1262:
992:
1257:{\displaystyle {\begin{matrix}{\bar {X}}_{w}&=&{\frac {1+p}{1-p}}\\{\bar {M}}_{n}&=&M_{o}{\frac {1}{1-p}}\\{\bar {M}}_{w}&=&M_{o}{\frac {1+p}{1-p}}\\PDI&=&{\frac {{\bar {M}}_{w}}{{\bar {M}}_{n}}}=1+p\\\end{matrix}}}
715:
Thus for a 1% excess of one monomer, r = 0.99 and the limiting degree of polymerization is 199, compared to infinity for the equimolar case. An excess of one reactant can be used to control the degree of polymerization.
829:
633:
is the stoichiometric ratio of reactants, the excess reactant is conventionally the denominator so that r < 1. If neither monomer is in excess, then r = 1 and the equation reduces to the equimolar case
976:
622:
710:
199:
900:
365:
298:
528:
484:
243:
396:
422:
728:
of a monomer molecule is the number of functional groups which participate in the polymerization. Monomers with functionality greater than two will introduce
986:
Related to the
Carothers equation are the following equations (for the simplest case of linear polymers formed from two monomers in equimolar quantities):
640:
The effect of the excess reactant is to reduce the degree of polymerization for a given value of p. In the limit of complete conversion of the
754:
1342:
is 2, which occurs at a monomer conversion of 100% (or p = 1). This is true for step-growth polymerization of linear polymers. For
1353:
In practice the average length of the polymer chain is limited by such things as the purity of the reactants, the absence of any
1306:
1293:
905:
550:
1347:
729:
650:
69:
The simplest case refers to the formation of a strictly linear polymer by the reaction (usually by condensation) of two
740:
molecules initially and equivalent numbers of two functional groups A and B, the total number of functional groups is N
1343:
147:
1459:
20:
844:
313:
252:
32:
248:
441:
258:
1398:, Lampe Frederick W. and Mark James E. "Contemporary Polymer Chemistry" (3rd ed., Pearson 2003) p.324
1377:
Cowie J.M.G. "Polymers: Chemistry & Physics of Modern
Materials (2nd edition, Blackie 1991), p.29
103:
497:
453:
255:, equal to the average number of monomer units in a polymer molecule. For the example of nylon-6,6
212:
1464:
54:
1417:
1395:
641:
444:
is required to achieve a high degree of polymerization. For example, a monomer conversion,
732:
into a polymer, and the degree of polymerization will depend on the average functionality f
374:
1386:
Rudin Alfred "The
Elements of Polymer Science and Engineering", Academic Press 1982, p.171
407:
1453:
1354:
541:
126:
1329:
1358:
74:
1421:
431:
1319:
70:
43:
824:{\displaystyle f_{av}={\frac {\sum N_{i}\cdot f_{i}}{\sum N_{i}}}}
58:
367:
is the extent of reaction (or conversion to polymer), defined by
430:. The total includes all degrees of polymerization: monomers,
1408:
Carothers, Wallace (1936). "Polymers and polyfunctionality".
1332:
index. (formerly known as polydispersity index, symbol PDI)
53:
There are several versions of this equation, proposed by
73:
in equimolar quantities. An example is the synthesis of
971:{\displaystyle {\frac {2(N_{0}-N)}{N_{0}\cdot f_{av}}}}
400:
is the number of molecules present initially as monomer
1338:
The last equation shows that the maximum value of the
997:
324:
995:
908:
847:
757:
653:
617:{\displaystyle {\bar {X}}_{n}={\frac {1+r}{1+r-2rp}}}
553:
500:
456:
410:
377:
316:
261:
215:
150:
65:
Linear polymers: two monomers in equimolar quantities
705:{\displaystyle {\bar {X}}_{n}\to {\frac {1+r}{1-r}}}
1256:
970:
894:
823:
704:
616:
522:
478:
416:
390:
359:
292:
237:
193:
194:{\displaystyle {\bar {X}}_{n}={\frac {1}{1-p}}}
426:is the number of molecules present after time
8:
895:{\displaystyle x_{n}={\frac {2}{2-pf_{av}}}}
736:per monomer unit. For a system containing N
720:Branched polymers: multifunctional monomers
360:{\displaystyle p={\tfrac {N_{0}-N}{N_{0}}}}
1318:is the molecular weight of the repeating
1230:
1219:
1218:
1211:
1200:
1199:
1196:
1150:
1144:
1127:
1116:
1115:
1092:
1086:
1069:
1058:
1057:
1026:
1013:
1002:
1001:
996:
994:
956:
943:
922:
909:
907:
880:
861:
852:
846:
812:
797:
784:
774:
762:
756:
676:
667:
656:
655:
652:
576:
567:
556:
555:
552:
514:
503:
502:
499:
470:
459:
458:
455:
409:
382:
376:
348:
331:
323:
315:
275:
264:
263:
260:
229:
218:
217:
214:
173:
164:
153:
152:
149:
440:This equation shows that a high monomer
136:
132:
121:
117:
113:
109:
96:
92:
88:
84:
80:
16:Formula for the degree of polymerization
1370:
1281:weight average degree of polymerization
536:Linear polymers: one monomer in excess
7:
1350:polymers, the Đ can be much higher.
1410:Transactions of the Faraday Society
544:excess, then the equation becomes
14:
293:{\displaystyle {\bar {X}}_{n}=2n}
1307:weight average molecular weight
1294:number average molecular weight
1224:
1205:
1121:
1063:
1007:
934:
915:
673:
661:
561:
523:{\displaystyle {\bar {X}}_{n}}
508:
479:{\displaystyle {\bar {X}}_{n}}
464:
269:
238:{\displaystyle {\bar {X}}_{n}}
223:
158:
1:
540:If one monomer is present in
1357:(i.e. high yield), and the
1344:chain-growth polymerization
836:modified Carothers equation
1481:
21:step-growth polymerization
448:, of 98% is required for
42:, for a given fractional
253:degree of polymerization
33:degree of polymerization
1258:
972:
896:
825:
706:
618:
524:
492:= 99% is required for
480:
418:
392:
361:
294:
239:
195:
1259:
973:
902:, where p equals to
897:
826:
707:
619:
525:
481:
419:
393:
391:{\displaystyle N_{0}}
362:
295:
240:
196:
1422:10.1039/TF9363200039
993:
906:
845:
755:
651:
644:monomer, p → 1 and
551:
498:
454:
408:
375:
314:
259:
213:
148:
104:hexamethylenediamine
29:Carothers' equation
1254:
1252:
968:
892:
821:
702:
614:
520:
476:
414:
388:
357:
355:
304:diamine units and
290:
235:
191:
125:, and one mole of
25:Carothers equation
1460:Polymer chemistry
1236:
1227:
1208:
1174:
1124:
1108:
1066:
1050:
1010:
982:Related equations
966:
890:
819:
700:
664:
612:
564:
511:
467:
417:{\displaystyle N}
354:
272:
226:
204:In this equation
189:
161:
102:from one mole of
77:whose formula is
55:Wallace Carothers
1472:
1444:
1441:
1435:
1432:
1426:
1425:
1405:
1399:
1396:Allcock Harry R.
1393:
1387:
1384:
1378:
1375:
1263:
1261:
1260:
1255:
1253:
1237:
1235:
1234:
1229:
1228:
1220:
1216:
1215:
1210:
1209:
1201:
1197:
1175:
1173:
1162:
1151:
1149:
1148:
1132:
1131:
1126:
1125:
1117:
1109:
1107:
1093:
1091:
1090:
1074:
1073:
1068:
1067:
1059:
1051:
1049:
1038:
1027:
1018:
1017:
1012:
1011:
1003:
977:
975:
974:
969:
967:
965:
964:
963:
948:
947:
937:
927:
926:
910:
901:
899:
898:
893:
891:
889:
888:
887:
862:
857:
856:
830:
828:
827:
822:
820:
818:
817:
816:
803:
802:
801:
789:
788:
775:
770:
769:
711:
709:
708:
703:
701:
699:
688:
677:
672:
671:
666:
665:
657:
642:limiting reagent
623:
621:
620:
615:
613:
611:
588:
577:
572:
571:
566:
565:
557:
531:
529:
527:
526:
521:
519:
518:
513:
512:
504:
491:
487:
485:
483:
482:
477:
475:
474:
469:
468:
460:
447:
429:
425:
423:
421:
420:
415:
399:
397:
395:
394:
389:
387:
386:
366:
364:
363:
358:
356:
353:
352:
343:
336:
335:
325:
307:
303:
299:
297:
296:
291:
280:
279:
274:
273:
265:
246:
244:
242:
241:
236:
234:
233:
228:
227:
219:
200:
198:
197:
192:
190:
188:
174:
169:
168:
163:
162:
154:
141:. For this case
140:
124:
101:
57:, who invented
49:
41:
1480:
1479:
1475:
1474:
1473:
1471:
1470:
1469:
1450:
1449:
1448:
1447:
1442:
1438:
1433:
1429:
1407:
1406:
1402:
1394:
1390:
1385:
1381:
1376:
1372:
1367:
1361:of the medium.
1317:
1304:
1291:
1278:
1251:
1250:
1217:
1198:
1194:
1189:
1177:
1176:
1163:
1152:
1140:
1138:
1133:
1114:
1111:
1110:
1097:
1082:
1080:
1075:
1056:
1053:
1052:
1039:
1028:
1024:
1019:
1000:
991:
990:
984:
952:
939:
938:
918:
911:
904:
903:
876:
866:
848:
843:
842:
808:
804:
793:
780:
776:
758:
753:
752:
747:
743:
739:
735:
722:
689:
678:
654:
649:
648:
589:
578:
554:
549:
548:
538:
501:
496:
495:
493:
489:
457:
452:
451:
449:
445:
427:
406:
405:
403:
378:
373:
372:
370:
344:
327:
326:
312:
311:
305:
301:
262:
257:
256:
216:
211:
210:
208:
178:
151:
146:
145:
138:
134:
130:
123:
119:
115:
111:
107:
100:
94:
90:
86:
82:
78:
67:
47:
40:
36:
17:
12:
11:
5:
1478:
1476:
1468:
1467:
1462:
1452:
1451:
1446:
1445:
1436:
1427:
1400:
1388:
1379:
1369:
1368:
1366:
1363:
1355:side reactions
1336:
1335:
1334:
1333:
1323:
1315:
1310:
1302:
1297:
1289:
1284:
1276:
1265:
1264:
1249:
1246:
1243:
1240:
1233:
1226:
1223:
1214:
1207:
1204:
1195:
1193:
1190:
1188:
1185:
1182:
1179:
1178:
1172:
1169:
1166:
1161:
1158:
1155:
1147:
1143:
1139:
1137:
1134:
1130:
1123:
1120:
1113:
1112:
1106:
1103:
1100:
1096:
1089:
1085:
1081:
1079:
1076:
1072:
1065:
1062:
1055:
1054:
1048:
1045:
1042:
1037:
1034:
1031:
1025:
1023:
1020:
1016:
1009:
1006:
999:
998:
983:
980:
979:
978:
962:
959:
955:
951:
946:
942:
936:
933:
930:
925:
921:
917:
914:
886:
883:
879:
875:
872:
869:
865:
860:
855:
851:
832:
831:
815:
811:
807:
800:
796:
792:
787:
783:
779:
773:
768:
765:
761:
745:
741:
737:
733:
721:
718:
713:
712:
698:
695:
692:
687:
684:
681:
675:
670:
663:
660:
638:
637:
636:
635:
625:
624:
610:
607:
604:
601:
598:
595:
592:
587:
584:
581:
575:
570:
563:
560:
542:stoichiometric
537:
534:
517:
510:
507:
473:
466:
463:
438:
437:
436:
435:
413:
401:
385:
381:
351:
347:
342:
339:
334:
330:
322:
319:
309:
308:diacid units).
289:
286:
283:
278:
271:
268:
249:number-average
232:
225:
222:
202:
201:
187:
184:
181:
177:
172:
167:
160:
157:
66:
63:
38:
15:
13:
10:
9:
6:
4:
3:
2:
1477:
1466:
1463:
1461:
1458:
1457:
1455:
1440:
1437:
1431:
1428:
1423:
1419:
1415:
1411:
1404:
1401:
1397:
1392:
1389:
1383:
1380:
1374:
1371:
1364:
1362:
1360:
1356:
1351:
1349:
1345:
1341:
1331:
1327:
1324:
1321:
1314:
1311:
1308:
1301:
1298:
1295:
1288:
1285:
1282:
1275:
1272:
1271:
1270:
1269:
1268:
1247:
1244:
1241:
1238:
1231:
1221:
1212:
1202:
1191:
1186:
1183:
1180:
1170:
1167:
1164:
1159:
1156:
1153:
1145:
1141:
1135:
1128:
1118:
1104:
1101:
1098:
1094:
1087:
1083:
1077:
1070:
1060:
1046:
1043:
1040:
1035:
1032:
1029:
1021:
1014:
1004:
989:
988:
987:
981:
960:
957:
953:
949:
944:
940:
931:
928:
923:
919:
912:
884:
881:
877:
873:
870:
867:
863:
858:
853:
849:
841:
840:
839:
837:
813:
809:
805:
798:
794:
790:
785:
781:
777:
771:
766:
763:
759:
751:
750:
749:
731:
727:
726:functionality
719:
717:
696:
693:
690:
685:
682:
679:
668:
658:
647:
646:
645:
643:
632:
629:
628:
627:
626:
608:
605:
602:
599:
596:
593:
590:
585:
582:
579:
573:
568:
558:
547:
546:
545:
543:
535:
533:
515:
505:
471:
461:
443:
434:and polymers.
433:
411:
402:
383:
379:
369:
368:
349:
345:
340:
337:
332:
328:
320:
317:
310:
287:
284:
281:
276:
266:
254:
251:value of the
250:
230:
220:
207:
206:
205:
185:
182:
179:
175:
170:
165:
155:
144:
143:
142:
128:
105:
99:
76:
72:
64:
62:
60:
56:
51:
45:
34:
30:
26:
22:
1439:
1430:
1413:
1409:
1403:
1391:
1382:
1373:
1352:
1339:
1337:
1325:
1312:
1299:
1286:
1280:
1273:
1266:
985:
835:
833:
725:
723:
714:
639:
630:
539:
439:
203:
97:
79:[−NH−(CH
68:
52:
46:conversion,
31:) gives the
28:
24:
18:
1443:Rudin p.170
127:adipic acid
1454:Categories
1434:Cowie p.40
1365:References
1330:dispersity
488:= 50, and
442:conversion
87:−NH−CO−(CH
1465:Equations
1416:: 39–49.
1359:viscosity
1225:¯
1206:¯
1168:−
1122:¯
1102:−
1064:¯
1044:−
1008:¯
950:⋅
929:−
871:−
806:∑
791:⋅
778:∑
730:branching
694:−
674:→
662:¯
600:−
562:¯
509:¯
465:¯
432:oligomers
338:−
270:¯
224:¯
183:−
159:¯
75:nylon-6,6
61:in 1935.
1348:branched
834:And the
131:HOOC−(CH
71:monomers
1346:or for
1328:is the
1320:monomer
1305:is the
1292:is the
1279:is the
1267:where:
532:= 100.
530:
494:
486:
450:
424:
404:
398:
371:
247:is the
245:
209:
44:monomer
634:above.
23:, the
1322:unit,
139:−COOH
95:−CO−]
59:nylon
724:The
112:N(CH
27:(or
1418:doi
838:is
748:.
19:In
1456::
1414:32
1412:.
746:av
734:av
129:,
120:NH
106:,
50:.
35:,
1424:.
1420::
1340:Đ
1326:Đ
1316:o
1313:M
1309:,
1303:w
1300:M
1296:,
1290:n
1287:M
1283:,
1277:w
1274:X
1248:p
1245:+
1242:1
1239:=
1232:n
1222:M
1213:w
1203:M
1192:=
1187:I
1184:D
1181:P
1171:p
1165:1
1160:p
1157:+
1154:1
1146:o
1142:M
1136:=
1129:w
1119:M
1105:p
1099:1
1095:1
1088:o
1084:M
1078:=
1071:n
1061:M
1047:p
1041:1
1036:p
1033:+
1030:1
1022:=
1015:w
1005:X
961:v
958:a
954:f
945:0
941:N
935:)
932:N
924:0
920:N
916:(
913:2
885:v
882:a
878:f
874:p
868:2
864:2
859:=
854:n
850:x
814:i
810:N
799:i
795:f
786:i
782:N
772:=
767:v
764:a
760:f
744:f
742:0
738:0
697:r
691:1
686:r
683:+
680:1
669:n
659:X
631:r
609:p
606:r
603:2
597:r
594:+
591:1
586:r
583:+
580:1
574:=
569:n
559:X
516:n
506:X
490:p
472:n
462:X
446:p
428:t
412:N
384:0
380:N
350:0
346:N
341:N
333:0
329:N
321:=
318:p
306:n
302:n
300:(
288:n
285:2
282:=
277:n
267:X
231:n
221:X
186:p
180:1
176:1
171:=
166:n
156:X
137:4
135:)
133:2
122:2
118:6
116:)
114:2
110:2
108:H
98:n
93:4
91:)
89:2
85:6
83:)
81:2
48:p
39:n
37:X
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