Cascading chemical reaction

Chemistry Level 1

The rate of a simple chemical reaction like \(\ce{A + 2B->C}\) depends on the input concentrations in a simple way. Here, for example, the output species \(\text{C}\) is made with a rate proportional to \(\left[\text{A}\right]\cdot\left[\text{B}\right]^2\) and, in general, the rates of one step reactions depend on the input concentrations raised to the power of their stoichiometric coefficient. But this isn't always the case.

Chain reactions occur when the output from one reaction \(\mathcal{R}_i\) is the input to another \(\mathcal{R}_j\) as in the following system of reactions: \[\begin{array}{rrr} \text{A} &\longleftrightarrow& \text{2B } \\ \text{B} &\longleftrightarrow& \text{3C } \\\ \text{C} &\longleftrightarrow& \text{4D } \\ \text{D} &\longrightarrow& \text{E}. \end{array}\] It's a lot less simple to find reaction order in a case like this, but as the concentration of \(\text{A}\) rises, a simple behavior emerges: in the asymptotic limit where \(\left[\text{A}\right]\) gets large, the rate of production of the output \(\left[\text{E}\right]\) is approximately equal to \(\text{(a constant)}\times\left[\text{A}\right]^\alpha.\)

What is \(\alpha?\)

Assume that fresh \(\text{A}\) is supplied continuously to the system so that its concentration is always \(\left[\text{A}_0\right],\) and that the conversion of \(\text{D}\) to \(\text{E}\) is irreversible. For simplicity, assume all the magnitudes of all rate constants are equal to \(1.\)

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