Chapter 9: Reaction Mechanisms, Pathways, Bioreactions and Bioreactors

Derive: dM/dt

Overall balance on all polymers \((j = 1\) to \(\infty\))

\(\frac{dP_j}{dt} = k \sum_{i=1}^{j-1} P_i P_{j-i} - 2kP_jM\)

\(\sum \left(\frac{dP_j}{dt} \right) = \frac{d}{dt} \sum P_j = \sum R_j = k \sum_{j=1}^{\infty} \sum_{i=1}^{j-1} P_i P_{j-i} - 2kM^2\)

\(= k \sum_{j=1}^{\infty} \left[ P_1 P_{j-1} + P_2 P_{j-2} + P_3 P_{j-3} + P_4 P_{j-4} \right] - 2kM^2\)

Let's look at the first term (1)

\(\sum_{j=1}^{\infty} P_i P_{j-1} = P_1 [P_0 + P_1 + P_2 + \dots]\)

\(= P_1 \left[ \sum_{j=1}^{\infty} P_j \right] = P_1 M\)

Let's look at the second term (2)

\(\sum_{j=1}^{\infty} P_2 P_{j-2} = P_2 [P_{-1} + P_0 + P_1 + P_2 + \dots]\)

\(= P_2 [P_1 + P_2 + \dots] = P_2 \sum P_j = P_2 M\)

Let's look at the third term (3)

\(\sum_{j=1}^{\infty} P_3 P_{j-2} = P_3 M\)

Now consider all the terms in brackets

\( k \left[ \right] = k [P_1 + P_2 + \dots] M = kM^2 \)

then

\(\sum \frac{dP_j}{dt} = \frac{dM}{dt} = kM^2 - 2kM^2 = -kM^2\)