Physical Chemistry Third Edition

13.4 The Reaction Kinetics of Polymer Formation 593

From Eqs. (13.4-7) and (13.4-15), we can write an equation for the time dependence

of〈M〉n:

〈M〉nM 0 (1+c 0 k′t) (13.4-16)

where we continue to neglect 18 amu compared with〈M〉n. For fairly large values of

the time,

1 +c 0 k′t≈c 0 k′t (13.4-17)

giving

〈M〉nM 0 c 0 k′t (13.4-18)

This relation can also be obtained using reaction kinetics.^26 The time dependence of

the number fraction of any degree of polymerization can be obtained by substituting

Eq. (13.4-6) into Eq. (13.4-13).

Exercise 13.17

Obtain an expression for the time dependence ofXx.

Themass fractionof molecules with degree of polymerizationxis

Wx

mass of molecules with degree of polymerizationx

mass of all molecules

(13.4-19a)



NxMx

∑∞

x 1 NxMx

(13.4-19b)

The mass-average molecular mass (often called the “weight-average molecular

weight”) is defined by

〈M〉w

∑∞

x 1

WxMx

∑∞

x 1 NxM

2

∑ x

∞

x 1 NxMx

(13.4-20)

In this average, each molecule is given an importance (weight) in the average propor-

tional to its mass. The required sum can be found in tables,^27 giving

〈M〉w(1−p)^2 M 0

∑∞

x 1

x^2 px−^1 M 0

1 +p

1 −p

(13.4-21)

The mass-average molecular mass is always equal to or larger than the number-average

molecular mass since the heavier molecules are given larger importance than lighter

molecules in the mass average. Ifpis nearly equal to unity, the mass-average molecular

mass is approximately twice as large as the number-average molecular mass. The

evolution in time of the mass-average molecular mass can be expressed as a function

of time, as was done for〈M〉nin Eq. (13.4-16). Figure 13.12a shows the distribution of

mass in a polyester, according to Eq. (13.4-13) and Figure 13.12b shows the evolution

in time of several mass fractions during a condensation polymerization.

(^26) C. Tanford,op. cit. (note 23).
(^27) H. B. Dwight,op. cit.(note 25).