An integral $\int_0^\infty\,xe^{-ax^2}dx$ $=$ $\frac{1}{2a}\, (5) $ Probability density of a component $v_x$ of the velocity of a molecule (see → here) $\mathscr{f}(v_x)=(\frac{m}{2\pi\cdot k\cdot T})^\frac{1}{2}e^{-\frac{mv_x^2}{2kT} }$ where $T$ is Kelvin temperature $k=1,38\cdot 10^{-23}$ is Boltzmann's constant $m$ is the mass of one molecule
A wall with surface $A$ perpendicular to the $x$ axis. A molecule whose velocity component along $x$ is $v_x\gt 0$ strikes the wall at a time $\Delta t$ if it is at a distance less than or equal to $v_x\Delta t$ of the wall. Similarly, all the molecules in a volume $Av_x\Delta t$. It is not important here that molecules leave this volume before hitting the wall their direction being oblique, because there will be as much that will compensate them coming from outside of this volume. If $\mathcal N$ is the number of molecules per unit volume, then there will be $\mathcal N\cdot Av_x\Delta t$ molecules in this volume. The average number of collisions against the wall during the time $\Delta t$ is equal to the average number of molecules in the volume $Av_x\Delta t$ multiplied by the probability density $\mathscr{f}(v_x)$ of this velocity. Average number of impacts against a wall surface $A$= $\int_0^\infty\, \mathcal N\cdot Av_x\Delta t \cdot \mathscr{f}(v_x)=$ $\mathcal N \cdot A \Delta t \int_0^\infty (\frac{m}{2\pi\cdot k\cdot T})^\frac{1}{2}v_xe^{-\frac{mv_x^2}{2kT} }=$ $\mathcal N \cdot A \Delta t (\frac{m}{2\pi\cdot k\cdot T})^\frac{1}{2}\frac{1}{2\cdot \frac{m}{2kT}}=$ $\mathcal N \cdot A \Delta t (\frac{kT}{2\pi \cdot m})^\frac{1}{2}$ On the other side, $N$ being the number of molecules in the corresponding volume $V$, we have: $\mathcal N$= $\frac{N}{V}=$ $\frac{P}{kT}$ (loi des gaz parfaits) Taking $A=1 m^2$ and $\Delta t=1s$, we have:
Frequency of collisions per surface unit = $Z= \frac{P}{\sqrt{2\pi \cdot m\cdot k\cdot T}}$ where $P$ is pressure $T$ is Kelvin temperature $k=1,38\cdot 10^{-23}$ is Boltzmann's constant $m$ is the mass of one molecule