Suppose we have one chemical species $V$ and two compartments in which $V$ can be, $A$ and $B$, with volumes $\Omega_A$ and $\Omega_B$ respectively, where $\Omega_A < \Omega_B$.
Compartments $A$ and $B$ are separated by a biological membrane. The permeability of the membrane with respect $V$ is $P$cm $\text{s}^{-1}$. Let us say the membrane has thickness $\varepsilon$ cm.
See diagram below.
I wish not to model diffusion or any spatial variation, I am assuming well mixed. I will represent the concentration of $V$ by $v$ with units $M=\text{mol}{L^{-1}}$. $v_A(t)$ and $v_B(t)$ represent the concentration of $V$ in $A$ and $B$ respectively, where $t$ is time.
I am going to represent movement from $A$ to $B$ by a 'reaction', similary for $B$ to $A$. My question is how from $P$ do I describe the rates of the reactions $Q_{AB}$ and $Q_{BA}$ from $P$ (and $\Omega_A,\Omega_B$ as these are concentrations?) so that the following makes sense:
$$\frac{dv_A}{dt}=Q_{BA}v_B-Q_{AB}v_A$$ $$\frac{dv_B}{dt}=-(Q_{BA}v_B-Q_{AB}v_A)$$