Htydjtk

Recently, I was told that in case of a particular step of a generic ligand substitution reaction:

$$ce{M(OH2)_{$N - n$}L_{n} + L <=> M(OH2)_{$N - n - 1$}L_{$n + 1$} + H2O}$$

The probability of the forward reaction and by extension, the equilibrium constant of this step, $K_n$ would be proportional to

$$frac{N - n}{n + 1}$$

by a purely statistical analysis. Now I have thought about this for quite a bit, but I can't understand the mathematical reasoning behind arriving at this expression. I suspect it has something to do with the numbers of the ligand being replaced and the ligand which is replacing the other one. Can anyone explain the process of arriving at this expression using simple (if possible) reasoning?

I think the easy way out is to invoke $S_mathrm m = R ln Omega$. If we assume that for a generic complex $ce{MA_nB_{$N-n$}}$,

$$Omega = {N choose n} = frac{N!}{n!(N-n)!} quad left[ = {N choose N-n} right]$$

and that for the individual molecules $ce{A}$ and $ce{B}$, $Omega = 1$, then the equilibrium constant $K$ for

$$ce{MA_nB_{$N-n$} + A <=> MA_{n+1}B_{$N-n-1$} + B}$$

is given by

$$begin{align}
K &= expleft(frac{-Delta_mathrm r G}{RT}right) \
&= expleft(frac{Delta_mathrm r S}{R}right) \
&= expleft(frac{S_mathrm{m}(ce{MA_{n+1}B_{$N-n-1$}}) + S_mathrm{m}(ce{B}) - S_mathrm{m}(ce{MA_nB_{$N-n$}}) - S_mathrm{m}(ce{A})}{R}right) \
&= exp[lnOmega(ce{MA_{n+1}B_{$N-n-1$}}) + lnOmega(ce{B}) - lnOmega(ce{MA_nB_{$N-n$}}) - lnOmega(ce{A})] \
&= expleft[lnleft(frac{Omega(ce{MA_{n+1}B_{$N-n-1$}})Omega(ce{B})}{Omega(ce{MA_nB_{$N-n$}})Omega(ce{A})}right)right] \
&= frac{Omega(ce{MA_{n+1}B_{$N-n-1$}})Omega(ce{B})}{Omega(ce{MA_nB_{$N-n$}})Omega(ce{A})} \
&= frac{N!}{(n+1)!(N-n-1)!} cdot frac{n!(N-n)!}{N!} \
&= frac{N-n}{n+1}
end{align}$$

The reason for ignoring $Delta_mathrm r H$ is because we are only interested in statistical effects, i.e. entropy, and we don't care about the actual stability of the complex or the strength of the M–L bonds. However, the exact justification for assuming this form for $Omega$ still eludes me. It makes intuitive sense (that there are $N!/(n!(N-n)!)$ ways to arrange $n$ different ligands in $N$ different coordination sites), but I can't convince myself (and don't want to attempt to convince you) that it's entirely rigorous. In particular, I feel like symmetry should play a role here; maybe it is simply that the effects of any symmetry eventually cancel out.