88:
732:. The limited accessibility of stationary phase pore volume for the polymer molecules results in shorter elution times for high-molecular-mass species. The use of low dispersity standards allows the user to correlate retention time with molecular mass, although the actual correlation is with the Hydrodynamic volume. If the relationship between molar mass and the hydrodynamic volume changes (i.e., the polymer is not exactly the same shape as the standard) then the calibration for mass is in error.
553:
203:
735:
The most common detectors used for size exclusion chromatography include online methods similar to the bench methods used above. By far the most common is the differential refractive index detector that measures the change in refractive index of the solvent. This detector is concentration-sensitive
548:{\displaystyle {\begin{aligned}M_{\mathrm {n} }&={\frac {\sum M_{i}N_{i}}{\sum N_{i}}}&&M_{\mathrm {w} }={\frac {\sum M_{i}^{2}N_{i}}{\sum M_{i}N_{i}}}\\M_{\mathrm {z} }&={\frac {\sum M_{i}^{3}N_{i}}{\sum M_{i}^{2}N_{i}}}&&M_{\mathrm {v} }=\left^{\frac {1}{a}}\end{aligned}}}
736:
and very molecular-mass-insensitive, so it is ideal for a single-detector GPC system, as it allows the generation of mass v's molecular mass curves. Less common but more accurate and reliable is a molecular-mass-sensitive detector using multi-angle laser-light scattering - see
618:. The quantity a in the expression for the viscosity average molar mass varies from 0.5 to 0.8 and depends on the interaction between solvent and polymer in a dilute solution. In a typical distribution curve, the average values are related to each other as follows:
740:. These detectors directly measure the molecular mass of the polymer and are most often used in conjunction with differential refractive index detectors. A further alternative is either low-angle light scattering, which uses a single low angle to determine the
1311:
1090:
1437:
908:
744:, or Right-angle-light laser scattering in combination with a viscometer, although this latter technique does not give an absolute measure of molar mass but one relative to the structural model used.
208:
1179:
685:
797:. Polymer molecules, even ones of the same type, come in different sizes (chain lengths, for linear polymers), so the average molecular mass will depend on the method of averaging. The
980:. Some properties are dependent on molecular size, so a larger molecule will have a larger contribution than a smaller molecule. The mass average molar mass is calculated by
716:
The most common technique for measuring molecular mass used in modern times is a variant of high-pressure liquid chromatography (HPLC) known by the interchangeable terms of
983:
1339:
816:
577:
These different definitions have true physical meaning because different techniques in physical polymer chemistry often measure just one of them. For instance,
759:
results in a dispersity of 1. By dissolving a polymer an insoluble high molar mass fraction may be filtered off resulting in a large reduction in
17:
100:
Different average values can be defined, depending on the statistical method applied. In practice, four averages are used, representing the
1122:
621:
805:
or average of the molecular masses of the individual macromolecules. It is determined by measuring the molecular mass of
1441:
The z-average molar mass can be determined with ultracentrifugation. The melt elasticity of a polymer is dependent on
1459:
1114:
1464:
911:
721:
717:
1306:{\displaystyle {\bar {X}}_{w}={\frac {1+p}{1-p}}\quad {\bar {M}}_{w}={\frac {M_{o}\left(1+p\right)}{1-p}},}
1118:
1577:
1169:
949:
752:
73:
1498:
I. Katime "QuĂmica FĂsica
Macromolecular". Servicio Editorial de la Universidad del PaĂs Vasco. Bilbao
1582:
919:
562:
1469:
1172:(for the simplest case of linear polymers formed from two monomers in equimolar quantities) as per
927:
69:
68:) of that species. In linear polymers, the individual polymer chains rarely have exactly the same
1523:
1522:
Stepto, R. F. T.; Gilbert, R. G.; Hess, M.; Jenkins, A. D.; Jones, R. G.; KratochvĂl P. (2009). "
1110:
737:
1173:
953:
923:
756:
81:
108:, the weight fraction, and two other functions which can be related to measured quantities:
1545:
Josephat U. Izunobi and
Clement L. Higginbotham J. Chem. Educ., 2011, 88 (8), pp 1098–1104
27:
Function describing the relation between a polymer species' molar mass and number of moles
8:
566:
585:
1474:
1085:{\displaystyle {\bar {M}}_{w}={\frac {\sum _{i}N_{i}M_{i}^{2}}{\sum _{i}N_{i}M_{i}}}}
748:
31:
1546:
615:
1432:{\displaystyle {\bar {M}}_{z}={\frac {\sum M_{i}^{3}N_{i}}{\sum M_{i}^{2}N_{i}}}}
802:
175:
790:
724:(GPC). These techniques involve forcing a polymer solution through a matrix of
1334:
is the third moment or third power average molar mass, which is calculated by
1571:
1479:
611:
105:
101:
903:{\displaystyle {\bar {M}}_{n}={\frac {\sum _{i}N_{i}M_{i}}{\sum _{i}N_{i}}}}
747:
The molar mass distribution of a polymer sample depends on factors such as
725:
91:
IUPAC definition for average degree of polymerization in polymer chemistry.
87:
43:
1561:
Seymore, R.B and
Caraher, C.E. Polymer Chemistry: An Introduction, 1992.
1137:
973:
935:
915:
741:
729:
688:
598:
58:
1550:
931:
578:
910:
The number average molecular mass of a polymer can be determined by
977:
794:
77:
47:
80:. The molar mass distribution of a polymer may be modified by
582:
713:
and gives an indication just how narrow a distribution is.
1510:
R.J. Young and P.A. Lovell, Introduction to
Polymers, 1991
728:
polymer particles at a pressure of up to several hundred
1543:
Polymer
Molecular Weight Analysis by 1H NMR Spectroscopy
809:
polymer molecules, summing the masses, and dividing by
1342:
1182:
1109:
The mass average molecular mass can be determined by
986:
819:
624:
206:
95:
581:measures number average molar mass and small-angle
42:) describes the relationship between the number of
1431:
1305:
1084:
902:
679:
547:
680:{\displaystyle M_{n}<M_{v}<M_{w}<M_{z}.}
1569:
147:stands for weight; also commonly referred to as
1322:is the molecular mass of the repeating unit.
1099:is the number of molecules of molecular mass
1506:
1504:
780:
755:gives a polymer with dispersity of 2. Ideal
1533:(2): 351–353. DOI:10.1351/PAC-REC-08-05-02.
942:High number-average molecular mass polymers
959:
1501:
86:
1518:
1516:
1325:
14:
1570:
72:and molar mass, and there is always a
1513:
174:stands for centrifugation (from
972:) is another way of describing the
24:
588:measures mass average molar mass.
447:
361:
283:
217:
25:
1594:
944:may be obtained only with a high
96:Definitions of molar mass average
1231:
801:molecular mass is the ordinary
153:weight average molecular weight
126:number average molecular weight
124:), also loosely referred to as
18:Number average molecular weight
1555:
1536:
1492:
1350:
1239:
1190:
1115:small angle neutron scattering
994:
827:
572:
13:
1:
1524:Dispersity in Polymer Science
1485:
1162:fractional monomer conversion
946:fractional monomer conversion
912:gel permeation chromatography
751:and work-up procedure. Ideal
722:gel permeation chromatography
718:size exclusion chromatography
40:molecular weight distribution
789:is a way of determining the
777:, thus reducing dispersity.
695:) of a sample is defined as
186:Viscosity average molar mass
7:
1453:
1149:mass-average molecular mass
10:
1599:
1170:step-growth polymerization
950:step-growth polymerization
753:step-growth polymerization
1465:Flory–Schulz distribution
1160:, is also related to the
970:weight average molar mass
787:number average molar mass
781:Number average molar mass
768:and a small reduction in
113:Number average molar mass
1470:Schulz-Zimm distribution
928:vapor pressure osmometry
70:degree of polymerization
1111:static light scattering
966:mass average molar mass
960:Mass average molar mass
738:static light scattering
561:is the exponent in the
132:Mass average molar mass
36:molar mass distribution
1433:
1307:
1123:sedimentation velocity
1086:
968:(often loosely termed
904:
681:
549:
92:
1460:Distribution function
1434:
1308:
1087:
920:Mark–Houwink equation
905:
757:living polymerization
682:
563:Mark–Houwink equation
550:
90:
82:polymer fractionation
1340:
1332:z-average molar mass
1326:Z-average molar mass
1180:
984:
817:
693:polydispersity index
622:
204:
159:Z-average molar mass
1415:
1385:
1174:Carothers' equation
1046:
954:Carothers' equation
924:colligative methods
691:(also known as the
567:intrinsic viscosity
487:
424:
394:
312:
1429:
1401:
1371:
1303:
1082:
1058:
1032:
1021:
900:
886:
854:
677:
545:
543:
467:
410:
380:
298:
93:
1578:Polymer chemistry
1551:10.1021/ed100461v
1475:Mass distribution
1427:
1353:
1298:
1242:
1229:
1193:
1128:The ratio of the
1080:
1049:
1012:
997:
934:determination or
898:
877:
845:
830:
749:chemical kinetics
614:in an analytical
597:is obtained from
565:that relates the
538:
524:
436:
349:
272:
32:polymer chemistry
16:(Redirected from
1590:
1583:Molar quantities
1562:
1559:
1553:
1540:
1534:
1528:Pure Appl. Chem.
1520:
1511:
1508:
1499:
1496:
1449:
1438:
1436:
1435:
1430:
1428:
1426:
1425:
1424:
1414:
1409:
1396:
1395:
1394:
1384:
1379:
1366:
1361:
1360:
1355:
1354:
1346:
1321:
1312:
1310:
1309:
1304:
1299:
1297:
1286:
1285:
1281:
1266:
1265:
1255:
1250:
1249:
1244:
1243:
1235:
1230:
1228:
1217:
1206:
1201:
1200:
1195:
1194:
1186:
1167:
1159:
1119:X-ray scattering
1105:
1098:
1091:
1089:
1088:
1083:
1081:
1079:
1078:
1077:
1068:
1067:
1057:
1047:
1045:
1040:
1031:
1030:
1020:
1010:
1005:
1004:
999:
998:
990:
909:
907:
906:
901:
899:
897:
896:
895:
885:
875:
874:
873:
864:
863:
853:
843:
838:
837:
832:
831:
823:
812:
808:
776:
767:
712:
703:
686:
684:
683:
678:
673:
672:
660:
659:
647:
646:
634:
633:
616:ultra-centrifuge
609:
596:
586:light scattering
560:
554:
552:
551:
546:
544:
540:
539:
531:
529:
525:
523:
522:
521:
512:
511:
498:
497:
496:
486:
475:
462:
452:
451:
450:
439:
437:
435:
434:
433:
423:
418:
405:
404:
403:
393:
388:
375:
366:
365:
364:
350:
348:
347:
346:
337:
336:
323:
322:
321:
311:
306:
293:
288:
287:
286:
275:
273:
271:
270:
269:
256:
255:
254:
245:
244:
231:
222:
221:
220:
196:
173:
169:
146:
142:
123:
67:
56:
21:
1598:
1597:
1593:
1592:
1591:
1589:
1588:
1587:
1568:
1567:
1566:
1565:
1560:
1556:
1541:
1537:
1521:
1514:
1509:
1502:
1497:
1493:
1488:
1456:
1448:
1442:
1420:
1416:
1410:
1405:
1397:
1390:
1386:
1380:
1375:
1367:
1365:
1356:
1345:
1344:
1343:
1341:
1338:
1337:
1328:
1320:
1314:
1287:
1271:
1267:
1261:
1257:
1256:
1254:
1245:
1234:
1233:
1232:
1218:
1207:
1205:
1196:
1185:
1184:
1183:
1181:
1178:
1177:
1165:
1158:
1152:
1104:
1100:
1097:
1093:
1073:
1069:
1063:
1059:
1053:
1048:
1041:
1036:
1026:
1022:
1016:
1011:
1009:
1000:
989:
988:
987:
985:
982:
981:
962:
948:in the case of
891:
887:
881:
876:
869:
865:
859:
855:
849:
844:
842:
833:
822:
821:
820:
818:
815:
814:
810:
806:
803:arithmetic mean
783:
775:
769:
766:
760:
711:
705:
702:
696:
668:
664:
655:
651:
642:
638:
629:
625:
623:
620:
619:
608:
602:
595:
589:
575:
569:to molar mass.
558:
542:
541:
530:
517:
513:
507:
503:
499:
492:
488:
476:
471:
463:
461:
457:
456:
446:
445:
441:
438:
429:
425:
419:
414:
406:
399:
395:
389:
384:
376:
374:
367:
360:
359:
355:
352:
351:
342:
338:
332:
328:
324:
317:
313:
307:
302:
294:
292:
282:
281:
277:
274:
265:
261:
257:
250:
246:
240:
236:
232:
230:
223:
216:
215:
211:
207:
205:
202:
201:
195:
189:
171:
168:
162:
144:
141:
135:
122:
116:
104:taken with the
98:
66:
62:
55:
51:
28:
23:
22:
15:
12:
11:
5:
1596:
1586:
1585:
1580:
1564:
1563:
1554:
1535:
1512:
1500:
1490:
1489:
1487:
1484:
1483:
1482:
1477:
1472:
1467:
1462:
1455:
1452:
1446:
1423:
1419:
1413:
1408:
1404:
1400:
1393:
1389:
1383:
1378:
1374:
1370:
1364:
1359:
1352:
1349:
1327:
1324:
1318:
1302:
1296:
1293:
1290:
1284:
1280:
1277:
1274:
1270:
1264:
1260:
1253:
1248:
1241:
1238:
1227:
1224:
1221:
1216:
1213:
1210:
1204:
1199:
1192:
1189:
1156:
1142:polydispersity
1136:is called the
1134:number average
1102:
1095:
1076:
1072:
1066:
1062:
1056:
1052:
1044:
1039:
1035:
1029:
1025:
1019:
1015:
1008:
1003:
996:
993:
961:
958:
894:
890:
884:
880:
872:
868:
862:
858:
852:
848:
841:
836:
829:
826:
799:number average
791:molecular mass
782:
779:
773:
764:
709:
700:
676:
671:
667:
663:
658:
654:
650:
645:
641:
637:
632:
628:
606:
593:
574:
571:
537:
534:
528:
520:
516:
510:
506:
502:
495:
491:
485:
482:
479:
474:
470:
466:
460:
455:
449:
444:
440:
432:
428:
422:
417:
413:
409:
402:
398:
392:
387:
383:
379:
373:
370:
368:
363:
358:
354:
353:
345:
341:
335:
331:
327:
320:
316:
310:
305:
301:
297:
291:
285:
280:
276:
268:
264:
260:
253:
249:
243:
239:
235:
229:
226:
224:
219:
214:
210:
209:
199:
198:
193:
183:
166:
156:
149:weight average
139:
129:
120:
97:
94:
64:
53:
26:
9:
6:
4:
3:
2:
1595:
1584:
1581:
1579:
1576:
1575:
1573:
1558:
1552:
1548:
1544:
1539:
1532:
1529:
1525:
1519:
1517:
1507:
1505:
1495:
1491:
1481:
1480:Sedimentation
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1457:
1451:
1445:
1439:
1421:
1417:
1411:
1406:
1402:
1398:
1391:
1387:
1381:
1376:
1372:
1368:
1362:
1357:
1347:
1335:
1333:
1323:
1317:
1300:
1294:
1291:
1288:
1282:
1278:
1275:
1272:
1268:
1262:
1258:
1251:
1246:
1236:
1225:
1222:
1219:
1214:
1211:
1208:
1202:
1197:
1187:
1175:
1171:
1163:
1155:
1150:
1145:
1143:
1139:
1135:
1131:
1126:
1124:
1120:
1116:
1112:
1107:
1074:
1070:
1064:
1060:
1054:
1050:
1042:
1037:
1033:
1027:
1023:
1017:
1013:
1006:
1001:
991:
979:
975:
971:
967:
957:
955:
952:, as per the
951:
947:
943:
939:
937:
933:
929:
925:
921:
917:
913:
892:
888:
882:
878:
870:
866:
860:
856:
850:
846:
839:
834:
824:
804:
800:
796:
792:
788:
778:
772:
763:
758:
754:
750:
745:
743:
739:
733:
731:
727:
723:
719:
714:
708:
699:
694:
690:
674:
669:
665:
661:
656:
652:
648:
643:
639:
635:
630:
626:
617:
613:
612:sedimentation
605:
600:
592:
587:
584:
580:
570:
568:
564:
555:
535:
532:
526:
518:
514:
508:
504:
500:
493:
489:
483:
480:
477:
472:
468:
464:
458:
453:
442:
430:
426:
420:
415:
411:
407:
400:
396:
390:
385:
381:
377:
371:
369:
356:
343:
339:
333:
329:
325:
318:
314:
308:
303:
299:
295:
289:
278:
266:
262:
258:
251:
247:
241:
237:
233:
227:
225:
212:
192:
187:
184:
181:
177:
165:
160:
157:
154:
150:
138:
133:
130:
127:
119:
114:
111:
110:
109:
107:
106:mole fraction
103:
102:weighted mean
89:
85:
83:
79:
78:average value
75:
71:
60:
49:
45:
41:
37:
33:
19:
1557:
1542:
1538:
1530:
1527:
1494:
1443:
1440:
1336:
1331:
1329:
1315:
1161:
1153:
1148:
1146:
1141:
1133:
1130:mass average
1129:
1127:
1108:
969:
965:
963:
945:
941:
940:
798:
786:
784:
770:
761:
746:
734:
726:cross-linked
715:
706:
697:
692:
603:
599:viscosimetry
590:
576:
556:
200:
190:
185:
179:
163:
158:
152:
148:
136:
131:
125:
117:
112:
99:
74:distribution
39:
35:
29:
704:divided by
573:Measurement
1572:Categories
1486:References
1138:dispersity
974:molar mass
936:proton NMR
916:viscometry
742:molar mass
720:(SEC) and
689:dispersity
180:Zentrifuge
76:around an
59:molar mass
57:) and the
1399:∑
1369:∑
1351:¯
1292:−
1240:¯
1223:−
1191:¯
1051:∑
1014:∑
995:¯
932:end-group
918:via the (
879:∑
847:∑
828:¯
579:osmometry
501:∑
465:∑
408:∑
378:∑
326:∑
296:∑
259:∑
234:∑
170:), where
143:), where
50:species (
1454:See also
926:such as
46:of each
1144:index.
1140:or the
1132:to the
978:polymer
795:polymer
155:(WAMW).
128:(NAMW).
48:polymer
1313:where
1121:, and
1092:where
557:Here,
176:German
34:, the
1168:, in
976:of a
793:of a
583:laser
178:
44:moles
1330:The
1147:The
964:The
785:The
687:The
662:<
649:<
636:<
601:and
38:(or
1547:doi
922:),
730:bar
610:by
151:or
30:In
1574::
1531:81
1526:"
1515:^
1503:^
1450:.
1176::
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1549::
1447:z
1444:M
1422:i
1418:N
1412:2
1407:i
1403:M
1392:i
1388:N
1382:3
1377:i
1373:M
1363:=
1358:z
1348:M
1319:o
1316:M
1301:,
1295:p
1289:1
1283:)
1279:p
1276:+
1273:1
1269:(
1263:o
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1252:=
1247:w
1237:M
1226:p
1220:1
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1203:=
1198:w
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1166:p
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1154:M
1103:i
1101:M
1096:i
1094:N
1075:i
1071:M
1065:i
1061:N
1055:i
1043:2
1038:i
1034:M
1028:i
1024:N
1018:i
1007:=
1002:w
992:M
893:i
889:N
883:i
871:i
867:M
861:i
857:N
851:i
840:=
835:n
825:M
811:n
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675:.
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536:a
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527:]
519:i
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494:i
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459:[
454:=
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391:3
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372:=
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309:2
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300:M
290:=
284:w
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242:i
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228:=
218:n
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188:(
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20:)
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