= Reverse reaction rate from equilibrium thermodynamics
{title2=$k_r=k_f/[K(c^\circ)^{\Delta\nu}]$}
For an elementary reversible reaction with concentration-based rates obeying the <law of mass action>, write
$$
r_f=k_f\prod_{\rm reactants}c_i^{-\nu_i},\qquad
r_r=k_r\prod_{\rm products}c_i^{\nu_i},
$$
where <stoichiometric coefficients> $\nu_i$ are positive for products. At <detailed balance>, $r_f=r_r$, so
$$
\frac{k_f}{k_r}=\prod_i c_{i,\rm eq}^{\nu_i}=K_c.
$$
This $K_c$ carries a power of <concentration> when $\Delta\nu=\sum_i\nu_i\ne0$.
For an ideal solution with activities $a_i=c_i/c^\circ$, the dimensionless thermodynamic <equilibrium constant> is $K=\prod_i(c_{i,\rm eq}/c^\circ)^{\nu_i}$. Consequently
$$
K_c=K(c^\circ)^{\Delta\nu},\qquad
k_r=\frac{k_f}{K(c^\circ)^{\Delta\nu}}.
$$
Keeping the <standard concentration> factor is what makes the units of both rate coefficients consistent. For $A+B\rightleftharpoons C$, $\Delta\nu=-1$ and $k_r=k_fc^\circ/K$: a second-order forward coefficient becomes a first-order reverse coefficient.
Thermodynamics fixes this ratio only under the stated elementary, reversible rate-law model. It does not determine either coefficient separately or replace a multistep <reaction mechanism> by one elementary step.
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