Z-average

We restrict ourselves to the case of the system at zero temperature. This is not relevant from the point of view of methodology. The case of nonzero temperatures may be considered in the completely analogous fashion. At zero temperature the coherent potential is chosen so that the Gmi(z) and G /(z) averaged over phonon degrees of freedom in phonon vacuum are equal... 

Table 1.5 also includes two additional types of molecular weight average besides those already discussed. The following remarks describe some features of these two, the z-average molecular weight, and the viscosity average ... 

L = line length, miles Z = average gas compressibility D = pipe inside diameter, in. h2 = elevation at terminus of line, ft h = elevation at origin of line, ft Pa, = average line pressure, psia E = efficiency factor... 

Z= Average compressibility factor using 1.0 will yield conservative results R = 1,544/mol. wt. 

Z= Average compressibility factor using 1.0 will yield conservative results R = 1,544/mol. wt T = Suction temperature, °R P, Pi = Suction, discharge pressures, psia K = Adiabatic exponent. Cp/C,. 

A further but less widely used average is the z-average molar mass, M, defined as ... 

Fig. 10. Dependence of z-average hydrodynamic radius z on weight-average molecular weight (M a,i ) of the branched PAAs obtained by SCVCP of f-BuA with the inimer 1 pH=3 (O) andpH=10 ( ). Linear PAAs (-A-) at pH=6-8 used as a reference are derived from [137]. (Reproduced with permission from [31]. Copyright 2001 American Chemical Society.)...

Note that the weight and Z average molecular weights so obtained are only approximate and less than the true values. This is a consequence of the universal calibration method giving the instantaneous number average molecular weight across the chromatogram. 

In principle, any type of sample can be analysed by SEC provided that it can be solubilised and that there are no enthalpic interactions between sample and packing material. By definition then, this technique cannot be carried out on vulcanisates and even unvulcanised fully compounded rubber samples can present problems due to filler-rubber interactions. The primary use of SEC is to determine the whole MWD of polymers and the various averages (number, viscosity, weight, z-average) based on a calibration curve and to allow qualitative comparisons of different samples. Many commercial polymers have a broad MWD leading to strong peak overlap in the chromatography of complex multicomponent systems. 

Ketones are generally not polymerizable, despite claims that acetone can be polymerized at low temperatures (1 ). A simple explanation for the lack of polymerizability of ketones compared to vinyl monomers can be deduced from consideration of Pauling (Z) average bond energies as shown in Equations 1 and 2, where AHP°l(est) is the estimated enthalpy of polymerization based upon the difference in bond energies of the two single bonds formed in the polymer compared to the double bond in the monomer ... 

Similarly, we calculate the weight average molecular weight (Mw), z average molecular weight (Mz), and z + 1 average molecular weight (Mz+ J from Eqs, 5.2,5.3, and 5.4, respectively. 

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