1344:
is slightly greater than that predicted by Raoult's law up to a concentration of 0.7 mol/kg, after which the vapor pressure is lower than Raoult's law predicts. For aqueous solutions, the osmotic coefficients can be calculated theoretically by
500:
1079:
1205:
161:
1572:
Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (2008). "Correlation and
Prediction of Thermodynamic Properties of Some Complex Aqueous Electrolytes by the Modified Three-Characteristic-Parameter Correlation Model".
1438:
253:
670:
1502:
Ge, Xinlei; Wang, Xidong; Zhang, Mei; Seetharaman, Seshadri (2007). "Correlation and
Prediction of Activity and Osmotic Coefficients of Aqueous Electrolytes at 298.15 K by the Modified TCPC Model".
917:
733:
807:
375:
1319:
1264:
845:
1109:
287:
1537:
Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (2008). "Correlation and
Prediction of Thermodynamic Properties of Nonaqueous Electrolytes by the Modified TCPC Model".
1336:
This means that, at least at low concentrations, the vapor pressure of the solvent will be greater than that predicted by Raoult's law. For instance, for solutions of
318:
550:
984:
964:
580:
40:
937:
71:
992:
174:
1407:
1117:
1608:
Ge, Xinlei; Wang, Xidong (2009). "A Simple Two-Parameter
Correlation Model for Aqueous Electrolyte Solutions across a Wide Range of Temperatures†".
588:
1470:
850:
676:
744:
532:
In a single solute solution, the (molality based) osmotic coefficient and the solute activity coefficient
1649:
1273:
1217:
1211:
820:
738:
and there is thus a differential relationship between them (temperature and pressure held constant):
553:
517:
1087:
260:
504:
the two definitions are similar, and in fact both approach 1 as the concentration goes to zero.
1383:
521:
495:{\displaystyle \ln x_{A}=-\ln \left(1+M_{A}\sum _{i}m_{i}\right)\approx -M_{A}\sum _{i}m_{i},}
296:
1373:
535:
1388:
969:
942:
558:
513:
55:
25:
922:
8:
1337:
524:, allows calculation of the salt activity coefficient through the osmotic coefficient.
321:
290:
54:. It can be also applied to solutes. Its definition depends on the ways of expressing
1625:
1590:
1554:
1519:
1466:
1074:{\displaystyle \phi =1+{\frac {1}{m}}\int _{0}^{m}md\left(\ln(\gamma _{\pm })\right)}
1417:
1617:
1582:
1546:
1511:
1490:
1421:
1412:
1358:
1346:
512:
For liquid solutions, the osmotic coefficient is often used to calculate the salt
1378:
1368:
1363:
51:
1341:
1326:
1200:{\displaystyle \ln(\gamma _{\pm })=\phi -1+\int _{0}^{m}{\frac {\phi -1}{m}}dm}
47:
1643:
1629:
1594:
1558:
1523:
1416:, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "
1333:
is the Debye–Hückel constant (equal to about 1.17 for water at 25 °C).
343:
166:
1425:
156:{\displaystyle \phi ={\frac {\mu _{A}^{*}-\mu _{A}}{RTM_{A}\sum _{i}m_{i}}}}
351:
359:
1267:
332:
1621:
1586:
1550:
1515:
16:
Quantity characterizing the deviation of a solvent from ideal behavior
1462:
520:
measurements, or measurements of deviations from ideality for other
248:{\displaystyle \phi =-{\frac {\mu _{A}^{*}-\mu _{A}}{RT\ln x_{A}}}}
62:
43:
363:
370:. The values for the two definitions are different, but since
986:
can be calculated from the salt activity coefficient via:
366:. The latter osmotic coefficient is sometimes called the
665:{\displaystyle RTm(1-\phi )=G^{E}-m{\frac {dG^{E}}{dm}}}
1487:
Guidelines for the extrapolation to zero ionic strength
912:{\displaystyle \phi ={\frac {-\ln(a_{A})}{\nu mM_{A}}}}
516:
from the solvent activity, or vice versa. For example,
1501:
1276:
1220:
1120:
1090:
995:
972:
945:
925:
853:
823:
747:
679:
591:
561:
538:
378:
299:
263:
177:
74:
42:
is a quantity which characterises the deviation of a
28:
1313:
1258:
1199:
1103:
1073:
978:
958:
931:
911:
839:
801:
728:{\displaystyle RT\ln \gamma ={\frac {dG^{E}}{dm}}}
727:
664:
574:
544:
494:
312:
281:
247:
155:
34:
1641:
1491:http://www.nea.fr/html/dbtdb/guidelines/tdb2.pdf
1214:, which is accurate only at low concentrations,
1084:Moreover, the activity coefficient of the salt
812:
527:
1571:
1536:
1459:Activity Coefficients in Electrolyte Solutions
1440:Activity Coefficients in Electrolyte Solutions
817:For a single salt solute with molal activity (
847:), the osmotic coefficient can be written as
802:{\displaystyle d((\phi -1)m)=md(\ln \gamma )}
1610:Journal of Chemical & Engineering Data
1575:Journal of Chemical & Engineering Data
1539:Journal of Chemical & Engineering Data
1504:Journal of Chemical & Engineering Data
1601:
939:is the stochiometric number of salt and
1642:
1607:
1456:
1436:
1479:
1314:{\textstyle -{\frac {2}{3}}AI^{3/2}}
1259:{\textstyle (\phi -1)\sum _{i}m_{i}}
13:
1413:Compendium of Chemical Terminology
14:
1661:
61:The osmotic coefficient based on
840:{\displaystyle \gamma _{\pm }m}
507:
1565:
1530:
1495:
1450:
1430:
1401:
1233:
1221:
1140:
1127:
1104:{\displaystyle \gamma _{\pm }}
1063:
1050:
885:
872:
796:
784:
772:
766:
754:
751:
613:
601:
324:of the solvent in a solution,
1:
1394:
966:the activity of the solvent.
813:Liquid electrolyte solutions
528:Relation to other quantities
368:rational osmotic coefficient
282:{\displaystyle \mu _{A}^{*}}
7:
1437:Pitzer, Kenneth S. (2018).
1352:
10:
1666:
1485:I. Grenthe and H. Wanner,
518:freezing point depression
1457:Pitzer, Kenneth (1991).
1111:can be calculated from:
554:excess Gibbs free energy
313:{\displaystyle \mu _{A}}
293:of the pure solvent and
1426:10.1351/goldbook.O04342
545:{\displaystyle \gamma }
1384:Thermodynamic activity
1315:
1260:
1201:
1105:
1075:
980:
960:
933:
913:
841:
803:
729:
666:
576:
546:
522:colligative properties
496:
314:
283:
249:
157:
36:
1316:
1261:
1202:
1106:
1076:
981:
979:{\displaystyle \phi }
961:
959:{\displaystyle a_{A}}
934:
914:
842:
804:
730:
667:
577:
575:{\displaystyle G^{E}}
547:
497:
315:
284:
250:
158:
37:
35:{\displaystyle \phi }
1389:Ion transport number
1274:
1218:
1118:
1088:
993:
970:
943:
932:{\displaystyle \nu }
923:
851:
821:
745:
677:
589:
559:
536:
514:activity coefficient
376:
297:
261:
175:
72:
56:chemical composition
26:
1418:osmotic coefficient
1212:Debye–Hückel theory
1172:
1032:
552:are related to the
278:
204:
98:
21:osmotic coefficient
1650:Physical chemistry
1374:van 't Hoff factor
1338:magnesium chloride
1311:
1256:
1245:
1197:
1158:
1101:
1071:
1018:
976:
956:
929:
909:
837:
799:
725:
662:
582:by the relations:
572:
542:
492:
478:
437:
322:chemical potential
310:
291:chemical potential
279:
264:
245:
190:
153:
139:
84:
32:
1622:10.1021/je800483q
1587:10.1021/je7006499
1551:10.1021/je700446q
1516:10.1021/je060451k
1472:978-1-315-89037-1
1288:
1236:
1189:
1016:
907:
723:
660:
469:
428:
243:
151:
130:
1657:
1634:
1633:
1605:
1599:
1598:
1569:
1563:
1562:
1534:
1528:
1527:
1499:
1493:
1483:
1477:
1476:
1454:
1448:
1447:
1445:
1434:
1428:
1405:
1359:Bromley equation
1347:Pitzer equations
1320:
1318:
1317:
1312:
1310:
1309:
1305:
1289:
1281:
1265:
1263:
1262:
1257:
1255:
1254:
1244:
1206:
1204:
1203:
1198:
1190:
1185:
1174:
1171:
1166:
1139:
1138:
1110:
1108:
1107:
1102:
1100:
1099:
1080:
1078:
1077:
1072:
1070:
1066:
1062:
1061:
1031:
1026:
1017:
1009:
985:
983:
982:
977:
965:
963:
962:
957:
955:
954:
938:
936:
935:
930:
918:
916:
915:
910:
908:
906:
905:
904:
888:
884:
883:
861:
846:
844:
843:
838:
833:
832:
808:
806:
805:
800:
734:
732:
731:
726:
724:
722:
714:
713:
712:
699:
671:
669:
668:
663:
661:
659:
651:
650:
649:
636:
628:
627:
581:
579:
578:
573:
571:
570:
551:
549:
548:
543:
501:
499:
498:
493:
488:
487:
477:
468:
467:
452:
448:
447:
446:
436:
427:
426:
394:
393:
319:
317:
316:
311:
309:
308:
288:
286:
285:
280:
277:
272:
254:
252:
251:
246:
244:
242:
241:
240:
218:
217:
216:
203:
198:
188:
162:
160:
159:
154:
152:
150:
149:
148:
138:
129:
128:
112:
111:
110:
97:
92:
82:
50:, referenced to
41:
39:
38:
33:
1665:
1664:
1660:
1659:
1658:
1656:
1655:
1654:
1640:
1639:
1638:
1637:
1606:
1602:
1570:
1566:
1535:
1531:
1500:
1496:
1484:
1480:
1473:
1455:
1451:
1443:
1435:
1431:
1406:
1402:
1397:
1379:Law of dilution
1369:Davies equation
1364:Pitzer equation
1355:
1349:or TCPC model.
1301:
1297:
1293:
1280:
1275:
1272:
1271:
1250:
1246:
1240:
1219:
1216:
1215:
1175:
1173:
1167:
1162:
1134:
1130:
1119:
1116:
1115:
1095:
1091:
1089:
1086:
1085:
1057:
1053:
1043:
1039:
1027:
1022:
1008:
994:
991:
990:
971:
968:
967:
950:
946:
944:
941:
940:
924:
921:
920:
900:
896:
889:
879:
875:
862:
860:
852:
849:
848:
828:
824:
822:
819:
818:
815:
746:
743:
742:
715:
708:
704:
700:
698:
678:
675:
674:
652:
645:
641:
637:
635:
623:
619:
590:
587:
586:
566:
562:
560:
557:
556:
537:
534:
533:
530:
510:
483:
479:
473:
463:
459:
442:
438:
432:
422:
418:
411:
407:
389:
385:
377:
374:
373:
341:
330:
304:
300:
298:
295:
294:
273:
268:
262:
259:
258:
236:
232:
219:
212:
208:
199:
194:
189:
187:
176:
173:
172:
144:
140:
134:
124:
120:
113:
106:
102:
93:
88:
83:
81:
73:
70:
69:
68:is defined by:
48:ideal behaviour
27:
24:
23:
17:
12:
11:
5:
1663:
1653:
1652:
1636:
1635:
1616:(2): 179–186.
1600:
1581:(4): 950–958.
1564:
1545:(1): 149–159.
1529:
1510:(2): 538–547.
1494:
1478:
1471:
1465:. p. 13.
1449:
1429:
1399:
1398:
1396:
1393:
1392:
1391:
1386:
1381:
1376:
1371:
1366:
1361:
1354:
1351:
1342:vapor pressure
1327:ionic strength
1308:
1304:
1300:
1296:
1292:
1287:
1284:
1279:
1253:
1249:
1243:
1239:
1235:
1232:
1229:
1226:
1223:
1208:
1207:
1196:
1193:
1188:
1184:
1181:
1178:
1170:
1165:
1161:
1157:
1154:
1151:
1148:
1145:
1142:
1137:
1133:
1129:
1126:
1123:
1098:
1094:
1082:
1081:
1069:
1065:
1060:
1056:
1052:
1049:
1046:
1042:
1038:
1035:
1030:
1025:
1021:
1015:
1012:
1007:
1004:
1001:
998:
975:
953:
949:
928:
903:
899:
895:
892:
887:
882:
878:
874:
871:
868:
865:
859:
856:
836:
831:
827:
814:
811:
810:
809:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
736:
735:
721:
718:
711:
707:
703:
697:
694:
691:
688:
685:
682:
672:
658:
655:
648:
644:
640:
634:
631:
626:
622:
618:
615:
612:
609:
606:
603:
600:
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594:
569:
565:
541:
529:
526:
509:
506:
491:
486:
482:
476:
472:
466:
462:
458:
455:
451:
445:
441:
435:
431:
425:
421:
417:
414:
410:
406:
403:
400:
397:
392:
388:
384:
381:
339:
328:
307:
303:
276:
271:
267:
239:
235:
231:
228:
225:
222:
215:
211:
207:
202:
197:
193:
186:
183:
180:
147:
143:
137:
133:
127:
123:
119:
116:
109:
105:
101:
96:
91:
87:
80:
77:
31:
15:
9:
6:
4:
3:
2:
1662:
1651:
1648:
1647:
1645:
1631:
1627:
1623:
1619:
1615:
1611:
1604:
1596:
1592:
1588:
1584:
1580:
1576:
1568:
1560:
1556:
1552:
1548:
1544:
1540:
1533:
1525:
1521:
1517:
1513:
1509:
1505:
1498:
1492:
1488:
1482:
1474:
1468:
1464:
1460:
1453:
1442:
1441:
1433:
1427:
1423:
1419:
1415:
1414:
1409:
1404:
1400:
1390:
1387:
1385:
1382:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1356:
1350:
1348:
1343:
1339:
1334:
1332:
1328:
1324:
1306:
1302:
1298:
1294:
1290:
1285:
1282:
1277:
1269:
1251:
1247:
1241:
1237:
1230:
1227:
1224:
1213:
1210:According to
1194:
1191:
1186:
1182:
1179:
1176:
1168:
1163:
1159:
1155:
1152:
1149:
1146:
1143:
1135:
1131:
1124:
1121:
1114:
1113:
1112:
1096:
1092:
1067:
1058:
1054:
1047:
1044:
1040:
1036:
1033:
1028:
1023:
1019:
1013:
1010:
1005:
1002:
999:
996:
989:
988:
987:
973:
951:
947:
926:
901:
897:
893:
890:
880:
876:
869:
866:
863:
857:
854:
834:
829:
825:
793:
790:
787:
781:
778:
775:
769:
763:
760:
757:
748:
741:
740:
739:
719:
716:
709:
705:
701:
695:
692:
689:
686:
683:
680:
673:
656:
653:
646:
642:
638:
632:
629:
624:
620:
616:
610:
607:
604:
598:
595:
592:
585:
584:
583:
567:
563:
555:
539:
525:
523:
519:
515:
505:
502:
489:
484:
480:
474:
470:
464:
460:
456:
453:
449:
443:
439:
433:
429:
423:
419:
415:
412:
408:
404:
401:
398:
395:
390:
386:
382:
379:
371:
369:
365:
361:
357:
353:
349:
345:
344:mole fraction
338:
334:
327:
323:
305:
301:
292:
274:
269:
265:
255:
237:
233:
229:
226:
223:
220:
213:
209:
205:
200:
195:
191:
184:
181:
178:
170:
168:
167:mole fraction
163:
145:
141:
135:
131:
125:
121:
117:
114:
107:
103:
99:
94:
89:
85:
78:
75:
67:
64:
59:
58:of mixtures.
57:
53:
49:
45:
29:
22:
1613:
1609:
1603:
1578:
1574:
1567:
1542:
1538:
1532:
1507:
1503:
1497:
1486:
1481:
1458:
1452:
1446:. CRC Press.
1439:
1432:
1411:
1403:
1335:
1330:
1322:
1209:
1083:
816:
737:
531:
511:
508:Applications
503:
372:
367:
355:
352:gas constant
347:
336:
325:
256:
171:
164:
65:
60:
52:Raoult's law
20:
18:
360:temperature
1395:References
1268:asymptotic
333:molar mass
169:basis by:
1630:0021-9568
1595:0021-9568
1559:0021-9568
1524:0021-9568
1463:CRC Press
1278:−
1238:∑
1228:−
1225:ϕ
1180:−
1177:ϕ
1160:∫
1150:−
1147:ϕ
1136:±
1132:γ
1125:
1097:±
1093:γ
1059:±
1055:γ
1048:
1020:∫
997:ϕ
974:ϕ
927:ν
891:ν
870:
864:−
855:ϕ
830:±
826:γ
794:γ
791:
761:−
758:ϕ
693:γ
690:
630:−
611:ϕ
608:−
540:γ
471:∑
457:−
454:≈
430:∑
405:
399:−
383:
302:μ
275:∗
266:μ
230:
210:μ
206:−
201:∗
192:μ
185:−
179:ϕ
165:and on a
132:∑
104:μ
100:−
95:∗
86:μ
76:ϕ
30:ϕ
1644:Category
1353:See also
1321:, where
63:molality
331:is its
320:is the
289:is the
44:solvent
1628:
1593:
1557:
1522:
1469:
1340:, the
919:where
364:Kelvin
257:where
1444:(PDF)
1408:IUPAC
46:from
1626:ISSN
1591:ISSN
1555:ISSN
1520:ISSN
1467:ISBN
1329:and
358:the
354:and
350:the
342:its
1618:doi
1583:doi
1547:doi
1512:doi
1489:,
1422:doi
1420:".
1325:is
1270:to
1266:is
362:in
19:An
1646::
1624:.
1614:54
1612:.
1589:.
1579:53
1577:.
1553:.
1543:53
1541:.
1518:.
1508:52
1506:.
1461:.
1410:,
1122:ln
1045:ln
867:ln
788:ln
687:ln
402:ln
380:ln
346:,
335:,
227:ln
1632:.
1620::
1597:.
1585::
1561:.
1549::
1526:.
1514::
1475:.
1424::
1331:A
1323:I
1307:2
1303:/
1299:3
1295:I
1291:A
1286:3
1283:2
1252:i
1248:m
1242:i
1234:)
1231:1
1222:(
1195:m
1192:d
1187:m
1183:1
1169:m
1164:0
1156:+
1153:1
1144:=
1141:)
1128:(
1068:)
1064:)
1051:(
1041:(
1037:d
1034:m
1029:m
1024:0
1014:m
1011:1
1006:+
1003:1
1000:=
952:A
948:a
902:A
898:M
894:m
886:)
881:A
877:a
873:(
858:=
835:m
797:)
785:(
782:d
779:m
776:=
773:)
770:m
767:)
764:1
755:(
752:(
749:d
720:m
717:d
710:E
706:G
702:d
696:=
684:T
681:R
657:m
654:d
647:E
643:G
639:d
633:m
625:E
621:G
617:=
614:)
605:1
602:(
599:m
596:T
593:R
568:E
564:G
490:,
485:i
481:m
475:i
465:A
461:M
450:)
444:i
440:m
434:i
424:A
420:M
416:+
413:1
409:(
396:=
391:A
387:x
356:T
348:R
340:A
337:x
329:A
326:M
306:A
270:A
238:A
234:x
224:T
221:R
214:A
196:A
182:=
146:i
142:m
136:i
126:A
122:M
118:T
115:R
108:A
90:A
79:=
66:m
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