2252:
1935:
36:
2247:{\displaystyle {\begin{aligned}T_{r}&={0.34\sigma ^{3} \over G^{2}m\rho \ln \Lambda }\\&\approx 0.95\times 10^{10}\!\left({\sigma \over 200\,\mathrm {km\,s} ^{-1}}\right)^{\!3}\!\!\left({\rho \over 10^{6}\,M_{\odot }\,\mathrm {pc} ^{-3}}\right)^{\!-1}\!\!\left({m \over M_{\odot }}\right)^{\!-1}\!\!\left({\ln \Lambda \over 15}\right)^{\!-1}\!\mathrm {yr} \end{aligned}}}
1838:
to reduce and become just saturation (relative humidity = 100%), which is the equilibrium state. The time it takes for the supersaturation to dissipate is called relaxation time. It will happen as ice crystals or liquid water content grow within the cloud and will thus consume the contained moisture.
1578:
1918:. The relaxation time is a measure of the time it takes for one object in the system (the "test star") to be significantly perturbed by other objects in the system (the "field stars"). It is most commonly defined as the time for the test star's velocity to change by of order itself.
851:
1850:
In water clouds where the concentrations are larger (hundreds per cm) and the temperatures are warmer (thus allowing for much lower supersaturation rates as compared to ice clouds), the relaxation times will be very low (seconds to minutes).
1327:
1819:
1695:
1833:
Consider a supersaturated portion of a cloud. Then shut off the updrafts, entrainment, and any other vapor sources/sinks and things that would induce the growth of the particles (ice or water). Then wait for this
287:
1342:
1858:
the concentrations are lower (just a few per liter) and the temperatures are colder (very high supersaturation rates) and so the relaxation times can be as long as several hours. Relaxation time is given as
1099:
1940:
503:
381:
787:
1176:
915:
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959:
720:
The term "structural relaxation" was introduced in the scientific literature in 1947/48 without any explanation, applied to NMR, and meaning the same as "thermal relaxation".
1208:
415:
538:
574:
to a small external perturbation. Since the underlying microscopic processes are active even in the absence of external perturbations, one can also study "relaxation
1203:
764:. A system initially at equilibrium is perturbed by a rapid change in a parameter such as the temperature (most commonly), the pressure, the electric field or the
768:
of the solvent. The return to equilibrium is then observed, usually by spectroscopic means, and the relaxation time measured. In combination with the chemical
1178:, and applying the law of conservation of mass, we can say that at any time, the sum of the concentrations of A and B must be equal to the concentration of
1583:
1702:
197:
784:
A monomolecular, first order reversible reaction which is close to equilibrium can be visualized by the following symbolic structure:
694:
displays a temperature dependence of molecular motion, which can be quantified as the average relaxation time for the solid in a
100:
1573:{\displaystyle {d{\ce {}} \over dt}=-k{\ce {}}+k'{\ce {}}=-k{\ce {}}+k'({\ce {}}_{0}-{\ce {}})=-(k+k'){\ce {}}+k'{\ce {}}_{0},}
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72:
53:
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79:
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circuit which generates a repeating waveform by the repetitive discharge of a capacitor through a resistance is called a
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710:
119:
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86:
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it is a measure of how long it takes to become neutralized by conduction process. This relaxation time is small in
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1107:
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In other words, reactant A and product B are forming into one another based on reaction rate constants k and k'.
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17:
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68:
57:
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846:{\displaystyle {\ce {A}}~{\overset {k}{\rightarrow }}~{\ce {B}}~{\overset {k'}{\rightarrow }}~{\ce {A}}}
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2409:. International Series in Natural Philosophy. Vol. 113 (3rd ed.). Elsevier Science.
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417:) is called the relaxation time of the system and the constant μ is the quasi-frequency.
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1322:{\displaystyle {\ce {}}+{\ce {}}={\ce {}}_{0}\Rightarrow {\ce {}}={\ce {}}_{0}-{\ce {}}}
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961:) causes the concentration of A to decrease over time, whereas the reverse reaction (
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2330:
Kittel, Charles (1947-01-01). "Ultrasonics research and the properties of matter".
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729:
545:
1925:. As the star moves along its orbit, its motion will be randomly perturbed by the
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towards a new equilibrium, i.e., the surface charges equalize. It is important in
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1835:
687:
429:
containing a charged capacitor and a resistor, the voltage decays exponentially:
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Various events occur on timescales relating to the relaxation time, including
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To solve for the concentration of A, recognize that the forward reaction (
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154:. The simplest theoretical description of relaxation as function of time
1205:, assuming the volume into which A and B are dissolved does not change:
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621:
426:
1690:{\displaystyle {\frac {d{\ce {}}}{-(k+k'){\ce {}}+k'{\ce {}}_{0}}}=dt}
282:{\displaystyle m{\frac {d^{2}y}{dt^{2}}}+\gamma {\frac {dy}{dt}}+ky=0}
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1855:
977:
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35:
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1814:{\displaystyle {\ce {}}={k'-ke^{-(k+k')t} \over k+k'}{\ce {}}_{0}}
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779:
738:(NMR), various relaxations are the properties that it measures.
1915:
760:, relaxation methods are used for the measurement of very fast
132:"Characteristic time" redirects here. Not to be confused with
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671:
1094:{\displaystyle {d{\ce {}} \over dt}=-k{\ce {}}+k'{\ce {}}}
1101:, where brackets around A and B indicate concentrations.
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The dielectric relaxation time is closely related to the
151:
1929:
of nearby stars. The relaxation time can be shown to be
182:
1005:) causes the concentration of A to increase over time.
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2269:
is the 1d velocity dispersion of the field stars, and
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This equation can be solved by substitution to yield
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of the system, this enables the determination of the
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to approach the molecular motion characteristic of a
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1885:= concentration (of ice crystals or water droplets)
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usually means the return of a perturbed system into
60:. Unsourced material may be challenged and removed.
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1813:
1689:
1580:which becomes the separable differential equation
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147:. Each relaxation process can be categorized by a
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1839:The dynamics of relaxation are very important in
892:
891:
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873:
655:. Very long relaxation times are responsible for
498:{\displaystyle V(t)=V_{0}e^{-{\frac {t}{RC}}}\ ,}
376:{\displaystyle y(t)=Ae^{-t/T}\cos(\mu t-\delta )}
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717:change due to molecular structural relaxation.
681:
741:
578:equilibrium" instead of the usual "relaxation
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1914:interacting bodies, for instance, stars in a
780:Monomolecular first-order reversible reaction
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27:Return of a perturbed system into equilibrium
1171:{\displaystyle t=0,{\ce {}}(t)={\ce {}}_{0}}
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1823:
420:
303:The displacement will then be of the form
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1910:, relaxation time relates to clusters of
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1331:Substituting this value for in terms of
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120:Learn how and when to remove this message
2433:Dynamical evolution of globular clusters
1921:Suppose that the test star has velocity
2429:
867:
776:for the forward and reverse reactions.
674:and can be large in semiconductors and
14:
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2423:
2329:
2386:(8th ed., W.H.Freeman 2006) p.805-7,
910:{\displaystyle {\ce {A <=> B}}}
570:, relaxation is usually studied as a
183:Mechanics: Damped unforced oscillator
589:
58:adding citations to reliable sources
29:
612:medium after it has been deformed.
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998:{\displaystyle {\ce {B -> A}}}
954:{\displaystyle {\ce {A -> B}}}
711:Differential scanning calorimetry
2403:Rogers, R.R.; Yau, M.K. (1989).
604:is the gradual disappearance of
34:
2406:A Short Course in Cloud Physics
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584:fluctuation-dissipation theorem
45:needs additional citations for
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2332:Reports on Progress in Physics
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682:Liquids and amorphous solids
548:of the circuit. A nonlinear
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1891:= mean radius of particles
742:Chemical relaxation methods
647:) reacts: the polarization
562:In condensed matter physics
410:{\displaystyle =2m/\gamma }
10:
2502:
2438:Princeton University Press
2384:Atkins' Physical Chemistry
2382:Atkins P. and de Paula J.
2352:10.1088/0034-4885/11/1/308
745:
736:nuclear magnetic resonance
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624:materials, the dielectric
616:Dielectric relaxation time
533:{\displaystyle \tau =RC\ }
139:In the physical sciences,
131:
300:of a weight on a spring.
69:"Relaxation" physics
2373:Wintner Phys. Rev. 1948.
1879:= diffusion coefficient
713:can be used to quantify
568:condensed matter physics
178:In simple linear systems
2430:Spitzer, Lyman (1987).
2294:supermassive black hole
2265:is the test-star mass,
1824:In atmospheric sciences
664:electrical conductivity
653:dielectric spectroscopy
421:Electronics: RC circuit
2486:Equations of astronomy
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1845:mathematical modelling
1829:Desaturation of clouds
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724:Spin relaxation in NMR
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158:is an exponential law
2364:Hall, Phys. Rev. 1948
2306:Relaxation oscillator
2288:, and formation of a
2261:is the mean density,
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1198:{\displaystyle A_{0}}
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770:equilibrium constant
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187:Let the homogeneous
54:improve this article
2481:Celestial mechanics
2344:1947RPPh...11..205K
1927:gravitational field
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383:. The constant T (
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2476:Time in astronomy
2436:. Princeton, NJ:
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1635:
1630:
1626:
1623:
1619:
1616:
1613:
1610:
1603:
1597:
1592:
1569:
1564:
1558:
1552:
1545:
1542:
1538:
1533:
1527:
1522:
1518:
1515:
1511:
1508:
1505:
1502:
1499:
1496:
1491:
1485:
1480:
1475:
1469:
1463:
1457:
1453:
1450:
1446:
1441:
1435:
1430:
1427:
1424:
1419:
1413:
1407:
1404:
1400:
1395:
1389:
1384:
1381:
1378:
1372:
1369:
1362:
1356:
1351:
1332:
1316:
1310:
1305:
1300:
1294:
1288:
1282:
1277:
1271:
1266:
1261:
1255:
1249:
1243:
1238:
1232:
1227:
1222:
1216:
1192:
1188:
1165:
1159:
1153:
1147:
1144:
1141:
1138:
1133:
1127:
1122:
1119:
1116:
1113:
1088:
1082:
1076:
1073:
1069:
1064:
1058:
1053:
1050:
1047:
1041:
1038:
1031:
1025:
1020:
985:
982:
976:
942:
937:
896:
889:
878:
871:
831:
828:
824:
806:
803:
781:
778:
774:rate constants
762:reaction rates
743:
740:
728:Main article:
725:
722:
683:
680:
617:
614:
591:
588:
563:
560:
540:is called the
526:
523:
520:
517:
506:
505:
494:
483:
480:
476:
471:
467:
461:
457:
453:
450:
447:
444:
441:
422:
419:
406:
402:
398:
395:
392:
372:
369:
366:
363:
360:
357:
354:
351:
346:
342:
338:
335:
331:
327:
324:
321:
318:
315:
312:
290:
289:
278:
275:
272:
269:
266:
260:
257:
252:
249:
243:
240:
232:
228:
224:
219:
214:
210:
203:
184:
181:
179:
176:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
2498:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2468:
2466:
2459:
2449:
2443:
2439:
2435:
2434:
2426:
2418:
2412:
2408:
2407:
2399:
2393:
2392:0-7167-8759-8
2389:
2385:
2379:
2370:
2361:
2353:
2349:
2345:
2341:
2337:
2333:
2326:
2322:
2312:
2311:Time constant
2309:
2307:
2304:
2303:
2297:
2295:
2291:
2287:
2286:equipartition
2283:
2282:core collapse
2278:
2276:
2268:
2264:
2260:
2226:
2223:
2217:
2212:
2205:
2202:
2196:
2187:
2184:
2178:
2171:
2167:
2163:
2158:
2149:
2146:
2140:
2132:
2129:
2113:
2109:
2102:
2098:
2093:
2088:
2079:
2073:
2065:
2062:
2044:
2040:
2035:
2027:
2023:
2019:
2016:
2013:
2011:
1997:
1994:
1991:
1988:
1983:
1979:
1971:
1967:
1963:
1957:
1955:
1948:
1944:
1932:
1931:
1930:
1928:
1924:
1919:
1917:
1913:
1909:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1874:
1873:
1868:
1864:
1860:
1857:
1852:
1848:
1846:
1843:for accurate
1842:
1841:cloud physics
1837:
1821:
1806:
1784:
1781:
1777:
1774:
1767:
1760:
1757:
1753:
1750:
1744:
1740:
1736:
1733:
1729:
1726:
1719:
1697:
1684:
1681:
1678:
1670:
1651:
1648:
1644:
1624:
1621:
1617:
1614:
1608:
1590:
1567:
1562:
1543:
1540:
1536:
1516:
1513:
1509:
1506:
1500:
1497:
1478:
1473:
1451:
1448:
1444:
1428:
1425:
1422:
1405:
1402:
1398:
1382:
1379:
1376:
1370:
1367:
1349:
1338:
1329:
1303:
1298:
1280:
1259:
1241:
1225:
1190:
1186:
1163:
1145:
1139:
1120:
1117:
1114:
1111:
1102:
1074:
1071:
1067:
1051:
1048:
1045:
1039:
1036:
1018:
1006:
983:
980:
974:
940:
935:
920:
917:
887:
876:
829:
826:
804:
777:
775:
771:
767:
763:
759:
753:
752:Pressure jump
749:
739:
737:
731:
721:
718:
716:
712:
708:
704:
700:
697:
693:
689:
679:
677:
673:
669:
668:semiconductor
665:
660:
658:
654:
650:
646:
642:
638:
634:
630:
627:
623:
613:
611:
607:
603:
602:
597:
587:
585:
581:
577:
573:
569:
559:
557:
556:
551:
547:
543:
524:
521:
518:
515:
508:The constant
492:
481:
478:
474:
469:
465:
459:
455:
451:
445:
439:
432:
431:
430:
428:
418:
404:
400:
396:
393:
390:
367:
364:
361:
358:
352:
349:
344:
340:
336:
333:
329:
325:
322:
316:
310:
301:
299:
295:
276:
273:
270:
267:
264:
258:
255:
250:
247:
241:
238:
230:
226:
222:
217:
212:
208:
201:
194:
193:
192:
190:
175:
173:
167:
163:
157:
153:
150:
146:
142:
135:
134:Time constant
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
2457:
2432:
2425:
2405:
2398:
2383:
2378:
2369:
2360:
2335:
2331:
2325:
2279:
2266:
2262:
2258:
2256:
1922:
1920:
1905:
1902:In astronomy
1894:
1888:
1882:
1876:
1871:
1866:
1862:
1853:
1849:
1832:
1698:
1336:
1330:
1103:
1007:
921:
918:
783:
755:
733:
719:
692:indomethacin
685:
661:
648:
644:
640:
636:
632:
628:
626:polarization
619:
610:viscoelastic
599:
593:
579:
575:
565:
553:
541:
507:
424:
302:
298:oscillations
291:
186:
165:
161:
155:
148:
140:
138:
116:
110:January 2012
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
1865:= (4π
1008:Therefore,
699:supercooled
145:equilibrium
2465:Categories
2447:0691083096
2416:0750632151
2317:References
1869:) seconds,
1856:ice clouds
701:liquid or
696:metastable
676:insulators
622:dielectric
550:oscillator
427:RC circuit
141:relaxation
80:newspapers
2292:around a
2284:, energy
2224:−
2209:Λ
2206:
2185:−
2172:⊙
2147:−
2130:−
2114:⊙
2094:ρ
2063:−
2041:σ
2020:×
2014:≈
2001:Λ
1998:
1992:ρ
1968:σ
1908:astronomy
1745:−
1734:−
1609:−
1501:−
1479:−
1426:−
1380:−
1339:) yields
1304:−
1265:⇒
1049:−
895:⇀
888:−
877:−
870:↽
823:→
802:→
639:changes,
516:τ
470:−
405:γ
368:δ
365:−
359:μ
353:
334:−
296:unforced
242:γ
2300:See also
1785:′
1761:′
1730:′
1652:′
1625:′
1544:′
1517:′
1452:′
1406:′
1075:′
984:′
975:→
936:→
830:′
715:enthalpy
606:stresses
2340:Bibcode
2273:is the
1872:where:
707:crystal
666:. In a
649:relaxes
608:from a
94:scholar
2444:
2413:
2390:
2257:where
1916:galaxy
836:
818:
810:
797:
672:metals
528:
490:
425:In an
294:damped
292:model
166:τ
152:τ
96:
89:
82:
75:
67:
1335:and (
703:glass
635:. If
160:exp(−
101:JSTOR
87:books
2442:ISBN
2411:ISBN
2388:ISBN
2271:ln Λ
2017:0.95
1964:0.34
1867:DNRK
750:and
580:into
73:news
2348:doi
2045:200
1906:In
1854:In
756:In
734:In
686:An
620:In
594:In
586:).
566:In
544:or
350:cos
174:).
56:by
2467::
2346:.
2336:11
2334:.
2296:.
2277:.
2213:15
2203:ln
2099:10
2028:10
2024:10
1995:ln
1847:.
766:pH
709:.
678:.
659:.
598:,
576:in
558:.
191::
2450:.
2419:.
2354:.
2350::
2342::
2267:σ
2263:m
2259:ρ
2237:r
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2227:1
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2197:(
2188:1
2179:)
2168:M
2164:m
2159:(
2150:1
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2133:3
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2122:p
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2080:3
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2051:k
2036:(
1989:m
1984:2
1980:G
1972:3
1958:=
1949:r
1945:T
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1889:R
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1807:0
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1798:A
1795:[
1782:k
1778:+
1775:k
1768:t
1765:)
1758:k
1754:+
1751:k
1748:(
1741:e
1737:k
1727:k
1720:=
1715:]
1712:A
1709:[
1685:t
1682:d
1679:=
1671:0
1665:]
1662:A
1659:[
1649:k
1645:+
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1637:A
1634:[
1629:)
1622:k
1618:+
1615:k
1612:(
1602:]
1599:A
1596:[
1591:d
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1554:A
1551:[
1541:k
1537:+
1532:]
1529:A
1526:[
1521:)
1514:k
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1507:k
1504:(
1498:=
1495:)
1490:]
1487:A
1484:[
1474:0
1468:]
1465:A
1462:[
1456:(
1449:k
1445:+
1440:]
1437:A
1434:[
1429:k
1423:=
1418:]
1415:B
1412:[
1403:k
1399:+
1394:]
1391:A
1388:[
1383:k
1377:=
1371:t
1368:d
1361:]
1358:A
1355:[
1350:d
1337:t
1333:0
1315:]
1312:A
1309:[
1299:0
1293:]
1290:A
1287:[
1281:=
1276:]
1273:B
1270:[
1260:0
1254:]
1251:A
1248:[
1242:=
1237:]
1234:B
1231:[
1226:+
1221:]
1218:A
1215:[
1191:0
1187:A
1164:0
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1155:A
1152:[
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1140:t
1137:(
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1126:[
1121:,
1118:0
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1112:t
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1084:B
1081:[
1072:k
1068:+
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1060:A
1057:[
1052:k
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1030:]
1027:A
1024:[
1019:d
992:A
981:k
970:B
948:B
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931:A
904:B
861:A
840:A
827:k
814:B
805:k
793:A
645:t
643:(
641:P
637:E
633:E
629:P
525:C
522:R
519:=
493:,
482:C
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475:t
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456:V
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446:t
443:(
440:V
401:/
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323:=
320:)
317:t
314:(
311:y
277:0
274:=
271:y
268:k
265:+
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231:2
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170:(
168:)
164:/
162:t
156:t
136:.
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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