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 n M 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 n M 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 of X x .
The mass fraction of molecules with degree of polymerization x is
W x
mass of molecules with degree of polymerization x
mass of all molecules
(13.4-19a)
N x M x
∞
x1 N x M x
(13.4-19b)
The mass-average molecular mass (often called the “weight-average molecular
weight”) is defined by
M w
∞
x1
W x M x
∞
x1 N x M 2
x
∞
x1 N x M x
(13.4-20)
In this average, each molecule is given an importance (weight) in the average proportional to its mass. The required sum can be found in tables, 27 giving
M w (1 − p)
2 M 0
∞
x1
x
2 p
x−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. If p is 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 n in 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).
593
From Eqs. (13.4-7) and (13.4-15), we can write an equation for the time dependence
of M n :
M n M 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 n M 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 of X x .
The mass fraction of molecules with degree of polymerization x is
W x
mass of molecules with degree of polymerization x
mass of all molecules
(13.4-19a)
N x M x
∞
x1 N x M x
(13.4-19b)
The mass-average molecular mass (often called the “weight-average molecular
weight”) is defined by
M w
∞
x1
W x M x
∞
x1 N x M 2
x
∞
x1 N x M x
(13.4-20)
In this average, each molecule is given an importance (weight) in the average proportional to its mass. The required sum can be found in tables, 27 giving
M w (1 − p)
2 M 0
∞
x1
x
2 p
x−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. If p is 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 n in 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).