I am considering the following chain reaction: $$\begin{align} \ce{A_0 + B <=>[$N\cdot k_+$][$k_-$]A_1\\ A_1 + B <=>[$(N-1)\cdot k_+$][$2\cdot k_-$]A_2\\ A_2 + B <=>[$(N-2)\cdot k_+$][$3\cdot k_-$]A_3\\ ...\\ A_{N-2} + B<=>[$2\cdot k_+$][$(N-1)\cdot k_-$]A_{N-1}\\ A_{N-1} + B<=>[$k_+$][$N\cdot k_-$]A_N} \\ \end{align}$$ Initial condition are $A_0 = 1, A_1=0, ... ,A_N=0$ and $B$ can vary from 0 to 1, $k_+$ is faster than $k_-$.

I have noticed (trough extensive numerical simulations) that result of this chain: $A_N$ is identical with the product of the following reaction in the power of N. $$\begin{align} \ce{A_0 +B&<=>[$k_+$][$k_-$]C} \\ \end{align}$$

Meaning that
$$\begin{align} \ce{ $ [A_N]=[C]^N$} \\ \end{align}$$

I am wondering how to check this result analytically. I would like to know what to start with.


PS. I have looked at the ODEs and so far my calculations failed to prove the point, but I am not sure about my directions.

  • $\begingroup$ Just a product of the reaction between $A_0$ and $B$, where $A_0$ and $B$ are exactly the same as in the initial condition for the reaction chain above. $\endgroup$ – Sp_J Dec 4 '17 at 14:20
  • $\begingroup$ The problem is that you seem to be using capital letters to denote both chemical species and their concentrations. $\endgroup$ – Rodrigo de Azevedo Dec 4 '17 at 14:23

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