Exact Asymptotic Relaxation of Pseudo-First-Order Reversible Reactions

The relaxation kinetics of the diffusion-influenced reversible reaction $A+B\rightleftharpoons C$ is studied in the pseudo-first-order limit $\left(\right[B\left]\gg \right[A\left]\right)$ when A and C are static and the B's move independently with diffusion coefficient D. For the initial condition $\left[A\right(0\left)\right]=1\mathrm{}$, $\left[C\right(0\left)\right]=0\mathrm{}$, it is shown that the asymptotics of $\left[A\right(t\left)\right]$ for $t\to \infty$ is given in d dimensions by $\left({1+K}_{\mathrm{eq}}\right[B\left]{)}^{-1\mathrm{}}{+K}_{\mathrm{eq}}^2\right[B]/({1+K}_{\mathrm{eq}}\left[B\right]{)}^3{f}_d\left(t\right)$ with $f_1\left(t\right)=(\pi \mathrm{Dt}{)}^{-1/2\mathrm{}}$, $f_2\left(t\right)=(4\pi \mathrm{Dt}{)}^{-1\mathrm{}}$, and $f_3\left(t\right)=(4\pi \mathrm{Dt}{)}^{-3/2\mathrm{}}$, and where $K_{\mathrm{eq}}$ is the equilibrium constant. By comparing with accurate simulations, this result is found to be exact for $d=1\mathrm{}$, and we predict that it is exact for higher dimensions.