Statistics of Linear Polycondensation

In reactions of type I, we can count the number of molecules by measuring the concentration (i.e., number/unit volume) of one of the end groups, say, the acid, by titration (m the good old days) or by spectroscopic methods. 

It is easy to verify that measuring the number of end groups also gives a measure of the number of molecules in a type n polycondensation, providing that you start with exactly equal numbers of molecules. Check it out by drawing pictures—you ll find that the number of molecules that end up with A s on both ends is equal to the number with B s 

Number of COOH groups reacted P Number of COOH groups originally present EQUATION 5-2 

To do all this, we first have to revisit an old friend, the parameter, p, which we defined in our discussion of kinetics as the extent of reaction, equal to the fraction of functional groups of a particular type that have reacted. For example, if we are measuring acid groups we would use Equation 5-2. We also saw that the number average degree of polymerization was given by (Equation 5-3)  

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Actually, for the subject we will cover first, the statistics of linear polycondensation, all you really need is a simple definition, some common sense and the ability to reason. OK, in the common sense department some of you are already in trouble I All we can do is... 

This last equation is the statistical number-distribution function for a linear polycondensation reaction at the extent of reaction p. 

This is the statistical weight-distribution function for a linear polycondensation reaction at the extent of reaction p. The number-distribution and weight-distribution functions are illustrated in Figs. 1 and 2 for values of p. 

Due to their statistical build-up, hb polymers exhibit very broad molar mass distributions, which become even broader with the DP. This effect can be ascribed to the greater probability of larger molecules reacting with the monomer due to the higher number of reactive groups. Compared to linear polycondensates, which in theory approach a polydispersity (M, /M ) of 2, the molar mass distribution of statistically hb samples will depend directly on the DP, as M /Mn DP/2 [125]. 

Use of statistical-probabilistic methods for describing polycondensations started with Flory, who used them to compute the equilibrium chain length distribution of linear systems and later was able to predict gelation conditions for multifunctional monomers. Stockmayer [7] could extend this method to the computation of chain length distributions and average molecular weights of nonlinear polymers. 

The two solutions are identical. Hence, for a long time no importance was attributed to the use of a kinetic approach for describing batch polycondensations starting from monomers, and the statistical approach was preferred. Of course, chemical engineers had to deal with semi-batch and continuous stirred tank reactors where the statistical approach, although possible, is cumbersome and error-prone, so a few papers appeared in the 1960s dealing with kinetically controlled linear polycondensations [274—276]. 

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