Rate Equation - Consecutive Reactions

Consecutive Reactions

If the rate constants for the following reaction are and ;, then the rate equation is:

For reactant A:

For reactant B:

For product C:

With the individual concentrations scaled by the total population of reactants to become probabilities, linear systems of differential equations such as these can be formulated as a master equation. The differential equations can be solved analytically and the integrated rate equations are

$\left=\left\{ \begin{array}{*{35}l} \left_{0}\frac{k_{1}}{k_{2}-k_{1}}\left( e^{-k_{1}t}-e^{-k_{2}t} \right)+\left_{0}e^{-k_{2}t} & k_{1}\ne k_{2} \\ \left_{0}k_{1}te^{-k_{1}t}+\left_{0}e^{-k_{1}t} & \text{otherwise} \\ \end{array} \right.$

$\left=\left\{ \begin{array}{*{35}l} \left_{0}\left( 1+\frac{k_{1}e^{-k_{2}t}-k_{2}e^{-k_{1}t}}{k_{2}-k_{1}} \right)+\left_{0}\left( 1-e^{-k_{2}t} \right)+\left_{0} & k_{1}\ne k_{2} \\ \left_{0}\left( 1-e^{-k_{1}t}-k_{1}te^{-k_{1}t} \right)+\left_{0}\left( 1-e^{-k_{1}t} \right)+\left_{0} & \text{otherwise} \\ \end{array} \right.$

The steady state approximation leads to very similar results in an easier way.

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