Number Average Molar Mass. Mass Average Molar Mass. Z-Average Molar Mass

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1 17 Molar mass: There are dfferent ways to report a molar mass lke (a) Number average molar mass, (b) mass average molar mass, (c) Vscosty average molar mass, (d) Z- Average molar mass Number Average Molar Mass Collgatve propertes (lke vapor pressure lowerng, freezng pont depresson, osmotc pressure etc) depend on the number of molecules and not on ther sze. Mn = ΣxM = ΣNM (Eq 1) ΣN where, x = Number fracton = mole fracton = N ΣN Mass Average Molar Mass Mass average molar mass depends on sze (mass/volume) as well as number of molecules. Propertes lke dffuson, sedmentaton, lght scatterng depends both on sze and mass of the molecules. Mw = ΣyM = ΣwM ΣNM = 2 (Eq 2)) Σw ΣNM where, y = mass or weght fracton = w Σw w = NM Z-Average Molar Mass The molar mass depends both on sze and mass of the molecules. However, the contrbuton of the mass of the partcle are weghted further n ths type of molar mass. M z can be determned by ultracentrfugaton technque. Mz = ΣNM 3 (Eq 3) ΣNM 2 E vsdom

2 18 Vscosty Average Molar Mass Molar mass obtaned from vscosty measurements s known as Vscosty Average Molar Mass. Mv = ΣNM1.75 to 2 (Eq 4) ΣNM0.75 to 1 If a =1, M v = Σy M = Mw; f a = -1 (not possble) then t can be shown that M v = M n. It has been reported that 0.5< a <1 corresponds to 0.75 < β <1. General Equaton, M β = ΣN M β+1 ΣNM β Mn ; β = 0; Μw ; β = 1; Mv: 0.75<β <1 ; Mz: β = 2 Mn < Mv < Mw < Mz [η] = KMv a =KΣyM a Mv = (ΣyM a )1/a These equatons provde the bass for molecular weght determnaton by GPC. It s possble to obtan polymer samples n whch the dstrbuton of molecular weght s very narrow. In these especal crcumstances the two averages M n and M w are approxmately equal and hence the rato M w /M n approxmates to unty. Generally, however, M w s larger than M n and the rato M w /M n can be used to gve some ndcaton of the Polydspersty of the sample (.e.) the dstrbuton of molecular weghts wthn the sample. Mass fr. y Mn Mv Mw Mz Molar mass

3 19 If a sample contans 95% and 5% molecules by weght havng a molar mass of 10,000 and 100, respectvely, then usng above equatons the M w and M n can be caculated to be 9505 and 1681, respectvely. The value of M n, gves an naccurate mpresson of the molar mass whereas the M w s a better ndcator of the molar mass n ths example. Whle n experments related to measurement of collgatve propertes one has to use a value of 1681, the experment lke lght scatterng wll nvolve the molar mass of PI =M w /M n Polydspersty, an ndcator of spread n the molar mass can be expressed as above. For polydsperse samples, molar mass determned from collgatve propertes, lght scatterng, and approprate data treatment of ultracentrfugaton are referred to as absolute molar mass whle those determned from GPC and vscometry are referred to as relatve molecular weghts. An absolute molar mass s determned by relatng an expermental parameter wth molar mass n an equaton, whereas GPC and vscometry requre calbraton employng polymers of known molar mass determned by an absolute method. Typcal technques for molecular weght determnaton are gven n the Table. 109 * log M Eluton volume (cm 3) Fg. 2. Molecular weght of monodsperse polystyrene standards as a functon of eluton volume n Tetrahydrofuran log[η].m 106 * + + x x Eluton volume (cm 3 ) Fg. 3. Unversal calbraton n gel-permeaton chromatography for a varety of polymers n THF 32

4 20 Table. Typcal Molecular Weght Determnaton Methods a Method Type of mol. Applcable wt. ther nformaton wt. average range Lght scatterng M w to nfnty Can also gve shape Membrane osmometry M n 2x10 4-2x10 6 Vapor phase osmometry M n to 40,000 Electron & X-ray mcroscopy M n,w,z 10 2 to nfnty Shape and dstrbuton Isopestc method (sothermal M n to 20,000 dstllaton) Ebullometry (bolng pt M n to 40,000 elevaton) Cryoscopy (meltng pt M n to 50,000 depreeson) End-group analyss M n to 20,000 smodalyss M n 500 to 25,000 Centrfugaton: Sedmentaton equlbrum Archbald modfcaton Trautman s method Sedmentaton velocty M z M z,w M w M b to nfnty to nfnty to nfnty to nfnty n the early portons of effluent soluton. The effluent passes through one compartment of a dfferental refractometer cell and then nto a syphon before gong to waste. The estmaton of the mass of the speces elutng over a partcular perod of tme s the purpose of the dfferental refractometer. The dfferental refractometer conssts of a dvded cell whch s carefully thermostated. Effluent from the sample set of columns passes through one compartment and effluent from a referenc

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