689:) curve will have two minima and a maximum in between. This results in phase separation. In general there will be a temperature where the three extremes coalesce and the system becomes fully miscible. This point is known as the
159:
599:
521:
678:= 0.5 and the enthalpy term that has a maximum there. At high temperatures, the entropic term in the free energy of mixing dominates and the system is fully miscible, but at lower temperatures the
438:
362:
608:
that if the first
Margules expression holds, then the other one must have the same shape. A regular solutions internal energy will vary during mixing or during process.
32:. Such a solution is formed by random mixing of components of similar molar volume and without strong specific interactions, and its behavior diverges from that of an
629:
284:
240:
208:
183:
667:
term. If the unlike interactions are more unfavorable than the like ones, we get competition between an entropy of mixing term that produces a minimum in the
48:
is equal to that of an ideal solution with the same composition, due to random mixing without strong specific interactions. For two components
696:
In contrast to ideal solutions, the volumes in the case of regular solutions are no longer strictly additive but must be calculated from
54:
604:
Notice that the
Margules function for each component contains the mole fraction of the other component. It can also be shown using the
527:
449:
690:
368:
292:
250:
is non-zero, unlike for an ideal solution, while the volume of the solution equals the sum of volumes of components.
811:
663:
In contrast to ideal solutions, regular solutions do possess a non-zero enthalpy of mixing, due to the
787:
28:
is equal to that of an ideal solution with the same composition, but is non-ideal due to a nonzero
806:
711:
614:
269:
217:
697:
605:
8:
192:
167:
247:
29:
668:
263:
45:
25:
660:
represents the difference in interaction energy between like and unlike neighbors.
37:
259:
41:
723:
33:
800:
243:
186:
211:
17:
154:{\displaystyle \Delta S_{mix}=-nR(x_{1}\ln x_{1}+x_{2}\ln x_{2})\,}
710:
The term was introduced in 1927 by the
American physical chemist
778:
Simon & McQuarrie
Physical Chemistry: A molecular approach
594:{\displaystyle \ f_{2,M}={\rm {exp}}(\alpha x_{1}^{2})\,}
516:{\displaystyle \ f_{1,M}={\rm {exp}}(\alpha x_{2}^{2})\,}
617:
530:
452:
371:
295:
272:
220:
195:
170:
57:
767:
623:
593:
515:
432:
356:
278:
234:
202:
177:
153:
40:at intermediate compositions and temperatures (a
798:
433:{\displaystyle \ P_{2}=x_{2}P_{2}^{*}f_{2,M}\,}
357:{\displaystyle \ P_{1}=x_{1}P_{1}^{*}f_{1,M}\,}
258:A regular solution can also be described by
590:
512:
429:
353:
231:
199:
174:
150:
743:
741:
739:
799:
761:
759:
757:
736:
693:or the upper consolute temperature.
754:
691:upper critical solution temperature
13:
561:
558:
555:
483:
480:
477:
58:
14:
823:
751:(8th ed. W.H. Freeman 2006) p.149
443:where the Margules function is
781:
772:
587:
566:
509:
488:
147:
89:
1:
729:
246:of each component. Only the
7:
790:Nature, v.168, p.868 (1951)
788:The Term 'Regular Solution'
747:P. Atkins and J. de Paula,
717:
253:
10:
828:
749:Atkins' Physical Chemistry
266:with only one parameter
812:Chemical thermodynamics
700:that are a function of
624:{\displaystyle \alpha }
279:{\displaystyle \alpha }
235:{\displaystyle x_{i}\,}
769:(Macmillan 1969) p.263
631:can be interpreted as
625:
595:
517:
434:
358:
280:
236:
204:
179:
155:
712:Joel Henry Hildebrand
698:partial molar volumes
626:
596:
518:
435:
359:
281:
237:
205:
180:
156:
615:
606:Gibbs-Duhem relation
528:
450:
369:
293:
270:
218:
210:the total number of
193:
168:
55:
24:is a solution whose
586:
508:
412:
336:
203:{\displaystyle n\,}
178:{\displaystyle R\,}
621:
591:
572:
513:
494:
430:
398:
354:
322:
276:
248:enthalpy of mixing
232:
200:
175:
151:
30:enthalpy of mixing
669:Gibbs free energy
533:
455:
374:
298:
264:Margules function
46:entropy of mixing
26:entropy of mixing
819:
791:
785:
779:
776:
770:
763:
752:
745:
630:
628:
627:
622:
600:
598:
597:
592:
585:
580:
565:
564:
549:
548:
531:
522:
520:
519:
514:
507:
502:
487:
486:
471:
470:
453:
439:
437:
436:
431:
428:
427:
411:
406:
397:
396:
384:
383:
372:
363:
361:
360:
355:
352:
351:
335:
330:
321:
320:
308:
307:
296:
285:
283:
282:
277:
262:modified with a
241:
239:
238:
233:
230:
229:
209:
207:
206:
201:
184:
182:
181:
176:
160:
158:
157:
152:
146:
145:
130:
129:
117:
116:
101:
100:
76:
75:
38:phase separation
22:regular solution
827:
826:
822:
821:
820:
818:
817:
816:
797:
796:
795:
794:
786:
782:
777:
773:
764:
755:
746:
737:
732:
720:
706:
688:
677:
659:
652:
645:
616:
613:
612:
581:
576:
554:
553:
538:
534:
529:
526:
525:
503:
498:
476:
475:
460:
456:
451:
448:
447:
417:
413:
407:
402:
392:
388:
379:
375:
370:
367:
366:
341:
337:
331:
326:
316:
312:
303:
299:
294:
291:
290:
271:
268:
267:
256:
225:
221:
219:
216:
215:
194:
191:
190:
169:
166:
165:
141:
137:
125:
121:
112:
108:
96:
92:
65:
61:
56:
53:
52:
42:miscibility gap
12:
11:
5:
825:
815:
814:
809:
807:Thermodynamics
793:
792:
780:
771:
753:
734:
733:
731:
728:
727:
726:
724:Solid solution
719:
716:
704:
686:
675:
657:
650:
643:
620:
602:
601:
589:
584:
579:
575:
571:
568:
563:
560:
557:
552:
547:
544:
541:
537:
523:
511:
506:
501:
497:
493:
490:
485:
482:
479:
474:
469:
466:
463:
459:
441:
440:
426:
423:
420:
416:
410:
405:
401:
395:
391:
387:
382:
378:
364:
350:
347:
344:
340:
334:
329:
325:
319:
315:
311:
306:
302:
275:
255:
252:
228:
224:
198:
173:
162:
161:
149:
144:
140:
136:
133:
128:
124:
120:
115:
111:
107:
104:
99:
95:
91:
88:
85:
82:
79:
74:
71:
68:
64:
60:
34:ideal solution
9:
6:
4:
3:
2:
824:
813:
810:
808:
805:
804:
802:
789:
784:
775:
768:
762:
760:
758:
750:
744:
742:
740:
735:
725:
722:
721:
715:
713:
708:
703:
699:
694:
692:
685:
681:
674:
670:
666:
661:
656:
649:
642:
638:
634:
618:
611:The value of
609:
607:
582:
577:
573:
569:
550:
545:
542:
539:
535:
524:
504:
499:
495:
491:
472:
467:
464:
461:
457:
446:
445:
444:
424:
421:
418:
414:
408:
403:
399:
393:
389:
385:
380:
376:
365:
348:
345:
342:
338:
332:
327:
323:
317:
313:
309:
304:
300:
289:
288:
287:
273:
265:
261:
251:
249:
245:
244:mole fraction
226:
222:
213:
196:
188:
171:
142:
138:
134:
131:
126:
122:
118:
113:
109:
105:
102:
97:
93:
86:
83:
80:
77:
72:
69:
66:
62:
51:
50:
49:
47:
43:
39:
35:
31:
27:
23:
19:
783:
774:
766:
748:
709:
701:
695:
683:
679:
672:
664:
662:
654:
647:
640:
636:
632:
610:
603:
442:
260:Raoult's law
257:
187:gas constant
163:
21:
15:
765:P.A. Rock,
36:by showing
801:Categories
730:References
619:α
570:α
492:α
409:∗
333:∗
274:α
135:
106:
81:−
59:Δ
18:chemistry
718:See also
635:, where
254:Features
185:is the
44:). Its
532:
454:
373:
297:
214:, and
164:where
212:moles
633:W/RT
242:the
20:, a
671:at
639:= 2
16:In
803::
756:^
738:^
714:.
707:.
658:22
653:-
651:11
646:-
644:12
286::
189:,
132:ln
103:ln
705:1
702:x
687:1
684:x
682:(
680:G
676:1
673:x
665:W
655:U
648:U
641:U
637:W
588:)
583:2
578:1
574:x
567:(
562:p
559:x
556:e
551:=
546:M
543:,
540:2
536:f
510:)
505:2
500:2
496:x
489:(
484:p
481:x
478:e
473:=
468:M
465:,
462:1
458:f
425:M
422:,
419:2
415:f
404:2
400:P
394:2
390:x
386:=
381:2
377:P
349:M
346:,
343:1
339:f
328:1
324:P
318:1
314:x
310:=
305:1
301:P
227:i
223:x
197:n
172:R
148:)
143:2
139:x
127:2
123:x
119:+
114:1
110:x
98:1
94:x
90:(
87:R
84:n
78:=
73:x
70:i
67:m
63:S
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