of acid and glycol, the functional group concentration can be written simply as After integration and substitution from Carothers equation, the final form is the following For a self-catalyzed system, the number average degree of polymerization (Xn) grows proportionally with . External catalyzed Polyesterification The uncatalyzed reaction is rather slow, and a high Xn is not readily attained. In the presence of a catalyst, there is an acceleration of the rate, and the kinetic expression is altered to which is kinetically first order in each functional group. Hence, and integration gives finally For an externally catalyzed system, the number average degree of polymerization grows proportionally with . Molecular weight distribution in linear polymerization The product of a polymerization is a mixture of polymer molecules of different molecular weights. For theoretical and practical reasons it is of interest to discuss the distribution of molecular weights in a polymerization. The molecular weight distribution (MWD) had been derived by Flory by a statistical approach based on the concept of equal reactivity of functional groups. Probability Step-growth polymerization is a random process so we can use statistics to calculate the probability of finding a chain with x-structural units ("x-mer") as a function of time or conversion. {\mathit{x}AA} + \mathit{x}BB -> AA-(BB-AA)_{\mathit{x}-1}-BB \mathit{x}AB -> A-(B-A)_{\mathit{x}-1}-B Probability that an 'A' functional group has reacted Probability of finding an 'A' unreacted Combining the above two equations leads to. Where Px is the probability of finding a chain that is x-units long and has an unreacted 'A'. As x increases the probability decreases. Number fraction distribution The number fraction distribution is the fraction of x-mers in any system and equals the probability of finding it in solution. Where N is the total number of polymer molecules present in the reaction. Weight fraction distribution The weight fraction distribution is the fraction of x-mers in a system and the probability of finding them in terms of mass fraction. Notes: Mo is the molar mass of the repeat unit, No is the initial number of monomer molecules, and N is the number of unreacted functional groups Substituting from the Carothers equation We can now obtain: PDI The polydispersity index (PDI), is a measure of the distribution of molecular mass in a given polymer sample. However, for step-growth polymerization the Carothers equation can be used to substitute and rearrange this formula into the following. Therefore, in step-growth when p=1, then the PDI=2.