Chapter 9: Reaction Mechanisms, Pathways, Bioreactions and Bioreactors

Derive: R_t

k_a are defined wrt reactant.

\( R_1 + R_1 \longrightarrow P_2 \)

\( -r_{t1} = k_a R_1^2, \quad r_{p2} = \frac{-r_a}{2} = \frac{k_a}{2} R_1^2 \)

\( R_1 + R_2 \longrightarrow P_3 \)

\( R_1 + R_3 \longrightarrow P_4 \)

\( R_2 + R_2 \longrightarrow P_4 \)

\( R_2 + R_3 \longrightarrow P_5 \)

\( r_{t1} = k_a R_1^2 + k_a R_1 R_2 + k_a R_1 R_3 \)

\( r_{t2} = k_a R_1 R_2 + k_a R_2^2 + k_a R_2 R_3 \)

\( r_{t3} = k_a R_1 R_3 + k_a R_2 R_3 + k_a R_3^2 \)

\( r_t = r_{t1} + r_{t2} + r_{t3} = k_a R_1^2 + 2 k_a R_1 R_2 + 2 k_a R_1 R_3 + k_a R_2^2 + 2 k_a R_2 R_3 \)

\( r_t = k_a (R^*)^2 = k_a (R_1 + R_2 + R_3) (R_1 + R_2 + R_3) \)

\( = k_a R_1^2 + 2 k_a R_1 R_2 + 2 k_a R_1 R_3 + k_a R_2^2 + 2 k_a R_2 R_3 \)

The net rate of termination of all radicals is:

\( \boxed{ R_{ta} = k_a (R^*)^2 } \)