Chapter 8: Multiple Reactions

Derivation for Series Reaction:

\((1) \quad \frac{dC_B}{dt} + k_2 C_B = k_1 C_{A0} \exp(-k_1 t)\)

\(\text{i.f.} = \exp(k_2 t)\)

\((2) \quad \frac{d\left[C_B \exp(k_2 t)\right]}{dt} = k_1 C_{A0} \exp\left[(k_2 - k_1)t\right]\)

Differentiating the rhs of Equation (2):

\(\exp(k_2 t) \frac{dC_B}{dt} + C_B k_2 \exp(k_2 t) = k_1 C_{A0} \exp\left[-k_1 t\right] \exp(k_2 t)\)

\(\exp(k_2 t)\) we see we arrive back at Equation (1):

\(\frac{dC_B}{dt} + k_2 C_B = k_1 C_{A0} \exp(-k_1 t)\)

Now integrating Equation (2):

\(\frac{d\left[C_B \exp(k_2 t)\right]}{dt} = k_1 C_{A0} \exp\left[(k_2 - k_1)t\right]\)

\(C_B \exp(k_2 t) = \frac{k_1 C_{A0} \exp(-k_1 t)}{k_2 - k_1} + K_1 \exp(-k_2 t)\)

Evaluating the constant of integration \(K_1:\)

When \(t = 0, \quad C_B = 0 \quad \therefore \quad K_1 = -\frac{k_1 C_{A0}}{k_2 - k_1}\)

Substituting for \(K_1\) and rearranging:

\(C_B = \frac{k_1 C_{A0}}{k_2 - k_1} \left[\exp(-k_1 t) - \exp(-k_2 t)\right]\)

Back to Chapter 8