b_{12}=(b_{11}+b_{22})/2}
16515:
16382:
15287:or in terms of the critical temperature
15216:
14292:
13855:was the critical compressibility factor
13613:
13077:
10160:
9745:{\displaystyle \kappa _{T}\rightarrow 0}
9067:The specific heat at constant pressure,
7924:so combining the previous results gives
7871:{\displaystyle s_{r}-{\mbox{C}}_{s}=\ln}
6456:this condition can be written simply as
4444:, and using these in the equation gives
4005:
3457:
1421:, and is characteristic of all isobars
1285:
457:
27720:van der Waals, Rowlinson ED, pp.191–194
27438:, John Wiley & Sons, NY (1937), p 9
22127:is a composition variable, for example
7011:{\displaystyle u_{r}=u/(RT_{\text{c}})}
5951:, and a constant volume specific heat,
5502:{\displaystyle p/p^{*},v/v^{*},T/T^{*}}
3999:Critical point and corresponding states
2179:, is sufficiently small), Specifically
1017:, those using specific volume contain
28693:
28549:
28211:van der Waals, Rowlinson ED, pp. 20-22
28184:Moran, M.J., and Shapiro, H.N., p. 113
27759:
27582:, Lat. Am. J. Heat and Mass Transfer,
27486:, Springer-Verlag, NY (1977), pp 13–15
20523:represents a straight line with slope
18732:{\displaystyle (p+a/v^{2})(v-b/3)=RT.}
17663:{\displaystyle p_{r}(\rho _{r},T_{r})}
17085:{\displaystyle \rho _{r}\rightarrow 0}
16632:{\displaystyle \rho =1/v\rightarrow 0}
16505:{\displaystyle v^{*}\rho =\rho _{r}/3}
15864:{\displaystyle T_{\rm {max}}=8T^{*}/9}
14506:{\displaystyle -0.4<\omega <0.9}
11090:{\displaystyle v_{g}>v_{\rm {max}}}
11042:{\displaystyle v_{f}<v_{\rm {min}}}
7696:{\displaystyle \partial _{T}p=R/(v-b)}
5870:The properties molar internal energy,
5235:which Boltzmann described as follows:
4488:{\displaystyle p_{\text{c}}=a/27b^{2}}
3439:Newton's laws. The work, known as the
2665:its specific heat at constant volume,
2508:Its coefficient of thermal expansion,
2354:is numerically indistinguishable from
2235:is numerically indistinguishable from
2063:{\displaystyle 0<p<p_{\text{c}}}
308:is the number of molecules (the ratio
28613:
28516:from the original on 15 November 2023
28506:"J. D. van der Waals – Nobel lecture"
28381:Molecular Theory of Gases and Liquids
28250:
28248:
28083:Tien, C.L., and Lienhard, J.H., p.254
25994:Statistical thermodynamics derivation
15920:{\displaystyle p_{\rm {max}}=p^{*}/3}
15324:{\displaystyle T<27T_{\text{c}}/4}
10241:which lie between the local minimum,
9618:{\displaystyle \alpha =\kappa _{T}=0}
9094:is defined as the partial derivative
5701:{\displaystyle T_{r}=(27/8)(T/T^{*})}
5247:This "law" is just a special case of
3477:In 1869 Irish professor of chemistry
1998:, and a mixture of liquid and gas at
560:{\displaystyle r>r_{0}>\sigma }
28534:
28432:
28154:Van Wylen, G.J., and Sonntag, R.E.,
24573:{\displaystyle b\propto \sigma ^{3}}
23226:direction as well as the one in the
19636:{\displaystyle p\approx p_{1}+p_{2}}
17710:Statistical mechanics suggests that
13678:{\displaystyle p_{r}=p(v_{r},T_{r})}
13228:{\displaystyle y=(s_{g}-s_{f})/(2R)}
9504:{\displaystyle v\rightarrow \infty }
9149:. However, it is not independent of
8896:{\displaystyle v\rightarrow \infty }
8540:{\displaystyle -(v\kappa _{T})^{-1}}
7880:
6699:{\displaystyle \partial _{v}c_{v}=0}
6449:{\displaystyle c_{v}=\partial _{T}u}
5630:{\displaystyle v_{r}=(1/3)(v/v^{*})}
3443:, was done principally by Clausius,
1293:The graph on the right is a plot of
37:to include the non-zero size of gas
28572:from the original on 18 August 2016
28027:Shamsundar, N. and Lienhard, J.H.,
27659:, Ph.D. Thesis, Leiden Univ. (1873)
27398:Van der Waals constants (data page)
18830:and also gave infinite pressure at
17705:
16790:, or in terms of reduced variables
16708:{\displaystyle p=Z(\rho ,T)\rho RT}
15927:. The zeros of the inversion curve
14284:{\displaystyle 1\leq T_{r}\leq 0.3}
13737:{\displaystyle p_{rs}=p_{s}(T_{r})}
13454:according to the equation of state
10555:{\displaystyle \kappa _{T}=\infty }
8491:{\displaystyle \alpha /\kappa _{T}}
4518:, which at the critical point is,
3357:{\displaystyle {\overline {v^{2}}}}
2916:{\displaystyle g=u+pv-Ts=f+pv=h-Ts}
2692:, can be shown to be a function of
1640:{\displaystyle v\geq v_{\rm {min}}}
1487:{\displaystyle p=0.2\,p_{\text{c}}}
887:In his book (see references and )
13:
28588:
28539:. Courier Corporation. p. 77.
28494:van der Waals, Rowlinson ED, p. 70
28485:van der Waals, Rowlinson, p. 69-70
28463:van der Waals, Rowlinson ED, p. 69
28426:
28414:from the original on 11 March 2020
28401:
28395:
28370:van der Waals, Rowlinson ED, p. 66
28245:
28242:van der Waals, Rowlinson ED, p. 68
28101:van der Waals, Rowlinson ED, p. 22
28070:, Nuclear Engineering and Design,
27756:van der Waals, Rowlinson ED, p. 19
27044:
27030:
26962:
26643:
26590:
26250:
26133:
26106:
26075:
25828:
25801:
25763:
25734:
25674:
25642:
25630:
25557:
25540:
25166:
25081:
23206:or fold in the literature) in the
22771:
22559:
22528:
22503:
22388:
22382:
22368:
22340:
22326:
22306:
22292:
22265:
22251:
22224:
22210:
21574:
21543:
21513:
21482:
21082:
20995:{\displaystyle f_{r}(0.875,v_{r})}
20295:
20284:
20253:
20236:
18459:
18279:{\displaystyle Z(\rho _{r},T_{r})}
18190:
18040:{\displaystyle (3-\rho _{r})^{-1}}
17958:{\displaystyle Z(\rho _{r},T_{r})}
17780:
17607:{\displaystyle Z(\rho _{r},T_{r})}
16448:
16368:{\displaystyle T_{r{\rm {max}}}=3}
16352:
16349:
16346:
16322:{\displaystyle p_{r{\rm {max}}}=9}
16306:
16303:
16300:
15890:
15887:
15884:
15831:
15828:
15825:
15769:
15766:
15763:
14669:
13159:{\displaystyle 0\leq y<\infty }
13153:
11592:
11569:
11472:
11469:
11466:
11402:
11399:
11396:
11381:energy required to vaporize a mole
11081:
11078:
11075:
11033:
11030:
11027:
10985:
10982:
10979:
10951:
10948:
10945:
10717:
10714:
10578:
10575:
10549:
10512:
10509:
10506:
10491:
10488:
10485:
10459:{\displaystyle v<v_{\rm {min}}}
10450:
10447:
10444:
10409:
10406:
10403:
10382:
10379:
10376:
10329:
10318:
10295:
10292:
10289:
10260:
10257:
10254:
10201:{\displaystyle T>T_{\text{c}}c}
10120:
10109:
10022:{\displaystyle \kappa _{T}^{-1}=0}
9706:
9670:
9652:
9633:
9586:In this liquid limit we also find
9498:
9251:
9223:
9115:
8979:{\displaystyle \kappa _{T}=v/(RT)}
8890:
8633:
8584:
8377:
8369:
8327:
8319:
7655:
7585:
7577:
6671:
6603:
6518:
6495:
6464:
6434:
6383:
6359:
6217:
6195:
6089:
6081:
5334:is a characteristic molar volume,
4209:
4192:
4170:
4159:
4066:
4055:
3212:In 1857 Rudolf Clausius published
2597:
2525:
1693:
1690:
1687:
1672:
1669:
1666:
1631:
1628:
1625:
1590:
1587:
1584:
1569:
1566:
1563:
1534:
1531:
1528:
1513:
1510:
1507:
1450:{\displaystyle p>p_{\text{c}}.}
1394:{\displaystyle b\leq v<\infty }
1388:
14:
28732:
28504:van der Waals, Johannes Diderik.
28445:from the original on 2 April 2024
27566:, Dover, N.Y., pp 443–452, (1975)
24438:{\displaystyle \epsilon ,\sigma }
23113:{\displaystyle \mu _{2}-\mu _{1}}
22986:{\displaystyle \mu _{2}-\mu _{1}}
20372:{\displaystyle f=FN_{\mbox{A}}/N}
19570:{\displaystyle p\neq p_{1}+p_{2}}
18575:{\displaystyle (5/8)(b\rho )^{2}}
17916:If we note in the expression for
17867:are the virial coefficients; the
16974:{\displaystyle T_{r}=\rho _{r}=1}
12090:{\displaystyle v\,dp=d(pv)-p\,dv}
10234:{\displaystyle T<T_{\text{c}}}
5559:{\displaystyle p_{r}=27(p/p^{*})}
2645:it explains the existence of the
2121:Relationship to the ideal gas law
1991:{\displaystyle T<T_{\text{s}}}
1958:{\displaystyle T>T_{\text{s}}}
1271:{\displaystyle T>T_{\text{c}}}
1238:{\displaystyle p>p_{\text{c}}}
222:{\displaystyle v=VN_{\text{A}}/N}
28110:Dong, W.G., and Lienhard, J.H.,
27884:4th Edition, pp. 574, 580 (2000)
27880:Moran, M.J., and Shapiro, H.N.,
27850:Dong, W.G., and Lienhard, J.H.,
27482:Truesdale, C. and Bharatha, S.,
27469:Tien, C.L., and Lienhard, J.H.,
25500:{\displaystyle b=N_{\text{A}}b'}
22007:{\displaystyle f_{r}=f/(RT_{c})}
15614:{\displaystyle z^{2}=T/(2T^{*})}
7176:{\displaystyle pv=RTv/(v-b)-a/v}
3216:. In it he derived the relation
2306:{\displaystyle v\gg (a/p)^{1/2}}
61:{\displaystyle \leftrightarrow }
28543:
28528:
28497:
28488:
28479:
28466:
28457:
28386:
28373:
28364:
28355:
28338:
28329:
28320:
28307:
28298:
28289:
28236:
28223:
28214:
28205:
28196:
28193:Tien and Lienhard, pp. 247–248
28187:
28178:
28161:
28148:
28139:
28130:
28121:
28104:
28095:
28086:
28077:
28060:
28051:
28038:
28021:
28004:
27981:
27972:
27963:
27954:
27941:
27932:
27923:
27914:
27905:
27896:
27887:
27874:
27861:
27844:
27815:
27806:
27797:
27788:
27779:
27750:
27741:
27732:
27723:
27714:
27697:
27680:
27671:
27662:
27649:
27636:
27623:
27606:
27589:
27556:
27547:
27534:
27525:
27516:
27403:Redlich–Kwong equation of state
27393:Theorem of corresponding states
26781:
26773:
26497:{\displaystyle b'=2\pi d^{3}/3}
26319:
26262:
26062:
26054:
25386:
25382:
24724:
24716:
24347:
24346:
24196:
24188:
23839:
23831:
23617:
23609:
23445:
23437:
22287:
22246:
20943:
20641:{\displaystyle T_{R}\geq T_{c}}
19772:{\displaystyle b_{11},\,b_{22}}
19731:{\displaystyle a_{11},\,a_{22}}
19230:
19222:
18952:
18944:
15148:
15140:
14648:The Joule–Thomson coefficient,
13512:
13271:are discontinuous. Considering
12998:
12997:
12810:
12809:
12808:
12807:
12602:
11958:is a multivalued function, the
11187:
11179:
10707:
9057:{\displaystyle \kappa _{T}=1/p}
8777:
8769:
8365:
8357:
7437:
7429:
7290:{\displaystyle h-C_{u}=RT-2a/v}
6863:{\displaystyle u-C_{u}=cRT-a/v}
5382:a characteristic pressure, and
4887:
4847:
3109:{\displaystyle \omega =-\log-1}
2446:
2425:
1362:{\displaystyle p=2p_{\text{c}}}
954:{\displaystyle VN_{\text{A}}/N}
751:{\displaystyle b=4N_{\text{A}}}
380:{\displaystyle R=N_{\text{A}}k}
28721:Johannes Diderik van der Waals
28701:Eponymous equations of physics
28044:Barrufet, M.A., Eubank, P.T.,
27507:
27498:
27489:
27476:
27463:
27454:
27441:
27428:
27415:
27325:
27310:
27262:
27247:
27227:
27174:
26971:
26958:
26903:
26883:
26657:
26651:
26583:
26571:
26551:
26531:
26236:
26230:
26005:of an ideal gas consisting of
25955:
25940:
25840:
25819:
25776:
25431:
25404:
25383:
25323:
25296:
25003:
24974:
24580:(see the introduction to this
24321:
24301:
24078:
24019:
23974:
23960:
23794:
23764:
23657:
23631:
23592:
23568:
23524:
23472:
23420:
23396:
22721:
22695:
22612:, can be written in the form
22580:
22555:
22094:
22069:
22001:
21985:
21944:
21928:
21911:
21908:
21886:
21868:
21334:
21315:
21155:
21139:
20989:
20970:
20892:as well as the same values of
20859:
20846:
20816:
20803:
20773:
20754:
20717:{\displaystyle p=p_{s}(T_{R})}
20711:
20698:
20595:
20576:
20510:
20491:
20330:
20303:
20268:
20130:contact with a heat reservoir
19989:
19963:
18811:
18796:
18767:
18746:
18723:
18720:
18704:
18680:
18671:
18651:
18639:
18619:
18616:
18589:
18563:
18553:
18550:
18536:
18510:
18500:
18474:
18464:
18428:
18412:
18273:
18247:
18218:
18196:
18168:
18147:
18144:
18132:
18128:
18112:
18095:
18083:
18057:
18025:
18005:
17991:{\displaystyle \rho _{r}<3}
17952:
17926:
17854:
17848:
17812:
17805:
17802:
17796:
17752:
17740:
17657:
17631:
17601:
17575:
17528:{\displaystyle Z(\rho ,T_{r})}
17522:
17503:
17294:
17274:
17232:
17216:
17196:
17176:
17076:
16777:
16768:
16751:
16736:
16693:
16681:
16623:
16014:
15991:
15758:
15733:
15693:
15668:
15608:
15592:
15479:
15455:
15432:
15407:
15168:
15155:
15122:
15109:
15051:
15038:
14822:
14807:
14765:
14759:
14683:
14564:{\displaystyle \omega =-0.267}
14535:{\displaystyle \omega =-0.302}
14216:
14146:
14137:
14110:
14065:
14046:
13994:
13991:
13972:
13959:
13916:
13900:
13799:
13767:
13731:
13718:
13672:
13646:
13560:
13534:
13509:
13483:
13222:
13213:
13205:
13179:
13057:
13050:
13032:
13026:
13008:
13002:
12935:
12929:
12890:
12884:
12863:
12857:
12791:
12785:
12767:
12761:
12687:
12680:
12672:
12663:
12656:
12644:
12635:
12628:
12590:
12583:
12575:
12572:
12566:
12548:
12545:
12539:
12506:
12500:
12183:
12171:
12071:
12062:
12000:
11988:
11919:
11893:
11440:
11414:
10801:
10779:
10747:
10732:
10675:
10652:
10609:
10596:
10337:
10128:
9971:
9950:
9936:
9913:
9897:
9881:
9867:
9854:
9835:are all infinite on the curve
9736:
9576:{\displaystyle v\rightarrow b}
9567:
9495:
9457:
9444:
9385:
9379:
9363:
9351:
9236:
9219:
9129:
8973:
8964:
8887:
8855:
8842:
8812:
8800:
8757:
8744:
8708:
8695:
8525:
8508:
8416:
8403:
8272:
8256:
8229:
8226:
8204:
8186:
8157:
8135:
8032:
8029:
8017:
8004:
7979:
7967:
7865:
7862:
7840:
7822:
7766:
7763:
7751:
7738:
7690:
7678:
7548:
7542:
7478:
7459:
7426:
7410:
7386:
7383:
7361:
7334:
7267:
7264:
7252:
7235:
7156:
7144:
7005:
6989:
6948:
6932:
6748:{\displaystyle c_{v}=c_{v}(T)}
6742:
6736:
6629:
6626:
6612:
6544:
6541:
6527:
6504:
6488:
6476:
6338:
6326:
6303:
6291:
6212:
6198:
6160:
6148:
6116:
6104:
6053:
6041:
5980:
5968:
5938:
5926:
5772:
5756:
5695:
5674:
5671:
5657:
5624:
5603:
5600:
5586:
5553:
5532:
5419:
5410:
4657:
4637:
4583:
4551:
4431:
4419:
4224:
4178:
4074:
4041:{\displaystyle T=T_{\text{c}}}
4015:at some critical temperature,
3978:
3972:
3943:
3937:
3837:
3811:
3803:
3757:
3721:
3718:
3692:
3684:
3658:
3188:
3161:
3097:
3079:
3052:
3039:
2996:
2980:
2621:
2611:
2593:
2549:
2539:
2521:
2286:
2271:
2024:{\displaystyle T=T_{\text{s}}}
1791:
1765:
1745:
1719:
1067:
1038:
871:
865:
786:{\displaystyle N/N_{\text{A}}}
735:
718:
703:
694:
336:{\displaystyle N/N_{\text{A}}}
55:
27:Johannes Diderik van der Waals
1:
28655:10.1016/S0065-2199(08)60216-X
28229:Lorentz, H. A., Ann, Physik,
27631:Foundations of Modern Physics
27408:
26121:thermal de Broglie wavelength
25031:
22448:{\displaystyle 0\leq x\leq 1}
21233:{\displaystyle p_{rs}=0.5730}
20403:that describe stable states.
20049:{\displaystyle x_{1}+x_{2}=1}
19852:{\displaystyle b_{12}=b_{21}}
19812:{\displaystyle a_{12}=a_{21}}
19524:{\displaystyle x_{1}+x_{2}=1}
19478:{\displaystyle N=N_{1}+N_{2}}
19432:{\displaystyle x_{2}=N_{2}/N}
19384:{\displaystyle x_{1}=N_{1}/N}
17422:is greater than or less than
16454:{\displaystyle T_{r}=\infty }
15971:{\displaystyle 3z^{2}-4z+1=0}
14960:{\displaystyle \mu _{J}>0}
14593:{\displaystyle \omega =-0.21}
13598:{\displaystyle 0\leq x\leq 1}
13130:, are shown dotted in Fig. 1.
10835:
10304:{\displaystyle v_{\rm {max}}}
10269:{\displaystyle v_{\rm {min}}}
9801:{\displaystyle c_{p},\alpha }
9550:{\displaystyle c_{p}-c_{v}=R}
7044:is a dimensionless constant.
5799:, while for most real fluids
5375:{\displaystyle p^{*}=a/b^{2}}
3472:van der Waals cohesive forces
1797:{\displaystyle (T_{s},v_{g})}
1751:{\displaystyle (T_{s},v_{f})}
671:{\displaystyle -\varepsilon }
28439:Computational Thermodynamics
27729:Goodstein, D.L., pp. 450–451
26185:with the usual definitions:
25872:for the number of moles and
22100:{\displaystyle f(T_{R},v,x)}
21031:{\displaystyle v_{rf}=0.576}
20865:{\displaystyle v_{g}(T_{R})}
20822:{\displaystyle v_{f}(T_{R})}
20744:the line will be tangent to
18782:{\displaystyle (v-b/3)^{-1}}
18522:{\displaystyle (b\rho )^{2}}
16139:
14995:{\displaystyle \alpha T-1=0}
12121:{\displaystyle p_{\text{s}}}
10032:
7772:{\displaystyle s-C_{s}=R\ln}
5425:{\displaystyle T^{*}=a/(Rb)}
4783:{\displaystyle p_{\text{c}}}
3386:
3349:
3247:
2090:{\displaystyle T_{\text{s}}}
1205:{\displaystyle T_{\text{c}}}
1178:{\displaystyle p_{\text{c}}}
1047:{\displaystyle R/{\bar {m}}}
521:{\displaystyle r<\sigma }
253:{\displaystyle N_{\text{A}}}
7:
27356:
24504:{\displaystyle T_{c},p_{c}}
21723:{\displaystyle v\geq v_{g}}
21373:{\displaystyle v\leq v_{f}}
21067:{\displaystyle v_{rg}=2.71}
20221:{\displaystyle D^{2}F>0}
18861:
18312:{\displaystyle \rho _{r}=0}
17347:{\displaystyle \rho _{r}=0}
16207:{\displaystyle T_{r},p_{r}}
15557:{\displaystyle p^{*},T^{*}}
15002:. Using the expression for
14736:{\displaystyle h_{2}=h_{1}}
13115:{\displaystyle p_{r},T_{r}}
11720:{\displaystyle p_{s},v_{g}}
11680:{\displaystyle p_{s},v_{f}}
11372:{\displaystyle g_{f}=g_{g}}
10095:{\displaystyle d^{2}S<0}
9828:{\displaystyle \kappa _{T}}
8929:{\displaystyle \alpha =1/T}
8288:Thermodynamic derivatives:
7047:
6585:for the vdW equation gives
5294:{\displaystyle p,v,T,a,b,R}
5233:law of corresponding states
4763:Then dividing the first by
3629:{\displaystyle p_{2},T_{2}}
3589:{\displaystyle p_{1},T_{1}}
2661:law of corresponding states
1154:at any given pressure, the
877:{\displaystyle \varphi (r)}
10:
28737:
28716:Engineering thermodynamics
27471:Statistical Thermodynamics
27436:Textbook of Thermodynamics
27425:, Dover, NY (1988), p. 174
26422:{\displaystyle Z=z^{N}/N!}
21340:{\displaystyle f(T_{R},v)}
20779:{\displaystyle f(T_{R},v)}
20601:{\displaystyle f(T_{R},v)}
20516:{\displaystyle f(T_{R},v)}
19889:is the arithmetic mean of
17388:{\displaystyle T_{r}=27/8}
16421:{\displaystyle T_{r}=27/8}
15252:there will be cooling for
14901:{\displaystyle \mu _{J}=0}
14229:which has an rms error of
13264:{\displaystyle T_{r}<1}
9176:, they are related by the
7493:
5986:{\displaystyle c_{v}(v,T)}
5711:The vdW equation produces
3207:
1647:, and a denser liquid for
1076:{\displaystyle {\bar {m}}}
961:(molar volume) used here,
28535:Hill, Terrell L. (1960).
27938:Moran and Shapiro, p. 580
27920:Moran and Shapiro, p. 574
27841:, arXiv:1402.1205, (2014)
27451:, Dover, NY (1995), p 231
26372:{\displaystyle \epsilon }
23013:direction, and intercept
22608:For a binary mixture the
22192:{\displaystyle x_{1}=1-x}
22048:{\displaystyle T_{r}=7/8}
21274:{\displaystyle T_{r}=7/8}
18823:{\displaystyle (b/v)^{2}}
17690:{\displaystyle \rho _{r}}
17555:{\displaystyle \rho _{r}}
16935:. At the critical point,
16512:which varies from 0 to 1.
16248:{\displaystyle T_{r}=3/4}
15492:, where, for simplicity,
15280:{\displaystyle 2a>RTb}
13361:{\displaystyle T_{r}=0.8}
13305:{\displaystyle T_{r}=7/8}
12032:{\displaystyle ^{\circ }}
10897:{\displaystyle T_{r}=7/8}
9984:. This curve, called the
2928:Joule–Thomson coefficient
2347:{\displaystyle p+a/v^{2}}
2172:{\displaystyle \rho =1/v}
28595:Chandler, David (1987).
28512:. Nobel Prize Outreach.
27902:Moran and Shapiro, p 574
26197:the mass of a particle,
26112:{\displaystyle \Lambda }
23194:{\displaystyle \mu _{2}}
23167:{\displaystyle \mu _{1}}
23140:{\displaystyle \mu _{1}}
23033:{\displaystyle \mu _{1}}
22903:{\displaystyle \mu _{2}}
22876:{\displaystyle \mu _{1}}
20724:and a suitable value of
17860:{\displaystyle V_{k}(T)}
15871:, and the corresponding
14868:{\displaystyle \mu _{J}}
14358:. for various values of
14245:{\displaystyle \pm 0.42}
13835:The first candidate for
12189:{\displaystyle p=p(v,T)}
7640:{\displaystyle c_{v}=cR}
6784:{\displaystyle c_{v}=cR}
5944:{\displaystyle p=p(v,T)}
5866:Thermodynamic properties
3330:, and mean square speed
2849:, and Gibbs free energy
2814:, Helmholtz free energy
1054:(the substance specific
655:, and a minimum energy,
349:base unit mole in the SI
28614:Cross, Michael (2004),
28018:(#279), 357–359, (1875)
26441:is available, but only
25045:Conventional derivation
24445:, which are related to
23835:which can be written as
22153:{\displaystyle x=x_{2}}
20829:, and saturated vapor,
20156:{\displaystyle T=T_{R}}
16461:. The abscissa here is
16283:, while the maximum is
15015:{\displaystyle \alpha }
14633:{\displaystyle \omega }
14613:{\displaystyle \omega }
14441:{\displaystyle \omega }
14398:) are more pronounced.
14391:{\displaystyle \omega }
14371:{\displaystyle \omega }
13166:is given physically by
10158:described as follows:
9765:{\displaystyle \alpha }
6309:{\displaystyle du(v,T)}
5855:{\displaystyle \omega }
5327:{\displaystyle v^{*}=b}
4048:the slope is negative,
3554:low temperature physics
3544:By 1877 sprays of both
3441:kinetic theory of gases
2653:including the observed
2500:Utility of the equation
648:{\displaystyle \sigma }
628:{\displaystyle \sigma }
28202:Boltzmann, pp. 353-356
28057:Boltzmann, pp. 248–250
27738:Boltzmann, pp. 232–233
27599:, Can. J. Chem. Eng.,
27449:Lectures on Gas Theory
27347:
27081:
27057:
26996:
26842:
26699:
26608:
26498:
26423:
26373:
26343:
26176:
26113:
26090:
25977:
25859:
25691:
25567:
25501:
25438:
25330:
25211:
25188:
25096:
25022:
24946:
24888:
24867:
24784:
24763:
24676:
24655:
24610:for mixtures produces
24604:
24574:
24541:
24505:
24465:
24439:
24410:
24262:
24233:
24164:
24111:
24018:
23997:
23935:
23880:
23763:
23742:
23696:
23675:
23542:
23370:
23369:{\displaystyle b_{12}}
23343:
23342:{\displaystyle a_{12}}
23316:
23254:
23240:
23220:
23195:
23168:
23141:
23114:
23074:
23054:
23034:
23007:
22987:
22947:
22927:
22904:
22877:
22850:
22830:
22794:
22744:
22599:
22489:
22469:
22449:
22417:
22193:
22154:
22121:
22101:
22049:
22008:
21951:
21744:
21724:
21691:
21664:
21637:
21616:
21596:
21529:
21468:
21448:
21421:
21401:
21374:
21341:
21302:
21295:
21275:
21234:
21198:
21197:{\displaystyle -0.215}
21175:
21068:
21032:
20996:
20936:
20935:{\displaystyle T_{R}.}
20906:
20886:
20866:
20823:
20780:
20738:
20718:
20669:
20642:
20602:
20560:
20540:
20517:
20475:
20455:
20435:
20434:{\displaystyle f=g-pv}
20397:
20373:
20323:
20222:
20186:
20157:
20116:
20050:
20004:
19937:
19936:{\displaystyle b_{22}}
19910:
19909:{\displaystyle b_{11}}
19883:
19882:{\displaystyle b_{12}}
19853:
19813:
19773:
19732:
19691:
19664:
19637:
19591:
19571:
19525:
19479:
19433:
19385:
19337:
19107:
19027:
18852:
18824:
18783:
18733:
18576:
18523:
18487:
18463:
18393:
18364:
18313:
18280:
18231:
18194:
18041:
17992:
17959:
17913:particle interaction.
17907:
17881:
17861:
17825:
17784:
17724:
17691:
17664:
17608:
17556:
17529:
17484:
17464:
17444:
17416:
17389:
17348:
17315:
17245:
17132:
17112:
17086:
17050:
16975:
16929:
16869:
16784:
16709:
16659:
16633:
16593:
16573:
16553:
16552:{\displaystyle pv=ZRT}
16521:
16513:
16506:
16455:
16422:
16379:Compressibility factor
16369:
16323:
16277:
16249:
16208:
16168:
16148:
16083:
16063:
16029:
15972:
15921:
15865:
15806:
15720:
15615:
15558:
15524:have been replaced by
15518:
15486:
15361:
15325:
15281:
15246:
15245:{\displaystyle v\gg b}
15223:
15209:
15016:
14996:
14961:
14928:
14902:
14869:
14842:
14772:
14771:{\displaystyle h=h(T)}
14737:
14697:
14634:
14614:
14594:
14565:
14536:
14507:
14472:
14471:{\displaystyle p_{rs}}
14448:for various values of
14442:
14422:
14392:
14372:
14352:
14325:
14324:{\displaystyle p_{rs}}
14301:Figure 3 is a plot of
14298:
14285:
14246:
14223:
14072:
14007:
13923:
13849:
13826:
13806:
13738:
13679:
13620:
13599:
13567:
13448:
13421:
13394:
13362:
13306:
13265:
13229:
13160:
13131:
13116:
13070:
12913:
12703:
12470:
12443:
12416:
12190:
12149:
12122:
12091:
12033:
12007:
12006:{\displaystyle v(p,T)}
11972:
11952:
11932:
11780:
11721:
11681:
11641:
11543:
11542:{\displaystyle g=h-Ts}
11508:
11447:
11373:
11333:
11267:
11091:
11043:
10995:
10926:
10898:
10854:
10826:
10556:
10522:
10460:
10419:
10357:
10305:
10270:
10235:
10202:
10166:
10148:
10096:
10060:
10023:
9978:
9829:
9802:
9766:
9746:
9713:
9619:
9577:
9551:
9505:
9479:
9328:
9301:
9170:
9143:
9088:
9058:
9017:
9016:{\displaystyle p=RT/v}
8980:
8930:
8897:
8871:
8656:
8600:
8541:
8498:, while the second is
8492:
8463:The first equation is
8457:
8279:
8065:
7918:
7917:{\displaystyle g=h-Ts}
7872:
7773:
7697:
7641:
7605:
7485:
7291:
7197:
7177:
7108:
7085:
7084:{\displaystyle h=u+pv}
7038:
7012:
6955:
6864:
6805:
6785:
6749:
6700:
6657:
6579:
6551:
6450:
6407:
6345:
6344:{\displaystyle u(v,T)}
6310:
6269:
6242:
5987:
5945:
5904:
5884:
5856:
5832:
5793:
5702:
5631:
5560:
5503:
5426:
5376:
5328:
5295:
5245:
5225:
5198:
5103:
4984:
4931:
4784:
4757:
4624:
4512:
4489:
4438:
4356:
4336:
4316:
4293:
4273:
4244:
4146:
4094:
4042:
4011:
3985:
3901:
3884:
3864:
3844:
3728:
3630:
3590:
3531:
3505:
3463:
3432:
3412:
3358:
3324:
3304:
3284:
3264:
3195:
3110:
3003:
2917:
2843:
2842:{\displaystyle f=u-Ts}
2808:
2807:{\displaystyle h=u+pv}
2773:
2746:
2726:
2706:
2686:
2636:
2564:
2469:
2392:
2368:
2348:
2307:
2249:
2229:
2203:
2202:{\displaystyle v\gg b}
2173:
2139:
2111:
2091:
2064:
2025:
1992:
1965:, a stable liquid for
1959:
1926:
1899:
1872:
1845:
1818:
1798:
1752:
1703:
1641:
1600:
1544:
1488:
1451:
1415:
1395:
1363:
1327:
1307:
1290:
1272:
1239:
1206:
1179:
1144:
1124:
1101:
1077:
1048:
1011:
987:
955:
913:
891:wrote equations using
878:
849:
829:
787:
752:
672:
649:
629:
609:
585:
568:
561:
528:, and attractive when
522:
496:
476:
449:
429:
405:
389:universal gas constant
381:
337:
302:
278:
254:
223:
175:
151:
131:
62:
23:van der Waals equation
28556:Modern Thermodynamics
28550:Denker, John (2014).
28315:Differential Geometry
28175:, pp. 803–806, (1946)
28127:Sommerfeld, pp. 61–63
28074:, pp. 297–314, (1986)
27812:Boltzmann, pp 239–240
27803:Boltzmann, pp 237-238
27785:Sommerfeld, pp. 56–57
27655:van der Waals, J.D.,
27616:, J. Am. Chem. Soc.,
27553:Sommerfeld, pp. 55–68
27460:Boltzmann, p. 221–224
27373:Inversion temperature
27348:
27082:
27058:
26997:
26843:
26700:
26609:
26499:
26424:
26374:
26344:
26177:
26114:
26091:
25978:
25860:
25692:
25568:
25502:
25439:
25331:
25212:
25189:
25097:
25023:
24947:
24868:
24847:
24764:
24743:
24656:
24635:
24605:
24575:
24542:
24506:
24466:
24440:
24411:
24263:
24213:
24144:
24112:
23998:
23977:
23936:
23860:
23743:
23722:
23697:
23676:
23543:
23371:
23344:
23317:
23249:
23241:
23221:
23196:
23169:
23142:
23115:
23075:
23055:
23035:
23008:
22988:
22948:
22928:
22905:
22878:
22851:
22831:
22829:{\displaystyle j=1,2}
22795:
22745:
22600:
22490:
22470:
22450:
22418:
22194:
22155:
22122:
22102:
22050:
22009:
21952:
21745:
21725:
21692:
21690:{\displaystyle p_{s}}
21670:, where the slope is
21665:
21663:{\displaystyle v_{g}}
21638:
21617:
21597:
21530:
21469:
21449:
21447:{\displaystyle v_{f}}
21422:
21402:
21400:{\displaystyle v_{f}}
21375:
21342:
21296:
21276:
21235:
21199:
21176:
21069:
21033:
20997:
20952:
20937:
20907:
20887:
20867:
20824:
20781:
20739:
20719:
20670:
20668:{\displaystyle T_{R}}
20643:
20603:
20561:
20541:
20518:
20476:
20456:
20436:
20398:
20374:
20324:
20223:
20187:
20158:
20117:
20051:
20005:
19938:
19911:
19884:
19854:
19814:
19774:
19733:
19692:
19690:{\displaystyle b_{x}}
19665:
19663:{\displaystyle a_{x}}
19638:
19592:
19572:
19526:
19480:
19434:
19386:
19338:
19108:
19028:
18853:
18825:
18784:
18734:
18577:
18524:
18488:
18443:
18394:
18365:
18314:
18281:
18232:
18174:
18042:
17993:
17960:
17908:
17887:th term represents a
17882:
17862:
17826:
17764:
17725:
17692:
17665:
17609:
17557:
17530:
17485:
17465:
17445:
17417:
17415:{\displaystyle T_{r}}
17390:
17349:
17316:
17246:
17133:
17113:
17087:
17051:
16976:
16930:
16870:
16785:
16710:
16660:
16634:
16594:
16574:
16554:
16519:
16507:
16456:
16423:
16386:
16370:
16324:
16278:
16250:
16209:
16169:
16149:
16084:
16064:
16062:{\displaystyle z=1/3}
16030:
15973:
15922:
15866:
15807:
15721:
15616:
15559:
15519:
15517:{\displaystyle a,b,R}
15487:
15362:
15326:
15282:
15247:
15220:
15210:
15017:
14997:
14962:
14929:
14903:
14870:
14843:
14773:
14738:
14698:
14635:
14615:
14595:
14566:
14537:
14508:
14473:
14443:
14423:
14421:{\displaystyle T_{r}}
14393:
14373:
14353:
14351:{\displaystyle T_{r}}
14326:
14296:
14286:
14247:
14224:
14073:
14008:
13924:
13850:
13848:{\displaystyle \phi }
13827:
13825:{\displaystyle \phi }
13807:
13739:
13680:
13617:
13600:
13568:
13449:
13447:{\displaystyle v_{g}}
13422:
13420:{\displaystyle v_{f}}
13395:
13363:
13307:
13266:
13230:
13161:
13117:
13081:
13071:
12914:
12704:
12471:
12469:{\displaystyle p_{s}}
12444:
12442:{\displaystyle T_{s}}
12417:
12191:
12150:
12148:{\displaystyle p_{s}}
12123:
12092:
12034:
12008:
11973:
11953:
11933:
11760:
11722:
11682:
11642:
11549:produces the result.
11544:
11509:
11448:
11374:
11334:
11268:
11092:
11044:
10996:
10927:
10925:{\displaystyle T_{s}}
10899:
10855:
10827:
10557:
10523:
10461:
10420:
10358:
10306:
10276:, and local maximum,
10271:
10236:
10203:
10172:While on an isotherm
10164:
10149:
10097:
10061:
10024:
9988:curve, is defined by
9979:
9830:
9803:
9767:
9747:
9714:
9620:
9578:
9552:
9506:
9480:
9329:
9327:{\displaystyle c_{p}}
9302:
9171:
9169:{\displaystyle c_{v}}
9144:
9089:
9087:{\displaystyle c_{p}}
9059:
9018:
8981:
8931:
8898:
8872:
8657:
8601:
8542:
8493:
8458:
8280:
8066:
7919:
7873:
7774:
7698:
7642:
7606:
7486:
7292:
7198:
7178:
7109:
7086:
7039:
7013:
6956:
6865:
6806:
6786:
6750:
6701:
6658:
6580:
6552:
6451:
6408:
6346:
6311:
6270:
6268:{\displaystyle C_{u}}
6243:
5988:
5946:
5905:
5885:
5857:
5833:
5794:
5703:
5632:
5561:
5504:
5427:
5377:
5329:
5296:
5237:
5226:
5224:{\displaystyle T_{r}}
5199:
5104:
4985:
4932:
4785:
4758:
4625:
4513:
4490:
4439:
4357:
4337:
4317:
4294:
4274:
4245:
4147:
4095:
4043:
4009:
3986:
3896:
3885:
3865:
3845:
3729:
3631:
3591:
3532:
3506:
3461:
3453:statistical mechanics
3433:
3413:
3359:
3325:
3305:
3285:
3265:
3196:
3111:
3004:
2918:
2844:
2809:
2774:
2772:{\displaystyle c_{p}}
2747:
2727:
2707:
2687:
2685:{\displaystyle c_{v}}
2649:and the liquid–vapor
2637:
2565:
2470:
2393:
2369:
2349:
2308:
2250:
2230:
2204:
2174:
2140:
2112:
2092:
2065:
2026:
1993:
1960:
1927:
1925:{\displaystyle T_{s}}
1900:
1898:{\displaystyle T_{s}}
1873:
1871:{\displaystyle v_{g}}
1846:
1844:{\displaystyle v_{f}}
1819:
1799:
1753:
1704:
1642:
1601:
1550:, and local maximum,
1545:
1489:
1452:
1416:
1396:
1364:
1328:
1308:
1289:
1273:
1240:
1207:
1180:
1145:
1125:
1102:
1078:
1049:
1012:
988:
956:
914:
879:
850:
830:
788:
753:
673:
650:
630:
610:
586:
562:
523:
497:
477:
461:
450:
430:
406:
382:
338:
303:
279:
255:
224:
176:
152:
132:
63:
28435:"Mixture properties"
28136:Sommerfeld, pp 60-62
28114:, Canad J Chem Eng,
28031:, Canad J Chem Eng,
27854:, Canad J Chem Eng,
27522:Epstein, P.S., p. 10
27383:Maxwell construction
27094:
27071:
27012:
26858:
26740:
26624:
26519:
26508:Second, we insert a
26456:
26389:
26363:
26224:
26130:
26103:
26023:
25892:
25708:
25585:
25525:
25470:
25347:
25244:
25201:
25197:The excluded volume
25140:
25111:Van der Waals radius
25056:
24957:
24614:
24588:
24551:
24515:
24475:
24449:
24423:
24272:
24125:
23944:
23706:
23686:
23552:
23380:
23353:
23326:
23260:
23230:
23210:
23178:
23151:
23124:
23084:
23064:
23044:
23017:
22997:
22957:
22937:
22914:
22887:
22860:
22840:
22808:
22754:
22616:
22499:
22479:
22459:
22427:
22203:
22164:
22131:
22111:
22063:
22018:
21961:
21757:
21734:
21701:
21674:
21647:
21627:
21606:
21539:
21478:
21458:
21431:
21411:
21384:
21351:
21309:
21285:
21244:
21208:
21185:
21078:
21042:
21006:
20957:
20916:
20896:
20876:
20833:
20790:
20748:
20728:
20679:
20652:
20612:
20570:
20550:
20527:
20485:
20465:
20445:
20410:
20387:
20337:
20232:
20196:
20185:{\displaystyle DF=0}
20167:
20134:
20060:
20014:
19947:
19920:
19893:
19866:
19823:
19783:
19742:
19701:
19674:
19647:
19601:
19581:
19535:
19489:
19443:
19395:
19347:
19117:
19037:
18871:
18834:
18793:
18743:
18586:
18533:
18497:
18403:
18377:
18323:
18290:
18241:
18051:
18002:
17969:
17920:
17891:
17871:
17835:
17734:
17714:
17674:
17618:
17569:
17539:
17497:
17474:
17454:
17443:{\displaystyle 27/8}
17426:
17399:
17358:
17325:
17255:
17142:
17122:
17096:
17063:
16985:
16939:
16879:
16794:
16719:
16669:
16643:
16603:
16583:
16563:
16528:
16465:
16432:
16391:
16333:
16287:
16276:{\displaystyle 27/4}
16259:
16218:
16178:
16158:
16093:
16073:
16039:
15982:
15931:
15875:
15816:
15730:
15625:
15568:
15528:
15496:
15371:
15343:
15291:
15256:
15230:
15026:
15006:
14971:
14938:
14912:
14879:
14852:
14785:
14747:
14707:
14652:
14624:
14604:
14575:
14546:
14517:
14482:
14452:
14432:
14405:
14382:
14362:
14335:
14305:
14256:
14233:
14082:
14017:
13933:
13859:
13839:
13816:
13748:
13689:
13627:
13577:
13458:
13431:
13404:
13384:
13339:
13275:
13242:
13170:
13138:
13086:
12923:
12712:
12484:
12478:Maxwell Construction
12453:
12426:
12200:
12159:
12132:
12105:
12043:
12017:
11982:
11962:
11942:
11731:
11691:
11651:
11556:
11518:
11457:
11387:
11343:
11277:
11101:
11053:
11005:
10936:
10909:
10867:
10844:
10840:Although the gap in
10566:
10533:
10470:
10429:
10367:
10315:
10280:
10245:
10212:
10176:
10106:
10070:
10059:{\displaystyle dS=0}
10041:
9992:
9839:
9812:
9779:
9756:
9723:
9629:
9590:
9561:
9515:
9489:
9338:
9311:
9184:
9153:
9098:
9071:
9027:
8990:
8940:
8906:
8881:
8666:
8610:
8551:
8502:
8467:
8313:
8075:
7928:
7893:
7783:
7707:
7651:
7615:
7506:
7301:
7207:
7187:
7118:
7095:
7060:
7037:{\displaystyle _{u}}
7022:
6965:
6875:
6815:
6795:
6759:
6710:
6667:
6589:
6561:
6460:
6417:
6355:
6320:
6282:
6252:
6005:
5955:
5914:
5894:
5874:
5846:
5803:
5715:
5641:
5570:
5513:
5436:
5386:
5338:
5305:
5255:
5249:dimensional analysis
5208:
5113:
4997:
4941:
4794:
4767:
4634:
4522:
4502:
4448:
4370:
4346:
4326:
4306:
4283:
4257:
4156:
4104:
4052:
4019:
3993:Maxwell construction
3911:
3874:
3854:
3738:
3640:
3600:
3560:
3530:{\displaystyle _{3}}
3515:
3504:{\displaystyle _{2}}
3489:
3422:
3368:
3334:
3314:
3294:
3274:
3220:
3120:
3021:
2939:
2853:
2818:
2783:
2756:
2736:
2716:
2696:
2669:
2659:it establishes the
2574:
2512:
2429:or in terms of
2402:
2382:
2358:
2317:
2262:
2239:
2213:
2187:
2149:
2129:
2101:
2074:
2035:
2002:
1969:
1936:
1909:
1882:
1855:
1828:
1808:
1762:
1716:
1651:
1610:
1554:
1498:
1461:
1425:
1405:
1373:
1337:
1317:
1297:
1249:
1216:
1189:
1162:
1134:
1114:
1091:
1058:
1021:
1001:
969:
927:
895:
859:
839:
797:
762:
682:
659:
639:
619:
599:
575:
532:
506:
486:
466:
439:
419:
395:
355:
312:
292:
268:
237:
189:
165:
141:
75:
52:
28647:1966AdAMP...2....1D
28276:10.1021/cr60137a013
28171:, Ind. Eng. Chem.,
28001:Callen, pp. 146–156
26647:
25811:
25773:
25684:
25373:
25259:
24907:
24815:
24695:
24631:
24603:{\displaystyle a,b}
24464:{\displaystyle a,b}
24076:
24047:
23906:
23523:
23497:
23468:
22804:of each substance,
22802:chemical potentials
22575:
22548:
22523:
21885:
21563:
21502:
20566:. Since the curve,
19597:sufficiently large
19332:
19268:
19221:
19157:
18851:{\displaystyle b/3}
18392:{\displaystyle a=0}
17906:{\displaystyle k+1}
17111:{\displaystyle Z=1}
16658:{\displaystyle Z=1}
15787:
15360:{\displaystyle b/v}
14927:{\displaystyle T,p}
14193:
14172:
11868:
11815:
11260:
11176:
10819:
10703:
10466:are liquid and for
10012:
9970:
8203:
7839:
6578:{\displaystyle p/T}
6399:
6375:
5191:
4926:
4886:
4746:
4725:
4272:{\displaystyle p,T}
3445:James Clerk Maxwell
2228:{\displaystyle v-b}
1281:superheated liquids
986:{\displaystyle V/N}
912:{\displaystyle V/M}
28706:Equations of state
27987:Callen, pp. 85–101
27821:Barenblatt, G.I.,
27620:, 3433–3440 (1955)
27343:
27077:
27053:
26992:
26851:Hence, we obtain:
26838:
26779:
26695:
26633:
26604:
26494:
26419:
26369:
26339:
26334:
26317:
26260:
26203:Boltzmann constant
26172:
26109:
26086:
26060:
26000:partition function
25973:
25855:
25795:
25757:
25687:
25668:
25563:
25497:
25434:
25361:
25326:
25247:
25207:
25184:
25092:
25018:
24942:
24890:
24798:
24722:
24678:
24617:
24600:
24570:
24537:
24501:
24461:
24435:
24406:
24258:
24194:
24107:
24051:
24022:
23931:
23881:
23837:
23692:
23671:
23615:
23538:
23501:
23475:
23446:
23443:
23366:
23339:
23312:
23236:
23216:
23191:
23164:
23137:
23110:
23070:
23050:
23030:
23003:
22983:
22943:
22926:{\displaystyle -p}
22923:
22900:
22873:
22846:
22826:
22790:
22740:
22595:
22558:
22527:
22502:
22485:
22465:
22445:
22413:
22189:
22150:
22117:
22097:
22045:
22004:
21947:
21871:
21851:
21815:
21740:
21720:
21687:
21660:
21633:
21612:
21592:
21542:
21525:
21481:
21464:
21444:
21417:
21397:
21370:
21337:
21303:
21291:
21271:
21230:
21194:
21171:
21064:
21028:
20992:
20932:
20902:
20882:
20862:
20819:
20776:
20734:
20714:
20665:
20638:
20598:
20556:
20539:{\displaystyle -p}
20536:
20513:
20471:
20451:
20431:
20393:
20369:
20358:
20319:
20218:
20182:
20153:
20112:
20046:
20000:
19933:
19906:
19879:
19849:
19809:
19769:
19728:
19687:
19660:
19633:
19587:
19567:
19521:
19475:
19429:
19381:
19333:
19318:
19254:
19228:
19207:
19143:
19103:
19023:
18950:
18848:
18820:
18779:
18729:
18572:
18519:
18483:
18389:
18360:
18344:
18309:
18276:
18227:
18037:
17988:
17955:
17903:
17877:
17857:
17821:
17720:
17687:
17660:
17604:
17552:
17525:
17480:
17460:
17440:
17412:
17385:
17344:
17311:
17241:
17128:
17108:
17082:
17046:
16971:
16925:
16865:
16780:
16705:
16655:
16629:
16589:
16569:
16549:
16522:
16514:
16502:
16451:
16418:
16365:
16319:
16273:
16245:
16204:
16164:
16144:
16079:
16059:
16025:
15968:
15917:
15861:
15802:
15757:
15716:
15611:
15554:
15514:
15482:
15357:
15321:
15277:
15242:
15224:
15205:
15146:
15012:
14992:
14957:
14924:
14898:
14865:
14838:
14768:
14733:
14693:
14630:
14610:
14600:) whose values of
14590:
14561:
14532:
14503:
14468:
14438:
14418:
14388:
14368:
14348:
14321:
14299:
14281:
14242:
14219:
14179:
14158:
14068:
14003:
13951:
13919:
13845:
13822:
13802:
13734:
13675:
13621:
13595:
13563:
13444:
13417:
13390:
13358:
13330:superheated liquid
13315:superheated liquid
13302:
13261:
13225:
13156:
13134:and the parameter
13132:
13124:superheated liquid
13112:
13066:
12909:
12699:
12480:; the results are
12466:
12439:
12412:
12186:
12145:
12118:
12087:
12029:
12003:
11968:
11948:
11928:
11840:
11781:
11717:
11677:
11637:
11539:
11504:
11443:
11369:
11329:
11263:
11246:
11185:
11162:
11087:
11039:
10991:
10922:
10894:
10850:
10822:
10805:
10689:
10552:
10518:
10456:
10415:
10353:
10301:
10266:
10231:
10198:
10167:
10144:
10092:
10056:
10019:
9995:
9974:
9956:
9825:
9798:
9762:
9742:
9709:
9615:
9573:
9547:
9501:
9485:Here in the limit
9475:
9324:
9297:
9166:
9139:
9084:
9054:
9013:
8976:
8926:
8893:
8867:
8775:
8652:
8596:
8537:
8488:
8453:
8363:
8275:
8189:
8169:
8097:
8061:
7914:
7868:
7825:
7805:
7769:
7693:
7637:
7601:
7481:
7435:
7323:
7287:
7193:
7173:
7107:{\displaystyle pv}
7104:
7091:, and the product
7081:
7034:
7008:
6951:
6897:
6860:
6801:
6781:
6745:
6696:
6653:
6575:
6557:. Differentiating
6547:
6446:
6403:
6382:
6358:
6341:
6306:
6265:
6238:
5983:
5941:
5900:
5880:
5852:
5828:
5789:
5698:
5627:
5556:
5499:
5422:
5372:
5324:
5291:
5221:
5194:
5177:
5099:
4980:
4927:
4912:
4872:
4780:
4753:
4732:
4711:
4620:
4508:
4485:
4434:
4352:
4332:
4312:
4289:
4269:
4240:
4142:
4090:
4038:
4012:
3981:
3880:
3860:
3840:
3724:
3626:
3586:
3527:
3501:
3464:
3428:
3408:
3354:
3320:
3300:
3280:
3270:for the pressure,
3260:
3191:
3106:
2999:
2913:
2839:
2804:
2769:
2742:
2722:
2702:
2682:
2632:
2560:
2465:
2441:
2431:
2388:
2364:
2344:
2303:
2245:
2225:
2199:
2169:
2135:
2107:
2087:
2060:
2021:
1988:
1955:
1922:
1895:
1868:
1841:
1814:
1794:
1748:
1699:
1637:
1596:
1540:
1484:
1457:The green isobar,
1447:
1411:
1391:
1359:
1323:
1303:
1291:
1268:
1235:
1202:
1175:
1140:
1120:
1097:
1073:
1044:
1007:
993:the reciprocal of
983:
951:
909:
874:
845:
825:
783:
748:
668:
645:
625:
605:
581:
569:
557:
518:
492:
472:
445:
425:
413:Boltzmann constant
401:
377:
333:
298:
274:
250:
219:
171:
147:
127:
58:
28326:Callen, pp. 47-48
28118:, 159–161, (1986)
28035:, 876–880, (1983)
27969:Callen, pp. 37–44
27858:, 159–161, (1986)
27776:, 161–163, (1982)
27747:Goodstein, p. 452
27694:, 575–590, (1869)
27686:Andrews, Thomas,
27562:Goodstein, D.L.,
27303:
27220:
27172:
27135:
27080:{\displaystyle V}
27051:
26943:
26819:
26778:
26771:
26733:integral we get:
26668:
26599:
26316:
26298:
26259:
26170:
26169:
26084:
26059:
26052:
26016:) particles, is:
25933:
25812:
25774:
25747:
25685:
25648:
25486:
25210:{\displaystyle b}
25179:
25087:
24721:
24404:
24369:
24344:
24296:
24193:
23836:
23695:{\displaystyle n}
23614:
23442:
23239:{\displaystyle v}
23219:{\displaystyle x}
23073:{\displaystyle p}
23053:{\displaystyle T}
23006:{\displaystyle x}
22946:{\displaystyle v}
22849:{\displaystyle p}
22488:{\displaystyle x}
22468:{\displaystyle v}
22395:
22354:
22320:
22279:
22238:
22120:{\displaystyle x}
21850:
21814:
21743:{\displaystyle v}
21636:{\displaystyle v}
21615:{\displaystyle v}
21474:(proportional to
21467:{\displaystyle f}
21454:the curvature of
21420:{\displaystyle v}
21294:{\displaystyle g}
21204:corresponding to
20905:{\displaystyle p}
20885:{\displaystyle g}
20737:{\displaystyle g}
20559:{\displaystyle g}
20474:{\displaystyle g}
20454:{\displaystyle p}
20396:{\displaystyle v}
20357:
19590:{\displaystyle v}
19227:
19101:
19074:
19021:
18994:
18949:
18942:
18915:
18343:
17880:{\displaystyle k}
17723:{\displaystyle Z}
17483:{\displaystyle 0}
17463:{\displaystyle T}
17131:{\displaystyle T}
17001:
16863:
16826:
16592:{\displaystyle p}
16572:{\displaystyle Z}
16214:the zeros are at
16167:{\displaystyle 2}
16142:
16082:{\displaystyle 1}
16011:
15310:
15145:
15132:
14836:
14428:as a function of
13950:
13913:
13892:
13882:
13869:
13393:{\displaystyle v}
12995:
12907:
12839:
12805:
12743:
12697:
12600:
12524:
12399:
12379:
12339:
12286:
12251:
12115:
12097:, is the Maxwell
11971:{\displaystyle p}
11951:{\displaystyle v}
11300:
11287:
11261:
11236:
11184:
11177:
11152:
10853:{\displaystyle v}
10820:
10773:
10761:
10705:
10646:
10634:
10228:
10192:
10165:Figure 1 repeated
9910:
9467:
8865:
8774:
8767:
8451:
8426:
8384:
8362:
8355:
8334:
8168:
8096:
7887:Gibbs free energy
7881:Gibbs free energy
7804:
7703:we obtain simply
7592:
7565:
7472:
7434:
7322:
7196:{\displaystyle h}
7002:
6896:
6804:{\displaystyle c}
6278:Now in order for
6224:
6096:
5903:{\displaystyle s}
5883:{\displaystyle u}
5819:
5769:
5748:
5738:
5725:
5192:
5167:
5096:
5062:
5028:
4977:
4964:
4951:
4919:
4906:
4879:
4863:
4844:
4828:
4813:
4777:
4739:
4718:
4692:
4653:
4580:
4561:
4532:
4511:{\displaystyle v}
4458:
4402:
4380:
4355:{\displaystyle v}
4335:{\displaystyle T}
4315:{\displaystyle p}
4292:{\displaystyle v}
4139:
4120:
4035:
3883:{\displaystyle b}
3863:{\displaystyle a}
3431:{\displaystyle k}
3389:
3352:
3323:{\displaystyle m}
3303:{\displaystyle n}
3283:{\displaystyle p}
3250:
3179:
3145:
3094:
3070:
2993:
2972:
2962:
2949:
2745:{\displaystyle s}
2725:{\displaystyle u}
2705:{\displaystyle T}
2655:metastable states
2440:
2430:
2391:{\displaystyle v}
2367:{\displaystyle p}
2248:{\displaystyle v}
2138:{\displaystyle v}
2110:{\displaystyle p}
2097:is a function of
2084:
2057:
2018:
1985:
1952:
1817:{\displaystyle v}
1481:
1441:
1414:{\displaystyle T}
1356:
1326:{\displaystyle v}
1306:{\displaystyle T}
1265:
1232:
1199:
1172:
1143:{\displaystyle b}
1123:{\displaystyle a}
1100:{\displaystyle k}
1070:
1041:
1010:{\displaystyle R}
940:
848:{\displaystyle I}
819:
780:
745:
608:{\displaystyle b}
584:{\displaystyle a}
495:{\displaystyle r}
475:{\displaystyle r}
448:{\displaystyle b}
428:{\displaystyle a}
404:{\displaystyle k}
371:
345:physical quantity
330:
301:{\displaystyle N}
277:{\displaystyle V}
262:Avogadro constant
247:
208:
174:{\displaystyle T}
150:{\displaystyle p}
125:
105:
33:that extends the
31:equation of state
28728:
28687:
28668:
28628:
28620:
28610:
28582:
28581:
28579:
28577:
28547:
28541:
28540:
28532:
28526:
28525:
28523:
28521:
28501:
28495:
28492:
28486:
28483:
28477:
28470:
28464:
28461:
28455:
28454:
28452:
28450:
28433:Niemeyer, Kyle.
28430:
28424:
28423:
28421:
28419:
28408:Maths for Divers
28399:
28393:
28390:
28384:
28377:
28371:
28368:
28362:
28359:
28353:
28342:
28336:
28333:
28327:
28324:
28318:
28311:
28305:
28302:
28296:
28293:
28287:
28286:
28284:
28282:
28264:Chemical Reviews
28261:
28252:
28243:
28240:
28234:
28227:
28221:
28218:
28212:
28209:
28203:
28200:
28194:
28191:
28185:
28182:
28176:
28165:
28159:
28152:
28146:
28145:Sommerfeld, p 61
28143:
28137:
28134:
28128:
28125:
28119:
28108:
28102:
28099:
28093:
28092:Sommerfeld, p 66
28090:
28084:
28081:
28075:
28064:
28058:
28055:
28049:
28042:
28036:
28025:
28019:
28008:
28002:
27999:
27988:
27985:
27979:
27976:
27970:
27967:
27961:
27958:
27952:
27945:
27939:
27936:
27930:
27927:
27921:
27918:
27912:
27909:
27903:
27900:
27894:
27891:
27885:
27878:
27872:
27865:
27859:
27848:
27842:
27837:Johnston, D.C.,
27835:
27826:
27819:
27813:
27810:
27804:
27801:
27795:
27794:Goodstein, p 449
27792:
27786:
27783:
27777:
27772:, Am. J. Phys.,
27766:
27757:
27754:
27748:
27745:
27739:
27736:
27730:
27727:
27721:
27718:
27712:
27701:
27695:
27684:
27678:
27677:Boltzmann, p 218
27675:
27669:
27666:
27660:
27653:
27647:
27640:
27634:
27627:
27621:
27610:
27604:
27603:, 446–449 (1982)
27593:
27587:
27586:, 169–187 (1986)
27578:Lienhard, J.H.,
27576:
27567:
27564:States of Matter
27560:
27554:
27551:
27545:
27540:Sommerfeld, A.,
27538:
27532:
27529:
27523:
27520:
27514:
27511:
27505:
27502:
27496:
27493:
27487:
27480:
27474:
27467:
27461:
27458:
27452:
27445:
27439:
27432:
27426:
27419:
27352:
27350:
27349:
27344:
27309:
27305:
27304:
27302:
27301:
27292:
27288:
27287:
27277:
27246:
27226:
27222:
27221:
27219:
27218:
27209:
27208:
27200:
27199:
27189:
27173:
27171:
27170:
27161:
27160:
27152:
27151:
27141:
27136:
27134:
27133:
27115:
27104:
27086:
27084:
27083:
27078:
27062:
27060:
27059:
27054:
27052:
27050:
27042:
27028:
27001:
26999:
26998:
26993:
26991:
26974:
26970:
26969:
26944:
26942:
26931:
26930:
26922:
26921:
26911:
26906:
26902:
26847:
26845:
26844:
26839:
26834:
26820:
26815:
26814:
26813:
26797:
26789:
26780:
26776:
26772:
26764:
26762:
26731:
26704:
26702:
26701:
26696:
26685:
26684:
26669:
26661:
26646:
26641:
26613:
26611:
26610:
26605:
26600:
26598:
26597:
26588:
26587:
26586:
26570:
26550:
26529:
26510:Boltzmann factor
26503:
26501:
26500:
26495:
26490:
26485:
26484:
26466:
26428:
26426:
26425:
26420:
26412:
26407:
26406:
26378:
26376:
26375:
26370:
26348:
26346:
26345:
26340:
26338:
26337:
26318:
26314:
26309:
26308:
26303:
26299:
26291:
26261:
26257:
26181:
26179:
26178:
26173:
26171:
26168:
26151:
26150:
26141:
26140:
26118:
26116:
26115:
26110:
26095:
26093:
26092:
26087:
26085:
26083:
26082:
26070:
26061:
26057:
26053:
26051:
26043:
26042:
26033:
25982:
25980:
25979:
25974:
25939:
25935:
25934:
25932:
25931:
25922:
25918:
25917:
25907:
25884:
25864:
25862:
25861:
25856:
25833:
25832:
25831:
25818:
25814:
25813:
25810:
25805:
25804:
25791:
25775:
25772:
25767:
25766:
25753:
25748:
25746:
25739:
25738:
25737:
25726:
25718:
25696:
25694:
25693:
25688:
25686:
25683:
25678:
25677:
25664:
25659:
25658:
25653:
25649:
25647:
25646:
25645:
25635:
25634:
25633:
25623:
25616:
25605:
25604:
25595:
25572:
25570:
25569:
25564:
25562:
25561:
25560:
25550:
25545:
25544:
25543:
25506:
25504:
25503:
25498:
25496:
25488:
25487:
25484:
25460:
25453:
25443:
25441:
25440:
25435:
25427:
25422:
25421:
25394:
25378:
25369:
25357:
25335:
25333:
25332:
25327:
25319:
25314:
25313:
25283:
25278:
25277:
25255:
25216:
25214:
25213:
25208:
25193:
25191:
25190:
25185:
25180:
25178:
25171:
25170:
25169:
25158:
25150:
25101:
25099:
25098:
25093:
25088:
25086:
25085:
25084:
25074:
25066:
25027:
25025:
25024:
25019:
25011:
25010:
25001:
25000:
24991:
24986:
24985:
24967:
24951:
24949:
24948:
24943:
24941:
24940:
24932:
24928:
24927:
24926:
24917:
24916:
24906:
24901:
24887:
24882:
24866:
24861:
24840:
24836:
24835:
24834:
24825:
24824:
24814:
24809:
24797:
24796:
24783:
24778:
24762:
24757:
24734:
24733:
24723:
24719:
24715:
24714:
24705:
24704:
24694:
24689:
24675:
24670:
24654:
24649:
24630:
24625:
24609:
24607:
24606:
24601:
24579:
24577:
24576:
24571:
24569:
24568:
24546:
24544:
24543:
24538:
24536:
24535:
24510:
24508:
24507:
24502:
24500:
24499:
24487:
24486:
24470:
24468:
24467:
24462:
24444:
24442:
24441:
24436:
24415:
24413:
24412:
24407:
24405:
24403:
24402:
24390:
24389:
24388:
24372:
24370:
24362:
24357:
24356:
24345:
24343:
24342:
24330:
24329:
24328:
24319:
24318:
24299:
24297:
24289:
24284:
24283:
24267:
24265:
24264:
24259:
24257:
24256:
24247:
24246:
24232:
24227:
24209:
24208:
24195:
24191:
24187:
24186:
24177:
24176:
24163:
24158:
24140:
24139:
24116:
24114:
24113:
24108:
24106:
24105:
24096:
24095:
24086:
24085:
24075:
24071:
24062:
24046:
24042:
24033:
24017:
24012:
23996:
23991:
23970:
23956:
23955:
23940:
23938:
23937:
23932:
23930:
23929:
23924:
23923:
23916:
23915:
23905:
23901:
23892:
23879:
23874:
23859:
23858:
23849:
23848:
23838:
23834:
23830:
23829:
23820:
23819:
23810:
23809:
23805:
23792:
23791:
23779:
23778:
23762:
23757:
23741:
23736:
23718:
23717:
23701:
23699:
23698:
23693:
23680:
23678:
23677:
23672:
23664:
23656:
23655:
23643:
23642:
23627:
23626:
23616:
23612:
23608:
23607:
23603:
23590:
23589:
23580:
23579:
23564:
23563:
23547:
23545:
23544:
23539:
23531:
23522:
23518:
23509:
23496:
23492:
23483:
23467:
23463:
23454:
23444:
23440:
23436:
23435:
23431:
23418:
23417:
23408:
23407:
23392:
23391:
23375:
23373:
23372:
23367:
23365:
23364:
23348:
23346:
23345:
23340:
23338:
23337:
23321:
23319:
23318:
23313:
23311:
23310:
23298:
23297:
23285:
23284:
23272:
23271:
23245:
23243:
23242:
23237:
23225:
23223:
23222:
23217:
23200:
23198:
23197:
23192:
23190:
23189:
23173:
23171:
23170:
23165:
23163:
23162:
23146:
23144:
23143:
23138:
23136:
23135:
23119:
23117:
23116:
23111:
23109:
23108:
23096:
23095:
23079:
23077:
23076:
23071:
23059:
23057:
23056:
23051:
23039:
23037:
23036:
23031:
23029:
23028:
23012:
23010:
23009:
23004:
22992:
22990:
22989:
22984:
22982:
22981:
22969:
22968:
22952:
22950:
22949:
22944:
22932:
22930:
22929:
22924:
22909:
22907:
22906:
22901:
22899:
22898:
22882:
22880:
22879:
22874:
22872:
22871:
22855:
22853:
22852:
22847:
22835:
22833:
22832:
22827:
22799:
22797:
22796:
22791:
22786:
22785:
22784:
22783:
22766:
22765:
22749:
22747:
22746:
22741:
22739:
22738:
22720:
22719:
22707:
22706:
22679:
22678:
22669:
22668:
22656:
22655:
22646:
22645:
22604:
22602:
22601:
22596:
22588:
22587:
22574:
22569:
22547:
22542:
22541:
22540:
22522:
22517:
22516:
22515:
22494:
22492:
22491:
22486:
22474:
22472:
22471:
22466:
22454:
22452:
22451:
22446:
22422:
22420:
22419:
22414:
22406:
22405:
22400:
22396:
22394:
22380:
22376:
22375:
22365:
22355:
22353:
22352:
22351:
22338:
22334:
22333:
22323:
22321:
22319:
22318:
22317:
22304:
22300:
22299:
22289:
22280:
22278:
22277:
22276:
22263:
22259:
22258:
22248:
22239:
22237:
22236:
22235:
22222:
22218:
22217:
22207:
22198:
22196:
22195:
22190:
22176:
22175:
22159:
22157:
22156:
22151:
22149:
22148:
22126:
22124:
22123:
22118:
22106:
22104:
22103:
22098:
22081:
22080:
22054:
22052:
22051:
22046:
22041:
22030:
22029:
22013:
22011:
22010:
22005:
22000:
21999:
21984:
21973:
21972:
21956:
21954:
21953:
21948:
21943:
21942:
21927:
21901:
21900:
21884:
21879:
21858:
21857:
21852:
21848:
21835:
21834:
21822:
21821:
21816:
21812:
21805:
21804:
21795:
21794:
21782:
21781:
21769:
21768:
21749:
21747:
21746:
21741:
21729:
21727:
21726:
21721:
21719:
21718:
21696:
21694:
21693:
21688:
21686:
21685:
21669:
21667:
21666:
21661:
21659:
21658:
21642:
21640:
21639:
21634:
21621:
21619:
21618:
21613:
21601:
21599:
21598:
21593:
21582:
21581:
21562:
21557:
21556:
21555:
21534:
21532:
21531:
21526:
21521:
21520:
21501:
21496:
21495:
21494:
21473:
21471:
21470:
21465:
21453:
21451:
21450:
21445:
21443:
21442:
21426:
21424:
21423:
21418:
21406:
21404:
21403:
21398:
21396:
21395:
21379:
21377:
21376:
21371:
21369:
21368:
21346:
21344:
21343:
21338:
21327:
21326:
21300:
21298:
21297:
21292:
21280:
21278:
21277:
21272:
21267:
21256:
21255:
21239:
21237:
21236:
21231:
21223:
21222:
21203:
21201:
21200:
21195:
21180:
21178:
21177:
21172:
21170:
21169:
21154:
21153:
21138:
21133:
21132:
21123:
21122:
21107:
21106:
21097:
21096:
21095:
21094:
21073:
21071:
21070:
21065:
21057:
21056:
21037:
21035:
21034:
21029:
21021:
21020:
21001:
20999:
20998:
20993:
20988:
20987:
20969:
20968:
20941:
20939:
20938:
20933:
20928:
20927:
20911:
20909:
20908:
20903:
20891:
20889:
20888:
20883:
20871:
20869:
20868:
20863:
20858:
20857:
20845:
20844:
20828:
20826:
20825:
20820:
20815:
20814:
20802:
20801:
20785:
20783:
20782:
20777:
20766:
20765:
20743:
20741:
20740:
20735:
20723:
20721:
20720:
20715:
20710:
20709:
20697:
20696:
20674:
20672:
20671:
20666:
20664:
20663:
20647:
20645:
20644:
20639:
20637:
20636:
20624:
20623:
20607:
20605:
20604:
20599:
20588:
20587:
20565:
20563:
20562:
20557:
20546:, and intercept
20545:
20543:
20542:
20537:
20522:
20520:
20519:
20514:
20503:
20502:
20480:
20478:
20477:
20472:
20460:
20458:
20457:
20452:
20440:
20438:
20437:
20432:
20402:
20400:
20399:
20394:
20378:
20376:
20375:
20370:
20365:
20360:
20359:
20355:
20328:
20326:
20325:
20320:
20312:
20311:
20306:
20294:
20277:
20276:
20271:
20265:
20264:
20252:
20244:
20243:
20227:
20225:
20224:
20219:
20208:
20207:
20191:
20189:
20188:
20183:
20162:
20160:
20159:
20154:
20152:
20151:
20121:
20119:
20118:
20113:
20111:
20110:
20101:
20100:
20088:
20087:
20078:
20077:
20055:
20053:
20052:
20047:
20039:
20038:
20026:
20025:
20009:
20007:
20006:
20001:
19996:
19988:
19987:
19975:
19974:
19959:
19958:
19942:
19940:
19939:
19934:
19932:
19931:
19915:
19913:
19912:
19907:
19905:
19904:
19888:
19886:
19885:
19880:
19878:
19877:
19858:
19856:
19855:
19850:
19848:
19847:
19835:
19834:
19818:
19816:
19815:
19810:
19808:
19807:
19795:
19794:
19778:
19776:
19775:
19770:
19768:
19767:
19754:
19753:
19737:
19735:
19734:
19729:
19727:
19726:
19713:
19712:
19696:
19694:
19693:
19688:
19686:
19685:
19669:
19667:
19666:
19661:
19659:
19658:
19642:
19640:
19639:
19634:
19632:
19631:
19619:
19618:
19596:
19594:
19593:
19588:
19576:
19574:
19573:
19568:
19566:
19565:
19553:
19552:
19530:
19528:
19527:
19522:
19514:
19513:
19501:
19500:
19484:
19482:
19481:
19476:
19474:
19473:
19461:
19460:
19438:
19436:
19435:
19430:
19425:
19420:
19419:
19407:
19406:
19390:
19388:
19387:
19382:
19377:
19372:
19371:
19359:
19358:
19342:
19340:
19339:
19334:
19331:
19326:
19317:
19316:
19304:
19303:
19294:
19293:
19284:
19283:
19267:
19262:
19253:
19252:
19240:
19239:
19229:
19225:
19220:
19215:
19206:
19205:
19193:
19192:
19183:
19182:
19173:
19172:
19156:
19151:
19142:
19141:
19129:
19128:
19112:
19110:
19109:
19104:
19102:
19100:
19099:
19090:
19089:
19080:
19075:
19073:
19072:
19071:
19055:
19047:
19032:
19030:
19029:
19024:
19022:
19020:
19019:
19010:
19009:
19000:
18995:
18993:
18992:
18991:
18975:
18967:
18962:
18961:
18951:
18947:
18943:
18941:
18940:
18931:
18930:
18921:
18916:
18914:
18913:
18912:
18896:
18888:
18883:
18882:
18857:
18855:
18854:
18849:
18844:
18829:
18827:
18826:
18821:
18819:
18818:
18806:
18788:
18786:
18785:
18780:
18778:
18777:
18762:
18738:
18736:
18735:
18730:
18719:
18718:
18703:
18698:
18697:
18670:
18635:
18615:
18614:
18605:
18581:
18579:
18578:
18573:
18571:
18570:
18546:
18528:
18526:
18525:
18520:
18518:
18517:
18492:
18490:
18489:
18484:
18482:
18481:
18462:
18457:
18439:
18438:
18398:
18396:
18395:
18390:
18369:
18367:
18366:
18361:
18356:
18345:
18341:
18335:
18334:
18318:
18316:
18315:
18310:
18302:
18301:
18285:
18283:
18282:
18277:
18272:
18271:
18259:
18258:
18236:
18234:
18233:
18228:
18226:
18225:
18213:
18208:
18207:
18193:
18188:
18164:
18159:
18158:
18143:
18142:
18127:
18126:
18111:
18082:
18081:
18069:
18068:
18046:
18044:
18043:
18038:
18036:
18035:
18023:
18022:
17997:
17995:
17994:
17989:
17981:
17980:
17964:
17962:
17961:
17956:
17951:
17950:
17938:
17937:
17912:
17910:
17909:
17904:
17886:
17884:
17883:
17878:
17866:
17864:
17863:
17858:
17847:
17846:
17830:
17828:
17827:
17822:
17820:
17819:
17795:
17794:
17783:
17778:
17729:
17727:
17726:
17721:
17706:Virial expansion
17696:
17694:
17693:
17688:
17686:
17685:
17669:
17667:
17666:
17661:
17656:
17655:
17643:
17642:
17630:
17629:
17613:
17611:
17610:
17605:
17600:
17599:
17587:
17586:
17561:
17559:
17558:
17553:
17551:
17550:
17534:
17532:
17531:
17526:
17521:
17520:
17489:
17487:
17486:
17481:
17469:
17467:
17466:
17461:
17449:
17447:
17446:
17441:
17436:
17421:
17419:
17418:
17413:
17411:
17410:
17394:
17392:
17391:
17386:
17381:
17370:
17369:
17353:
17351:
17350:
17345:
17337:
17336:
17320:
17318:
17317:
17312:
17307:
17302:
17301:
17292:
17291:
17267:
17266:
17250:
17248:
17247:
17242:
17231:
17230:
17215:
17204:
17203:
17194:
17193:
17175:
17161:
17160:
17159:
17158:
17137:
17135:
17134:
17129:
17117:
17115:
17114:
17109:
17091:
17089:
17088:
17083:
17075:
17074:
17055:
17053:
17052:
17047:
17042:
17028:
17014:
17003:
17002:
16999:
16980:
16978:
16977:
16972:
16964:
16963:
16951:
16950:
16934:
16932:
16931:
16926:
16918:
16917:
16908:
16897:
16896:
16874:
16872:
16871:
16866:
16864:
16862:
16861:
16860:
16847:
16846:
16845:
16832:
16827:
16825:
16824:
16823:
16804:
16789:
16787:
16786:
16781:
16767:
16735:
16714:
16712:
16711:
16706:
16664:
16662:
16661:
16656:
16638:
16636:
16635:
16630:
16619:
16598:
16596:
16595:
16590:
16578:
16576:
16575:
16570:
16558:
16556:
16555:
16550:
16511:
16509:
16508:
16503:
16498:
16493:
16492:
16477:
16476:
16460:
16458:
16457:
16452:
16444:
16443:
16427:
16425:
16424:
16419:
16414:
16403:
16402:
16374:
16372:
16371:
16366:
16358:
16357:
16356:
16355:
16329:, and occurs at
16328:
16326:
16325:
16320:
16312:
16311:
16310:
16309:
16282:
16280:
16279:
16274:
16269:
16254:
16252:
16251:
16246:
16241:
16230:
16229:
16213:
16211:
16210:
16205:
16203:
16202:
16190:
16189:
16173:
16171:
16170:
16165:
16153:
16151:
16150:
16145:
16143:
16135:
16124:
16113:
16112:
16103:
16088:
16086:
16085:
16080:
16068:
16066:
16065:
16060:
16055:
16034:
16032:
16031:
16026:
16021:
16012:
16001:
15977:
15975:
15974:
15969:
15946:
15945:
15926:
15924:
15923:
15918:
15913:
15908:
15907:
15895:
15894:
15893:
15870:
15868:
15867:
15862:
15857:
15852:
15851:
15836:
15835:
15834:
15811:
15809:
15808:
15803:
15798:
15786:
15782:
15773:
15772:
15756:
15755:
15743:
15725:
15723:
15722:
15717:
15709:
15708:
15704:
15691:
15690:
15678:
15655:
15654:
15645:
15637:
15636:
15620:
15618:
15617:
15612:
15607:
15606:
15591:
15580:
15579:
15563:
15561:
15560:
15555:
15553:
15552:
15540:
15539:
15523:
15521:
15520:
15515:
15491:
15489:
15488:
15483:
15478:
15477:
15465:
15448:
15447:
15443:
15430:
15429:
15417:
15391:
15390:
15381:
15366:
15364:
15363:
15358:
15353:
15330:
15328:
15327:
15322:
15317:
15312:
15311:
15308:
15286:
15284:
15283:
15278:
15251:
15249:
15248:
15243:
15214:
15212:
15211:
15206:
15195:
15194:
15176:
15175:
15147:
15143:
15133:
15131:
15130:
15129:
15099:
15098:
15082:
15078:
15077:
15059:
15058:
15030:
15021:
15019:
15018:
15013:
15001:
14999:
14998:
14993:
14966:
14964:
14963:
14958:
14950:
14949:
14933:
14931:
14930:
14925:
14907:
14905:
14904:
14899:
14891:
14890:
14874:
14872:
14871:
14866:
14864:
14863:
14847:
14845:
14844:
14839:
14837:
14835:
14834:
14825:
14802:
14797:
14796:
14777:
14775:
14774:
14769:
14742:
14740:
14739:
14734:
14732:
14731:
14719:
14718:
14702:
14700:
14699:
14694:
14692:
14691:
14686:
14677:
14676:
14664:
14663:
14639:
14637:
14636:
14631:
14619:
14617:
14616:
14611:
14599:
14597:
14596:
14591:
14570:
14568:
14567:
14562:
14541:
14539:
14538:
14533:
14512:
14510:
14509:
14504:
14477:
14475:
14474:
14469:
14467:
14466:
14447:
14445:
14444:
14439:
14427:
14425:
14424:
14419:
14417:
14416:
14397:
14395:
14394:
14389:
14377:
14375:
14374:
14369:
14357:
14355:
14354:
14349:
14347:
14346:
14330:
14328:
14327:
14322:
14320:
14319:
14290:
14288:
14287:
14282:
14274:
14273:
14251:
14249:
14248:
14243:
14228:
14226:
14225:
14220:
14215:
14214:
14192:
14187:
14171:
14166:
14136:
14135:
14126:
14103:
14102:
14077:
14075:
14074:
14069:
14058:
14057:
14045:
14044:
14032:
14031:
14012:
14010:
14009:
14004:
13984:
13983:
13971:
13970:
13958:
13957:
13952:
13948:
13928:
13926:
13925:
13920:
13915:
13914:
13911:
13899:
13894:
13893:
13890:
13884:
13883:
13880:
13871:
13870:
13867:
13854:
13852:
13851:
13846:
13831:
13829:
13828:
13823:
13811:
13809:
13808:
13803:
13792:
13791:
13779:
13778:
13760:
13759:
13743:
13741:
13740:
13735:
13730:
13729:
13717:
13716:
13704:
13703:
13684:
13682:
13681:
13676:
13671:
13670:
13658:
13657:
13639:
13638:
13604:
13602:
13601:
13596:
13572:
13570:
13569:
13564:
13559:
13558:
13546:
13545:
13533:
13528:
13527:
13508:
13507:
13495:
13494:
13476:
13475:
13453:
13451:
13450:
13445:
13443:
13442:
13426:
13424:
13423:
13418:
13416:
13415:
13399:
13397:
13396:
13391:
13367:
13365:
13364:
13359:
13351:
13350:
13311:
13309:
13308:
13303:
13298:
13287:
13286:
13270:
13268:
13267:
13262:
13254:
13253:
13234:
13232:
13231:
13226:
13212:
13204:
13203:
13191:
13190:
13165:
13163:
13162:
13157:
13121:
13119:
13118:
13113:
13111:
13110:
13098:
13097:
13075:
13073:
13072:
13067:
13065:
13064:
12996:
12994:
12968:
12942:
12918:
12916:
12915:
12910:
12908:
12906:
12905:
12904:
12879:
12878:
12877:
12846:
12844:
12840:
12832:
12823:
12822:
12806:
12804:
12803:
12802:
12780:
12779:
12778:
12750:
12748:
12744:
12736:
12727:
12726:
12708:
12706:
12705:
12700:
12698:
12696:
12695:
12694:
12675:
12671:
12670:
12643:
12642:
12623:
12615:
12614:
12601:
12599:
12598:
12597:
12578:
12531:
12529:
12525:
12517:
12499:
12498:
12475:
12473:
12472:
12467:
12465:
12464:
12448:
12446:
12445:
12440:
12438:
12437:
12421:
12419:
12418:
12413:
12405:
12401:
12400:
12398:
12397:
12385:
12380:
12378:
12377:
12365:
12349:
12345:
12344:
12340:
12338:
12331:
12330:
12320:
12313:
12312:
12302:
12287:
12285:
12278:
12277:
12267:
12266:
12257:
12252:
12250:
12243:
12242:
12232:
12231:
12222:
12215:
12214:
12195:
12193:
12192:
12187:
12154:
12152:
12151:
12146:
12144:
12143:
12127:
12125:
12124:
12119:
12117:
12116:
12113:
12096:
12094:
12093:
12088:
12038:
12036:
12035:
12030:
12028:
12027:
12012:
12010:
12009:
12004:
11977:
11975:
11974:
11969:
11957:
11955:
11954:
11949:
11937:
11935:
11934:
11929:
11918:
11917:
11905:
11904:
11892:
11891:
11867:
11866:
11865:
11855:
11854:
11853:
11826:
11825:
11814:
11807:
11806:
11796:
11795:
11794:
11779:
11774:
11756:
11755:
11743:
11742:
11726:
11724:
11723:
11718:
11716:
11715:
11703:
11702:
11686:
11684:
11683:
11678:
11676:
11675:
11663:
11662:
11646:
11644:
11643:
11638:
11600:
11599:
11577:
11576:
11548:
11546:
11545:
11540:
11513:
11511:
11510:
11505:
11503:
11502:
11490:
11489:
11477:
11476:
11475:
11452:
11450:
11449:
11444:
11439:
11438:
11426:
11425:
11407:
11406:
11405:
11378:
11376:
11375:
11370:
11368:
11367:
11355:
11354:
11338:
11336:
11335:
11330:
11328:
11327:
11315:
11314:
11302:
11301:
11298:
11289:
11288:
11285:
11272:
11270:
11269:
11264:
11262:
11259:
11254:
11242:
11237:
11235:
11228:
11227:
11217:
11216:
11215:
11202:
11197:
11196:
11186:
11182:
11178:
11175:
11170:
11158:
11153:
11151:
11144:
11143:
11133:
11132:
11131:
11118:
11113:
11112:
11096:
11094:
11093:
11088:
11086:
11085:
11084:
11065:
11064:
11048:
11046:
11045:
11040:
11038:
11037:
11036:
11017:
11016:
11000:
10998:
10997:
10992:
10990:
10989:
10988:
10969:
10968:
10956:
10955:
10954:
10932:, by a pressure
10931:
10929:
10928:
10923:
10921:
10920:
10903:
10901:
10900:
10895:
10890:
10879:
10878:
10859:
10857:
10856:
10851:
10831:
10829:
10828:
10823:
10821:
10818:
10813:
10804:
10794:
10793:
10777:
10775:
10774:
10771:
10762:
10760:
10759:
10750:
10727:
10722:
10721:
10720:
10706:
10704:
10702:
10697:
10684:
10683:
10682:
10667:
10666:
10650:
10648:
10647:
10644:
10635:
10633:
10632:
10631:
10618:
10617:
10616:
10594:
10583:
10582:
10581:
10561:
10559:
10558:
10553:
10545:
10544:
10527:
10525:
10524:
10519:
10517:
10516:
10515:
10496:
10495:
10494:
10465:
10463:
10462:
10457:
10455:
10454:
10453:
10424:
10422:
10421:
10416:
10414:
10413:
10412:
10387:
10386:
10385:
10362:
10360:
10359:
10354:
10346:
10345:
10340:
10328:
10310:
10308:
10307:
10302:
10300:
10299:
10298:
10275:
10273:
10272:
10267:
10265:
10264:
10263:
10240:
10238:
10237:
10232:
10230:
10229:
10226:
10207:
10205:
10204:
10199:
10194:
10193:
10190:
10153:
10151:
10150:
10145:
10137:
10136:
10131:
10119:
10101:
10099:
10098:
10093:
10082:
10081:
10065:
10063:
10062:
10057:
10028:
10026:
10025:
10020:
10011:
10003:
9983:
9981:
9980:
9975:
9969:
9964:
9949:
9944:
9943:
9928:
9927:
9912:
9911:
9908:
9896:
9895:
9880:
9875:
9874:
9834:
9832:
9831:
9826:
9824:
9823:
9807:
9805:
9804:
9799:
9791:
9790:
9771:
9769:
9768:
9763:
9751:
9749:
9748:
9743:
9735:
9734:
9718:
9716:
9715:
9710:
9702:
9701:
9692:
9678:
9677:
9668:
9660:
9659:
9641:
9640:
9624:
9622:
9621:
9616:
9608:
9607:
9582:
9580:
9579:
9574:
9556:
9554:
9553:
9548:
9540:
9539:
9527:
9526:
9510:
9508:
9507:
9502:
9484:
9482:
9481:
9476:
9468:
9466:
9465:
9464:
9434:
9433:
9417:
9416:
9415:
9403:
9402:
9392:
9378:
9377:
9350:
9349:
9333:
9331:
9330:
9325:
9323:
9322:
9306:
9304:
9303:
9298:
9296:
9295:
9286:
9281:
9280:
9259:
9258:
9249:
9244:
9243:
9231:
9230:
9209:
9208:
9196:
9195:
9175:
9173:
9172:
9167:
9165:
9164:
9148:
9146:
9145:
9140:
9138:
9137:
9132:
9123:
9122:
9110:
9109:
9093:
9091:
9090:
9085:
9083:
9082:
9063:
9061:
9060:
9055:
9050:
9039:
9038:
9022:
9020:
9019:
9014:
9009:
8985:
8983:
8982:
8977:
8963:
8952:
8951:
8935:
8933:
8932:
8927:
8922:
8902:
8900:
8899:
8894:
8876:
8874:
8873:
8868:
8866:
8864:
8863:
8862:
8832:
8831:
8815:
8799:
8798:
8785:
8776:
8772:
8768:
8766:
8765:
8764:
8734:
8733:
8717:
8716:
8715:
8694:
8693:
8683:
8678:
8677:
8661:
8659:
8658:
8653:
8651:
8650:
8641:
8640:
8631:
8630:
8605:
8603:
8602:
8597:
8592:
8591:
8582:
8581:
8563:
8562:
8546:
8544:
8543:
8538:
8536:
8535:
8523:
8522:
8497:
8495:
8494:
8489:
8487:
8486:
8477:
8462:
8460:
8459:
8454:
8452:
8450:
8449:
8440:
8432:
8427:
8425:
8424:
8423:
8401:
8393:
8385:
8383:
8375:
8367:
8364:
8360:
8356:
8354:
8340:
8335:
8333:
8325:
8317:
8284:
8282:
8281:
8276:
8271:
8270:
8255:
8244:
8243:
8219:
8218:
8202:
8197:
8176:
8175:
8170:
8166:
8150:
8149:
8134:
8129:
8128:
8104:
8103:
8098:
8094:
8087:
8086:
8070:
8068:
8067:
8062:
8057:
8016:
8015:
7994:
7993:
7966:
7946:
7945:
7923:
7921:
7920:
7915:
7877:
7875:
7874:
7869:
7855:
7854:
7838:
7833:
7812:
7811:
7806:
7802:
7795:
7794:
7778:
7776:
7775:
7770:
7750:
7749:
7725:
7724:
7702:
7700:
7699:
7694:
7677:
7663:
7662:
7646:
7644:
7643:
7638:
7627:
7626:
7610:
7608:
7607:
7602:
7593:
7591:
7583:
7575:
7566:
7561:
7553:
7541:
7540:
7524:
7523:
7490:
7488:
7487:
7482:
7474:
7473:
7470:
7458:
7447:
7446:
7436:
7432:
7425:
7424:
7409:
7398:
7397:
7376:
7375:
7360:
7355:
7354:
7330:
7329:
7324:
7320:
7313:
7312:
7296:
7294:
7293:
7288:
7283:
7251:
7225:
7224:
7202:
7200:
7199:
7194:
7182:
7180:
7179:
7174:
7169:
7143:
7113:
7111:
7110:
7105:
7090:
7088:
7087:
7082:
7043:
7041:
7040:
7035:
7033:
7032:
7017:
7015:
7014:
7009:
7004:
7003:
7000:
6988:
6977:
6976:
6960:
6958:
6957:
6952:
6947:
6946:
6931:
6920:
6919:
6904:
6903:
6898:
6894:
6887:
6886:
6869:
6867:
6866:
6861:
6856:
6833:
6832:
6810:
6808:
6807:
6802:
6790:
6788:
6787:
6782:
6771:
6770:
6754:
6752:
6751:
6746:
6735:
6734:
6722:
6721:
6705:
6703:
6702:
6697:
6689:
6688:
6679:
6678:
6662:
6660:
6659:
6654:
6652:
6651:
6642:
6622:
6611:
6610:
6601:
6600:
6584:
6582:
6581:
6576:
6571:
6556:
6554:
6553:
6548:
6537:
6526:
6525:
6516:
6515:
6503:
6502:
6472:
6471:
6455:
6453:
6452:
6447:
6442:
6441:
6429:
6428:
6412:
6410:
6409:
6404:
6398:
6393:
6374:
6369:
6350:
6348:
6347:
6342:
6315:
6313:
6312:
6307:
6274:
6272:
6271:
6266:
6264:
6263:
6247:
6245:
6244:
6239:
6230:
6226:
6225:
6223:
6215:
6208:
6193:
6191:
6190:
6147:
6146:
6123:
6119:
6097:
6095:
6087:
6079:
6040:
6039:
6023:
6022:
5992:
5990:
5989:
5984:
5967:
5966:
5950:
5948:
5947:
5942:
5909:
5907:
5906:
5901:
5889:
5887:
5886:
5881:
5861:
5859:
5858:
5853:
5837:
5835:
5834:
5829:
5821:
5820:
5817:
5798:
5796:
5795:
5790:
5785:
5771:
5770:
5767:
5755:
5750:
5749:
5746:
5740:
5739:
5736:
5727:
5726:
5723:
5707:
5705:
5704:
5699:
5694:
5693:
5684:
5667:
5653:
5652:
5636:
5634:
5633:
5628:
5623:
5622:
5613:
5596:
5582:
5581:
5565:
5563:
5562:
5557:
5552:
5551:
5542:
5525:
5524:
5508:
5506:
5505:
5500:
5498:
5497:
5488:
5477:
5476:
5467:
5456:
5455:
5446:
5431:
5429:
5428:
5423:
5409:
5398:
5397:
5381:
5379:
5378:
5373:
5371:
5370:
5361:
5350:
5349:
5333:
5331:
5330:
5325:
5317:
5316:
5300:
5298:
5297:
5292:
5230:
5228:
5227:
5222:
5220:
5219:
5203:
5201:
5200:
5195:
5193:
5190:
5185:
5173:
5168:
5166:
5159:
5158:
5145:
5144:
5143:
5130:
5125:
5124:
5108:
5106:
5105:
5100:
5098:
5097:
5094:
5088:
5077:
5076:
5064:
5063:
5060:
5054:
5043:
5042:
5030:
5029:
5026:
5020:
5009:
5008:
4989:
4987:
4986:
4981:
4979:
4978:
4975:
4966:
4965:
4962:
4953:
4952:
4949:
4936:
4934:
4933:
4928:
4925:
4920:
4917:
4908:
4907:
4904:
4898:
4885:
4880:
4877:
4865:
4864:
4861:
4855:
4846:
4845:
4842:
4830:
4829:
4826:
4820:
4815:
4814:
4811:
4789:
4787:
4786:
4781:
4779:
4778:
4775:
4762:
4760:
4759:
4754:
4745:
4740:
4737:
4724:
4719:
4716:
4704:
4703:
4694:
4693:
4690:
4678:
4677:
4665:
4664:
4655:
4654:
4651:
4629:
4627:
4626:
4621:
4595:
4594:
4582:
4581:
4578:
4563:
4562:
4559:
4547:
4546:
4534:
4533:
4530:
4517:
4515:
4514:
4509:
4494:
4492:
4491:
4486:
4484:
4483:
4471:
4460:
4459:
4456:
4443:
4441:
4440:
4435:
4418:
4404:
4403:
4400:
4382:
4381:
4378:
4361:
4359:
4358:
4353:
4341:
4339:
4338:
4333:
4321:
4319:
4318:
4313:
4298:
4296:
4295:
4290:
4278:
4276:
4275:
4270:
4249:
4247:
4246:
4241:
4233:
4232:
4227:
4221:
4220:
4208:
4200:
4199:
4187:
4186:
4181:
4169:
4151:
4149:
4148:
4143:
4141:
4140:
4137:
4122:
4121:
4118:
4099:
4097:
4096:
4091:
4083:
4082:
4077:
4065:
4047:
4045:
4044:
4039:
4037:
4036:
4033:
3990:
3988:
3987:
3982:
3971:
3970:
3958:
3957:
3936:
3935:
3923:
3922:
3889:
3887:
3886:
3881:
3869:
3867:
3866:
3861:
3849:
3847:
3846:
3841:
3836:
3835:
3823:
3822:
3810:
3802:
3801:
3792:
3791:
3779:
3778:
3769:
3768:
3756:
3755:
3733:
3731:
3730:
3725:
3717:
3716:
3704:
3703:
3691:
3683:
3682:
3670:
3669:
3635:
3633:
3632:
3627:
3625:
3624:
3612:
3611:
3595:
3593:
3592:
3587:
3585:
3584:
3572:
3571:
3536:
3534:
3533:
3528:
3526:
3525:
3510:
3508:
3507:
3502:
3500:
3499:
3449:J. Willard Gibbs
3437:
3435:
3434:
3429:
3417:
3415:
3414:
3409:
3395:
3390:
3385:
3384:
3375:
3363:
3361:
3360:
3355:
3353:
3348:
3347:
3338:
3329:
3327:
3326:
3321:
3309:
3307:
3306:
3301:
3290:, in a gas with
3289:
3287:
3286:
3281:
3269:
3267:
3266:
3261:
3256:
3251:
3246:
3245:
3236:
3200:
3198:
3197:
3192:
3181:
3180:
3177:
3171:
3160:
3159:
3147:
3146:
3143:
3137:
3132:
3131:
3115:
3113:
3112:
3107:
3096:
3095:
3092:
3086:
3072:
3071:
3068:
3062:
3051:
3050:
3008:
3006:
3005:
3000:
2995:
2994:
2991:
2979:
2974:
2973:
2970:
2964:
2963:
2960:
2951:
2950:
2947:
2922:
2920:
2919:
2914:
2848:
2846:
2845:
2840:
2813:
2811:
2810:
2805:
2778:
2776:
2775:
2770:
2768:
2767:
2751:
2749:
2748:
2743:
2731:
2729:
2728:
2723:
2711:
2709:
2708:
2703:
2691:
2689:
2688:
2683:
2681:
2680:
2651:phase transition
2641:
2639:
2638:
2633:
2628:
2620:
2619:
2614:
2605:
2604:
2586:
2585:
2569:
2567:
2566:
2561:
2556:
2548:
2547:
2542:
2533:
2532:
2474:
2472:
2471:
2466:
2442:
2438:
2432:
2428:
2421:
2397:
2395:
2394:
2389:
2373:
2371:
2370:
2365:
2353:
2351:
2350:
2345:
2343:
2342:
2333:
2312:
2310:
2309:
2304:
2302:
2301:
2297:
2281:
2254:
2252:
2251:
2246:
2234:
2232:
2231:
2226:
2208:
2206:
2205:
2200:
2178:
2176:
2175:
2170:
2165:
2144:
2142:
2141:
2136:
2116:
2114:
2113:
2108:
2096:
2094:
2093:
2088:
2086:
2085:
2082:
2069:
2067:
2066:
2061:
2059:
2058:
2055:
2030:
2028:
2027:
2022:
2020:
2019:
2016:
1997:
1995:
1994:
1989:
1987:
1986:
1983:
1964:
1962:
1961:
1956:
1954:
1953:
1950:
1931:
1929:
1928:
1923:
1921:
1920:
1904:
1902:
1901:
1896:
1894:
1893:
1877:
1875:
1874:
1869:
1867:
1866:
1850:
1848:
1847:
1842:
1840:
1839:
1824:, an average of
1823:
1821:
1820:
1815:
1803:
1801:
1800:
1795:
1790:
1789:
1777:
1776:
1757:
1755:
1754:
1749:
1744:
1743:
1731:
1730:
1708:
1706:
1705:
1700:
1698:
1697:
1696:
1677:
1676:
1675:
1646:
1644:
1643:
1638:
1636:
1635:
1634:
1605:
1603:
1602:
1597:
1595:
1594:
1593:
1574:
1573:
1572:
1549:
1547:
1546:
1541:
1539:
1538:
1537:
1518:
1517:
1516:
1493:
1491:
1490:
1485:
1483:
1482:
1479:
1456:
1454:
1453:
1448:
1443:
1442:
1439:
1420:
1418:
1417:
1412:
1400:
1398:
1397:
1392:
1368:
1366:
1365:
1360:
1358:
1357:
1354:
1332:
1330:
1329:
1324:
1312:
1310:
1309:
1304:
1277:
1275:
1274:
1269:
1267:
1266:
1263:
1244:
1242:
1241:
1236:
1234:
1233:
1230:
1211:
1209:
1208:
1203:
1201:
1200:
1197:
1184:
1182:
1181:
1176:
1174:
1173:
1170:
1149:
1147:
1146:
1141:
1129:
1127:
1126:
1121:
1106:
1104:
1103:
1098:
1082:
1080:
1079:
1074:
1072:
1071:
1063:
1053:
1051:
1050:
1045:
1043:
1042:
1034:
1031:
1016:
1014:
1013:
1008:
992:
990:
989:
984:
979:
960:
958:
957:
952:
947:
942:
941:
938:
918:
916:
915:
910:
905:
883:
881:
880:
875:
854:
852:
851:
846:
834:
832:
831:
826:
821:
820:
817:
792:
790:
789:
784:
782:
781:
778:
772:
757:
755:
754:
749:
747:
746:
743:
731:
726:
725:
713:
677:
675:
674:
669:
654:
652:
651:
646:
634:
632:
631:
626:
614:
612:
611:
606:
590:
588:
587:
582:
566:
564:
563:
558:
550:
549:
527:
525:
524:
519:
501:
499:
498:
493:
481:
479:
478:
473:
454:
452:
451:
446:
434:
432:
431:
426:
410:
408:
407:
402:
386:
384:
383:
378:
373:
372:
369:
342:
340:
339:
334:
332:
331:
328:
322:
307:
305:
304:
299:
283:
281:
280:
275:
259:
257:
256:
251:
249:
248:
245:
228:
226:
225:
220:
215:
210:
209:
206:
180:
178:
177:
172:
156:
154:
153:
148:
136:
134:
133:
128:
126:
124:
123:
111:
106:
104:
93:
85:
67:
65:
64:
59:
28736:
28735:
28731:
28730:
28729:
28727:
28726:
28725:
28691:
28690:
28684:
28674:Thermal Physics
28665:
28618:
28607:
28591:
28589:Further reading
28586:
28585:
28575:
28573:
28566:
28548:
28544:
28533:
28529:
28519:
28517:
28510:The Nobel Prize
28502:
28498:
28493:
28489:
28484:
28480:
28471:
28467:
28462:
28458:
28448:
28446:
28431:
28427:
28417:
28415:
28402:Hewitt, Nigel.
28400:
28396:
28391:
28387:
28378:
28374:
28369:
28365:
28360:
28356:
28343:
28339:
28334:
28330:
28325:
28321:
28312:
28308:
28303:
28299:
28294:
28290:
28280:
28278:
28259:
28253:
28246:
28241:
28237:
28228:
28224:
28219:
28215:
28210:
28206:
28201:
28197:
28192:
28188:
28183:
28179:
28166:
28162:
28153:
28149:
28144:
28140:
28135:
28131:
28126:
28122:
28109:
28105:
28100:
28096:
28091:
28087:
28082:
28078:
28065:
28061:
28056:
28052:
28043:
28039:
28026:
28022:
28009:
28005:
28000:
27991:
27986:
27982:
27977:
27973:
27968:
27964:
27959:
27955:
27946:
27942:
27937:
27933:
27929:Whitman, p. 204
27928:
27924:
27919:
27915:
27911:Whitman, p. 203
27910:
27906:
27901:
27897:
27892:
27888:
27879:
27875:
27867:Whitman, A.M.,
27866:
27862:
27849:
27845:
27836:
27829:
27820:
27816:
27811:
27807:
27802:
27798:
27793:
27789:
27784:
27780:
27767:
27760:
27755:
27751:
27746:
27742:
27737:
27733:
27728:
27724:
27719:
27715:
27702:
27698:
27685:
27681:
27676:
27672:
27667:
27663:
27654:
27650:
27641:
27637:
27628:
27624:
27611:
27607:
27594:
27590:
27577:
27570:
27561:
27557:
27552:
27548:
27539:
27535:
27530:
27526:
27521:
27517:
27512:
27508:
27503:
27499:
27494:
27490:
27481:
27477:
27468:
27464:
27459:
27455:
27446:
27442:
27434:Epstein, P.S.,
27433:
27429:
27420:
27416:
27411:
27359:
27297:
27293:
27283:
27279:
27278:
27276:
27269:
27265:
27239:
27214:
27210:
27201:
27195:
27191:
27190:
27188:
27181:
27177:
27166:
27162:
27153:
27147:
27143:
27142:
27140:
27126:
27116:
27105:
27103:
27095:
27092:
27091:
27072:
27069:
27068:
27043:
27029:
27027:
27013:
27010:
27009:
26984:
26965:
26961:
26957:
26932:
26923:
26917:
26913:
26912:
26910:
26895:
26882:
26859:
26856:
26855:
26827:
26809:
26805:
26798:
26796:
26782:
26774:
26763:
26755:
26741:
26738:
26737:
26713:
26680:
26676:
26660:
26642:
26637:
26625:
26622:
26621:
26593:
26589:
26566:
26559:
26555:
26543:
26530:
26528:
26520:
26517:
26516:
26486:
26480:
26476:
26459:
26457:
26454:
26453:
26408:
26402:
26398:
26390:
26387:
26386:
26364:
26361:
26360:
26333:
26332:
26312:
26310:
26304:
26290:
26286:
26285:
26276:
26275:
26255:
26253:
26243:
26242:
26225:
26222:
26221:
26191:Planck constant
26152:
26146:
26142:
26139:
26131:
26128:
26127:
26104:
26101:
26100:
26078:
26074:
26069:
26055:
26044:
26038:
26034:
26032:
26024:
26021:
26020:
26014:non-interacting
26011:
25996:
25927:
25923:
25913:
25909:
25908:
25906:
25899:
25895:
25893:
25890:
25889:
25879:
25873:
25827:
25826:
25822:
25806:
25800:
25799:
25790:
25783:
25779:
25768:
25762:
25761:
25752:
25733:
25732:
25728:
25727:
25719:
25717:
25709:
25706:
25705:
25679:
25673:
25672:
25663:
25654:
25641:
25640:
25636:
25629:
25628:
25624:
25622:
25618:
25617:
25609:
25600:
25596:
25588:
25586:
25583:
25582:
25556:
25555:
25551:
25546:
25539:
25538:
25534:
25526:
25523:
25522:
25518:
25489:
25483:
25479:
25471:
25468:
25467:
25458:
25451:
25423:
25417:
25413:
25387:
25374:
25365:
25350:
25348:
25345:
25344:
25315:
25309:
25305:
25279:
25273:
25269:
25251:
25245:
25242:
25241:
25233:or radius
25202:
25199:
25198:
25165:
25164:
25160:
25159:
25151:
25149:
25141:
25138:
25137:
25131:excluded volume
25080:
25079:
25075:
25067:
25065:
25057:
25054:
25053:
25047:
25034:
25006:
25002:
24996:
24992:
24987:
24981:
24977:
24963:
24958:
24955:
24954:
24933:
24922:
24918:
24912:
24908:
24902:
24894:
24883:
24872:
24862:
24851:
24846:
24842:
24841:
24830:
24826:
24820:
24816:
24810:
24802:
24789:
24785:
24779:
24768:
24758:
24747:
24742:
24738:
24729:
24725:
24717:
24710:
24706:
24700:
24696:
24690:
24682:
24671:
24660:
24650:
24639:
24626:
24621:
24615:
24612:
24611:
24589:
24586:
24585:
24564:
24560:
24552:
24549:
24548:
24531:
24527:
24516:
24513:
24512:
24495:
24491:
24482:
24478:
24476:
24473:
24472:
24450:
24447:
24446:
24424:
24421:
24420:
24395:
24391:
24381:
24377:
24373:
24371:
24361:
24352:
24348:
24335:
24331:
24324:
24320:
24311:
24307:
24300:
24298:
24288:
24279:
24275:
24273:
24270:
24269:
24252:
24248:
24239:
24235:
24228:
24217:
24201:
24197:
24189:
24182:
24178:
24169:
24165:
24159:
24148:
24132:
24128:
24126:
24123:
24122:
24101:
24097:
24091:
24087:
24081:
24077:
24067:
24063:
24055:
24038:
24034:
24026:
24013:
24002:
23992:
23981:
23966:
23951:
23947:
23945:
23942:
23941:
23925:
23919:
23918:
23917:
23911:
23907:
23897:
23893:
23885:
23875:
23864:
23854:
23853:
23844:
23840:
23832:
23825:
23821:
23815:
23811:
23801:
23797:
23793:
23784:
23780:
23771:
23767:
23758:
23747:
23737:
23726:
23713:
23709:
23707:
23704:
23703:
23687:
23684:
23683:
23660:
23651:
23647:
23638:
23634:
23622:
23618:
23610:
23599:
23595:
23591:
23585:
23581:
23575:
23571:
23559:
23555:
23553:
23550:
23549:
23527:
23514:
23510:
23505:
23488:
23484:
23479:
23459:
23455:
23450:
23438:
23427:
23423:
23419:
23413:
23409:
23403:
23399:
23387:
23383:
23381:
23378:
23377:
23360:
23356:
23354:
23351:
23350:
23333:
23329:
23327:
23324:
23323:
23306:
23302:
23293:
23289:
23280:
23276:
23267:
23263:
23261:
23258:
23257:
23231:
23228:
23227:
23211:
23208:
23207:
23185:
23181:
23179:
23176:
23175:
23158:
23154:
23152:
23149:
23148:
23131:
23127:
23125:
23122:
23121:
23104:
23100:
23091:
23087:
23085:
23082:
23081:
23065:
23062:
23061:
23045:
23042:
23041:
23024:
23020:
23018:
23015:
23014:
22998:
22995:
22994:
22977:
22973:
22964:
22960:
22958:
22955:
22954:
22938:
22935:
22934:
22915:
22912:
22911:
22894:
22890:
22888:
22885:
22884:
22867:
22863:
22861:
22858:
22857:
22841:
22838:
22837:
22809:
22806:
22805:
22779:
22775:
22774:
22770:
22761:
22757:
22755:
22752:
22751:
22734:
22730:
22715:
22711:
22702:
22698:
22674:
22670:
22664:
22660:
22651:
22647:
22641:
22637:
22617:
22614:
22613:
22583:
22579:
22570:
22562:
22543:
22536:
22532:
22531:
22518:
22511:
22507:
22506:
22500:
22497:
22496:
22480:
22477:
22476:
22460:
22457:
22456:
22428:
22425:
22424:
22401:
22381:
22371:
22367:
22366:
22364:
22360:
22359:
22347:
22343:
22339:
22329:
22325:
22324:
22322:
22313:
22309:
22305:
22295:
22291:
22290:
22288:
22272:
22268:
22264:
22254:
22250:
22249:
22247:
22231:
22227:
22223:
22213:
22209:
22208:
22206:
22204:
22201:
22200:
22171:
22167:
22165:
22162:
22161:
22144:
22140:
22132:
22129:
22128:
22112:
22109:
22108:
22076:
22072:
22064:
22061:
22060:
22037:
22025:
22021:
22019:
22016:
22015:
21995:
21991:
21980:
21968:
21964:
21962:
21959:
21958:
21938:
21934:
21923:
21896:
21892:
21880:
21875:
21853:
21846:
21845:
21830:
21826:
21817:
21810:
21809:
21800:
21796:
21790:
21786:
21777:
21773:
21764:
21760:
21758:
21755:
21754:
21735:
21732:
21731:
21714:
21710:
21702:
21699:
21698:
21681:
21677:
21675:
21672:
21671:
21654:
21650:
21648:
21645:
21644:
21628:
21625:
21624:
21607:
21604:
21603:
21577:
21573:
21558:
21551:
21547:
21546:
21540:
21537:
21536:
21516:
21512:
21497:
21490:
21486:
21485:
21479:
21476:
21475:
21459:
21456:
21455:
21438:
21434:
21432:
21429:
21428:
21427:increases past
21412:
21409:
21408:
21391:
21387:
21385:
21382:
21381:
21364:
21360:
21352:
21349:
21348:
21322:
21318:
21310:
21307:
21306:
21286:
21283:
21282:
21263:
21251:
21247:
21245:
21242:
21241:
21215:
21211:
21209:
21206:
21205:
21186:
21183:
21182:
21162:
21158:
21149:
21145:
21134:
21128:
21124:
21118:
21114:
21102:
21098:
21090:
21086:
21085:
21081:
21079:
21076:
21075:
21049:
21045:
21043:
21040:
21039:
21013:
21009:
21007:
21004:
21003:
20983:
20979:
20964:
20960:
20958:
20955:
20954:
20923:
20919:
20917:
20914:
20913:
20897:
20894:
20893:
20877:
20874:
20873:
20853:
20849:
20840:
20836:
20834:
20831:
20830:
20810:
20806:
20797:
20793:
20791:
20788:
20787:
20761:
20757:
20749:
20746:
20745:
20729:
20726:
20725:
20705:
20701:
20692:
20688:
20680:
20677:
20676:
20659:
20655:
20653:
20650:
20649:
20632:
20628:
20619:
20615:
20613:
20610:
20609:
20583:
20579:
20571:
20568:
20567:
20551:
20548:
20547:
20528:
20525:
20524:
20498:
20494:
20486:
20483:
20482:
20466:
20463:
20462:
20446:
20443:
20442:
20411:
20408:
20407:
20388:
20385:
20384:
20361:
20353:
20349:
20338:
20335:
20334:
20307:
20302:
20301:
20290:
20272:
20267:
20266:
20260:
20256:
20248:
20239:
20235:
20233:
20230:
20229:
20203:
20199:
20197:
20194:
20193:
20168:
20165:
20164:
20147:
20143:
20135:
20132:
20131:
20106:
20102:
20096:
20092:
20083:
20079:
20073:
20069:
20061:
20058:
20057:
20034:
20030:
20021:
20017:
20015:
20012:
20011:
19992:
19983:
19979:
19970:
19966:
19954:
19950:
19948:
19945:
19944:
19927:
19923:
19921:
19918:
19917:
19900:
19896:
19894:
19891:
19890:
19873:
19869:
19867:
19864:
19863:
19843:
19839:
19830:
19826:
19824:
19821:
19820:
19803:
19799:
19790:
19786:
19784:
19781:
19780:
19763:
19759:
19749:
19745:
19743:
19740:
19739:
19722:
19718:
19708:
19704:
19702:
19699:
19698:
19681:
19677:
19675:
19672:
19671:
19654:
19650:
19648:
19645:
19644:
19627:
19623:
19614:
19610:
19602:
19599:
19598:
19582:
19579:
19578:
19577:, although for
19561:
19557:
19548:
19544:
19536:
19533:
19532:
19509:
19505:
19496:
19492:
19490:
19487:
19486:
19469:
19465:
19456:
19452:
19444:
19441:
19440:
19421:
19415:
19411:
19402:
19398:
19396:
19393:
19392:
19373:
19367:
19363:
19354:
19350:
19348:
19345:
19344:
19327:
19322:
19312:
19308:
19299:
19295:
19289:
19285:
19279:
19275:
19263:
19258:
19248:
19244:
19235:
19231:
19223:
19216:
19211:
19201:
19197:
19188:
19184:
19178:
19174:
19168:
19164:
19152:
19147:
19137:
19133:
19124:
19120:
19118:
19115:
19114:
19095:
19091:
19085:
19081:
19079:
19067:
19063:
19056:
19048:
19046:
19038:
19035:
19034:
19015:
19011:
19005:
19001:
18999:
18987:
18983:
18976:
18968:
18966:
18957:
18953:
18945:
18936:
18932:
18926:
18922:
18920:
18908:
18904:
18897:
18889:
18887:
18878:
18874:
18872:
18869:
18868:
18864:
18840:
18835:
18832:
18831:
18814:
18810:
18802:
18794:
18791:
18790:
18770:
18766:
18758:
18744:
18741:
18740:
18714:
18710:
18699:
18693:
18689:
18666:
18631:
18610:
18606:
18601:
18587:
18584:
18583:
18566:
18562:
18542:
18534:
18531:
18530:
18513:
18509:
18498:
18495:
18494:
18477:
18473:
18458:
18447:
18431:
18427:
18404:
18401:
18400:
18378:
18375:
18374:
18352:
18339:
18330:
18326:
18324:
18321:
18320:
18297:
18293:
18291:
18288:
18287:
18267:
18263:
18254:
18250:
18242:
18239:
18238:
18221:
18217:
18209:
18203:
18199:
18189:
18178:
18160:
18154:
18150:
18135:
18131:
18122:
18118:
18107:
18077:
18073:
18064:
18060:
18052:
18049:
18048:
18028:
18024:
18018:
18014:
18003:
18000:
17999:
17976:
17972:
17970:
17967:
17966:
17946:
17942:
17933:
17929:
17921:
17918:
17917:
17892:
17889:
17888:
17872:
17869:
17868:
17842:
17838:
17836:
17833:
17832:
17815:
17811:
17790:
17786:
17779:
17768:
17735:
17732:
17731:
17715:
17712:
17711:
17708:
17681:
17677:
17675:
17672:
17671:
17651:
17647:
17638:
17634:
17625:
17621:
17619:
17616:
17615:
17595:
17591:
17582:
17578:
17570:
17567:
17566:
17546:
17542:
17540:
17537:
17536:
17516:
17512:
17498:
17495:
17494:
17475:
17472:
17471:
17455:
17452:
17451:
17432:
17427:
17424:
17423:
17406:
17402:
17400:
17397:
17396:
17377:
17365:
17361:
17359:
17356:
17355:
17332:
17328:
17326:
17323:
17322:
17303:
17297:
17293:
17287:
17283:
17262:
17258:
17256:
17253:
17252:
17226:
17222:
17211:
17199:
17195:
17189:
17185:
17171:
17154:
17150:
17149:
17145:
17143:
17140:
17139:
17123:
17120:
17119:
17097:
17094:
17093:
17070:
17066:
17064:
17061:
17060:
17038:
17024:
17010:
16998:
16994:
16986:
16983:
16982:
16959:
16955:
16946:
16942:
16940:
16937:
16936:
16913:
16909:
16904:
16892:
16888:
16880:
16877:
16876:
16856:
16852:
16848:
16841:
16837:
16833:
16831:
16819:
16815:
16808:
16803:
16795:
16792:
16791:
16763:
16731:
16720:
16717:
16716:
16670:
16667:
16666:
16644:
16641:
16640:
16615:
16604:
16601:
16600:
16584:
16581:
16580:
16564:
16561:
16560:
16529:
16526:
16525:
16494:
16488:
16484:
16472:
16468:
16466:
16463:
16462:
16439:
16435:
16433:
16430:
16429:
16410:
16398:
16394:
16392:
16389:
16388:
16381:
16345:
16344:
16340:
16336:
16334:
16331:
16330:
16299:
16298:
16294:
16290:
16288:
16285:
16284:
16265:
16260:
16257:
16256:
16237:
16225:
16221:
16219:
16216:
16215:
16198:
16194:
16185:
16181:
16179:
16176:
16175:
16159:
16156:
16155:
16134:
16120:
16108:
16104:
16099:
16094:
16091:
16090:
16074:
16071:
16070:
16051:
16040:
16037:
16036:
16017:
16000:
15983:
15980:
15979:
15941:
15937:
15932:
15929:
15928:
15909:
15903:
15899:
15883:
15882:
15878:
15876:
15873:
15872:
15853:
15847:
15843:
15824:
15823:
15819:
15817:
15814:
15813:
15794:
15778:
15774:
15762:
15761:
15751:
15747:
15739:
15731:
15728:
15727:
15700:
15696:
15692:
15686:
15682:
15674:
15650:
15646:
15641:
15632:
15628:
15626:
15623:
15622:
15602:
15598:
15587:
15575:
15571:
15569:
15566:
15565:
15548:
15544:
15535:
15531:
15529:
15526:
15525:
15497:
15494:
15493:
15473:
15469:
15461:
15439:
15435:
15431:
15425:
15421:
15413:
15386:
15382:
15377:
15372:
15369:
15368:
15349:
15344:
15341:
15340:
15338:
15313:
15307:
15303:
15292:
15289:
15288:
15257:
15254:
15253:
15231:
15228:
15227:
15190:
15186:
15171:
15167:
15141:
15125:
15121:
15094:
15090:
15083:
15073:
15069:
15054:
15050:
15031:
15029:
15027:
15024:
15023:
15007:
15004:
15003:
14972:
14969:
14968:
14945:
14941:
14939:
14936:
14935:
14913:
14910:
14909:
14886:
14882:
14880:
14877:
14876:
14859:
14855:
14853:
14850:
14849:
14830:
14826:
14803:
14801:
14792:
14788:
14786:
14783:
14782:
14748:
14745:
14744:
14727:
14723:
14714:
14710:
14708:
14705:
14704:
14687:
14682:
14681:
14672:
14668:
14659:
14655:
14653:
14650:
14649:
14646:
14625:
14622:
14621:
14605:
14602:
14601:
14576:
14573:
14572:
14571:) and mercury (
14547:
14544:
14543:
14518:
14515:
14514:
14483:
14480:
14479:
14459:
14455:
14453:
14450:
14449:
14433:
14430:
14429:
14412:
14408:
14406:
14403:
14402:
14383:
14380:
14379:
14363:
14360:
14359:
14342:
14338:
14336:
14333:
14332:
14312:
14308:
14306:
14303:
14302:
14269:
14265:
14257:
14254:
14253:
14252:over the range
14234:
14231:
14230:
14210:
14206:
14188:
14183:
14167:
14162:
14131:
14127:
14122:
14095:
14091:
14083:
14080:
14079:
14053:
14049:
14040:
14036:
14024:
14020:
14018:
14015:
14014:
13979:
13975:
13966:
13962:
13953:
13946:
13945:
13934:
13931:
13930:
13910:
13906:
13895:
13889:
13885:
13879:
13875:
13866:
13862:
13860:
13857:
13856:
13840:
13837:
13836:
13817:
13814:
13813:
13787:
13783:
13774:
13770:
13755:
13751:
13749:
13746:
13745:
13725:
13721:
13712:
13708:
13696:
13692:
13690:
13687:
13686:
13666:
13662:
13653:
13649:
13634:
13630:
13628:
13625:
13624:
13612:
13578:
13575:
13574:
13554:
13550:
13541:
13537:
13529:
13523:
13519:
13503:
13499:
13490:
13486:
13471:
13467:
13459:
13456:
13455:
13438:
13434:
13432:
13429:
13428:
13411:
13407:
13405:
13402:
13401:
13385:
13382:
13381:
13346:
13342:
13340:
13337:
13336:
13326:subcooled vapor
13319:subcooled vapor
13294:
13282:
13278:
13276:
13273:
13272:
13249:
13245:
13243:
13240:
13239:
13208:
13199:
13195:
13186:
13182:
13171:
13168:
13167:
13139:
13136:
13135:
13128:subcooled vapor
13106:
13102:
13093:
13089:
13087:
13084:
13083:
13060:
13056:
12969:
12943:
12941:
12924:
12921:
12920:
12897:
12893:
12880:
12870:
12866:
12847:
12845:
12831:
12827:
12815:
12811:
12798:
12794:
12781:
12774:
12770:
12751:
12749:
12735:
12731:
12719:
12715:
12713:
12710:
12709:
12690:
12686:
12676:
12666:
12662:
12638:
12634:
12624:
12622:
12607:
12603:
12593:
12589:
12579:
12532:
12530:
12516:
12512:
12491:
12487:
12485:
12482:
12481:
12460:
12456:
12454:
12451:
12450:
12433:
12429:
12427:
12424:
12423:
12393:
12389:
12384:
12373:
12369:
12364:
12363:
12359:
12326:
12322:
12321:
12308:
12304:
12303:
12301:
12297:
12273:
12269:
12268:
12262:
12258:
12256:
12238:
12234:
12233:
12227:
12223:
12221:
12220:
12216:
12210:
12206:
12201:
12198:
12197:
12160:
12157:
12156:
12139:
12135:
12133:
12130:
12129:
12112:
12108:
12106:
12103:
12102:
12099:equal area rule
12044:
12041:
12040:
12023:
12020:
12018:
12015:
12014:
11983:
11980:
11979:
11963:
11960:
11959:
11943:
11940:
11939:
11913:
11909:
11900:
11896:
11887:
11883:
11861:
11857:
11856:
11849:
11845:
11844:
11821:
11817:
11802:
11798:
11797:
11790:
11786:
11785:
11775:
11764:
11751:
11747:
11738:
11734:
11732:
11729:
11728:
11711:
11707:
11698:
11694:
11692:
11689:
11688:
11671:
11667:
11658:
11654:
11652:
11649:
11648:
11595:
11591:
11572:
11568:
11557:
11554:
11553:
11519:
11516:
11515:
11498:
11494:
11485:
11481:
11465:
11464:
11460:
11458:
11455:
11454:
11434:
11430:
11421:
11417:
11395:
11394:
11390:
11388:
11385:
11384:
11363:
11359:
11350:
11346:
11344:
11341:
11340:
11323:
11319:
11310:
11306:
11297:
11293:
11284:
11280:
11278:
11275:
11274:
11255:
11250:
11241:
11223:
11219:
11218:
11211:
11207:
11203:
11201:
11192:
11188:
11180:
11171:
11166:
11157:
11139:
11135:
11134:
11127:
11123:
11119:
11117:
11108:
11104:
11102:
11099:
11098:
11074:
11073:
11069:
11060:
11056:
11054:
11051:
11050:
11026:
11025:
11021:
11012:
11008:
11006:
11003:
11002:
10978:
10977:
10973:
10964:
10960:
10944:
10943:
10939:
10937:
10934:
10933:
10916:
10912:
10910:
10907:
10906:
10905:at temperature
10886:
10874:
10870:
10868:
10865:
10864:
10845:
10842:
10841:
10838:
10814:
10809:
10789:
10785:
10778:
10776:
10770:
10766:
10755:
10751:
10728:
10726:
10713:
10712:
10708:
10698:
10693:
10685:
10678:
10674:
10662:
10658:
10651:
10649:
10643:
10639:
10627:
10623:
10619:
10612:
10608:
10595:
10593:
10574:
10573:
10569:
10567:
10564:
10563:
10540:
10536:
10534:
10531:
10530:
10505:
10504:
10500:
10484:
10483:
10479:
10471:
10468:
10467:
10443:
10442:
10438:
10430:
10427:
10426:
10402:
10401:
10397:
10375:
10374:
10370:
10368:
10365:
10364:
10341:
10336:
10335:
10324:
10316:
10313:
10312:
10288:
10287:
10283:
10281:
10278:
10277:
10253:
10252:
10248:
10246:
10243:
10242:
10225:
10221:
10213:
10210:
10209:
10189:
10185:
10177:
10174:
10173:
10132:
10127:
10126:
10115:
10107:
10104:
10103:
10077:
10073:
10071:
10068:
10067:
10042:
10039:
10038:
10035:
10004:
9999:
9993:
9990:
9989:
9965:
9960:
9945:
9939:
9935:
9923:
9919:
9907:
9903:
9891:
9887:
9876:
9870:
9866:
9840:
9837:
9836:
9819:
9815:
9813:
9810:
9809:
9786:
9782:
9780:
9777:
9776:
9757:
9754:
9753:
9730:
9726:
9724:
9721:
9720:
9697:
9693:
9688:
9673:
9669:
9664:
9655:
9651:
9636:
9632:
9630:
9627:
9626:
9603:
9599:
9591:
9588:
9587:
9562:
9559:
9558:
9535:
9531:
9522:
9518:
9516:
9513:
9512:
9490:
9487:
9486:
9460:
9456:
9429:
9425:
9418:
9411:
9407:
9398:
9394:
9393:
9391:
9373:
9369:
9345:
9341:
9339:
9336:
9335:
9318:
9314:
9312:
9309:
9308:
9291:
9287:
9282:
9276:
9272:
9254:
9250:
9245:
9239:
9235:
9226:
9222:
9204:
9200:
9191:
9187:
9185:
9182:
9181:
9160:
9156:
9154:
9151:
9150:
9133:
9128:
9127:
9118:
9114:
9105:
9101:
9099:
9096:
9095:
9078:
9074:
9072:
9069:
9068:
9046:
9034:
9030:
9028:
9025:
9024:
9005:
8991:
8988:
8987:
8959:
8947:
8943:
8941:
8938:
8937:
8918:
8907:
8904:
8903:
8882:
8879:
8878:
8858:
8854:
8827:
8823:
8816:
8794:
8790:
8786:
8784:
8770:
8760:
8756:
8729:
8725:
8718:
8711:
8707:
8689:
8685:
8684:
8682:
8673:
8669:
8667:
8664:
8663:
8646:
8642:
8636:
8632:
8623:
8619:
8611:
8608:
8607:
8587:
8583:
8574:
8570:
8558:
8554:
8552:
8549:
8548:
8528:
8524:
8518:
8514:
8503:
8500:
8499:
8482:
8478:
8473:
8468:
8465:
8464:
8445:
8441:
8433:
8431:
8419:
8415:
8402:
8394:
8392:
8376:
8368:
8366:
8358:
8344:
8339:
8326:
8318:
8316:
8314:
8311:
8310:
8307:
8304:
8297:
8266:
8262:
8251:
8239:
8235:
8214:
8210:
8198:
8193:
8171:
8164:
8163:
8145:
8141:
8130:
8124:
8120:
8099:
8092:
8091:
8082:
8078:
8076:
8073:
8072:
8053:
8011:
8007:
7989:
7985:
7962:
7941:
7937:
7929:
7926:
7925:
7894:
7891:
7890:
7883:
7850:
7846:
7834:
7829:
7807:
7800:
7799:
7790:
7786:
7784:
7781:
7780:
7745:
7741:
7720:
7716:
7708:
7705:
7704:
7673:
7658:
7654:
7652:
7649:
7648:
7622:
7618:
7616:
7613:
7612:
7584:
7576:
7574:
7554:
7552:
7536:
7532:
7519:
7515:
7507:
7504:
7503:
7496:
7469:
7465:
7454:
7442:
7438:
7430:
7420:
7416:
7405:
7393:
7389:
7371:
7367:
7356:
7350:
7346:
7325:
7318:
7317:
7308:
7304:
7302:
7299:
7298:
7279:
7247:
7220:
7216:
7208:
7205:
7204:
7188:
7185:
7184:
7165:
7139:
7119:
7116:
7115:
7096:
7093:
7092:
7061:
7058:
7057:
7050:
7028:
7025:
7023:
7020:
7019:
6999:
6995:
6984:
6972:
6968:
6966:
6963:
6962:
6942:
6938:
6927:
6915:
6911:
6899:
6892:
6891:
6882:
6878:
6876:
6873:
6872:
6852:
6828:
6824:
6816:
6813:
6812:
6796:
6793:
6792:
6766:
6762:
6760:
6757:
6756:
6730:
6726:
6717:
6713:
6711:
6708:
6707:
6706:. Consequently
6684:
6680:
6674:
6670:
6668:
6665:
6664:
6647:
6643:
6638:
6618:
6606:
6602:
6596:
6592:
6590:
6587:
6586:
6567:
6562:
6559:
6558:
6533:
6521:
6517:
6511:
6507:
6498:
6494:
6467:
6463:
6461:
6458:
6457:
6437:
6433:
6424:
6420:
6418:
6415:
6414:
6394:
6386:
6370:
6362:
6356:
6353:
6352:
6321:
6318:
6317:
6283:
6280:
6279:
6259:
6255:
6253:
6250:
6249:
6216:
6204:
6194:
6192:
6186:
6182:
6181:
6177:
6142:
6138:
6088:
6080:
6078:
6074:
6070:
6035:
6031:
6018:
6014:
6006:
6003:
6002:
5999:
5962:
5958:
5956:
5953:
5952:
5915:
5912:
5911:
5895:
5892:
5891:
5890:, and entropy,
5875:
5872:
5871:
5868:
5847:
5844:
5843:
5816:
5812:
5804:
5801:
5800:
5781:
5766:
5762:
5751:
5745:
5741:
5735:
5731:
5722:
5718:
5716:
5713:
5712:
5689:
5685:
5680:
5663:
5648:
5644:
5642:
5639:
5638:
5618:
5614:
5609:
5592:
5577:
5573:
5571:
5568:
5567:
5547:
5543:
5538:
5520:
5516:
5514:
5511:
5510:
5493:
5489:
5484:
5472:
5468:
5463:
5451:
5447:
5442:
5437:
5434:
5433:
5405:
5393:
5389:
5387:
5384:
5383:
5366:
5362:
5357:
5345:
5341:
5339:
5336:
5335:
5312:
5308:
5306:
5303:
5302:
5256:
5253:
5252:
5215:
5211:
5209:
5206:
5205:
5186:
5181:
5172:
5154:
5150:
5146:
5139:
5135:
5131:
5129:
5120:
5116:
5114:
5111:
5110:
5093:
5089:
5084:
5072:
5068:
5059:
5055:
5050:
5038:
5034:
5025:
5021:
5016:
5004:
5000:
4998:
4995:
4994:
4974:
4970:
4961:
4957:
4948:
4944:
4942:
4939:
4938:
4921:
4916:
4903:
4899:
4894:
4881:
4876:
4860:
4856:
4851:
4841:
4837:
4825:
4821:
4816:
4810:
4806:
4795:
4792:
4791:
4774:
4770:
4768:
4765:
4764:
4741:
4736:
4720:
4715:
4699:
4695:
4689:
4685:
4673:
4669:
4660:
4656:
4650:
4646:
4635:
4632:
4631:
4590:
4586:
4577:
4573:
4558:
4554:
4542:
4538:
4529:
4525:
4523:
4520:
4519:
4503:
4500:
4499:
4479:
4475:
4467:
4455:
4451:
4449:
4446:
4445:
4414:
4399:
4395:
4377:
4373:
4371:
4368:
4367:
4347:
4344:
4343:
4327:
4324:
4323:
4307:
4304:
4303:
4284:
4281:
4280:
4258:
4255:
4254:
4228:
4223:
4222:
4216:
4212:
4204:
4195:
4191:
4182:
4177:
4176:
4165:
4157:
4154:
4153:
4136:
4132:
4117:
4113:
4105:
4102:
4101:
4078:
4073:
4072:
4061:
4053:
4050:
4049:
4032:
4028:
4020:
4017:
4016:
4001:
3966:
3962:
3953:
3949:
3931:
3927:
3918:
3914:
3912:
3909:
3908:
3875:
3872:
3871:
3855:
3852:
3851:
3831:
3827:
3818:
3814:
3806:
3797:
3793:
3787:
3783:
3774:
3770:
3764:
3760:
3751:
3747:
3739:
3736:
3735:
3712:
3708:
3699:
3695:
3687:
3678:
3674:
3665:
3661:
3641:
3638:
3637:
3620:
3616:
3607:
3603:
3601:
3598:
3597:
3580:
3576:
3567:
3563:
3561:
3558:
3557:
3550:liquid nitrogen
3521:
3518:
3516:
3513:
3512:
3495:
3492:
3490:
3487:
3486:
3423:
3420:
3419:
3391:
3380:
3376:
3374:
3369:
3366:
3365:
3343:
3339:
3337:
3335:
3332:
3331:
3315:
3312:
3311:
3295:
3292:
3291:
3275:
3272:
3271:
3252:
3241:
3237:
3235:
3221:
3218:
3217:
3210:
3176:
3172:
3167:
3155:
3151:
3142:
3138:
3133:
3127:
3123:
3121:
3118:
3117:
3091:
3087:
3082:
3067:
3063:
3058:
3046:
3042:
3022:
3019:
3018:
2990:
2986:
2975:
2969:
2965:
2959:
2955:
2946:
2942:
2940:
2937:
2936:
2854:
2851:
2850:
2819:
2816:
2815:
2784:
2781:
2780:
2763:
2759:
2757:
2754:
2753:
2737:
2734:
2733:
2717:
2714:
2713:
2697:
2694:
2693:
2676:
2672:
2670:
2667:
2666:
2624:
2615:
2610:
2609:
2600:
2596:
2581:
2577:
2575:
2572:
2571:
2552:
2543:
2538:
2537:
2528:
2524:
2513:
2510:
2509:
2502:
2439: and
2436:
2426:
2417:
2403:
2400:
2399:
2383:
2380:
2379:
2359:
2356:
2355:
2338:
2334:
2329:
2318:
2315:
2314:
2293:
2289:
2285:
2277:
2263:
2260:
2259:
2240:
2237:
2236:
2214:
2211:
2210:
2188:
2185:
2184:
2161:
2150:
2147:
2146:
2130:
2127:
2126:
2123:
2102:
2099:
2098:
2081:
2077:
2075:
2072:
2071:
2054:
2050:
2036:
2033:
2032:
2015:
2011:
2003:
2000:
1999:
1982:
1978:
1970:
1967:
1966:
1949:
1945:
1937:
1934:
1933:
1916:
1912:
1910:
1907:
1906:
1889:
1885:
1883:
1880:
1879:
1862:
1858:
1856:
1853:
1852:
1835:
1831:
1829:
1826:
1825:
1809:
1806:
1805:
1785:
1781:
1772:
1768:
1763:
1760:
1759:
1739:
1735:
1726:
1722:
1717:
1714:
1713:
1686:
1685:
1681:
1665:
1664:
1660:
1652:
1649:
1648:
1624:
1623:
1619:
1611:
1608:
1607:
1583:
1582:
1578:
1562:
1561:
1557:
1555:
1552:
1551:
1527:
1526:
1522:
1506:
1505:
1501:
1499:
1496:
1495:
1478:
1474:
1462:
1459:
1458:
1438:
1434:
1426:
1423:
1422:
1406:
1403:
1402:
1374:
1371:
1370:
1353:
1349:
1338:
1335:
1334:
1318:
1315:
1314:
1298:
1295:
1294:
1262:
1258:
1250:
1247:
1246:
1229:
1225:
1217:
1214:
1213:
1196:
1192:
1190:
1187:
1186:
1169:
1165:
1163:
1160:
1159:
1135:
1132:
1131:
1115:
1112:
1111:
1092:
1089:
1088:
1062:
1061:
1059:
1056:
1055:
1033:
1032:
1027:
1022:
1019:
1018:
1002:
999:
998:
975:
970:
967:
966:
943:
937:
933:
928:
925:
924:
921:specific volume
901:
896:
893:
892:
860:
857:
856:
840:
837:
836:
816:
812:
798:
795:
794:
777:
773:
768:
763:
760:
759:
742:
738:
727:
721:
717:
709:
683:
680:
679:
660:
657:
656:
640:
637:
636:
620:
617:
616:
600:
597:
596:
576:
573:
572:
545:
541:
533:
530:
529:
507:
504:
503:
487:
484:
483:
467:
464:
463:
440:
437:
436:
420:
417:
416:
396:
393:
392:
368:
364:
356:
353:
352:
351:). In addition
327:
323:
318:
313:
310:
309:
293:
290:
289:
269:
266:
265:
244:
240:
238:
235:
234:
211:
205:
201:
190:
187:
186:
166:
163:
162:
142:
139:
138:
119:
115:
110:
94:
86:
84:
76:
73:
72:
53:
50:
49:
17:
12:
11:
5:
28734:
28724:
28723:
28718:
28713:
28708:
28703:
28689:
28688:
28682:
28669:
28663:
28630:
28611:
28605:
28590:
28587:
28584:
28583:
28565:978-1502530356
28564:
28542:
28527:
28496:
28487:
28478:
28476:, 1447, (1968)
28465:
28456:
28425:
28394:
28385:
28372:
28363:
28354:
28337:
28328:
28319:
28306:
28297:
28295:Callen, p. 105
28288:
28270:(1): 233–244.
28244:
28235:
28222:
28213:
28204:
28195:
28186:
28177:
28160:
28147:
28138:
28129:
28120:
28103:
28094:
28085:
28076:
28059:
28050:
28037:
28020:
28010:Maxwell, J.C.
28003:
27989:
27980:
27978:Callen, p. 153
27971:
27962:
27953:
27949:Thermodynamics
27947:Callen, H.B.,
27940:
27931:
27922:
27913:
27904:
27895:
27893:Whitman, p 155
27886:
27873:
27860:
27843:
27827:
27814:
27805:
27796:
27787:
27778:
27758:
27749:
27740:
27731:
27722:
27713:
27703:Klein, M. J.,
27696:
27679:
27670:
27661:
27648:
27635:
27629:Weinberg, S.,
27622:
27605:
27588:
27568:
27555:
27546:
27533:
27524:
27515:
27506:
27497:
27495:Epstein, p. 11
27488:
27475:
27462:
27453:
27440:
27427:
27413:
27412:
27410:
27407:
27406:
27405:
27400:
27395:
27390:
27385:
27380:
27375:
27370:
27365:
27358:
27355:
27354:
27353:
27342:
27339:
27336:
27333:
27330:
27327:
27324:
27321:
27318:
27315:
27312:
27308:
27300:
27296:
27291:
27286:
27282:
27275:
27272:
27268:
27264:
27261:
27258:
27255:
27252:
27249:
27245:
27242:
27238:
27235:
27232:
27229:
27225:
27217:
27213:
27207:
27204:
27198:
27194:
27187:
27184:
27180:
27176:
27169:
27165:
27159:
27156:
27150:
27146:
27139:
27132:
27129:
27125:
27122:
27119:
27114:
27111:
27108:
27102:
27099:
27076:
27065:
27064:
27049:
27046:
27041:
27038:
27035:
27032:
27026:
27023:
27020:
27017:
27003:
27002:
26990:
26987:
26983:
26980:
26977:
26973:
26968:
26964:
26960:
26956:
26953:
26950:
26947:
26941:
26938:
26935:
26929:
26926:
26920:
26916:
26909:
26905:
26901:
26898:
26894:
26891:
26888:
26885:
26881:
26878:
26875:
26872:
26869:
26866:
26863:
26849:
26848:
26837:
26833:
26830:
26826:
26823:
26818:
26812:
26808:
26804:
26801:
26795:
26792:
26788:
26785:
26770:
26767:
26761:
26758:
26754:
26751:
26748:
26745:
26706:
26705:
26694:
26691:
26688:
26683:
26679:
26675:
26672:
26667:
26664:
26659:
26656:
26653:
26650:
26645:
26640:
26636:
26632:
26629:
26615:
26614:
26603:
26596:
26592:
26585:
26582:
26579:
26576:
26573:
26569:
26565:
26562:
26558:
26553:
26549:
26546:
26542:
26539:
26536:
26533:
26527:
26524:
26506:
26505:
26493:
26489:
26483:
26479:
26475:
26472:
26469:
26465:
26462:
26431:
26430:
26418:
26415:
26411:
26405:
26401:
26397:
26394:
26368:
26350:
26349:
26336:
26331:
26328:
26325:
26322:
26311:
26307:
26302:
26297:
26294:
26289:
26284:
26281:
26278:
26277:
26274:
26271:
26268:
26265:
26254:
26252:
26249:
26248:
26246:
26241:
26238:
26235:
26232:
26229:
26183:
26182:
26167:
26164:
26161:
26158:
26155:
26149:
26145:
26138:
26135:
26108:
26097:
26096:
26081:
26077:
26073:
26068:
26065:
26050:
26047:
26041:
26037:
26031:
26028:
26009:
25998:The canonical
25995:
25992:
25984:
25983:
25972:
25969:
25966:
25963:
25960:
25957:
25954:
25951:
25948:
25945:
25942:
25938:
25930:
25926:
25921:
25916:
25912:
25905:
25902:
25898:
25877:
25866:
25865:
25854:
25851:
25848:
25845:
25842:
25839:
25836:
25830:
25825:
25821:
25817:
25809:
25803:
25798:
25794:
25789:
25786:
25782:
25778:
25771:
25765:
25760:
25756:
25751:
25745:
25742:
25736:
25731:
25725:
25722:
25716:
25713:
25699:
25698:
25682:
25676:
25671:
25667:
25662:
25657:
25652:
25644:
25639:
25632:
25627:
25621:
25615:
25612:
25608:
25603:
25599:
25594:
25591:
25575:
25574:
25559:
25554:
25549:
25542:
25537:
25533:
25530:
25516:
25509:
25508:
25495:
25492:
25482:
25478:
25475:
25446:
25445:
25433:
25430:
25426:
25420:
25416:
25412:
25409:
25406:
25403:
25400:
25397:
25393:
25390:
25385:
25381:
25377:
25372:
25368:
25364:
25360:
25356:
25353:
25338:
25337:
25325:
25322:
25318:
25312:
25308:
25304:
25301:
25298:
25295:
25292:
25289:
25286:
25282:
25276:
25272:
25268:
25265:
25262:
25258:
25254:
25250:
25206:
25195:
25194:
25183:
25177:
25174:
25168:
25163:
25157:
25154:
25148:
25145:
25129:is called the
25103:
25102:
25091:
25083:
25078:
25073:
25070:
25064:
25061:
25046:
25043:
25033:
25030:
25017:
25014:
25009:
25005:
24999:
24995:
24990:
24984:
24980:
24976:
24973:
24970:
24966:
24962:
24939:
24936:
24931:
24925:
24921:
24915:
24911:
24905:
24900:
24897:
24893:
24886:
24881:
24878:
24875:
24871:
24865:
24860:
24857:
24854:
24850:
24845:
24839:
24833:
24829:
24823:
24819:
24813:
24808:
24805:
24801:
24795:
24792:
24788:
24782:
24777:
24774:
24771:
24767:
24761:
24756:
24753:
24750:
24746:
24741:
24737:
24732:
24728:
24713:
24709:
24703:
24699:
24693:
24688:
24685:
24681:
24674:
24669:
24666:
24663:
24659:
24653:
24648:
24645:
24642:
24638:
24634:
24629:
24624:
24620:
24599:
24596:
24593:
24567:
24563:
24559:
24556:
24534:
24530:
24526:
24523:
24520:
24498:
24494:
24490:
24485:
24481:
24460:
24457:
24454:
24434:
24431:
24428:
24401:
24398:
24394:
24387:
24384:
24380:
24376:
24368:
24365:
24360:
24355:
24351:
24341:
24338:
24334:
24327:
24323:
24317:
24314:
24310:
24306:
24303:
24295:
24292:
24287:
24282:
24278:
24255:
24251:
24245:
24242:
24238:
24231:
24226:
24223:
24220:
24216:
24212:
24207:
24204:
24200:
24185:
24181:
24175:
24172:
24168:
24162:
24157:
24154:
24151:
24147:
24143:
24138:
24135:
24131:
24104:
24100:
24094:
24090:
24084:
24080:
24074:
24070:
24066:
24061:
24058:
24054:
24050:
24045:
24041:
24037:
24032:
24029:
24025:
24021:
24016:
24011:
24008:
24005:
24001:
23995:
23990:
23987:
23984:
23980:
23976:
23973:
23969:
23965:
23962:
23959:
23954:
23950:
23928:
23922:
23914:
23910:
23904:
23900:
23896:
23891:
23888:
23884:
23878:
23873:
23870:
23867:
23863:
23857:
23852:
23847:
23843:
23828:
23824:
23818:
23814:
23808:
23804:
23800:
23796:
23790:
23787:
23783:
23777:
23774:
23770:
23766:
23761:
23756:
23753:
23750:
23746:
23740:
23735:
23732:
23729:
23725:
23721:
23716:
23712:
23691:
23670:
23667:
23663:
23659:
23654:
23650:
23646:
23641:
23637:
23633:
23630:
23625:
23621:
23606:
23602:
23598:
23594:
23588:
23584:
23578:
23574:
23570:
23567:
23562:
23558:
23537:
23534:
23530:
23526:
23521:
23517:
23513:
23508:
23504:
23500:
23495:
23491:
23487:
23482:
23478:
23474:
23471:
23466:
23462:
23458:
23453:
23449:
23434:
23430:
23426:
23422:
23416:
23412:
23406:
23402:
23398:
23395:
23390:
23386:
23363:
23359:
23336:
23332:
23309:
23305:
23301:
23296:
23292:
23288:
23283:
23279:
23275:
23270:
23266:
23235:
23215:
23188:
23184:
23161:
23157:
23134:
23130:
23107:
23103:
23099:
23094:
23090:
23069:
23049:
23027:
23023:
23002:
22980:
22976:
22972:
22967:
22963:
22942:
22922:
22919:
22897:
22893:
22870:
22866:
22845:
22825:
22822:
22819:
22816:
22813:
22800:are the molar
22789:
22782:
22778:
22773:
22769:
22764:
22760:
22737:
22733:
22729:
22726:
22723:
22718:
22714:
22710:
22705:
22701:
22697:
22694:
22691:
22688:
22685:
22682:
22677:
22673:
22667:
22663:
22659:
22654:
22650:
22644:
22640:
22636:
22633:
22630:
22627:
22624:
22621:
22610:Euler equation
22594:
22591:
22586:
22582:
22578:
22573:
22568:
22565:
22561:
22557:
22554:
22551:
22546:
22539:
22535:
22530:
22526:
22521:
22514:
22510:
22505:
22484:
22464:
22444:
22441:
22438:
22435:
22432:
22412:
22409:
22404:
22399:
22393:
22390:
22387:
22384:
22379:
22374:
22370:
22363:
22358:
22350:
22346:
22342:
22337:
22332:
22328:
22316:
22312:
22308:
22303:
22298:
22294:
22286:
22283:
22275:
22271:
22267:
22262:
22257:
22253:
22245:
22242:
22234:
22230:
22226:
22221:
22216:
22212:
22188:
22185:
22182:
22179:
22174:
22170:
22147:
22143:
22139:
22136:
22116:
22096:
22093:
22090:
22087:
22084:
22079:
22075:
22071:
22068:
22044:
22040:
22036:
22033:
22028:
22024:
22003:
21998:
21994:
21990:
21987:
21983:
21979:
21976:
21971:
21967:
21946:
21941:
21937:
21933:
21930:
21926:
21922:
21919:
21916:
21913:
21910:
21907:
21904:
21899:
21895:
21891:
21888:
21883:
21878:
21874:
21870:
21867:
21864:
21861:
21856:
21844:
21841:
21838:
21833:
21829:
21825:
21820:
21808:
21803:
21799:
21793:
21789:
21785:
21780:
21776:
21772:
21767:
21763:
21739:
21717:
21713:
21709:
21706:
21684:
21680:
21657:
21653:
21632:
21611:
21591:
21588:
21585:
21580:
21576:
21572:
21569:
21566:
21561:
21554:
21550:
21545:
21524:
21519:
21515:
21511:
21508:
21505:
21500:
21493:
21489:
21484:
21463:
21441:
21437:
21416:
21394:
21390:
21367:
21363:
21359:
21356:
21336:
21333:
21330:
21325:
21321:
21317:
21314:
21290:
21270:
21266:
21262:
21259:
21254:
21250:
21229:
21226:
21221:
21218:
21214:
21193:
21190:
21168:
21165:
21161:
21157:
21152:
21148:
21144:
21141:
21137:
21131:
21127:
21121:
21117:
21113:
21110:
21105:
21101:
21093:
21089:
21084:
21063:
21060:
21055:
21052:
21048:
21027:
21024:
21019:
21016:
21012:
20991:
20986:
20982:
20978:
20975:
20972:
20967:
20963:
20931:
20926:
20922:
20901:
20881:
20861:
20856:
20852:
20848:
20843:
20839:
20818:
20813:
20809:
20805:
20800:
20796:
20775:
20772:
20769:
20764:
20760:
20756:
20753:
20733:
20713:
20708:
20704:
20700:
20695:
20691:
20687:
20684:
20662:
20658:
20635:
20631:
20627:
20622:
20618:
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20586:
20582:
20578:
20575:
20555:
20535:
20532:
20512:
20509:
20506:
20501:
20497:
20493:
20490:
20470:
20450:
20430:
20427:
20424:
20421:
20418:
20415:
20392:
20368:
20364:
20352:
20348:
20345:
20342:
20318:
20315:
20310:
20305:
20300:
20297:
20293:
20289:
20286:
20283:
20280:
20275:
20270:
20263:
20259:
20255:
20251:
20247:
20242:
20238:
20217:
20214:
20211:
20206:
20202:
20181:
20178:
20175:
20172:
20150:
20146:
20142:
20139:
20109:
20105:
20099:
20095:
20091:
20086:
20082:
20076:
20072:
20068:
20065:
20045:
20042:
20037:
20033:
20029:
20024:
20020:
19999:
19995:
19991:
19986:
19982:
19978:
19973:
19969:
19965:
19962:
19957:
19953:
19930:
19926:
19903:
19899:
19876:
19872:
19862:Assuming that
19846:
19842:
19838:
19833:
19829:
19806:
19802:
19798:
19793:
19789:
19766:
19762:
19757:
19752:
19748:
19725:
19721:
19716:
19711:
19707:
19684:
19680:
19657:
19653:
19630:
19626:
19622:
19617:
19613:
19609:
19606:
19586:
19564:
19560:
19556:
19551:
19547:
19543:
19540:
19520:
19517:
19512:
19508:
19504:
19499:
19495:
19472:
19468:
19464:
19459:
19455:
19451:
19448:
19428:
19424:
19418:
19414:
19410:
19405:
19401:
19380:
19376:
19370:
19366:
19362:
19357:
19353:
19330:
19325:
19321:
19315:
19311:
19307:
19302:
19298:
19292:
19288:
19282:
19278:
19274:
19271:
19266:
19261:
19257:
19251:
19247:
19243:
19238:
19234:
19219:
19214:
19210:
19204:
19200:
19196:
19191:
19187:
19181:
19177:
19171:
19167:
19163:
19160:
19155:
19150:
19146:
19140:
19136:
19132:
19127:
19123:
19098:
19094:
19088:
19084:
19078:
19070:
19066:
19062:
19059:
19054:
19051:
19045:
19042:
19018:
19014:
19008:
19004:
18998:
18990:
18986:
18982:
18979:
18974:
18971:
18965:
18960:
18956:
18939:
18935:
18929:
18925:
18919:
18911:
18907:
18903:
18900:
18895:
18892:
18886:
18881:
18877:
18863:
18860:
18847:
18843:
18839:
18817:
18813:
18809:
18805:
18801:
18798:
18776:
18773:
18769:
18765:
18761:
18757:
18754:
18751:
18748:
18728:
18725:
18722:
18717:
18713:
18709:
18706:
18702:
18696:
18692:
18688:
18685:
18682:
18679:
18676:
18673:
18669:
18665:
18662:
18659:
18656:
18653:
18650:
18647:
18644:
18641:
18638:
18634:
18630:
18627:
18624:
18621:
18618:
18613:
18609:
18604:
18600:
18597:
18594:
18591:
18569:
18565:
18561:
18558:
18555:
18552:
18549:
18545:
18541:
18538:
18516:
18512:
18508:
18505:
18502:
18480:
18476:
18472:
18469:
18466:
18461:
18456:
18453:
18450:
18446:
18442:
18437:
18434:
18430:
18426:
18423:
18420:
18417:
18414:
18411:
18408:
18388:
18385:
18382:
18359:
18355:
18351:
18348:
18338:
18333:
18329:
18308:
18305:
18300:
18296:
18275:
18270:
18266:
18262:
18257:
18253:
18249:
18246:
18224:
18220:
18216:
18212:
18206:
18202:
18198:
18192:
18187:
18184:
18181:
18177:
18173:
18170:
18167:
18163:
18157:
18153:
18149:
18146:
18141:
18138:
18134:
18130:
18125:
18121:
18117:
18114:
18110:
18106:
18103:
18100:
18097:
18094:
18091:
18088:
18085:
18080:
18076:
18072:
18067:
18063:
18059:
18056:
18034:
18031:
18027:
18021:
18017:
18013:
18010:
18007:
17987:
17984:
17979:
17975:
17954:
17949:
17945:
17941:
17936:
17932:
17928:
17925:
17902:
17899:
17896:
17876:
17856:
17853:
17850:
17845:
17841:
17831:The functions
17818:
17814:
17810:
17807:
17804:
17801:
17798:
17793:
17789:
17782:
17777:
17774:
17771:
17767:
17763:
17760:
17757:
17754:
17751:
17748:
17745:
17742:
17739:
17719:
17707:
17704:
17684:
17680:
17659:
17654:
17650:
17646:
17641:
17637:
17633:
17628:
17624:
17603:
17598:
17594:
17590:
17585:
17581:
17577:
17574:
17549:
17545:
17524:
17519:
17515:
17511:
17508:
17505:
17502:
17479:
17459:
17439:
17435:
17431:
17409:
17405:
17384:
17380:
17376:
17373:
17368:
17364:
17343:
17340:
17335:
17331:
17310:
17306:
17300:
17296:
17290:
17286:
17282:
17279:
17276:
17273:
17270:
17265:
17261:
17240:
17237:
17234:
17229:
17225:
17221:
17218:
17214:
17210:
17207:
17202:
17198:
17192:
17188:
17184:
17181:
17178:
17174:
17170:
17167:
17164:
17157:
17153:
17148:
17127:
17107:
17104:
17101:
17081:
17078:
17073:
17069:
17045:
17041:
17037:
17034:
17031:
17027:
17023:
17020:
17017:
17013:
17009:
17006:
16997:
16993:
16990:
16970:
16967:
16962:
16958:
16954:
16949:
16945:
16924:
16921:
16916:
16912:
16907:
16903:
16900:
16895:
16891:
16887:
16884:
16859:
16855:
16851:
16844:
16840:
16836:
16830:
16822:
16818:
16814:
16811:
16807:
16802:
16799:
16779:
16776:
16773:
16770:
16766:
16762:
16759:
16756:
16753:
16750:
16747:
16744:
16741:
16738:
16734:
16730:
16727:
16724:
16704:
16701:
16698:
16695:
16692:
16689:
16686:
16683:
16680:
16677:
16674:
16654:
16651:
16648:
16628:
16625:
16622:
16618:
16614:
16611:
16608:
16588:
16568:
16548:
16545:
16542:
16539:
16536:
16533:
16501:
16497:
16491:
16487:
16483:
16480:
16475:
16471:
16450:
16447:
16442:
16438:
16417:
16413:
16409:
16406:
16401:
16397:
16380:
16377:
16364:
16361:
16354:
16351:
16348:
16343:
16339:
16318:
16315:
16308:
16305:
16302:
16297:
16293:
16272:
16268:
16264:
16244:
16240:
16236:
16233:
16228:
16224:
16201:
16197:
16193:
16188:
16184:
16163:
16141:
16138:
16133:
16130:
16127:
16123:
16119:
16116:
16111:
16107:
16102:
16098:
16078:
16058:
16054:
16050:
16047:
16044:
16024:
16020:
16016:
16010:
16007:
16004:
15999:
15996:
15993:
15990:
15987:
15967:
15964:
15961:
15958:
15955:
15952:
15949:
15944:
15940:
15936:
15916:
15912:
15906:
15902:
15898:
15892:
15889:
15886:
15881:
15860:
15856:
15850:
15846:
15842:
15839:
15833:
15830:
15827:
15822:
15801:
15797:
15793:
15790:
15785:
15781:
15777:
15771:
15768:
15765:
15760:
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15750:
15746:
15742:
15738:
15735:
15715:
15712:
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15703:
15699:
15695:
15689:
15685:
15681:
15677:
15673:
15670:
15667:
15664:
15661:
15658:
15653:
15649:
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15640:
15635:
15631:
15610:
15605:
15601:
15597:
15594:
15590:
15586:
15583:
15578:
15574:
15551:
15547:
15543:
15538:
15534:
15513:
15510:
15507:
15504:
15501:
15481:
15476:
15472:
15468:
15464:
15460:
15457:
15454:
15451:
15446:
15442:
15438:
15434:
15428:
15424:
15420:
15416:
15412:
15409:
15406:
15403:
15400:
15397:
15394:
15389:
15385:
15380:
15376:
15356:
15352:
15348:
15320:
15316:
15306:
15302:
15299:
15296:
15276:
15273:
15270:
15267:
15264:
15261:
15241:
15238:
15235:
15226:Note that for
15204:
15201:
15198:
15193:
15189:
15185:
15182:
15179:
15174:
15170:
15166:
15163:
15160:
15157:
15154:
15151:
15139:
15136:
15128:
15124:
15120:
15117:
15114:
15111:
15108:
15105:
15102:
15097:
15093:
15089:
15086:
15081:
15076:
15072:
15068:
15065:
15062:
15057:
15053:
15049:
15046:
15043:
15040:
15037:
15034:
15011:
14991:
14988:
14985:
14982:
14979:
14976:
14956:
14953:
14948:
14944:
14923:
14920:
14917:
14897:
14894:
14889:
14885:
14862:
14858:
14833:
14829:
14824:
14821:
14818:
14815:
14812:
14809:
14806:
14800:
14795:
14791:
14767:
14764:
14761:
14758:
14755:
14752:
14730:
14726:
14722:
14717:
14713:
14690:
14685:
14680:
14675:
14671:
14667:
14662:
14658:
14645:
14642:
14629:
14609:
14589:
14586:
14583:
14580:
14560:
14557:
14554:
14551:
14531:
14528:
14525:
14522:
14502:
14499:
14496:
14493:
14490:
14487:
14465:
14462:
14458:
14437:
14415:
14411:
14387:
14367:
14345:
14341:
14318:
14315:
14311:
14280:
14277:
14272:
14268:
14264:
14261:
14241:
14238:
14218:
14213:
14209:
14205:
14202:
14199:
14196:
14191:
14186:
14182:
14178:
14175:
14170:
14165:
14161:
14157:
14154:
14151:
14148:
14145:
14142:
14139:
14134:
14130:
14125:
14121:
14118:
14115:
14112:
14109:
14106:
14101:
14098:
14094:
14090:
14087:
14067:
14064:
14061:
14056:
14052:
14048:
14043:
14039:
14035:
14030:
14027:
14023:
14002:
13999:
13996:
13993:
13990:
13987:
13982:
13978:
13974:
13969:
13965:
13961:
13956:
13944:
13941:
13938:
13918:
13909:
13905:
13902:
13898:
13888:
13878:
13874:
13865:
13844:
13821:
13801:
13798:
13795:
13790:
13786:
13782:
13777:
13773:
13769:
13766:
13763:
13758:
13754:
13733:
13728:
13724:
13720:
13715:
13711:
13707:
13702:
13699:
13695:
13674:
13669:
13665:
13661:
13656:
13652:
13648:
13645:
13642:
13637:
13633:
13611:
13608:
13594:
13591:
13588:
13585:
13582:
13562:
13557:
13553:
13549:
13544:
13540:
13536:
13532:
13526:
13522:
13518:
13515:
13511:
13506:
13502:
13498:
13493:
13489:
13485:
13482:
13479:
13474:
13470:
13466:
13463:
13441:
13437:
13414:
13410:
13389:
13357:
13354:
13349:
13345:
13301:
13297:
13293:
13290:
13285:
13281:
13260:
13257:
13252:
13248:
13224:
13221:
13218:
13215:
13211:
13207:
13202:
13198:
13194:
13189:
13185:
13181:
13178:
13175:
13155:
13152:
13149:
13146:
13143:
13109:
13105:
13101:
13096:
13092:
13063:
13059:
13055:
13052:
13049:
13046:
13043:
13040:
13037:
13034:
13031:
13028:
13025:
13022:
13019:
13016:
13013:
13010:
13007:
13004:
13001:
12993:
12990:
12987:
12984:
12981:
12978:
12975:
12972:
12967:
12964:
12961:
12958:
12955:
12952:
12949:
12946:
12940:
12937:
12934:
12931:
12928:
12903:
12900:
12896:
12892:
12889:
12886:
12883:
12876:
12873:
12869:
12865:
12862:
12859:
12856:
12853:
12850:
12843:
12838:
12835:
12830:
12826:
12821:
12818:
12814:
12801:
12797:
12793:
12790:
12787:
12784:
12777:
12773:
12769:
12766:
12763:
12760:
12757:
12754:
12747:
12742:
12739:
12734:
12730:
12725:
12722:
12718:
12693:
12689:
12685:
12682:
12679:
12674:
12669:
12665:
12661:
12658:
12655:
12652:
12649:
12646:
12641:
12637:
12633:
12630:
12627:
12621:
12618:
12613:
12610:
12606:
12596:
12592:
12588:
12585:
12582:
12577:
12574:
12571:
12568:
12565:
12562:
12559:
12556:
12553:
12550:
12547:
12544:
12541:
12538:
12535:
12528:
12523:
12520:
12515:
12511:
12508:
12505:
12502:
12497:
12494:
12490:
12463:
12459:
12436:
12432:
12411:
12408:
12404:
12396:
12392:
12388:
12383:
12376:
12372:
12368:
12362:
12358:
12355:
12352:
12348:
12343:
12337:
12334:
12329:
12325:
12319:
12316:
12311:
12307:
12300:
12296:
12293:
12290:
12284:
12281:
12276:
12272:
12265:
12261:
12255:
12249:
12246:
12241:
12237:
12230:
12226:
12219:
12213:
12209:
12205:
12185:
12182:
12179:
12176:
12173:
12170:
12167:
12164:
12142:
12138:
12111:
12086:
12083:
12079:
12076:
12073:
12070:
12067:
12064:
12061:
12058:
12055:
12052:
12048:
12026:
12022:
12002:
11999:
11996:
11993:
11990:
11987:
11967:
11947:
11927:
11924:
11921:
11916:
11912:
11908:
11903:
11899:
11895:
11890:
11886:
11882:
11879:
11876:
11872:
11864:
11860:
11852:
11848:
11843:
11839:
11836:
11833:
11830:
11824:
11820:
11813:
11810:
11805:
11801:
11793:
11789:
11784:
11778:
11773:
11770:
11767:
11763:
11759:
11754:
11750:
11746:
11741:
11737:
11714:
11710:
11706:
11701:
11697:
11674:
11670:
11666:
11661:
11657:
11636:
11633:
11629:
11626:
11623:
11620:
11616:
11613:
11610:
11607:
11603:
11598:
11594:
11590:
11587:
11584:
11580:
11575:
11571:
11567:
11564:
11561:
11538:
11535:
11532:
11529:
11526:
11523:
11501:
11497:
11493:
11488:
11484:
11480:
11474:
11471:
11468:
11463:
11442:
11437:
11433:
11429:
11424:
11420:
11416:
11413:
11410:
11404:
11401:
11398:
11393:
11366:
11362:
11358:
11353:
11349:
11326:
11322:
11318:
11313:
11309:
11305:
11296:
11292:
11283:
11258:
11253:
11249:
11245:
11240:
11234:
11231:
11226:
11222:
11214:
11210:
11206:
11200:
11195:
11191:
11174:
11169:
11165:
11161:
11156:
11150:
11147:
11142:
11138:
11130:
11126:
11122:
11116:
11111:
11107:
11083:
11080:
11077:
11072:
11068:
11063:
11059:
11035:
11032:
11029:
11024:
11020:
11015:
11011:
10987:
10984:
10981:
10976:
10972:
10967:
10963:
10959:
10953:
10950:
10947:
10942:
10919:
10915:
10893:
10889:
10885:
10882:
10877:
10873:
10849:
10837:
10834:
10817:
10812:
10808:
10803:
10800:
10797:
10792:
10788:
10784:
10781:
10769:
10765:
10758:
10754:
10749:
10746:
10743:
10740:
10737:
10734:
10731:
10725:
10719:
10716:
10711:
10701:
10696:
10692:
10688:
10681:
10677:
10673:
10670:
10665:
10661:
10657:
10654:
10642:
10638:
10630:
10626:
10622:
10615:
10611:
10607:
10604:
10601:
10598:
10592:
10589:
10586:
10580:
10577:
10572:
10551:
10548:
10543:
10539:
10514:
10511:
10508:
10503:
10499:
10493:
10490:
10487:
10482:
10478:
10475:
10452:
10449:
10446:
10441:
10437:
10434:
10411:
10408:
10405:
10400:
10396:
10393:
10390:
10384:
10381:
10378:
10373:
10352:
10349:
10344:
10339:
10334:
10331:
10327:
10323:
10320:
10297:
10294:
10291:
10286:
10262:
10259:
10256:
10251:
10224:
10220:
10217:
10197:
10188:
10184:
10181:
10143:
10140:
10135:
10130:
10125:
10122:
10118:
10114:
10111:
10091:
10088:
10085:
10080:
10076:
10055:
10052:
10049:
10046:
10034:
10031:
10018:
10015:
10010:
10007:
10002:
9998:
9973:
9968:
9963:
9959:
9955:
9952:
9948:
9942:
9938:
9934:
9931:
9926:
9922:
9918:
9915:
9906:
9902:
9899:
9894:
9890:
9886:
9883:
9879:
9873:
9869:
9865:
9862:
9859:
9856:
9853:
9850:
9847:
9844:
9822:
9818:
9797:
9794:
9789:
9785:
9761:
9741:
9738:
9733:
9729:
9708:
9705:
9700:
9696:
9691:
9687:
9684:
9681:
9676:
9672:
9667:
9663:
9658:
9654:
9650:
9647:
9644:
9639:
9635:
9614:
9611:
9606:
9602:
9598:
9595:
9572:
9569:
9566:
9546:
9543:
9538:
9534:
9530:
9525:
9521:
9500:
9497:
9494:
9474:
9471:
9463:
9459:
9455:
9452:
9449:
9446:
9443:
9440:
9437:
9432:
9428:
9424:
9421:
9414:
9410:
9406:
9401:
9397:
9390:
9387:
9384:
9381:
9376:
9372:
9368:
9365:
9362:
9359:
9356:
9353:
9348:
9344:
9321:
9317:
9294:
9290:
9285:
9279:
9275:
9271:
9268:
9265:
9262:
9257:
9253:
9248:
9242:
9238:
9234:
9229:
9225:
9221:
9218:
9215:
9212:
9207:
9203:
9199:
9194:
9190:
9163:
9159:
9136:
9131:
9126:
9121:
9117:
9113:
9108:
9104:
9081:
9077:
9053:
9049:
9045:
9042:
9037:
9033:
9012:
9008:
9004:
9001:
8998:
8995:
8975:
8972:
8969:
8966:
8962:
8958:
8955:
8950:
8946:
8925:
8921:
8917:
8914:
8911:
8892:
8889:
8886:
8861:
8857:
8853:
8850:
8847:
8844:
8841:
8838:
8835:
8830:
8826:
8822:
8819:
8814:
8811:
8808:
8805:
8802:
8797:
8793:
8789:
8783:
8780:
8763:
8759:
8755:
8752:
8749:
8746:
8743:
8740:
8737:
8732:
8728:
8724:
8721:
8714:
8710:
8706:
8703:
8700:
8697:
8692:
8688:
8681:
8676:
8672:
8649:
8645:
8639:
8635:
8629:
8626:
8622:
8618:
8615:
8595:
8590:
8586:
8580:
8577:
8573:
8569:
8566:
8561:
8557:
8534:
8531:
8527:
8521:
8517:
8513:
8510:
8507:
8485:
8481:
8476:
8472:
8448:
8444:
8439:
8436:
8430:
8422:
8418:
8414:
8411:
8408:
8405:
8400:
8397:
8391:
8388:
8382:
8379:
8374:
8371:
8353:
8350:
8347:
8343:
8338:
8332:
8329:
8324:
8321:
8306:
8302:
8295:
8286:
8274:
8269:
8265:
8261:
8258:
8254:
8250:
8247:
8242:
8238:
8234:
8231:
8228:
8225:
8222:
8217:
8213:
8209:
8206:
8201:
8196:
8192:
8188:
8185:
8182:
8179:
8174:
8162:
8159:
8156:
8153:
8148:
8144:
8140:
8137:
8133:
8127:
8123:
8119:
8116:
8113:
8110:
8107:
8102:
8090:
8085:
8081:
8060:
8056:
8052:
8049:
8046:
8043:
8040:
8037:
8034:
8031:
8028:
8025:
8022:
8019:
8014:
8010:
8006:
8003:
8000:
7997:
7992:
7988:
7984:
7981:
7978:
7975:
7972:
7969:
7965:
7961:
7958:
7955:
7952:
7949:
7944:
7940:
7936:
7933:
7913:
7910:
7907:
7904:
7901:
7898:
7882:
7879:
7867:
7864:
7861:
7858:
7853:
7849:
7845:
7842:
7837:
7832:
7828:
7824:
7821:
7818:
7815:
7810:
7798:
7793:
7789:
7768:
7765:
7762:
7759:
7756:
7753:
7748:
7744:
7740:
7737:
7734:
7731:
7728:
7723:
7719:
7715:
7712:
7692:
7689:
7686:
7683:
7680:
7676:
7672:
7669:
7666:
7661:
7657:
7636:
7633:
7630:
7625:
7621:
7600:
7597:
7590:
7587:
7582:
7579:
7572:
7569:
7564:
7560:
7557:
7550:
7547:
7544:
7539:
7535:
7530:
7527:
7522:
7518:
7514:
7511:
7495:
7492:
7480:
7477:
7468:
7464:
7461:
7457:
7453:
7450:
7445:
7441:
7428:
7423:
7419:
7415:
7412:
7408:
7404:
7401:
7396:
7392:
7388:
7385:
7382:
7379:
7374:
7370:
7366:
7363:
7359:
7353:
7349:
7345:
7342:
7339:
7336:
7333:
7328:
7316:
7311:
7307:
7286:
7282:
7278:
7275:
7272:
7269:
7266:
7263:
7260:
7257:
7254:
7250:
7246:
7243:
7240:
7237:
7234:
7231:
7228:
7223:
7219:
7215:
7212:
7192:
7172:
7168:
7164:
7161:
7158:
7155:
7152:
7149:
7146:
7142:
7138:
7135:
7132:
7129:
7126:
7123:
7103:
7100:
7080:
7077:
7074:
7071:
7068:
7065:
7049:
7046:
7031:
7027:
7007:
6998:
6994:
6991:
6987:
6983:
6980:
6975:
6971:
6950:
6945:
6941:
6937:
6934:
6930:
6926:
6923:
6918:
6914:
6910:
6907:
6902:
6890:
6885:
6881:
6859:
6855:
6851:
6848:
6845:
6842:
6839:
6836:
6831:
6827:
6823:
6820:
6800:
6780:
6777:
6774:
6769:
6765:
6744:
6741:
6738:
6733:
6729:
6725:
6720:
6716:
6695:
6692:
6687:
6683:
6677:
6673:
6650:
6646:
6641:
6637:
6634:
6631:
6628:
6625:
6621:
6617:
6614:
6609:
6605:
6599:
6595:
6574:
6570:
6566:
6546:
6543:
6540:
6536:
6532:
6529:
6524:
6520:
6514:
6510:
6506:
6501:
6497:
6493:
6490:
6487:
6484:
6481:
6478:
6475:
6470:
6466:
6445:
6440:
6436:
6432:
6427:
6423:
6402:
6397:
6392:
6389:
6385:
6381:
6378:
6373:
6368:
6365:
6361:
6340:
6337:
6334:
6331:
6328:
6325:
6305:
6302:
6299:
6296:
6293:
6290:
6287:
6262:
6258:
6237:
6234:
6229:
6222:
6219:
6214:
6211:
6207:
6203:
6200:
6197:
6189:
6185:
6180:
6175:
6172:
6169:
6166:
6162:
6159:
6156:
6153:
6150:
6145:
6141:
6136:
6133:
6130:
6127:
6122:
6118:
6115:
6112:
6109:
6106:
6103:
6100:
6094:
6091:
6086:
6083:
6077:
6073:
6068:
6065:
6062:
6059:
6055:
6052:
6049:
6046:
6043:
6038:
6034:
6029:
6026:
6021:
6017:
6013:
6010:
5998:
5995:
5982:
5979:
5976:
5973:
5970:
5965:
5961:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5919:
5899:
5879:
5867:
5864:
5851:
5827:
5824:
5815:
5811:
5808:
5788:
5784:
5780:
5777:
5774:
5765:
5761:
5758:
5754:
5744:
5734:
5730:
5721:
5697:
5692:
5688:
5683:
5679:
5676:
5673:
5670:
5666:
5662:
5659:
5656:
5651:
5647:
5626:
5621:
5617:
5612:
5608:
5605:
5602:
5599:
5595:
5591:
5588:
5585:
5580:
5576:
5555:
5550:
5546:
5541:
5537:
5534:
5531:
5528:
5523:
5519:
5496:
5492:
5487:
5483:
5480:
5475:
5471:
5466:
5462:
5459:
5454:
5450:
5445:
5441:
5421:
5418:
5415:
5412:
5408:
5404:
5401:
5396:
5392:
5369:
5365:
5360:
5356:
5353:
5348:
5344:
5323:
5320:
5315:
5311:
5290:
5287:
5284:
5281:
5278:
5275:
5272:
5269:
5266:
5263:
5260:
5218:
5214:
5189:
5184:
5180:
5176:
5171:
5165:
5162:
5157:
5153:
5149:
5142:
5138:
5134:
5128:
5123:
5119:
5092:
5087:
5083:
5080:
5075:
5071:
5067:
5058:
5053:
5049:
5046:
5041:
5037:
5033:
5024:
5019:
5015:
5012:
5007:
5003:
4973:
4969:
4960:
4956:
4947:
4924:
4915:
4911:
4902:
4897:
4893:
4890:
4884:
4875:
4871:
4868:
4859:
4854:
4850:
4840:
4836:
4833:
4824:
4819:
4809:
4805:
4802:
4799:
4773:
4752:
4749:
4744:
4735:
4731:
4728:
4723:
4714:
4710:
4707:
4702:
4698:
4688:
4684:
4681:
4676:
4672:
4668:
4663:
4659:
4649:
4645:
4642:
4639:
4619:
4616:
4613:
4610:
4607:
4604:
4601:
4598:
4593:
4589:
4585:
4576:
4572:
4569:
4566:
4557:
4553:
4550:
4545:
4541:
4537:
4528:
4507:
4482:
4478:
4474:
4470:
4466:
4463:
4454:
4433:
4430:
4427:
4424:
4421:
4417:
4413:
4410:
4407:
4398:
4394:
4391:
4388:
4385:
4376:
4364:
4363:
4351:
4331:
4311:
4300:
4288:
4268:
4265:
4262:
4251:
4239:
4236:
4231:
4226:
4219:
4215:
4211:
4207:
4203:
4198:
4194:
4190:
4185:
4180:
4175:
4172:
4168:
4164:
4161:
4135:
4131:
4128:
4125:
4116:
4112:
4109:
4089:
4086:
4081:
4076:
4071:
4068:
4064:
4060:
4057:
4031:
4027:
4024:
4000:
3997:
3980:
3977:
3974:
3969:
3965:
3961:
3956:
3952:
3948:
3945:
3942:
3939:
3934:
3930:
3926:
3921:
3917:
3879:
3859:
3839:
3834:
3830:
3826:
3821:
3817:
3813:
3809:
3805:
3800:
3796:
3790:
3786:
3782:
3777:
3773:
3767:
3763:
3759:
3754:
3750:
3746:
3743:
3723:
3720:
3715:
3711:
3707:
3702:
3698:
3694:
3690:
3686:
3681:
3677:
3673:
3668:
3664:
3660:
3657:
3654:
3651:
3648:
3645:
3623:
3619:
3615:
3610:
3606:
3583:
3579:
3575:
3570:
3566:
3524:
3520:
3498:
3494:
3479:Thomas Andrews
3427:
3407:
3404:
3401:
3398:
3394:
3388:
3383:
3379:
3373:
3351:
3346:
3342:
3319:
3299:
3279:
3259:
3255:
3249:
3244:
3240:
3234:
3231:
3228:
3225:
3209:
3206:
3190:
3187:
3184:
3175:
3170:
3166:
3163:
3158:
3154:
3150:
3141:
3136:
3130:
3126:
3105:
3102:
3099:
3090:
3085:
3081:
3078:
3075:
3066:
3061:
3057:
3054:
3049:
3045:
3041:
3038:
3035:
3032:
3029:
3026:
2998:
2989:
2985:
2982:
2978:
2968:
2958:
2954:
2945:
2932:
2931:
2924:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2838:
2835:
2832:
2829:
2826:
2823:
2803:
2800:
2797:
2794:
2791:
2788:
2766:
2762:
2741:
2721:
2701:
2679:
2675:
2663:
2657:
2647:critical point
2643:
2631:
2627:
2623:
2618:
2613:
2608:
2603:
2599:
2595:
2592:
2589:
2584:
2580:
2559:
2555:
2551:
2546:
2541:
2536:
2531:
2527:
2523:
2520:
2517:
2501:
2498:
2464:
2461:
2458:
2455:
2452:
2449:
2445:
2435:
2424:
2420:
2416:
2413:
2410:
2407:
2387:
2376:
2375:
2363:
2341:
2337:
2332:
2328:
2325:
2322:
2300:
2296:
2292:
2288:
2284:
2280:
2276:
2273:
2270:
2267:
2256:
2244:
2224:
2221:
2218:
2198:
2195:
2192:
2168:
2164:
2160:
2157:
2154:
2134:
2122:
2119:
2106:
2080:
2053:
2049:
2046:
2043:
2040:
2014:
2010:
2007:
1981:
1977:
1974:
1948:
1944:
1941:
1919:
1915:
1892:
1888:
1865:
1861:
1838:
1834:
1813:
1793:
1788:
1784:
1780:
1775:
1771:
1767:
1747:
1742:
1738:
1734:
1729:
1725:
1721:
1695:
1692:
1689:
1684:
1680:
1674:
1671:
1668:
1663:
1659:
1656:
1633:
1630:
1627:
1622:
1618:
1615:
1592:
1589:
1586:
1581:
1577:
1571:
1568:
1565:
1560:
1536:
1533:
1530:
1525:
1521:
1515:
1512:
1509:
1504:
1477:
1472:
1469:
1466:
1446:
1437:
1433:
1430:
1410:
1390:
1387:
1384:
1381:
1378:
1352:
1348:
1345:
1342:
1322:
1302:
1261:
1257:
1254:
1228:
1224:
1221:
1195:
1168:
1156:critical point
1139:
1119:
1096:
1069:
1066:
1040:
1037:
1030:
1026:
1006:
995:number density
982:
978:
974:
950:
946:
936:
932:
923:) in place of
908:
904:
900:
873:
870:
867:
864:
844:
824:
815:
811:
808:
805:
802:
776:
771:
767:
741:
737:
734:
730:
724:
720:
716:
712:
708:
705:
702:
699:
696:
693:
690:
687:
667:
664:
644:
624:
604:
580:
556:
553:
548:
544:
540:
537:
517:
514:
511:
491:
471:
444:
424:
400:
376:
367:
363:
360:
326:
321:
317:
297:
273:
243:
218:
214:
204:
200:
197:
194:
170:
146:
122:
118:
114:
109:
103:
100:
97:
92:
89:
83:
80:
57:
15:
9:
6:
4:
3:
2:
28733:
28722:
28719:
28717:
28714:
28712:
28709:
28707:
28704:
28702:
28699:
28698:
28696:
28685:
28679:
28675:
28670:
28666:
28664:9780120038022
28660:
28656:
28652:
28648:
28644:
28640:
28636:
28631:
28626:
28625:
28617:
28612:
28608:
28602:
28598:
28593:
28592:
28571:
28567:
28561:
28557:
28553:
28546:
28538:
28531:
28515:
28511:
28507:
28500:
28491:
28482:
28475:
28469:
28460:
28444:
28440:
28436:
28429:
28413:
28409:
28405:
28398:
28389:
28382:
28376:
28367:
28358:
28352:, 1-27 (1974)
28351:
28347:
28341:
28332:
28323:
28316:
28313:Kreysig, E.,
28310:
28301:
28292:
28277:
28273:
28269:
28265:
28258:
28251:
28249:
28239:
28233:, 134, (1881)
28232:
28226:
28217:
28208:
28199:
28190:
28181:
28174:
28170:
28164:
28157:
28151:
28142:
28133:
28124:
28117:
28113:
28107:
28098:
28089:
28080:
28073:
28069:
28063:
28054:
28047:
28041:
28034:
28030:
28024:
28017:
28013:
28007:
27998:
27996:
27994:
27984:
27975:
27966:
27960:Epstein, p 10
27957:
27950:
27944:
27935:
27926:
27917:
27908:
27899:
27890:
27883:
27877:
27870:
27864:
27857:
27853:
27847:
27840:
27834:
27832:
27824:
27818:
27809:
27800:
27791:
27782:
27775:
27771:
27765:
27763:
27753:
27744:
27735:
27726:
27717:
27711:, 31, (1974)
27710:
27706:
27700:
27693:
27689:
27683:
27674:
27665:
27658:
27652:
27645:
27639:
27632:
27626:
27619:
27615:
27609:
27602:
27598:
27592:
27585:
27581:
27575:
27573:
27565:
27559:
27550:
27543:
27537:
27528:
27519:
27513:Epstein, p.10
27510:
27504:Epstein, p.10
27501:
27492:
27485:
27479:
27472:
27466:
27457:
27450:
27444:
27437:
27431:
27424:
27418:
27414:
27404:
27401:
27399:
27396:
27394:
27391:
27389:
27386:
27384:
27381:
27379:
27376:
27374:
27371:
27369:
27366:
27364:
27361:
27360:
27340:
27337:
27334:
27331:
27328:
27322:
27319:
27316:
27313:
27306:
27298:
27294:
27289:
27284:
27280:
27273:
27270:
27266:
27259:
27256:
27253:
27250:
27243:
27240:
27236:
27233:
27230:
27223:
27215:
27211:
27205:
27202:
27196:
27192:
27185:
27182:
27178:
27167:
27163:
27157:
27154:
27148:
27144:
27137:
27130:
27127:
27123:
27120:
27117:
27112:
27109:
27106:
27100:
27097:
27090:
27089:
27088:
27074:
27047:
27039:
27036:
27033:
27024:
27021:
27018:
27015:
27008:
27007:
27006:
26988:
26985:
26981:
26978:
26975:
26966:
26954:
26951:
26948:
26945:
26939:
26936:
26933:
26927:
26924:
26918:
26914:
26907:
26899:
26896:
26892:
26889:
26886:
26879:
26876:
26873:
26870:
26867:
26864:
26861:
26854:
26853:
26852:
26835:
26831:
26828:
26824:
26821:
26816:
26810:
26806:
26802:
26799:
26793:
26790:
26786:
26783:
26768:
26765:
26759:
26756:
26752:
26749:
26746:
26743:
26736:
26735:
26734:
26730:
26727:
26724:
26720:
26716:
26711:
26692:
26689:
26686:
26681:
26677:
26673:
26670:
26665:
26662:
26654:
26648:
26638:
26634:
26630:
26627:
26620:
26619:
26618:
26601:
26594:
26580:
26577:
26574:
26567:
26563:
26560:
26556:
26547:
26544:
26540:
26537:
26534:
26525:
26522:
26515:
26514:
26513:
26511:
26491:
26487:
26481:
26477:
26473:
26470:
26467:
26463:
26460:
26452:
26451:
26450:
26448:
26444:
26440:
26436:
26416:
26413:
26409:
26403:
26399:
26395:
26392:
26385:
26384:
26383:
26380:
26366:
26358:
26354:
26329:
26326:
26323:
26320:
26305:
26300:
26295:
26292:
26287:
26282:
26279:
26272:
26269:
26266:
26263:
26244:
26239:
26233:
26227:
26220:
26219:
26218:
26216:
26212:
26208:
26204:
26200:
26196:
26192:
26188:
26165:
26162:
26159:
26156:
26153:
26147:
26143:
26136:
26126:
26125:
26124:
26122:
26079:
26071:
26066:
26063:
26048:
26045:
26039:
26035:
26029:
26026:
26019:
26018:
26017:
26015:
26008:
26004:
26001:
25991:
25989:
25970:
25967:
25964:
25961:
25958:
25952:
25949:
25946:
25943:
25936:
25928:
25924:
25919:
25914:
25910:
25903:
25900:
25896:
25888:
25887:
25886:
25883:
25876:
25871:
25868:Upon writing
25852:
25849:
25846:
25843:
25837:
25834:
25823:
25815:
25807:
25796:
25792:
25787:
25784:
25780:
25769:
25758:
25754:
25749:
25743:
25740:
25729:
25723:
25720:
25714:
25711:
25704:
25703:
25702:
25680:
25669:
25665:
25660:
25655:
25650:
25637:
25625:
25619:
25613:
25610:
25606:
25601:
25597:
25592:
25589:
25581:
25580:
25579:
25552:
25547:
25535:
25531:
25528:
25521:
25520:
25519:
25515:
25493:
25490:
25480:
25476:
25473:
25466:
25465:
25464:
25461:
25454:
25428:
25424:
25418:
25414:
25410:
25407:
25401:
25398:
25395:
25391:
25388:
25379:
25375:
25370:
25366:
25362:
25358:
25354:
25351:
25343:
25342:
25341:
25320:
25316:
25310:
25306:
25302:
25299:
25293:
25290:
25287:
25284:
25280:
25274:
25270:
25266:
25263:
25260:
25256:
25252:
25248:
25240:
25239:
25238:
25236:
25232:
25227:
25225:
25221:
25204:
25181:
25175:
25172:
25161:
25155:
25152:
25146:
25143:
25136:
25135:
25134:
25132:
25128:
25124:
25121: −
25120:
25116:
25112:
25108:
25089:
25076:
25071:
25068:
25062:
25059:
25052:
25051:
25050:
25042:
25039:
25029:
25015:
25012:
25007:
24997:
24993:
24988:
24982:
24978:
24971:
24968:
24964:
24960:
24937:
24934:
24929:
24923:
24919:
24913:
24909:
24903:
24898:
24895:
24891:
24884:
24879:
24876:
24873:
24869:
24863:
24858:
24855:
24852:
24848:
24843:
24837:
24831:
24827:
24821:
24817:
24811:
24806:
24803:
24799:
24793:
24790:
24786:
24780:
24775:
24772:
24769:
24765:
24759:
24754:
24751:
24748:
24744:
24739:
24735:
24730:
24726:
24711:
24707:
24701:
24697:
24691:
24686:
24683:
24679:
24672:
24667:
24664:
24661:
24657:
24651:
24646:
24643:
24640:
24636:
24632:
24627:
24622:
24618:
24597:
24594:
24591:
24583:
24565:
24561:
24557:
24554:
24532:
24528:
24524:
24521:
24518:
24496:
24492:
24488:
24483:
24479:
24458:
24455:
24452:
24432:
24429:
24426:
24417:
24399:
24396:
24392:
24385:
24382:
24378:
24374:
24366:
24363:
24358:
24353:
24349:
24339:
24336:
24332:
24325:
24315:
24312:
24308:
24304:
24293:
24290:
24285:
24280:
24276:
24253:
24249:
24243:
24240:
24236:
24229:
24224:
24221:
24218:
24214:
24210:
24205:
24202:
24198:
24183:
24179:
24173:
24170:
24166:
24160:
24155:
24152:
24149:
24145:
24141:
24136:
24133:
24129:
24119:
24102:
24098:
24092:
24088:
24082:
24072:
24068:
24064:
24059:
24056:
24052:
24048:
24043:
24039:
24035:
24030:
24027:
24023:
24014:
24009:
24006:
24003:
23999:
23993:
23988:
23985:
23982:
23978:
23971:
23967:
23963:
23957:
23952:
23948:
23926:
23912:
23908:
23902:
23898:
23894:
23889:
23886:
23882:
23876:
23871:
23868:
23865:
23861:
23850:
23845:
23841:
23826:
23822:
23816:
23812:
23806:
23802:
23798:
23788:
23785:
23781:
23775:
23772:
23768:
23759:
23754:
23751:
23748:
23744:
23738:
23733:
23730:
23727:
23723:
23719:
23714:
23710:
23689:
23668:
23665:
23661:
23652:
23648:
23644:
23639:
23635:
23628:
23623:
23619:
23604:
23600:
23596:
23586:
23582:
23576:
23572:
23565:
23560:
23556:
23535:
23532:
23528:
23519:
23515:
23511:
23506:
23502:
23498:
23493:
23489:
23485:
23480:
23476:
23469:
23464:
23460:
23456:
23451:
23447:
23432:
23428:
23424:
23414:
23410:
23404:
23400:
23393:
23388:
23384:
23361:
23357:
23334:
23330:
23307:
23303:
23299:
23294:
23290:
23286:
23281:
23277:
23273:
23268:
23264:
23253:
23248:
23233:
23213:
23205:
23186:
23182:
23159:
23155:
23132:
23128:
23105:
23101:
23097:
23092:
23088:
23067:
23047:
23025:
23021:
23000:
22978:
22974:
22970:
22965:
22961:
22940:
22920:
22917:
22895:
22891:
22868:
22864:
22843:
22823:
22820:
22817:
22814:
22811:
22803:
22787:
22780:
22776:
22767:
22762:
22758:
22735:
22731:
22727:
22724:
22716:
22712:
22708:
22703:
22699:
22692:
22689:
22686:
22683:
22680:
22675:
22671:
22665:
22661:
22657:
22652:
22648:
22642:
22638:
22634:
22631:
22628:
22625:
22622:
22619:
22611:
22606:
22592:
22589:
22584:
22576:
22571:
22566:
22563:
22552:
22549:
22544:
22537:
22533:
22524:
22519:
22512:
22508:
22482:
22462:
22442:
22439:
22436:
22433:
22430:
22410:
22407:
22402:
22397:
22391:
22385:
22377:
22372:
22361:
22356:
22348:
22344:
22335:
22330:
22314:
22310:
22301:
22296:
22284:
22281:
22273:
22269:
22260:
22255:
22243:
22240:
22232:
22228:
22219:
22214:
22186:
22183:
22180:
22177:
22172:
22168:
22145:
22141:
22137:
22134:
22114:
22091:
22088:
22085:
22082:
22077:
22073:
22066:
22056:
22042:
22038:
22034:
22031:
22026:
22022:
21996:
21992:
21988:
21981:
21977:
21974:
21969:
21965:
21939:
21935:
21931:
21924:
21920:
21917:
21905:
21902:
21897:
21893:
21889:
21881:
21876:
21872:
21865:
21862:
21859:
21854:
21842:
21839:
21831:
21827:
21823:
21818:
21806:
21801:
21797:
21791:
21787:
21783:
21778:
21774:
21770:
21765:
21761:
21751:
21737:
21715:
21711:
21707:
21704:
21682:
21678:
21655:
21651:
21630:
21609:
21589:
21586:
21583:
21578:
21570:
21567:
21564:
21559:
21552:
21548:
21522:
21517:
21509:
21506:
21503:
21498:
21491:
21487:
21461:
21439:
21435:
21414:
21392:
21388:
21365:
21361:
21357:
21354:
21331:
21328:
21323:
21319:
21312:
21288:
21268:
21264:
21260:
21257:
21252:
21248:
21227:
21224:
21219:
21216:
21212:
21191:
21188:
21166:
21163:
21159:
21150:
21146:
21142:
21135:
21129:
21125:
21119:
21115:
21111:
21108:
21103:
21099:
21091:
21087:
21061:
21058:
21053:
21050:
21046:
21025:
21022:
21017:
21014:
21010:
20984:
20980:
20976:
20973:
20965:
20961:
20951:
20947:
20945:
20929:
20924:
20920:
20899:
20879:
20854:
20850:
20841:
20837:
20811:
20807:
20798:
20794:
20770:
20767:
20762:
20758:
20751:
20731:
20706:
20702:
20693:
20689:
20685:
20682:
20660:
20656:
20633:
20629:
20625:
20620:
20616:
20592:
20589:
20584:
20580:
20573:
20553:
20533:
20530:
20507:
20504:
20499:
20495:
20488:
20468:
20448:
20428:
20425:
20422:
20419:
20416:
20413:
20404:
20390:
20382:
20366:
20362:
20350:
20346:
20343:
20340:
20332:
20316:
20313:
20308:
20298:
20291:
20287:
20281:
20278:
20273:
20261:
20257:
20249:
20245:
20240:
20215:
20212:
20209:
20204:
20200:
20179:
20176:
20173:
20170:
20148:
20144:
20140:
20137:
20129:
20124:
20107:
20103:
20097:
20093:
20089:
20084:
20080:
20074:
20070:
20066:
20063:
20043:
20040:
20035:
20031:
20027:
20022:
20018:
19997:
19993:
19984:
19980:
19976:
19971:
19967:
19960:
19955:
19951:
19928:
19924:
19901:
19897:
19874:
19870:
19860:
19844:
19840:
19836:
19831:
19827:
19804:
19800:
19796:
19791:
19787:
19764:
19760:
19755:
19750:
19746:
19723:
19719:
19714:
19709:
19705:
19682:
19678:
19655:
19651:
19628:
19624:
19620:
19615:
19611:
19607:
19604:
19584:
19562:
19558:
19554:
19549:
19545:
19541:
19538:
19518:
19515:
19510:
19506:
19502:
19497:
19493:
19470:
19466:
19462:
19457:
19453:
19449:
19446:
19426:
19422:
19416:
19412:
19408:
19403:
19399:
19378:
19374:
19368:
19364:
19360:
19355:
19351:
19328:
19323:
19319:
19313:
19309:
19305:
19300:
19296:
19290:
19286:
19280:
19276:
19272:
19269:
19264:
19259:
19255:
19249:
19245:
19241:
19236:
19232:
19217:
19212:
19208:
19202:
19198:
19194:
19189:
19185:
19179:
19175:
19169:
19165:
19161:
19158:
19153:
19148:
19144:
19138:
19134:
19130:
19125:
19121:
19096:
19092:
19086:
19082:
19076:
19068:
19064:
19060:
19057:
19052:
19049:
19043:
19040:
19016:
19012:
19006:
19002:
18996:
18988:
18984:
18980:
18977:
18972:
18969:
18963:
18958:
18954:
18937:
18933:
18927:
18923:
18917:
18909:
18905:
18901:
18898:
18893:
18890:
18884:
18879:
18875:
18859:
18845:
18841:
18837:
18815:
18807:
18803:
18799:
18774:
18771:
18763:
18759:
18755:
18752:
18749:
18739:On expanding
18726:
18715:
18711:
18707:
18700:
18694:
18690:
18686:
18683:
18677:
18674:
18667:
18663:
18660:
18657:
18654:
18648:
18645:
18642:
18636:
18632:
18628:
18625:
18622:
18611:
18607:
18602:
18598:
18595:
18592:
18567:
18559:
18556:
18547:
18543:
18539:
18514:
18506:
18503:
18478:
18470:
18467:
18454:
18451:
18448:
18444:
18440:
18435:
18432:
18424:
18421:
18418:
18415:
18409:
18406:
18386:
18383:
18380:
18371:
18357:
18353:
18349:
18346:
18336:
18331:
18327:
18306:
18303:
18298:
18294:
18268:
18264:
18260:
18255:
18251:
18244:
18222:
18214:
18210:
18204:
18200:
18185:
18182:
18179:
18175:
18171:
18165:
18161:
18155:
18151:
18139:
18136:
18123:
18119:
18115:
18108:
18104:
18101:
18098:
18092:
18089:
18086:
18078:
18074:
18070:
18065:
18061:
18054:
18032:
18029:
18019:
18015:
18011:
18008:
17985:
17982:
17977:
17973:
17947:
17943:
17939:
17934:
17930:
17923:
17914:
17900:
17897:
17894:
17874:
17851:
17843:
17839:
17816:
17808:
17799:
17791:
17787:
17775:
17772:
17769:
17765:
17761:
17758:
17755:
17749:
17746:
17743:
17737:
17717:
17703:
17701:
17682:
17678:
17652:
17648:
17644:
17639:
17635:
17626:
17622:
17596:
17592:
17588:
17583:
17579:
17572:
17563:
17547:
17543:
17517:
17513:
17509:
17506:
17500:
17491:
17477:
17457:
17437:
17433:
17429:
17407:
17403:
17382:
17378:
17374:
17371:
17366:
17362:
17341:
17338:
17333:
17329:
17308:
17304:
17298:
17288:
17284:
17280:
17277:
17271:
17268:
17263:
17259:
17238:
17235:
17227:
17223:
17219:
17212:
17208:
17205:
17200:
17190:
17186:
17182:
17179:
17172:
17168:
17165:
17162:
17155:
17151:
17146:
17125:
17105:
17102:
17099:
17079:
17071:
17067:
17059:In the limit
17057:
17043:
17039:
17035:
17032:
17029:
17025:
17021:
17018:
17015:
17011:
17007:
17004:
16995:
16991:
16988:
16968:
16965:
16960:
16956:
16952:
16947:
16943:
16922:
16919:
16914:
16910:
16905:
16901:
16898:
16893:
16889:
16885:
16882:
16857:
16853:
16849:
16842:
16838:
16834:
16828:
16820:
16816:
16812:
16809:
16805:
16800:
16797:
16774:
16771:
16764:
16760:
16757:
16754:
16748:
16745:
16742:
16739:
16732:
16728:
16725:
16722:
16702:
16699:
16696:
16690:
16687:
16684:
16678:
16675:
16672:
16652:
16649:
16646:
16626:
16620:
16616:
16612:
16609:
16606:
16586:
16566:
16546:
16543:
16540:
16537:
16534:
16531:
16518:
16499:
16495:
16489:
16485:
16481:
16478:
16473:
16469:
16445:
16440:
16436:
16415:
16411:
16407:
16404:
16399:
16395:
16385:
16376:
16362:
16359:
16341:
16337:
16316:
16313:
16295:
16291:
16270:
16266:
16262:
16242:
16238:
16234:
16231:
16226:
16222:
16199:
16195:
16191:
16186:
16182:
16161:
16136:
16131:
16128:
16125:
16121:
16117:
16114:
16109:
16105:
16100:
16096:
16076:
16056:
16052:
16048:
16045:
16042:
16022:
16018:
16008:
16005:
16002:
15997:
15994:
15988:
15985:
15965:
15962:
15959:
15956:
15953:
15950:
15947:
15942:
15938:
15934:
15914:
15910:
15904:
15900:
15896:
15879:
15858:
15854:
15848:
15844:
15840:
15837:
15820:
15799:
15795:
15791:
15788:
15783:
15779:
15775:
15752:
15748:
15744:
15740:
15736:
15713:
15710:
15705:
15701:
15697:
15687:
15683:
15679:
15675:
15671:
15665:
15662:
15659:
15656:
15651:
15647:
15642:
15638:
15633:
15629:
15603:
15599:
15595:
15588:
15584:
15581:
15576:
15572:
15549:
15545:
15541:
15536:
15532:
15511:
15508:
15505:
15502:
15499:
15474:
15470:
15466:
15462:
15458:
15452:
15449:
15444:
15440:
15436:
15426:
15422:
15418:
15414:
15410:
15404:
15401:
15398:
15395:
15392:
15387:
15383:
15378:
15374:
15354:
15350:
15346:
15336:
15334:
15318:
15314:
15304:
15300:
15297:
15294:
15274:
15271:
15268:
15265:
15262:
15259:
15239:
15236:
15233:
15219:
15215:
15202:
15199:
15196:
15191:
15187:
15183:
15180:
15177:
15172:
15164:
15161:
15158:
15152:
15149:
15137:
15134:
15126:
15118:
15115:
15112:
15106:
15103:
15100:
15095:
15091:
15087:
15084:
15079:
15074:
15070:
15066:
15063:
15060:
15055:
15047:
15044:
15041:
15035:
15032:
15009:
14989:
14986:
14983:
14980:
14977:
14974:
14954:
14951:
14946:
14942:
14921:
14918:
14915:
14895:
14892:
14887:
14883:
14860:
14856:
14831:
14827:
14819:
14816:
14813:
14810:
14804:
14798:
14793:
14789:
14779:
14762:
14756:
14753:
14750:
14728:
14724:
14720:
14715:
14711:
14688:
14678:
14673:
14665:
14660:
14656:
14641:
14627:
14607:
14587:
14584:
14581:
14578:
14558:
14555:
14552:
14549:
14529:
14526:
14523:
14520:
14500:
14497:
14494:
14491:
14488:
14485:
14463:
14460:
14456:
14435:
14413:
14409:
14399:
14385:
14365:
14343:
14339:
14316:
14313:
14309:
14295:
14291:
14278:
14275:
14270:
14266:
14262:
14259:
14239:
14236:
14211:
14207:
14203:
14200:
14197:
14194:
14189:
14184:
14180:
14176:
14173:
14168:
14163:
14159:
14155:
14152:
14149:
14143:
14140:
14132:
14128:
14123:
14119:
14116:
14113:
14107:
14104:
14099:
14096:
14092:
14088:
14085:
14062:
14059:
14054:
14050:
14041:
14037:
14033:
14028:
14025:
14021:
14000:
13997:
13988:
13985:
13980:
13976:
13967:
13963:
13954:
13942:
13939:
13936:
13907:
13903:
13896:
13886:
13876:
13872:
13863:
13842:
13833:
13819:
13796:
13793:
13788:
13784:
13780:
13775:
13771:
13764:
13761:
13756:
13752:
13726:
13722:
13713:
13709:
13705:
13700:
13697:
13693:
13667:
13663:
13659:
13654:
13650:
13643:
13640:
13635:
13631:
13616:
13607:
13592:
13589:
13586:
13583:
13580:
13555:
13551:
13547:
13542:
13538:
13530:
13524:
13520:
13516:
13513:
13504:
13500:
13496:
13491:
13487:
13480:
13477:
13472:
13468:
13464:
13461:
13439:
13435:
13412:
13408:
13387:
13378:
13375:
13371:
13355:
13352:
13347:
13343:
13333:
13331:
13327:
13322:
13320:
13316:
13299:
13295:
13291:
13288:
13283:
13279:
13258:
13255:
13250:
13246:
13236:
13219:
13216:
13209:
13200:
13196:
13192:
13187:
13183:
13176:
13173:
13150:
13147:
13144:
13141:
13129:
13125:
13107:
13103:
13099:
13094:
13090:
13080:
13076:
13061:
13053:
13047:
13044:
13041:
13038:
13035:
13029:
13023:
13020:
13017:
13014:
13011:
13005:
12999:
12991:
12988:
12985:
12982:
12979:
12976:
12973:
12970:
12965:
12962:
12959:
12956:
12953:
12950:
12947:
12944:
12938:
12932:
12926:
12901:
12898:
12894:
12887:
12881:
12874:
12871:
12867:
12860:
12854:
12851:
12848:
12841:
12836:
12833:
12828:
12824:
12819:
12816:
12812:
12799:
12795:
12788:
12782:
12775:
12771:
12764:
12758:
12755:
12752:
12745:
12740:
12737:
12732:
12728:
12723:
12720:
12716:
12691:
12683:
12677:
12667:
12659:
12653:
12650:
12647:
12639:
12631:
12625:
12619:
12616:
12611:
12608:
12604:
12594:
12586:
12580:
12569:
12563:
12560:
12557:
12554:
12551:
12542:
12536:
12533:
12526:
12521:
12518:
12513:
12509:
12503:
12495:
12492:
12488:
12479:
12461:
12457:
12434:
12430:
12409:
12406:
12402:
12394:
12390:
12386:
12381:
12374:
12370:
12366:
12360:
12356:
12353:
12350:
12346:
12341:
12335:
12332:
12327:
12323:
12317:
12314:
12309:
12305:
12298:
12294:
12291:
12288:
12282:
12279:
12274:
12270:
12263:
12259:
12253:
12247:
12244:
12239:
12235:
12228:
12224:
12217:
12211:
12207:
12203:
12180:
12177:
12174:
12168:
12165:
12162:
12140:
12136:
12109:
12100:
12084:
12081:
12077:
12074:
12068:
12065:
12059:
12056:
12053:
12050:
12046:
12024:
12021:
11997:
11994:
11991:
11985:
11965:
11945:
11938:Here because
11925:
11922:
11914:
11910:
11906:
11901:
11897:
11888:
11884:
11880:
11877:
11874:
11870:
11862:
11858:
11850:
11846:
11841:
11837:
11834:
11831:
11828:
11822:
11818:
11811:
11808:
11803:
11799:
11791:
11787:
11782:
11776:
11771:
11768:
11765:
11761:
11757:
11752:
11748:
11744:
11739:
11735:
11712:
11708:
11704:
11699:
11695:
11672:
11668:
11664:
11659:
11655:
11634:
11631:
11627:
11624:
11621:
11618:
11614:
11611:
11608:
11605:
11601:
11596:
11588:
11585:
11582:
11578:
11573:
11565:
11562:
11559:
11550:
11536:
11533:
11530:
11527:
11524:
11521:
11499:
11495:
11491:
11486:
11482:
11478:
11461:
11435:
11431:
11427:
11422:
11418:
11411:
11408:
11391:
11382:
11364:
11360:
11356:
11351:
11347:
11324:
11320:
11316:
11311:
11307:
11303:
11294:
11290:
11281:
11256:
11251:
11247:
11243:
11238:
11232:
11229:
11224:
11220:
11212:
11208:
11204:
11198:
11193:
11189:
11172:
11167:
11163:
11159:
11154:
11148:
11145:
11140:
11136:
11128:
11124:
11120:
11114:
11109:
11105:
11070:
11066:
11061:
11057:
11022:
11018:
11013:
11009:
10974:
10970:
10965:
10961:
10957:
10940:
10917:
10913:
10891:
10887:
10883:
10880:
10875:
10871:
10863:
10847:
10833:
10815:
10810:
10806:
10798:
10795:
10790:
10786:
10782:
10767:
10763:
10756:
10752:
10744:
10741:
10738:
10735:
10729:
10723:
10709:
10699:
10694:
10690:
10686:
10679:
10671:
10668:
10663:
10659:
10655:
10640:
10636:
10628:
10624:
10620:
10613:
10605:
10602:
10599:
10590:
10587:
10584:
10570:
10546:
10541:
10537:
10501:
10497:
10480:
10476:
10473:
10439:
10435:
10432:
10398:
10394:
10391:
10388:
10371:
10350:
10347:
10342:
10332:
10325:
10321:
10284:
10249:
10222:
10218:
10215:
10195:
10186:
10182:
10179:
10170:
10163:
10159:
10157:
10141:
10138:
10133:
10123:
10116:
10112:
10089:
10086:
10083:
10078:
10074:
10053:
10050:
10047:
10044:
10030:
10016:
10013:
10008:
10005:
10000:
9996:
9987:
9966:
9961:
9957:
9953:
9946:
9940:
9932:
9929:
9924:
9920:
9916:
9904:
9900:
9892:
9888:
9884:
9877:
9871:
9863:
9860:
9857:
9851:
9848:
9845:
9842:
9820:
9816:
9795:
9792:
9787:
9783:
9773:
9759:
9739:
9731:
9727:
9703:
9698:
9694:
9689:
9685:
9682:
9679:
9674:
9665:
9661:
9656:
9648:
9645:
9642:
9637:
9612:
9609:
9604:
9600:
9596:
9593:
9584:
9570:
9564:
9544:
9541:
9536:
9532:
9528:
9523:
9519:
9492:
9472:
9469:
9461:
9453:
9450:
9447:
9441:
9438:
9435:
9430:
9426:
9422:
9419:
9412:
9408:
9404:
9399:
9395:
9388:
9382:
9374:
9370:
9366:
9360:
9357:
9354:
9346:
9342:
9319:
9315:
9292:
9288:
9283:
9277:
9273:
9269:
9266:
9263:
9260:
9255:
9246:
9240:
9232:
9227:
9216:
9213:
9210:
9205:
9201:
9197:
9192:
9188:
9179:
9161:
9157:
9134:
9124:
9119:
9111:
9106:
9102:
9079:
9075:
9065:
9051:
9047:
9043:
9040:
9035:
9031:
9010:
9006:
9002:
8999:
8996:
8993:
8970:
8967:
8960:
8956:
8953:
8948:
8944:
8923:
8919:
8915:
8912:
8909:
8884:
8877:In the limit
8859:
8851:
8848:
8845:
8839:
8836:
8833:
8828:
8824:
8820:
8817:
8809:
8806:
8803:
8795:
8791:
8787:
8781:
8778:
8761:
8753:
8750:
8747:
8741:
8738:
8735:
8730:
8726:
8722:
8719:
8712:
8704:
8701:
8698:
8690:
8686:
8679:
8674:
8670:
8647:
8643:
8637:
8627:
8624:
8620:
8616:
8613:
8593:
8588:
8578:
8575:
8571:
8567:
8564:
8559:
8555:
8532:
8529:
8519:
8515:
8511:
8505:
8483:
8479:
8474:
8470:
8446:
8442:
8437:
8434:
8428:
8420:
8412:
8409:
8406:
8398:
8395:
8389:
8386:
8380:
8372:
8351:
8348:
8345:
8341:
8336:
8330:
8322:
8305:
8298:
8291:
8285:
8267:
8263:
8259:
8252:
8248:
8245:
8240:
8236:
8223:
8220:
8215:
8211:
8207:
8199:
8194:
8190:
8183:
8180:
8177:
8172:
8160:
8154:
8151:
8146:
8142:
8138:
8131:
8125:
8121:
8117:
8114:
8111:
8105:
8100:
8088:
8083:
8079:
8058:
8054:
8050:
8047:
8044:
8041:
8038:
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5809:
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5778:
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5742:
5732:
5728:
5719:
5709:
5690:
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5677:
5668:
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5619:
5615:
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5574:
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5521:
5517:
5494:
5490:
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5439:
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5342:
5321:
5318:
5313:
5309:
5288:
5285:
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5279:
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5273:
5270:
5267:
5264:
5261:
5258:
5250:
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5236:
5234:
5216:
5212:
5187:
5182:
5178:
5174:
5169:
5163:
5160:
5155:
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5147:
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5132:
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5121:
5117:
5090:
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5047:
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5022:
5017:
5013:
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4945:
4922:
4913:
4909:
4900:
4895:
4891:
4888:
4882:
4873:
4869:
4866:
4857:
4852:
4848:
4838:
4834:
4831:
4822:
4817:
4807:
4803:
4800:
4797:
4771:
4750:
4747:
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4733:
4729:
4726:
4721:
4712:
4708:
4705:
4700:
4696:
4686:
4682:
4679:
4674:
4670:
4666:
4661:
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4640:
4617:
4614:
4611:
4608:
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4599:
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4587:
4574:
4570:
4567:
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4555:
4548:
4543:
4539:
4535:
4526:
4505:
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4468:
4464:
4461:
4452:
4428:
4425:
4422:
4415:
4411:
4408:
4405:
4396:
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4389:
4386:
4383:
4374:
4349:
4329:
4309:
4301:
4286:
4266:
4263:
4260:
4252:
4237:
4234:
4229:
4217:
4213:
4205:
4201:
4196:
4188:
4183:
4173:
4166:
4162:
4133:
4129:
4126:
4123:
4114:
4110:
4107:
4087:
4084:
4079:
4069:
4062:
4058:
4029:
4025:
4022:
4014:
4013:
4008:
4004:
3996:
3994:
3975:
3967:
3963:
3959:
3954:
3950:
3946:
3940:
3932:
3928:
3924:
3919:
3915:
3904:
3900:
3895:
3893:
3877:
3857:
3832:
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3824:
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3807:
3798:
3794:
3788:
3784:
3780:
3775:
3771:
3765:
3761:
3752:
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3744:
3741:
3713:
3709:
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3700:
3696:
3688:
3679:
3675:
3671:
3666:
3662:
3655:
3652:
3649:
3646:
3643:
3621:
3617:
3613:
3608:
3604:
3581:
3577:
3573:
3568:
3564:
3555:
3551:
3547:
3546:liquid oxygen
3542:
3540:
3522:
3519:
3496:
3493:
3484:
3480:
3475:
3473:
3469:
3460:
3456:
3454:
3450:
3446:
3442:
3425:
3405:
3402:
3399:
3396:
3392:
3381:
3377:
3371:
3344:
3340:
3317:
3297:
3277:
3257:
3253:
3242:
3238:
3232:
3229:
3226:
3223:
3215:
3205:
3202:
3185:
3182:
3173:
3168:
3164:
3156:
3152:
3148:
3139:
3134:
3128:
3124:
3103:
3100:
3088:
3083:
3076:
3073:
3064:
3059:
3055:
3047:
3043:
3036:
3033:
3030:
3027:
3024:
3016:
3012:
2987:
2983:
2976:
2966:
2956:
2952:
2943:
2929:
2925:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2836:
2833:
2830:
2827:
2824:
2821:
2801:
2798:
2795:
2792:
2789:
2786:
2764:
2760:
2739:
2719:
2699:
2677:
2673:
2664:
2662:
2658:
2656:
2652:
2648:
2644:
2629:
2625:
2616:
2606:
2601:
2590:
2587:
2582:
2578:
2557:
2553:
2544:
2534:
2529:
2518:
2515:
2507:
2506:
2505:
2497:
2495:
2491:
2487:
2482:
2477:
2462:
2459:
2456:
2453:
2450:
2447:
2443:
2433:
2422:
2418:
2414:
2411:
2408:
2405:
2385:
2361:
2339:
2335:
2330:
2326:
2323:
2320:
2298:
2294:
2290:
2282:
2278:
2274:
2268:
2265:
2257:
2242:
2222:
2219:
2216:
2196:
2193:
2190:
2182:
2181:
2180:
2166:
2162:
2158:
2155:
2152:
2132:
2118:
2104:
2078:
2051:
2047:
2044:
2041:
2038:
2012:
2008:
2005:
1979:
1975:
1972:
1946:
1942:
1939:
1917:
1913:
1890:
1886:
1863:
1859:
1836:
1832:
1811:
1786:
1782:
1778:
1773:
1769:
1740:
1736:
1732:
1727:
1723:
1710:
1682:
1678:
1661:
1657:
1654:
1620:
1616:
1613:
1579:
1575:
1558:
1523:
1519:
1502:
1475:
1470:
1467:
1464:
1444:
1435:
1431:
1428:
1408:
1385:
1382:
1379:
1376:
1350:
1346:
1343:
1340:
1320:
1300:
1288:
1284:
1282:
1259:
1255:
1252:
1226:
1222:
1219:
1193:
1166:
1157:
1153:
1152:boiling point
1137:
1117:
1108:
1094:
1086:
1064:
1035:
1028:
1024:
1004:
996:
980:
976:
972:
964:
948:
944:
934:
930:
922:
906:
902:
898:
890:
885:
868:
862:
842:
822:
813:
809:
806:
803:
800:
774:
769:
765:
739:
732:
728:
722:
714:
710:
706:
700:
697:
691:
688:
685:
665:
662:
642:
622:
602:
594:
578:
571:The constant
554:
551:
546:
542:
538:
535:
515:
512:
509:
489:
469:
460:
456:
442:
422:
414:
398:
390:
374:
365:
361:
358:
350:
346:
324:
319:
315:
295:
287:
271:
263:
241:
232:
216:
212:
202:
198:
195:
192:
184:
168:
160:
144:
120:
116:
112:
107:
101:
98:
95:
90:
87:
81:
78:
70:
48:
44:
40:
36:
35:ideal gas law
32:
28:
24:
19:
28673:
28638:
28634:
28623:
28596:
28574:. Retrieved
28555:
28545:
28536:
28530:
28518:. Retrieved
28509:
28499:
28490:
28481:
28473:
28468:
28459:
28447:. Retrieved
28438:
28428:
28416:. Retrieved
28407:
28397:
28388:
28380:
28375:
28366:
28357:
28349:
28345:
28344:DeBoer, J.,
28340:
28331:
28322:
28314:
28309:
28300:
28291:
28279:. Retrieved
28267:
28263:
28238:
28230:
28225:
28216:
28207:
28198:
28189:
28180:
28172:
28168:
28163:
28155:
28150:
28141:
28132:
28123:
28115:
28111:
28106:
28097:
28088:
28079:
28071:
28067:
28062:
28053:
28045:
28040:
28032:
28028:
28023:
28015:
28011:
28006:
27983:
27974:
27965:
27956:
27948:
27943:
27934:
27925:
27916:
27907:
27898:
27889:
27881:
27876:
27868:
27863:
27855:
27851:
27846:
27838:
27822:
27817:
27808:
27799:
27790:
27781:
27773:
27769:
27752:
27743:
27734:
27725:
27716:
27708:
27704:
27699:
27691:
27687:
27682:
27673:
27664:
27656:
27651:
27643:
27642:Gibbs, J.W.
27638:
27630:
27625:
27617:
27613:
27608:
27600:
27596:
27595:Peck, R.E.,
27591:
27583:
27579:
27563:
27558:
27549:
27541:
27536:
27527:
27518:
27509:
27500:
27491:
27483:
27478:
27470:
27465:
27456:
27448:
27443:
27435:
27430:
27422:
27417:
27066:
27004:
26850:
26728:
26725:
26722:
26718:
26714:
26709:
26707:
26616:
26507:
26446:
26442:
26438:
26434:
26432:
26381:
26356:
26352:
26351:
26214:
26210:
26206:
26198:
26194:
26186:
26184:
26098:
26013:
26006:
26002:
25997:
25985:
25881:
25874:
25869:
25867:
25700:
25576:
25513:
25510:
25456:
25449:
25447:
25339:
25234:
25230:
25228:
25223:
25219:
25196:
25130:
25126:
25122:
25118:
25114:
25106:
25104:
25048:
25035:
24581:
24418:
24120:
23255:
23250:
22607:
22057:
21752:
21304:
20441:. Thus when
20405:
20125:
19861:
18865:
18372:
17915:
17709:
17565:By plotting
17564:
17492:
17118:(for finite
17058:
16523:
15726:which gives
15337:
15332:
15225:
14934:plane where
14780:
14647:
14400:
14300:
13834:
13622:
13379:
13334:
13323:
13237:
13133:
11551:
10861:
10839:
10311:, for which
10171:
10168:
10036:
9774:
9752:faster than
9585:
9066:
8308:
8300:
8293:
8289:
7884:
7497:
7051:
6413:. Then with
6277:
6000:
5869:
5710:
5246:
5242:
5238:
5232:
4992:
4497:
4365:
4002:
3905:
3902:
3897:
3543:
3482:
3476:
3468:Pieter Rijke
3465:
3213:
3211:
3203:
2933:
2503:
2478:
2377:
2124:
1711:
1292:
1109:
886:
570:
231:molar volume
43:phase change
22:
20:
18:
28348:, Physica,
27768:Lekner, J.
27707:, Physica,
26012:identical (
25038:derivations
22953:direction,
17998:, the term
17470:approaches
17321:, and when
183:temperature
28695:Categories
28683:0716710889
28606:0195042778
28520:5 December
28418:1 February
28167:Su, G.J.,
28014:, Nature,
27409:References
27087:. We get:
26712:there are
25032:Derivation
20944:previously
20128:diathermal
11049:and vapor
10836:Saturation
9180:equation,
9023:, finally
5842:) factor,
3017:) factor,
2732:, entropy
2494:real gases
2486:Sommerfeld
1085:molar mass
28576:23 August
27378:Iteration
27368:Ideal gas
27317:−
27263:⇒
27234:−
27175:⇒
27138:−
27121:−
27045:∂
27037:
27031:∂
26982:
26976:−
26963:Λ
26955:
26946:−
26890:−
26880:
26865:
26825:ϵ
26803:π
26794:ϵ
26750:−
26744:ϕ
26674:π
26644:∞
26635:∫
26628:ϕ
26591:Λ
26564:ϕ
26561:−
26538:−
26474:π
26367:ϵ
26324:≥
26283:ϵ
26280:−
26251:∞
26157:π
26134:Λ
26107:Λ
26076:Λ
25947:−
25835:−
25777:⇒
25750:−
25741:−
25701:so that:
25411:π
25402:×
25384:→
25303:π
25294:×
25267:π
25173:−
24994:σ
24979:σ
24935:−
24892:σ
24870:∑
24849:∑
24800:σ
24787:ϵ
24766:∑
24745:∑
24727:ϵ
24680:σ
24658:∑
24637:∑
24619:σ
24562:σ
24558:∝
24529:σ
24525:ϵ
24522:∝
24433:σ
24427:ϵ
24215:∑
24146:∑
24000:∑
23979:∑
23862:∑
23745:∑
23724:∑
23649:σ
23636:σ
23620:σ
23583:ϵ
23573:ϵ
23557:ϵ
23183:μ
23156:μ
23129:μ
23102:μ
23098:−
23089:μ
23022:μ
22975:μ
22971:−
22962:μ
22918:−
22892:μ
22865:μ
22772:∂
22759:μ
22732:μ
22713:μ
22709:−
22700:μ
22684:−
22662:μ
22639:μ
22626:−
22560:∂
22553:−
22529:∂
22504:∂
22440:≤
22434:≤
22389:∂
22383:∂
22369:∂
22357:−
22341:∂
22327:∂
22307:∂
22293:∂
22266:∂
22252:∂
22225:∂
22211:∂
22184:−
21918:−
21903:−
21866:
21860:−
21843:−
21784:−
21708:≥
21575:∂
21571:−
21544:∂
21514:∂
21510:−
21483:∂
21358:≤
21189:−
21112:−
21083:∂
20626:≥
20531:−
20423:−
20296:∂
20285:∂
20282:−
20254:∂
20237:∂
20056:produces
19608:≈
19542:≠
19485:(so that
19077:−
19061:−
18997:−
18981:−
18918:−
18902:−
18772:−
18753:−
18626:−
18560:ρ
18507:ρ
18471:ρ
18460:∞
18445:∑
18433:−
18425:ρ
18419:−
18295:ρ
18252:ρ
18201:ρ
18191:∞
18176:∑
18152:ρ
18137:−
18102:−
18062:ρ
18030:−
18016:ρ
18012:−
17974:ρ
17965:that for
17931:ρ
17809:ρ
17781:∞
17766:∑
17744:ρ
17679:ρ
17636:ρ
17580:ρ
17544:ρ
17507:ρ
17330:ρ
17285:ρ
17281:−
17206:−
17187:ρ
17183:−
17152:ρ
17077:→
17068:ρ
17019:−
16957:ρ
16920:≤
16890:ρ
16886:≤
16839:ρ
16829:−
16817:ρ
16813:−
16761:ρ
16755:−
16749:ρ
16743:−
16697:ρ
16685:ρ
16624:→
16607:ρ
16486:ρ
16479:ρ
16474:∗
16449:∞
16140:¯
16110:∗
16006:−
15998:±
15948:−
15905:∗
15849:∗
15753:∗
15688:∗
15663:−
15652:∗
15604:∗
15550:∗
15537:∗
15475:∗
15450:−
15427:∗
15396:−
15388:∗
15237:≫
15178:−
15162:−
15116:−
15101:−
15061:−
15045:−
15010:α
14981:−
14975:α
14943:μ
14884:μ
14857:μ
14817:−
14811:α
14790:μ
14670:∂
14657:μ
14628:ω
14608:ω
14585:−
14579:ω
14556:−
14550:ω
14527:−
14521:ω
14495:ω
14486:−
14436:ω
14386:ω
14366:ω
14276:≤
14263:≤
14237:±
14204:
14156:11.181777
14153:−
14144:ω
14117:−
14089:
14063:ω
13998:−
13943:−
13937:ω
13843:ϕ
13820:ϕ
13797:ϕ
13590:≤
13584:≤
13497:−
13193:−
13154:∞
13145:≤
13039:
12989:−
12983:
12974:
12963:
12957:−
12951:
12899:−
12872:−
12651:−
12555:
12382:−
12351:−
12333:−
12315:−
12295:
12289:−
12280:−
12254:−
12245:−
12075:−
12025:∘
11907:−
11842:∫
11838:−
11783:∫
11762:∑
11745:−
11625:−
11593:∂
11570:∂
11531:−
11492:−
11428:−
11239:−
11230:−
11155:−
11146:−
10796:−
10739:−
10669:−
10603:−
10550:∞
10538:κ
10395:≤
10389:≤
10330:∂
10319:∂
10121:∂
10110:∂
10033:Stability
10006:−
9997:κ
9930:−
9861:−
9817:κ
9796:α
9760:α
9737:→
9728:κ
9707:∞
9695:κ
9686:α
9671:∂
9653:∂
9649:−
9634:∂
9601:κ
9594:α
9568:→
9529:−
9499:∞
9496:→
9470:≥
9451:−
9436:−
9367:−
9289:κ
9274:α
9252:∂
9224:∂
9214:−
9198:−
9116:∂
9032:κ
8945:κ
8910:α
8891:∞
8888:→
8849:−
8834:−
8807:−
8779:α
8751:−
8736:−
8702:−
8671:κ
8634:∂
8625:−
8614:α
8585:∂
8576:−
8568:−
8556:κ
8530:−
8516:κ
8506:−
8480:κ
8471:α
8410:−
8390:−
8378:∂
8370:∂
8349:−
8328:∂
8320:∂
8246:−
8221:−
8184:
8178:−
8161:−
8152:−
8089:−
8045:−
8024:−
8002:
7996:−
7983:−
7974:−
7935:−
7906:−
7857:−
7820:
7797:−
7758:−
7736:
7714:−
7685:−
7656:∂
7586:∂
7578:∂
7571:∫
7529:∫
7513:−
7400:−
7378:−
7315:−
7271:−
7259:−
7214:−
7203:is simply
7160:−
7151:−
6922:−
6889:−
6847:−
6822:−
6672:∂
6604:∂
6519:∂
6496:∂
6465:∂
6435:∂
6384:∂
6360:∂
6218:∂
6196:∂
6174:∫
6135:∫
6099:−
6090:∂
6082:∂
6067:∫
6028:∫
6012:−
5850:ω
5691:∗
5620:∗
5549:∗
5495:∗
5474:∗
5453:∗
5395:∗
5347:∗
5314:∗
5170:−
5161:−
4730:−
4680:−
4644:−
4606:−
4549:−
4210:∂
4193:∂
4171:∂
4160:∂
4067:∂
4056:∂
3892:Goodstein
3825:−
3781:−
3706:−
3672:−
3653:−
3387:¯
3350:¯
3248:¯
3186:ω
3101:−
3037:
3031:−
3025:ω
3009:, or the
2905:−
2875:−
2831:−
2598:∂
2591:−
2579:κ
2526:∂
2516:α
2269:≫
2258:and when
2220:−
2194:≫
2153:ρ
1679:≤
1658:≤
1617:≥
1389:∞
1380:≤
1068:¯
1039:¯
889:Boltzmann
863:φ
810:ε
707:σ
701:π
666:ε
663:−
643:σ
623:σ
593:dimension
555:σ
516:σ
108:−
99:−
56:↔
39:molecules
28711:Gas laws
28641:: 1–32.
28570:Archived
28514:Archived
28443:Archived
28412:Archived
27388:Real gas
27363:Gas laws
27357:See also
27244:′
27206:′
27158:′
27131:′
26928:′
26900:′
26832:′
26787:′
26760:′
26548:′
26464:′
25614:′
25593:′
25494:′
25392:′
25371:′
25355:′
25257:′
25125:, where
24471:through
22107:, where
20331:pressure
18862:Mixtures
18342: or
17354:this is
14848:so when
14198:17.92998
13374:Lienhard
9986:spinodal
9775:Finally
8547:, where
7114:is just
7054:enthalpy
7048:Enthalpy
5840:acentric
3116:, where
3015:acentric
2070:, where
159:pressure
29:, is an
28643:Bibcode
28449:2 April
28281:2 April
27823:Scaling
26189:is the
26119:is the
25988:Laplace
24582:article
22993:in the
22933:in the
20675:, with
20383:at all
19439:, with
14177:3.68769
14150:7.49408
14108:5.37270
10156:Epstein
7500:entropy
7494:Entropy
7183:. Then
3208:History
2481:Epstein
2313:, then
2209:, then
1083:is the
411:is the
387:is the
284:is the
260:is the
28680:
28661:
28603:
28562:
26099:where
26007:N = nN
23120:, and
22836:. For
21957:where
21228:0.5730
20381:convex
19391:, and
19113:where
17700:figure
17670:using
16875:where
16559:where
15621:, for
13812:where
13573:where
13328:, and
13126:, and
12919:where
9808:, and
8936:while
7611:Using
6961:where
6248:where
5637:, and
3011:Pitzer
2490:Newton
1758:, and
835:where
415:, and
288:, and
286:volume
185:, and
137:where
47:liquid
28619:(PDF)
28260:(PDF)
25237:) is
25109:(the
23204:plait
22750:Here
21192:0.215
21181:, is
21026:0.576
20974:0.875
20379:, is
19343:Here
16035:, or
15812:, or
14559:0.267
14530:0.302
13619:2006.
10562:) as
9178:Mayer
7433:where
7018:and C
6791:with
6663:, so
3418:with
2183:when
1110:Once
963:Gibbs
347:with
343:is a
69:vapor
28678:ISBN
28659:ISBN
28601:ISBN
28578:2016
28560:ISBN
28522:2023
28451:2024
28420:2019
28283:2024
26777:with
26315:when
26267:<
26258:when
26205:and
26201:the
26058:with
25013:<
24972:<
24547:and
23349:and
23174:and
22883:and
22475:and
22408:>
22282:>
22241:>
21347:for
21062:2.71
21038:and
20912:and
20461:and
20314:>
20213:>
20192:and
19916:and
19819:and
19738:and
19670:and
18347:<
18337:>
17983:<
17251:for
16255:and
16154:and
16069:and
15298:<
15266:>
14952:>
14588:0.21
14498:<
14492:<
14240:0.42
13372:and
13370:Tien
13317:and
13256:<
13151:<
13036:cosh
12980:cosh
12971:sinh
12960:sinh
12948:cosh
12552:cosh
11067:>
11019:<
10971:<
10958:<
10862:e.g.
10498:>
10477:>
10436:<
10348:>
10219:<
10183:>
10139:<
10087:<
10066:and
8299:and
7885:The
7498:The
7052:The
5826:0.31
5823:<
5810:<
5807:0.23
4085:<
3870:and
3734:and
3596:and
3548:and
3539:here
2926:its
2048:<
2042:<
1976:<
1943:>
1851:and
1432:>
1386:<
1256:>
1223:>
1130:and
552:>
539:>
513:<
435:and
21:The
28651:doi
28272:doi
27692:159
25448:So
25117:by
24720:and
24511:by
24192:and
23613:and
23441:and
22160:so
20946:.
19226:and
18948:and
18286:at
17614:vs
17535:vs
16599:or
14501:0.9
14489:0.4
14331:vs
14279:0.3
13989:0.7
13949:log
13427:to
13356:0.8
12449:or
11687:to
11183:and
9334:as
8773:and
8361:and
7889:is
7056:is
3077:0.7
3034:log
2492:of
1709:.
1471:0.2
1313:vs
229:is
181:is
157:is
28697::
28657:.
28649:.
28637:.
28621:,
28568:.
28554:.
28508:.
28474:65
28441:.
28437:.
28410:.
28406:.
28350:73
28268:44
28266:.
28262:.
28247:^
28231:12
28173:38
28116:64
28072:95
28033:61
28016:11
27992:^
27856:64
27830:^
27774:50
27761:^
27709:73
27618:77
27601:60
27584:10
27571:^
27034:ln
26979:ln
26952:ln
26877:ln
26862:ln
26729:dr
26447:Nb
26445:−
26379:.
26193:,
26123::
25880:=
25875:nV
24998:22
24983:11
24294:64
24291:27
23653:22
23640:11
23624:12
23587:22
23577:11
23561:12
23507:22
23481:11
23452:12
23415:22
23405:11
23389:12
23362:12
23335:12
23308:22
23295:11
23282:22
23269:11
23080:,
23060:,
22856:,
21863:ln
20163:,
20098:22
20075:11
19985:22
19972:11
19956:12
19943:,
19929:22
19902:11
19875:12
19845:21
19832:12
19805:21
19792:12
19765:22
19751:11
19724:22
19710:11
19314:22
19281:12
19250:11
19203:22
19170:12
19139:11
19007:22
18989:22
18928:11
18910:11
18708:24
18350:27
18105:27
17490:.
17430:27
17375:27
17092:,
17056:.
16981:,
16639:,
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16263:27
16132:0.
16009:12
16003:16
15335:"
15301:27
15144:or
14640:.
14201:ln
14086:ln
13955:10
12620:27
12519:27
12292:ln
9772:.
9511:,
8292:,
8181:ln
7999:ln
7817:ln
7733:ln
5993:.
5661:27
5566:,
5530:27
4990:.
4495:.
4473:27
4423:27
4322:,
4299:);
3995:.
3511:CO
3474:.
3455:.
1185:,
1107:.
884:.
391:,
264:,
233:,
161:,
45:,
28686:.
28667:.
28653::
28645::
28639:2
28629:.
28609:.
28580:.
28524:.
28453:.
28422:.
28285:.
28274::
27341:.
27338:T
27335:R
27332:n
27329:=
27326:)
27323:b
27320:n
27314:V
27311:(
27307:)
27299:2
27295:V
27290:a
27285:2
27281:n
27274:+
27271:p
27267:(
27260:T
27257:k
27254:N
27251:=
27248:)
27241:b
27237:N
27231:V
27228:(
27224:)
27216:2
27212:V
27203:a
27197:2
27193:N
27186:+
27183:p
27179:(
27168:2
27164:V
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27149:2
27145:N
27128:b
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27107:N
27101:=
27098:p
27075:V
27063:,
27048:V
27040:Z
27025:T
27022:k
27019:=
27016:p
26989:!
26986:N
26972:)
26967:3
26959:(
26949:N
26940:T
26937:k
26934:V
26925:a
26919:2
26915:N
26908:+
26904:)
26897:b
26893:N
26887:V
26884:(
26874:N
26871:=
26868:Z
26836:.
26829:b
26822:=
26817:3
26811:3
26807:d
26800:2
26791:=
26784:a
26769:V
26766:N
26757:a
26753:2
26747:=
26726:r
26723:π
26721:4
26719:V
26717:/
26715:N
26710:r
26693:,
26690:r
26687:d
26682:2
26678:r
26671:4
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26655:r
26652:(
26649:u
26639:d
26631:=
26602:.
26595:3
26584:)
26581:T
26578:k
26575:2
26572:(
26568:/
26557:e
26552:)
26545:b
26541:N
26535:V
26532:(
26526:=
26523:z
26504:.
26492:3
26488:/
26482:3
26478:d
26471:2
26468:=
26461:b
26443:V
26439:V
26435:z
26429:,
26417:!
26414:N
26410:/
26404:N
26400:z
26396:=
26393:Z
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26330:,
26327:d
26321:r
26306:6
26301:)
26296:r
26293:d
26288:(
26273:,
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26264:r
26245:{
26240:=
26237:)
26234:r
26231:(
26228:u
26215:V
26211:z
26207:T
26199:k
26195:m
26187:h
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26144:h
26137:=
26080:3
26072:V
26067:=
26064:z
26049:!
26046:N
26040:N
26036:z
26030:=
26027:Z
26010:A
26003:Z
25971:.
25968:T
25965:R
25962:n
25959:=
25956:)
25953:b
25950:n
25944:V
25941:(
25937:)
25929:2
25925:V
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25911:n
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25901:p
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25882:V
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25870:n
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25850:T
25847:R
25844:=
25841:)
25838:b
25829:m
25824:V
25820:(
25816:)
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25788:+
25785:p
25781:(
25770:2
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25715:=
25712:p
25697:,
25681:2
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25670:V
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25661:=
25656:2
25651:)
25643:m
25638:V
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25626:N
25620:(
25611:a
25607:=
25602:2
25598:C
25590:a
25573:.
25558:m
25553:V
25548:/
25541:A
25536:N
25532:=
25529:C
25517:A
25514:N
25507:.
25491:b
25485:A
25481:N
25477:=
25474:b
25459:′
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25450:b
25444:,
25432:)
25429:3
25425:/
25419:3
25415:r
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25405:(
25399:4
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25389:b
25380:2
25376:/
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25352:b
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25324:)
25321:3
25317:/
25311:3
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25300:4
25297:(
25291:8
25288:=
25285:3
25281:/
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25261:=
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25224:r
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25218:2
25205:b
25182:.
25176:b
25167:m
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25144:p
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25119:V
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25090:.
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25077:V
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25069:R
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25060:p
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24989:/
24975:(
24969:2
24965:/
24961:1
24938:1
24930:]
24924:j
24920:x
24914:i
24910:x
24904:3
24899:j
24896:i
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24880:1
24877:=
24874:j
24864:n
24859:i
24856:=
24853:i
24844:[
24838:]
24832:j
24828:x
24822:i
24818:x
24812:3
24807:j
24804:i
24794:j
24791:i
24781:n
24776:1
24773:=
24770:j
24760:n
24755:1
24752:=
24749:i
24740:[
24736:=
24731:x
24712:j
24708:x
24702:i
24698:x
24692:3
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24684:i
24673:n
24668:1
24665:=
24662:j
24652:n
24647:i
24644:=
24641:i
24633:=
24628:3
24623:x
24598:b
24595:,
24592:a
24566:3
24555:b
24533:3
24519:a
24497:c
24493:p
24489:,
24484:c
24480:T
24459:b
24456:,
24453:a
24430:,
24400:x
24397:c
24393:p
24386:x
24383:c
24379:T
24375:R
24367:8
24364:1
24359:=
24354:x
24350:b
24340:x
24337:c
24333:p
24326:2
24322:)
24316:x
24313:c
24309:T
24305:R
24302:(
24286:=
24281:x
24277:a
24254:i
24250:x
24244:i
24241:c
24237:p
24230:n
24225:1
24222:=
24219:i
24211:=
24206:x
24203:c
24199:p
24184:i
24180:x
24174:i
24171:c
24167:T
24161:n
24156:1
24153:=
24150:i
24142:=
24137:x
24134:c
24130:T
24103:j
24099:x
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24089:x
24083:3
24079:)
24073:3
24069:/
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24060:j
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24044:3
24040:/
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24024:b
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24015:n
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24004:j
23994:n
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23983:i
23975:)
23972:8
23968:/
23964:1
23961:(
23958:=
23953:x
23949:b
23927:2
23921:)
23913:i
23909:x
23903:2
23899:/
23895:1
23890:i
23887:i
23883:a
23877:n
23872:1
23869:=
23866:i
23856:(
23851:=
23846:x
23842:a
23827:j
23823:x
23817:i
23813:x
23807:2
23803:/
23799:1
23795:)
23789:j
23786:j
23782:a
23776:i
23773:i
23769:a
23765:(
23760:n
23755:1
23752:=
23749:j
23739:n
23734:1
23731:=
23728:i
23720:=
23715:x
23711:a
23690:n
23669:,
23666:2
23662:/
23658:)
23645:+
23632:(
23629:=
23605:2
23601:/
23597:1
23593:)
23569:(
23566:=
23536:,
23533:2
23529:/
23525:)
23520:3
23516:/
23512:1
23503:b
23499:+
23494:3
23490:/
23486:1
23477:b
23473:(
23470:=
23465:3
23461:/
23457:1
23448:b
23433:2
23429:/
23425:1
23421:)
23411:a
23401:a
23397:(
23394:=
23385:a
23358:b
23331:a
23304:b
23300:,
23291:b
23287:,
23278:a
23274:,
23265:a
23234:v
23214:x
23187:2
23160:1
23133:1
23106:1
23093:2
23068:p
23048:T
23026:1
23001:x
22979:1
22966:2
22941:v
22921:p
22896:2
22869:1
22844:p
22824:2
22821:,
22818:1
22815:=
22812:j
22788:f
22781:j
22777:x
22768:=
22763:j
22736:1
22728:+
22725:x
22722:)
22717:1
22704:2
22696:(
22693:+
22690:v
22687:p
22681:=
22676:2
22672:x
22666:2
22658:+
22653:1
22649:x
22643:1
22635:+
22632:v
22629:p
22623:=
22620:f
22593:0
22590:=
22585:2
22581:)
22577:f
22572:2
22567:v
22564:x
22556:(
22550:f
22545:2
22538:2
22534:v
22525:f
22520:2
22513:2
22509:x
22483:x
22463:v
22443:1
22437:x
22431:0
22411:0
22403:2
22398:)
22392:v
22386:x
22378:f
22373:2
22362:(
22349:2
22345:x
22336:f
22331:2
22315:2
22311:v
22302:f
22297:2
22285:0
22274:2
22270:x
22261:f
22256:2
22244:0
22233:2
22229:v
22220:f
22215:2
22187:x
22181:1
22178:=
22173:1
22169:x
22146:2
22142:x
22138:=
22135:x
22115:x
22095:)
22092:x
22089:,
22086:v
22083:,
22078:R
22074:T
22070:(
22067:f
22043:8
22039:/
22035:7
22032:=
22027:r
22023:T
22002:)
21997:c
21993:T
21989:R
21986:(
21982:/
21978:f
21975:=
21970:r
21966:f
21945:)
21940:r
21936:v
21932:8
21929:(
21925:/
21921:9
21915:}
21912:]
21909:)
21906:1
21898:r
21894:v
21890:3
21887:(
21882:c
21877:r
21873:T
21869:[
21855:s
21849:C
21840:c
21837:{
21832:r
21828:T
21824:+
21819:u
21813:C
21807:=
21802:r
21798:s
21792:r
21788:T
21779:r
21775:u
21771:=
21766:r
21762:f
21738:v
21716:g
21712:v
21705:v
21683:s
21679:p
21656:g
21652:v
21631:v
21610:v
21590:0
21587:=
21584:p
21579:v
21568:=
21565:f
21560:2
21553:2
21549:v
21523:p
21518:v
21507:=
21504:f
21499:2
21492:2
21488:v
21462:f
21440:f
21436:v
21415:v
21393:f
21389:v
21380:(
21366:f
21362:v
21355:v
21335:)
21332:v
21329:,
21324:R
21320:T
21316:(
21313:f
21289:g
21269:8
21265:/
21261:7
21258:=
21253:r
21249:T
21225:=
21220:s
21217:r
21213:p
21167:s
21164:r
21160:p
21156:)
21151:c
21147:T
21143:R
21140:(
21136:/
21130:c
21126:v
21120:c
21116:p
21109:=
21104:r
21100:f
21092:r
21088:v
21059:=
21054:g
21051:r
21047:v
21023:=
21018:f
21015:r
21011:v
20990:)
20985:r
20981:v
20977:,
20971:(
20966:r
20962:f
20930:.
20925:R
20921:T
20900:p
20880:g
20860:)
20855:R
20851:T
20847:(
20842:g
20838:v
20817:)
20812:R
20808:T
20804:(
20799:f
20795:v
20774:)
20771:v
20768:,
20763:R
20759:T
20755:(
20752:f
20732:g
20712:)
20707:R
20703:T
20699:(
20694:s
20690:p
20686:=
20683:p
20661:R
20657:T
20634:c
20630:T
20621:R
20617:T
20596:)
20593:v
20590:,
20585:R
20581:T
20577:(
20574:f
20554:g
20534:p
20511:)
20508:v
20505:,
20500:R
20496:T
20492:(
20489:f
20469:g
20449:p
20429:v
20426:p
20420:g
20417:=
20414:f
20391:v
20367:N
20363:/
20356:A
20351:N
20347:F
20344:=
20341:f
20317:0
20309:T
20304:|
20299:v
20292:/
20288:p
20279:=
20274:T
20269:|
20262:2
20258:v
20250:/
20246:f
20241:2
20216:0
20210:F
20205:2
20201:D
20180:0
20177:=
20174:F
20171:D
20149:R
20145:T
20141:=
20138:T
20108:2
20104:x
20094:b
20090:+
20085:1
20081:x
20071:b
20067:=
20064:b
20044:1
20041:=
20036:2
20032:x
20028:+
20023:1
20019:x
19998:2
19994:/
19990:)
19981:b
19977:+
19968:b
19964:(
19961:=
19952:b
19925:b
19898:b
19871:b
19841:b
19837:=
19828:b
19801:a
19797:=
19788:a
19761:b
19756:,
19747:b
19720:a
19715:,
19706:a
19683:x
19679:b
19656:x
19652:a
19629:2
19625:p
19621:+
19616:1
19612:p
19605:p
19585:v
19563:2
19559:p
19555:+
19550:1
19546:p
19539:p
19519:1
19516:=
19511:2
19507:x
19503:+
19498:1
19494:x
19471:2
19467:N
19463:+
19458:1
19454:N
19450:=
19447:N
19427:N
19423:/
19417:2
19413:N
19409:=
19404:2
19400:x
19379:N
19375:/
19369:1
19365:N
19361:=
19356:1
19352:x
19329:2
19324:2
19320:x
19310:b
19306:+
19301:2
19297:x
19291:1
19287:x
19277:b
19273:2
19270:+
19265:2
19260:1
19256:x
19246:b
19242:=
19237:x
19233:b
19218:2
19213:2
19209:x
19199:a
19195:+
19190:2
19186:x
19180:1
19176:x
19166:a
19162:2
19159:+
19154:2
19149:1
19145:x
19135:a
19131:=
19126:x
19122:a
19097:2
19093:v
19087:x
19083:a
19069:x
19065:b
19058:v
19053:T
19050:R
19044:=
19041:p
19017:2
19013:v
19003:a
18985:b
18978:v
18973:T
18970:R
18964:=
18959:2
18955:p
18938:2
18934:v
18924:a
18906:b
18899:v
18894:T
18891:R
18885:=
18880:1
18876:p
18846:3
18842:/
18838:b
18816:2
18812:)
18808:v
18804:/
18800:b
18797:(
18775:1
18768:)
18764:3
18760:/
18756:b
18750:v
18747:(
18727:.
18724:]
18721:)
18716:2
18712:v
18705:(
18701:/
18695:2
18691:b
18687:7
18684:+
18681:)
18678:v
18675:3
18672:(
18668:/
18664:b
18661:2
18658:+
18655:1
18652:[
18649:T
18646:R
18643:=
18640:)
18637:3
18633:/
18629:b
18623:v
18620:(
18617:)
18612:2
18608:v
18603:/
18599:a
18596:+
18593:p
18590:(
18568:2
18564:)
18557:b
18554:(
18551:)
18548:8
18544:/
18540:5
18537:(
18515:2
18511:)
18504:b
18501:(
18479:i
18475:)
18468:b
18465:(
18455:0
18452:=
18449:i
18441:=
18436:1
18429:)
18422:b
18416:1
18413:(
18410:=
18407:Z
18387:0
18384:=
18381:a
18358:8
18354:/
18332:r
18328:T
18307:0
18304:=
18299:r
18274:)
18269:r
18265:T
18261:,
18256:r
18248:(
18245:Z
18223:k
18219:)
18215:3
18211:/
18205:r
18197:(
18186:2
18183:=
18180:k
18172:+
18169:)
18166:3
18162:/
18156:r
18148:(
18145:]
18140:1
18133:]
18129:)
18124:r
18120:T
18116:8
18113:(
18109:/
18099:1
18096:[
18093:+
18090:1
18087:=
18084:)
18079:r
18075:T
18071:,
18066:r
18058:(
18055:Z
18033:1
18026:)
18020:r
18009:3
18006:(
17986:3
17978:r
17953:)
17948:r
17944:T
17940:,
17935:r
17927:(
17924:Z
17901:1
17898:+
17895:k
17875:k
17855:)
17852:T
17849:(
17844:k
17840:V
17817:k
17813:)
17806:(
17803:)
17800:T
17797:(
17792:k
17788:V
17776:1
17773:=
17770:k
17762:+
17759:1
17756:=
17753:)
17750:T
17747:,
17741:(
17738:Z
17718:Z
17683:r
17658:)
17653:r
17649:T
17645:,
17640:r
17632:(
17627:r
17623:p
17602:)
17597:r
17593:T
17589:,
17584:r
17576:(
17573:Z
17548:r
17523:)
17518:r
17514:T
17510:,
17504:(
17501:Z
17478:0
17458:T
17438:8
17434:/
17408:r
17404:T
17383:8
17379:/
17372:=
17367:r
17363:T
17342:0
17339:=
17334:r
17309:8
17305:/
17299:2
17295:)
17289:r
17278:3
17275:(
17272:3
17269:=
17264:r
17260:T
17239:0
17236:=
17233:)
17228:r
17224:T
17220:8
17217:(
17213:/
17209:9
17201:2
17197:)
17191:r
17180:3
17177:(
17173:/
17169:3
17166:=
17163:Z
17156:r
17147:D
17126:T
17106:1
17103:=
17100:Z
17080:0
17072:r
17044:8
17040:/
17036:3
17033:=
17030:8
17026:/
17022:9
17016:2
17012:/
17008:3
17005:=
17000:c
16996:Z
16992:=
16989:Z
16969:1
16966:=
16961:r
16953:=
16948:r
16944:T
16923:3
16915:r
16911:v
16906:/
16902:1
16899:=
16894:r
16883:0
16858:r
16854:T
16850:8
16843:r
16835:9
16821:r
16810:3
16806:3
16801:=
16798:Z
16778:)
16775:T
16772:R
16769:(
16765:/
16758:a
16752:)
16746:b
16740:1
16737:(
16733:/
16729:1
16726:=
16723:Z
16703:T
16700:R
16694:)
16691:T
16688:,
16682:(
16679:Z
16676:=
16673:p
16653:1
16650:=
16647:Z
16627:0
16621:v
16617:/
16613:1
16610:=
16587:p
16567:Z
16547:T
16544:R
16541:Z
16538:=
16535:v
16532:p
16500:3
16496:/
16490:r
16482:=
16470:v
16446:=
16441:r
16437:T
16416:8
16412:/
16405:=
16400:r
16396:T
16363:3
16360:=
16353:x
16350:a
16347:m
16342:r
16338:T
16317:9
16314:=
16307:x
16304:a
16301:m
16296:r
16292:p
16271:4
16267:/
16243:4
16239:/
16235:3
16232:=
16227:r
16223:T
16200:r
16196:p
16192:,
16187:r
16183:T
16162:2
16137:2
16129:=
16126:9
16122:/
16118:2
16115:=
16106:T
16101:/
16097:T
16089:(
16077:1
16057:3
16053:/
16049:1
16046:=
16043:z
16023:6
16019:/
16015:)
15995:4
15992:(
15989:=
15986:z
15966:0
15963:=
15960:1
15957:+
15954:z
15951:4
15943:2
15939:z
15935:3
15915:3
15911:/
15901:p
15897:=
15891:x
15888:a
15885:m
15880:p
15859:9
15855:/
15845:T
15841:8
15838:=
15832:x
15829:a
15826:m
15821:T
15800:3
15796:/
15792:2
15789:=
15784:2
15780:/
15776:1
15770:x
15767:a
15764:m
15759:)
15749:T
15745:2
15741:/
15737:T
15734:(
15714:0
15711:=
15706:2
15702:/
15698:1
15694:)
15684:T
15680:2
15676:/
15672:T
15669:(
15666:6
15660:4
15657:=
15648:p
15643:/
15639:p
15634:z
15630:D
15609:)
15600:T
15596:2
15593:(
15589:/
15585:T
15582:=
15577:2
15573:z
15546:T
15542:,
15533:p
15512:R
15509:,
15506:b
15503:,
15500:a
15480:)
15471:T
15467:2
15463:/
15459:T
15456:(
15453:3
15445:2
15441:/
15437:1
15433:)
15423:T
15419:2
15415:/
15411:T
15408:(
15405:4
15402:+
15399:1
15393:=
15384:p
15379:/
15375:p
15355:v
15351:/
15347:b
15319:4
15315:/
15309:c
15305:T
15295:T
15275:b
15272:T
15269:R
15263:a
15260:2
15240:b
15234:v
15203:0
15200:=
15197:b
15192:2
15188:v
15184:T
15181:R
15173:2
15169:)
15165:b
15159:v
15156:(
15153:a
15150:2
15138:0
15135:=
15127:2
15123:)
15119:b
15113:v
15110:(
15107:a
15104:2
15096:3
15092:v
15088:T
15085:R
15080:b
15075:2
15071:v
15067:T
15064:R
15056:2
15052:)
15048:b
15042:v
15039:(
15036:a
15033:2
14990:0
14987:=
14984:1
14978:T
14955:0
14947:J
14922:p
14919:,
14916:T
14896:0
14893:=
14888:J
14861:J
14832:p
14828:c
14823:)
14820:1
14814:T
14808:(
14805:v
14799:=
14794:J
14766:)
14763:T
14760:(
14757:h
14754:=
14751:h
14729:1
14725:h
14721:=
14716:2
14712:h
14689:h
14684:|
14679:T
14674:p
14666:=
14661:J
14582:=
14553:=
14524:=
14464:s
14461:r
14457:p
14414:r
14410:T
14344:r
14340:T
14317:s
14314:r
14310:p
14271:r
14267:T
14260:1
14217:)
14212:r
14208:T
14195:+
14190:6
14185:r
14181:T
14174:+
14169:3
14164:r
14160:T
14147:(
14141:+
14138:)
14133:r
14129:T
14124:/
14120:1
14114:1
14111:(
14105:=
14100:s
14097:r
14093:p
14066:)
14060:,
14055:r
14051:T
14047:(
14042:s
14038:p
14034:=
14029:s
14026:r
14022:p
14001:1
13995:]
13992:)
13986:=
13981:r
13977:T
13973:(
13968:r
13964:p
13960:[
13940:=
13917:)
13912:c
13908:T
13904:R
13901:(
13897:/
13891:c
13887:v
13881:c
13877:p
13873:=
13868:c
13864:Z
13800:)
13794:,
13789:r
13785:T
13781:,
13776:r
13772:v
13768:(
13765:p
13762:=
13757:r
13753:p
13732:)
13727:r
13723:T
13719:(
13714:s
13710:p
13706:=
13701:s
13698:r
13694:p
13673:)
13668:r
13664:T
13660:,
13655:r
13651:v
13647:(
13644:p
13641:=
13636:r
13632:p
13593:1
13587:x
13581:0
13561:)
13556:g
13552:N
13548:+
13543:f
13539:N
13535:(
13531:/
13525:g
13521:N
13517:=
13514:x
13510:)
13505:f
13501:v
13492:g
13488:v
13484:(
13481:x
13478:+
13473:f
13469:v
13465:=
13462:v
13440:g
13436:v
13413:f
13409:v
13388:v
13353:=
13348:r
13344:T
13300:8
13296:/
13292:7
13289:=
13284:r
13280:T
13259:1
13251:r
13247:T
13223:)
13220:R
13217:2
13214:(
13210:/
13206:)
13201:f
13197:s
13188:g
13184:s
13180:(
13177:=
13174:y
13148:y
13142:0
13108:r
13104:T
13100:,
13095:r
13091:p
13062:2
13058:)
13054:y
13051:(
13048:f
13045:+
13042:y
13033:)
13030:y
13027:(
13024:f
13021:2
13018:+
13015:1
13012:=
13009:)
13006:y
13003:(
13000:g
12992:y
12986:y
12977:y
12966:y
12954:y
12945:y
12939:=
12936:)
12933:y
12930:(
12927:f
12902:y
12895:e
12891:)
12888:y
12885:(
12882:f
12875:y
12868:e
12864:)
12861:y
12858:(
12855:f
12852:+
12849:1
12842:)
12837:3
12834:1
12829:(
12825:=
12820:g
12817:r
12813:v
12800:y
12796:e
12792:)
12789:y
12786:(
12783:f
12776:y
12772:e
12768:)
12765:y
12762:(
12759:f
12756:+
12753:1
12746:)
12741:3
12738:1
12733:(
12729:=
12724:f
12721:r
12717:v
12692:2
12688:)
12684:y
12681:(
12678:g
12673:]
12668:2
12664:)
12660:y
12657:(
12654:f
12648:1
12645:[
12640:2
12636:)
12632:y
12629:(
12626:f
12617:=
12612:s
12609:r
12605:p
12595:2
12591:)
12587:y
12584:(
12581:g
12576:]
12573:)
12570:y
12567:(
12564:f
12561:+
12558:y
12549:[
12546:)
12543:y
12540:(
12537:f
12534:2
12527:)
12522:8
12514:(
12510:=
12507:)
12504:y
12501:(
12496:s
12493:r
12489:T
12462:s
12458:p
12435:s
12431:T
12410:0
12407:=
12403:)
12395:f
12391:v
12387:1
12375:g
12371:v
12367:1
12361:(
12357:a
12354:2
12347:]
12342:)
12336:b
12328:f
12324:v
12318:b
12310:g
12306:v
12299:(
12283:b
12275:f
12271:v
12264:f
12260:v
12248:b
12240:g
12236:v
12229:g
12225:v
12218:[
12212:s
12208:T
12204:R
12184:)
12181:T
12178:,
12175:v
12172:(
12169:p
12166:=
12163:p
12141:s
12137:p
12114:s
12110:p
12085:v
12082:d
12078:p
12072:)
12069:v
12066:p
12063:(
12060:d
12057:=
12054:p
12051:d
12047:v
12001:)
11998:T
11995:,
11992:p
11989:(
11986:v
11966:p
11946:v
11926:0
11923:=
11920:)
11915:f
11911:v
11902:g
11898:v
11894:(
11889:s
11885:p
11881:+
11878:v
11875:d
11871:p
11863:g
11859:v
11851:f
11847:v
11835:=
11832:p
11829:d
11823:j
11819:v
11812:1
11809:+
11804:j
11800:p
11792:j
11788:p
11777:3
11772:1
11769:=
11766:j
11758:=
11753:f
11749:g
11740:g
11736:g
11713:g
11709:v
11705:,
11700:s
11696:p
11673:f
11669:v
11665:,
11660:s
11656:p
11635:T
11632:d
11628:s
11622:p
11619:d
11615:v
11612:=
11609:T
11606:d
11602:g
11597:T
11589:+
11586:p
11583:d
11579:g
11574:p
11566:=
11563:g
11560:d
11537:s
11534:T
11528:h
11525:=
11522:g
11500:f
11496:h
11487:g
11483:h
11479:=
11473:p
11470:a
11467:v
11462:q
11441:)
11436:f
11432:s
11423:g
11419:s
11415:(
11412:T
11409:=
11403:p
11400:a
11397:v
11392:q
11365:g
11361:g
11357:=
11352:f
11348:g
11325:g
11321:v
11317:,
11312:f
11308:v
11304:,
11299:s
11295:T
11291:,
11286:s
11282:p
11257:2
11252:g
11248:v
11244:a
11233:b
11225:g
11221:v
11213:s
11209:T
11205:R
11199:=
11194:s
11190:p
11173:2
11168:f
11164:v
11160:a
11149:b
11141:f
11137:v
11129:s
11125:T
11121:R
11115:=
11110:s
11106:p
11082:x
11079:a
11076:m
11071:v
11062:g
11058:v
11034:n
11031:i
11028:m
11023:v
11014:f
11010:v
10986:x
10983:a
10980:m
10975:p
10966:s
10962:p
10952:n
10949:i
10946:m
10941:p
10918:s
10914:T
10892:8
10888:/
10884:7
10881:=
10876:r
10872:T
10848:v
10816:3
10811:r
10807:v
10802:)
10799:2
10791:r
10787:v
10783:3
10780:(
10772:c
10768:p
10764:=
10757:3
10753:v
10748:)
10745:b
10742:2
10736:v
10733:(
10730:a
10724:=
10718:p
10715:s
10710:p
10700:3
10695:r
10691:v
10687:4
10680:2
10676:)
10672:1
10664:r
10660:v
10656:3
10653:(
10645:c
10641:T
10637:=
10629:3
10625:v
10621:R
10614:2
10610:)
10606:b
10600:v
10597:(
10591:a
10588:2
10585:=
10579:p
10576:s
10571:T
10547:=
10542:T
10513:n
10510:i
10507:m
10502:v
10492:x
10489:a
10486:m
10481:v
10474:v
10451:n
10448:i
10445:m
10440:v
10433:v
10410:x
10407:a
10404:m
10399:v
10392:v
10383:n
10380:i
10377:m
10372:v
10351:0
10343:T
10338:|
10333:v
10326:/
10322:p
10296:x
10293:a
10290:m
10285:v
10261:n
10258:i
10255:m
10250:v
10227:c
10223:T
10216:T
10196:c
10191:c
10187:T
10180:T
10142:0
10134:T
10129:|
10124:v
10117:/
10113:p
10090:0
10084:S
10079:2
10075:d
10054:0
10051:=
10048:S
10045:d
10017:0
10014:=
10009:1
10001:T
9972:)
9967:3
9962:r
9958:v
9954:4
9951:(
9947:/
9941:2
9937:)
9933:1
9925:r
9921:v
9917:3
9914:(
9909:c
9905:T
9901:=
9898:)
9893:3
9889:v
9885:R
9882:(
9878:/
9872:2
9868:)
9864:b
9858:v
9855:(
9852:a
9849:2
9846:=
9843:T
9821:T
9793:,
9788:p
9784:c
9740:0
9732:T
9704:=
9699:T
9690:/
9683:=
9680:v
9675:p
9666:/
9662:v
9657:T
9646:=
9643:p
9638:T
9613:0
9610:=
9605:T
9597:=
9571:b
9565:v
9545:R
9542:=
9537:v
9533:c
9524:p
9520:c
9493:v
9473:R
9462:2
9458:)
9454:b
9448:v
9445:(
9442:a
9439:2
9431:3
9427:v
9423:T
9420:R
9413:3
9409:v
9405:T
9400:2
9396:R
9389:=
9386:)
9383:T
9380:(
9375:v
9371:c
9364:)
9361:T
9358:,
9355:v
9352:(
9347:p
9343:c
9320:p
9316:c
9293:T
9284:/
9278:2
9270:v
9267:T
9264:=
9261:p
9256:v
9247:/
9241:2
9237:)
9233:p
9228:T
9220:(
9217:T
9211:=
9206:v
9202:c
9193:p
9189:c
9162:v
9158:c
9135:p
9130:|
9125:h
9120:T
9112:=
9107:p
9103:c
9080:p
9076:c
9052:p
9048:/
9044:1
9041:=
9036:T
9011:v
9007:/
9003:T
9000:R
8997:=
8994:p
8974:)
8971:T
8968:R
8965:(
8961:/
8957:v
8954:=
8949:T
8924:T
8920:/
8916:1
8913:=
8885:v
8860:2
8856:)
8852:b
8846:v
8843:(
8840:a
8837:2
8829:3
8825:v
8821:T
8818:R
8813:)
8810:b
8804:v
8801:(
8796:2
8792:v
8788:R
8782:=
8762:2
8758:)
8754:b
8748:v
8745:(
8742:a
8739:2
8731:3
8727:v
8723:T
8720:R
8713:2
8709:)
8705:b
8699:v
8696:(
8691:2
8687:v
8680:=
8675:T
8648:p
8644:v
8638:T
8628:1
8621:v
8617:=
8594:v
8589:p
8579:1
8572:v
8565:=
8560:T
8533:1
8526:)
8520:T
8512:v
8509:(
8484:T
8475:/
8447:3
8443:v
8438:a
8435:2
8429:+
8421:2
8417:)
8413:b
8407:v
8404:(
8399:T
8396:R
8387:=
8381:v
8373:p
8352:b
8346:v
8342:R
8337:=
8331:T
8323:p
8303:p
8301:c
8296:T
8294:κ
8290:α
8273:)
8268:r
8264:v
8260:4
8257:(
8253:/
8249:9
8241:r
8237:T
8233:}
8230:]
8227:)
8224:1
8216:r
8212:v
8208:3
8205:(
8200:c
8195:r
8191:T
8187:[
8173:s
8167:C
8158:)
8155:1
8147:r
8143:v
8139:3
8136:(
8132:/
8126:r
8122:v
8118:3
8115:+
8112:c
8109:{
8106:=
8101:u
8095:C
8084:r
8080:g
8059:v
8055:/
8051:a
8048:2
8042:T
8039:R
8036:}
8033:]
8030:)
8027:b
8021:v
8018:(
8013:c
8009:T
8005:[
7991:s
7987:C
7980:)
7977:b
7971:v
7968:(
7964:/
7960:v
7957:+
7954:c
7951:{
7948:=
7943:u
7939:C
7932:g
7912:s
7909:T
7903:h
7900:=
7897:g
7866:]
7863:)
7860:1
7852:r
7848:v
7844:3
7841:(
7836:c
7831:r
7827:T
7823:[
7814:=
7809:s
7803:C
7792:r
7788:s
7767:]
7764:)
7761:b
7755:v
7752:(
7747:c
7743:T
7739:[
7730:R
7727:=
7722:s
7718:C
7711:s
7691:)
7688:b
7682:v
7679:(
7675:/
7671:R
7668:=
7665:p
7660:T
7635:R
7632:c
7629:=
7624:v
7620:c
7599:v
7596:d
7589:T
7581:p
7568:+
7563:T
7559:T
7556:d
7549:)
7546:T
7543:(
7538:v
7534:c
7526:=
7521:s
7517:C
7510:s
7479:)
7476:c
7471:c
7467:T
7463:R
7460:(
7456:/
7452:h
7449:=
7444:r
7440:h
7427:)
7422:r
7418:v
7414:4
7411:(
7407:/
7403:9
7395:r
7391:T
7387:]
7384:)
7381:1
7373:r
7369:v
7365:3
7362:(
7358:/
7352:r
7348:v
7344:3
7341:+
7338:c
7335:[
7332:=
7327:u
7321:C
7310:r
7306:h
7285:v
7281:/
7277:a
7274:2
7268:]
7265:)
7262:b
7256:v
7253:(
7249:/
7245:v
7242:+
7239:c
7236:[
7233:T
7230:R
7227:=
7222:u
7218:C
7211:h
7191:h
7171:v
7167:/
7163:a
7157:)
7154:b
7148:v
7145:(
7141:/
7137:v
7134:T
7131:R
7128:=
7125:v
7122:p
7102:v
7099:p
7079:v
7076:p
7073:+
7070:u
7067:=
7064:h
7030:u
7006:)
7001:c
6997:T
6993:R
6990:(
6986:/
6982:u
6979:=
6974:r
6970:u
6949:)
6944:r
6940:v
6936:8
6933:(
6929:/
6925:9
6917:r
6913:T
6909:c
6906:=
6901:u
6895:C
6884:r
6880:u
6858:v
6854:/
6850:a
6844:T
6841:R
6838:c
6835:=
6830:u
6826:C
6819:u
6799:c
6779:R
6776:c
6773:=
6768:v
6764:c
6743:)
6740:T
6737:(
6732:v
6728:c
6724:=
6719:v
6715:c
6694:0
6691:=
6686:v
6682:c
6676:v
6649:2
6645:v
6640:/
6636:a
6633:=
6630:]
6627:)
6624:T
6620:/
6616:p
6613:(
6608:T
6598:2
6594:T
6573:T
6569:/
6565:p
6545:]
6542:)
6539:T
6535:/
6531:p
6528:(
6523:T
6513:2
6509:T
6505:[
6500:T
6492:=
6489:)
6486:T
6483:,
6480:v
6477:(
6474:c
6469:v
6444:u
6439:T
6431:=
6426:v
6422:c
6401:u
6396:2
6391:v
6388:T
6380:=
6377:u
6372:2
6367:T
6364:v
6339:)
6336:T
6333:,
6330:v
6327:(
6324:u
6304:)
6301:T
6298:,
6295:v
6292:(
6289:u
6286:d
6261:u
6257:C
6236:v
6233:d
6228:]
6221:T
6213:)
6210:T
6206:/
6202:p
6199:(
6188:2
6184:T
6179:[
6171:+
6168:T
6165:d
6161:)
6158:T
6155:,
6152:v
6149:(
6144:v
6140:c
6132:=
6129:v
6126:d
6121:]
6117:)
6114:T
6111:,
6108:v
6105:(
6102:p
6093:T
6085:p
6076:T
6072:[
6064:+
6061:T
6058:d
6054:)
6051:T
6048:,
6045:v
6042:(
6037:v
6033:c
6025:=
6020:u
6016:C
6009:u
5981:)
5978:T
5975:,
5972:v
5969:(
5964:v
5960:c
5939:)
5936:T
5933:,
5930:v
5927:(
5924:p
5921:=
5918:p
5898:s
5878:u
5818:c
5814:Z
5787:8
5783:/
5779:3
5776:=
5773:)
5768:c
5764:T
5760:R
5757:(
5753:/
5747:c
5743:v
5737:c
5733:p
5729:=
5724:c
5720:Z
5696:)
5687:T
5682:/
5678:T
5675:(
5672:)
5669:8
5665:/
5658:(
5655:=
5650:r
5646:T
5625:)
5616:v
5611:/
5607:v
5604:(
5601:)
5598:3
5594:/
5590:1
5587:(
5584:=
5579:r
5575:v
5554:)
5545:p
5540:/
5536:p
5533:(
5527:=
5522:r
5518:p
5491:T
5486:/
5482:T
5479:,
5470:v
5465:/
5461:v
5458:,
5449:p
5444:/
5440:p
5420:)
5417:b
5414:R
5411:(
5407:/
5403:a
5400:=
5391:T
5368:2
5364:b
5359:/
5355:a
5352:=
5343:p
5322:b
5319:=
5310:v
5289:R
5286:,
5283:b
5280:,
5277:a
5274:,
5271:T
5268:,
5265:v
5262:,
5259:p
5217:r
5213:T
5188:2
5183:r
5179:v
5175:3
5164:1
5156:r
5152:v
5148:3
5141:r
5137:T
5133:8
5127:=
5122:r
5118:p
5095:c
5091:v
5086:/
5082:v
5079:=
5074:r
5070:v
5066:,
5061:c
5057:T
5052:/
5048:T
5045:=
5040:r
5036:T
5032:,
5027:c
5023:p
5018:/
5014:p
5011:=
5006:r
5002:p
4976:c
4972:T
4968:,
4963:c
4959:v
4955:,
4950:c
4946:p
4923:3
4918:c
4914:v
4910:=
4905:c
4901:p
4896:/
4892:b
4889:a
4883:2
4878:c
4874:v
4870:3
4867:=
4862:c
4858:p
4853:/
4849:a
4843:c
4839:v
4835:3
4832:=
4827:c
4823:p
4818:/
4812:c
4808:T
4804:R
4801:+
4798:b
4776:c
4772:p
4751:0
4748:=
4743:3
4738:c
4734:v
4727:v
4722:2
4717:c
4713:v
4709:3
4706:+
4701:2
4697:v
4691:c
4687:v
4683:3
4675:3
4671:v
4667:=
4662:3
4658:)
4652:c
4648:v
4641:v
4638:(
4618:0
4615:=
4612:b
4609:a
4603:v
4600:a
4597:+
4592:2
4588:v
4584:)
4579:c
4575:T
4571:R
4568:+
4565:b
4560:c
4556:p
4552:(
4544:3
4540:v
4536:c
4531:c
4527:p
4506:v
4481:2
4477:b
4469:/
4465:a
4462:=
4457:c
4453:p
4432:)
4429:b
4426:R
4420:(
4416:/
4412:a
4409:8
4406:=
4401:c
4397:T
4393:,
4390:b
4387:3
4384:=
4379:c
4375:v
4362:)
4350:v
4330:T
4310:p
4287:v
4267:T
4264:,
4261:p
4250:;
4238:0
4235:=
4230:T
4225:|
4218:2
4214:v
4206:/
4202:p
4197:2
4189:=
4184:T
4179:|
4174:v
4167:/
4163:p
4138:c
4134:v
4130:=
4127:v
4124:,
4119:c
4115:p
4111:=
4108:p
4088:0
4080:T
4075:|
4070:v
4063:/
4059:p
4034:c
4030:T
4026:=
4023:T
3979:)
3976:y
3973:(
3968:T
3964:f
3960:=
3955:s
3951:T
3947:,
3944:)
3941:y
3938:(
3933:p
3929:f
3925:=
3920:s
3916:p
3878:b
3858:a
3838:)
3833:1
3829:T
3820:2
3816:T
3812:(
3808:/
3804:)
3799:2
3795:T
3789:1
3785:p
3776:1
3772:T
3766:2
3762:p
3758:(
3753:2
3749:v
3745:=
3742:a
3722:]
3719:)
3714:1
3710:p
3701:2
3697:p
3693:(
3689:/
3685:)
3680:1
3676:T
3667:2
3663:T
3659:(
3656:R
3650:v
3647:=
3644:b
3622:2
3618:T
3614:,
3609:2
3605:p
3582:1
3578:T
3574:,
3569:1
3565:p
3523:3
3497:2
3426:k
3406:T
3403:k
3400:=
3397:3
3393:/
3382:2
3378:v
3372:m
3345:2
3341:v
3318:m
3298:n
3278:p
3258:3
3254:/
3243:2
3239:v
3233:m
3230:n
3227:=
3224:p
3189:)
3183:,
3178:c
3174:T
3169:/
3165:T
3162:(
3157:s
3153:p
3149:=
3144:c
3140:p
3135:/
3129:s
3125:p
3104:1
3098:]
3093:c
3089:p
3084:/
3080:)
3074:=
3069:c
3065:T
3060:/
3056:T
3053:(
3048:s
3044:p
3040:[
3028:=
3013:(
2997:)
2992:c
2988:T
2984:R
2981:(
2977:/
2971:c
2967:v
2961:c
2957:p
2953:=
2948:c
2944:Z
2923:]
2911:s
2908:T
2902:h
2899:=
2896:v
2893:p
2890:+
2887:f
2884:=
2881:s
2878:T
2872:v
2869:p
2866:+
2863:u
2860:=
2857:g
2837:s
2834:T
2828:u
2825:=
2822:f
2802:v
2799:p
2796:+
2793:u
2790:=
2787:h
2765:p
2761:c
2740:s
2720:u
2700:T
2678:v
2674:c
2642:]
2630:v
2626:/
2622:)
2617:T
2612:|
2607:v
2602:p
2594:(
2588:=
2583:T
2558:v
2554:/
2550:)
2545:p
2540:|
2535:v
2530:T
2522:(
2519:=
2463:T
2460:k
2457:N
2454:=
2451:V
2448:p
2444:N
2434:V
2423:v
2419:/
2415:T
2412:R
2409:=
2406:p
2386:v
2374:.
2362:p
2340:2
2336:v
2331:/
2327:a
2324:+
2321:p
2299:2
2295:/
2291:1
2287:)
2283:p
2279:/
2275:a
2272:(
2266:v
2255:,
2243:v
2223:b
2217:v
2197:b
2191:v
2167:v
2163:/
2159:1
2156:=
2133:v
2105:p
2083:s
2079:T
2056:c
2052:p
2045:p
2039:0
2017:s
2013:T
2009:=
2006:T
1984:s
1980:T
1973:T
1951:s
1947:T
1940:T
1918:s
1914:T
1891:s
1887:T
1864:g
1860:v
1837:f
1833:v
1812:v
1792:)
1787:g
1783:v
1779:,
1774:s
1770:T
1766:(
1746:)
1741:f
1737:v
1733:,
1728:s
1724:T
1720:(
1694:n
1691:i
1688:m
1683:v
1673:x
1670:a
1667:m
1662:v
1655:v
1632:n
1629:i
1626:m
1621:v
1614:v
1591:x
1588:a
1585:m
1580:v
1576:,
1570:x
1567:a
1564:m
1559:T
1535:n
1532:i
1529:m
1524:v
1520:,
1514:n
1511:i
1508:m
1503:T
1480:c
1476:p
1468:=
1465:p
1445:.
1440:c
1436:p
1429:p
1409:T
1383:v
1377:b
1355:c
1351:p
1347:2
1344:=
1341:p
1321:v
1301:T
1264:c
1260:T
1253:T
1231:c
1227:p
1220:p
1198:c
1194:T
1171:c
1167:p
1138:b
1118:a
1095:k
1065:m
1036:m
1029:/
1025:R
1005:R
981:N
977:/
973:V
949:N
945:/
939:A
935:N
931:V
919:(
907:M
903:/
899:V
872:)
869:r
866:(
843:I
823:b
818:A
814:N
807:I
804:=
801:a
779:A
775:N
770:/
766:N
744:A
740:N
736:]
733:3
729:/
723:3
719:)
715:2
711:/
704:(
698:4
695:[
692:4
689:=
686:b
603:b
579:a
567:.
547:0
543:r
536:r
510:r
490:r
470:r
443:b
423:a
399:k
375:k
370:A
366:N
362:=
359:R
329:A
325:N
320:/
316:N
296:N
272:V
246:A
242:N
217:N
213:/
207:A
203:N
199:V
196:=
193:v
169:T
145:p
121:2
117:v
113:a
102:b
96:v
91:T
88:R
82:=
79:p
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