Chapter 8: Multiple Reactions

Derivation for Optimum Yield with Multiple Reactions:

To maximize the amount of B produced, we need to differentiate our function for C_B. Setting the differential equation equal to zero, we can solve for the time to reach this optimum, t_opt.

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

\(\frac{dC_B}{dt} = 0 = \frac{k_1 C_{A0}}{k_2 - k_1} \left(-k_1 e^{-k_1 t} + k_2 e^{-k_2 t}\right)\)

Solving for \(t_{\text{opt}}\) gives:

\(t_{\text{opt}} = \frac{1}{k_2 - k_1} \ln\left(\frac{k_2}{k_1}\right)\)

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