Physical Chemistry Third Edition

592 13 Chemical Reaction Mechanisms II: Catalysis and Miscellaneous Topics

Equation (13.4-9) agrees with experiment fairly well for the later stages of polymer-

ization, although not so well for the early stages. Figure 13.11 shows data for two

polyesterification reactions, each beginning with 82% of the carboxyl groups esterified.

0

100

98.8

98.3

97.5

95.5

80.0

80

60

40

20

20 40 60 80 100120 140

% Conversion of original COOH groups

112 p

Time/min

1,10-Decamethylene glycol

at 161 8 C [S.D. Hamann, D.H.

Solomon, and J.D. Swift, J.

Macromol Sci.Chem.,A2,

153 (1968)]

Diethylene glycol at 109 8 C

[P.J. Flory, J. Am. Chem.

Soc., 61 , 3334 (1939)]

Figure 13.11 Polyesterification of

Adipic Acid with Two Polyalcohols,

Catalyzed with p-Toluene Sulfonic

Acid.The left axis shows the value

of 1/( 1 −p) wherep is the fraction

of carboxyl groups that have reacted.

The right axis is labeled with the

percent of original carboxyl groups that

have reacted. From H. R. Allcock and

F. W. Lampe,Contemporary Polymer

Chemistry, Prentice Hall, Englewood

Cliffs, NJ, 1981, p. 254.

Since there is a distribution of degrees of polymerization, there is a distribution of

molecular masses. Ifxis even, the molecular mass of molecules with a given value

ofxis

Mx

x

2

Mr+18 amuxM 0 +18 amu≈xM 0 (13.4-10)

whereMris the mass of the repeating unit of the polymer in amu andM 0 is equal to

Mr/2. The term 18 amu is added because each molecule has an OH group at one end

and an H atom at the other end that are not part of the repeating unit. Equation (13.4-10)

can be modified for odd values ofx. For either odd or evenx, the final approximate

equality is valid for large values ofx.

LetNxbe the number of polymer molecules with degree of polymerization equal

toxand letXxdenote thenumber fractionof polymer molecules with this degree of

polymerization:

Xx

Nx

N

(13.4-11)

whereNis the total number of polymer molecules. Thenumber-average molecular

massis the mean molecular mass with each molecule given equal importance in the

averaging process:

〈M〉n

1

N

∑∞

x 1

NxMx

∑∞

x 1

XxMx (13.4-12)

where the sum is over all values ofx.We have used an infinite upper limit for the sum

for mathematical convenience. This introduces only a negligible error sinceNxrapidly

approaches zero for large values ofx. The number-average molecular mass is equal to

the total polymer mass divided by the total number of polymer molecules.

A formula for〈M〉ncan be derived from statistical reasoning.^24 We assume thatxis

even. The polymer molecule consists ofx−1 units with esterified carboxyl groups,

and one unit with a free carboxyl group. Let the fraction of all carboxyl groups that are

free be equal to 1−p, and the fraction that are esterified be equal top. The probability

of having a given value ofxis

Probability ofxpx−^1 (1−p) (13.4-13)

The number-average molecular mass can now be written

〈M〉n

∑∞

x 1

XxxM 0 M 0 (1−p)

∑∞

x 1

xpx−^1 (13.4-14)

where a term equal to 18 amu has been omitted. This sum can be found in tables,^25

giving

〈M〉n

M 0 (1−p)

(1−p)^2



M 0

1 −p

(13.4-15)

(^24) H. R. Allcock and F. W. Lampe,op. cit. (note 23).
(^25) See for example H. B. Dwight,Tables of Integrals and Other Mathematical Data, 4th ed., Macmillan,
New York, 1961, p. 8.