# Chain reaction vs single reaction

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.

Thanks!

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.

• 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. – Sp_J Dec 4 '17 at 14:20
• The problem is that you seem to be using capital letters to denote both chemical species and their concentrations. – Rodrigo de Azevedo Dec 4 '17 at 14:23