# Relationship between rate equation and equilibrium constant

I have read mannaia's answer on the question of 'rate order and confusion', as well as Nicolau Saker Neto's answer to a related question. They have both been very helpful.

As I understand, the equilibrium is actually derived from the rate constants of a reaction. At dynamic equilibrium, the rate of the forward reaction is equal to the rate of the backward reaction, hence in a hypothetical reaction

$$\ce{pA + qB <=> rC + sD}$$

with the rate equations $Rate_{fwd} = k_{fwd}[A]^{a}[B]^{b}$ and $Rate_{bck} = k_{bck}[C]^c[D]^d$, with $a, b, c,$ and $d$ not related to $p, q, r$ and $s$.

The equilibrium is derived as such: \begin{align} R_{fwd} &= R_{bck}\\ k_{fwd}[A]^{a}[B]^{b}&=k_{bck}[C]^c[D]^d\\ \frac{k_{fwd}}{k_{bck}} &= \frac{[C]^c[D]^d}{[A]^a[B]^b}\\ \therefore k&=\frac {k_{fwd}} {k_{bck}}\\ \end{align}

Based on the derivation, and the fact that the stoichiometric coefficients of the reactants in the rate-determining step (RDS) of the reaction is equal to the order of the reactants in the rate equation, is it correct to state that the reaction equations given as the equilibrium equation is actually the rate-determining step of two reactions?

For instance, in an S$_\mathrm{N}$1 reaction, obviously the RDS is the departure of the leaving group, but you don't have to (and usually are not in a position to) know the concentration of the intermediate carbocation in order to calculate the equilibrium of the overall reaction. You can simply write an equilibrium equation for the initial reactants and final products: $$K=\frac{[\ce{LG}][\ce{product}]}{[\ce{nucleophile}][\ce{substrate}]}.$$
Of course finding $K$ may be not be easy, even if you have access to good thermodynamic data for the reactants and products, because $\Delta G_0$ (and therefore $K$) depends on solvation, temperature and other conditions. But given identical conditions, $K$ is constant over a wide range of concentrations, so you might be able to find it in the literature or measure it with a single experiment.
Note that you can also write an equilibrium equation for the RDS alone: $$K'=\frac{[\ce{LG}][\ce{carbocation}]}{[\ce{substrate}]}$$ But since you seldom know the concentration of the carbocation, this equation isn't as useful in practice (it is of course extremely useful for mechanistic studies).