So imagine we are doing kinetic gas theory with spheres. To model the collisions however, we consider the potential used over here.
We choose the regularized potential as
$$ V_{\varepsilon}~=~\frac{(x_1-x_2)^2}{\varepsilon}\theta(|x_1 - x_2| - d).$$
where $V_{\varepsilon}$ is a family potentials of ${\varepsilon}$ as a label, $x_1$ is the position of the first gas molecule, $x_2$ is the position of the second gas molecule and $d$ is the diameter of our molecules.
A general observation for our potential is while $\epsilon \to 0^+$ the time spent in the potential $t \to 0$. We, presume the time spent $\tau_C$ in the potential is small but not $0$ and we realise if we vary the steep slope of our short ranged potential then it should not affect the equation of state. Thus, if we have a time averaged force:
$$ \frac{\partial \langle F \rangle_t}{\partial \rho} = \frac{\partial \langle F \rangle_t}{\partial T} = \frac{\partial \langle F \rangle_t}{\partial P} = 0$$
where the modulus of force is denoted by $F$. Now, we define this time averaged force as the force applied between $2$ collisions. Thus,
$$ \langle F \rangle_t \approx \frac{F_0 \tau_C}{\tau_F} $$
where $\tau_F$ is the mean free time and $F_0$ is a consequence of mean-value theorem. I strongly suspect (but have no means of proving) that:
$$F_0 = PA$$
where $A$ is the area of the molecule. Can someone prove/disprove this?
