Page 186 - Introduction to chemical reaction engineering and kinetics

168 Chapter 7: Homogeneous Reaction Mechanisms and Rate Laws

7.3.2 Step-Change Polymerization
Consider the following mechanism for step-change polymerization of monomer M (PI)
to P2, P,, . . .) P,, . . . . The mechanism corresponds to a complex series-parallel scheme:
series with respect to the growing polymer, and parallel with respect to M. Each step is
a second-order elementary reaction, and the rate constant k (defined for each step)’ is
the same for all steps.

M + MIP, (1)
k
M + P,+P, (2)

k
M + P,-, +P, (r - 1)

where r is the number of monomer units in the polymer. This mechanism differs from
a chain-mechanism polymerization in that there are no initiation or termination steps.
Furthermore, the species P,, P,, etc. are product species and not reactive intermedi-
ates. Therefore, we cannot apply the SSH to obtain a rate law for the disappearance of
monomer (as in the previous section for equation 7.3-l), independent of cp,, cp,, etc.
From the mechanism above, the rate of disappearance of monomer, (- rM), is

(-TM) = 2kcL + kcMcPz + . . . + kcMcp, + . . .

= kc,(2c, + 2 cP,) (7.3-3)
r=2
The rates of appearance of dimer, trimer, etc. correspondingly are

+2 = kCM(CM - CP,) (7.3-4)
(7.3-5)
+3 = kCM(CPZ - CP3 >
9 . .

+r = kcM(cp,-, - cpr), etc. (7.3-6)
These rate laws are coupled through the concentrations. When combined with the
material-balance equations in the context of a particular reactor, they lead to uncou-
pled equations for calculating the product distribution. For a constant-density system
in a CSTR operated at steady-state, they lead to algebraic equations, and in a BR or
a PFR at steady-state, to simultaneous nonlinear ordinary differential equations. We
demonstrate here the results for the CSTR case.
For the CSTR case, illustrated in Figure 7.2, suppose the feed concentration of
monomer 1s cMo, the feed rate is q, and the reactor volume is V. Using the material-
balance equation 2.3-4, we have, for the monomer:

cMoq -cMq+rMV = 0

‘The interpretation of k as a step rate constant (see equations 1.4-8 and 4.1-3) was used by Denbigh and Turner
(197 1, p. 123). The interpretation of k as the species rate constant kM was used subsequently by Denbigh and
Turner (1984, p. 125). Details of the consequences of the model, both here and in Chapter 18, differ according
to which interpretation is made. In any case, we focus on the use of the model in a general sense, and not on the
correctness of the interpretation of k.