Let's say I have an ideal gas of particles and the strike zone of area $\sigma=\pi d^2$ of a molecule of the molecule is what is the sub system considered. The end of the face of the cylinder is the strike zone. One may visualize it via 2 methods:
The continuum method:
Or the discrete method:
1. Continuum Mechanics
Resorting to continuum mechanics, then the fluid element (of a molecule) which will strike the target once every $\tau_F$ (the mean free time) on average. The magnitude of the force applied will be $P|\sigma|$. Intuitively, this makes sense as then the gas will have an internal pressure of $P$ (with each collision) and one will have a force frequency of $\bar F_{\nu}$
$$ \bar F_{\nu} = \frac{P |\sigma|}{\tau_F} $$
Note: $$\lim_{\tau_F \to 0}\bar F_{\nu} \tau_F = P |\sigma|$$ Does indeed give the correct equation $$|F| = P|\sigma|$$
Where $P$ is pressure and $\sigma$ is the area of a molecule.
2. Discrete Mechanics
Now, if I remain in the discrete case I have the gas exerting a force $F_C$ (which is renormalizable) during the interval of the collision $\tau_C$ a collision interval. In this case the force frequency is $\bar F_{\nu}' $
$$\bar F_{\nu}' = \frac{\langle |F_C| \rangle}{\tau_C} $$
Since both models are describing the same system, I was wondering if there prove/disprove this conjecture:
$$ \frac{P |\sigma|}{\tau_F} \approx \frac{\langle |F_C| \rangle}{\tau_C}$$
