Collisions in a real gas

Relative velocity

Consider two particles A and B : - At the beginning $t_i$ they are positioned in $A_i$ and $B_i$ respectively. These two points are characterized in the orthonormal coordinate system by position vectors $\vec r_{a,i}$ and $\vec r_{b,i}$. - They move with the respective velocities $\vec v_a$ and $\vec v_b$ - At the end $t_f$ they are positioned in $A_f$ and $B_f$ respectively. These two points are characterized in the orthonormal coordinate system by position vectors $\vec r_{a,f}$ et $\vec r_{b,f}$. - We have: $\vec{A_iA_f}$ $=$ $\vec v_a\Delta t$ and $\vec{B_iB_f}$ $=$ $\vec v_b\Delta t$ with $\Delta t$ $=$ $t_f-t_i$ - Vector addition gives us: $\vec r_{a,f}=\vec r_{a,i}+\vec v_a\Delta t$ $\vec r_{b,f}=\vec r_{b,i}+\vec v_b\Delta t$ Consider at what velocity the distance between particles changes. This velocity will be called relative velocity of the two particles. $\vec d_f-\vec d_i$ $=$ $(\vec v_b-\vec v_a)\Delta t$ $\frac{\vec d_f-\vec d_i}{\Delta t}$ $=$ $\vec v_b-\vec v_a$

Relative velocity: Two particles moving with velocities $\vec v_a$ and $\vec v_b$ have a relative velocity $\vec v=\vec v_b-\vec v_a$

Relative average velocity of particles of a gaseous substance

The relative velocity vector module is: $|\vec v|$ = $\sqrt{\vec v \cdot \vec v} =$ $\sqrt{(\vec v_b-\vec v_a)(\vec v_b-\vec v_a)} =$ $\sqrt{\vec v_b\vec v_b-2\vec v_b\vec v_a+ \vec v_a\vec v_a} =$ $\sqrt{|\vec v_b|^2-2\vec v_b\vec v_a+ |\vec v_a|^2}$ - The random motion of particles in a gas ensures that there are as much of occurrences where the product $\vec v_b\vec v_a$ takes the same absolute value as its opposite. On average over a large number of cases this product will annul. - $|\vec v_b|^2$ et $|\vec v_a|^2$ on average will be equal to the square of the average velocity $\lt v \gt$ →    introduced here . We can therefore conclude that the relative mean square velocity of two particles is: $\lt v\gt_{rel}=\sqrt{2u^2}=\sqrt{2}\lt v\gt$ where $\lt v\gt$ is the average velocity of the particles

Average relative velocity of two particles of a gaseous substance: $\lt v\gt_{rel}=\sqrt{2}\lt v\gt$ where $\lt v\gt$ is the average velocity

Particle collision frequency of a gaseous substance

Here is a supposed compact molecule of a mobile gaseous substance of diameter $d$ that moves with the average relative square velocity $u_{rel}=\sqrt{2}u$ (compared to other molecules)

For a time $\Delta T $ it scans a "collision" tube with length $L=\sqrt{2}\lt v \gt\Delta T$ and section $\pi d^2$ where it can meet with other molecules whose center is located in the tube. - The volume of this tube is $\sqrt{2}\pi d^2 \lt v\gt\Delta T$ and if $N$ is the number of molecules whose center is in the tube, the number of collisions of our molecule during the time $\Delta T$ is $N-1\approx N$ - As $P\cdot V=N\cdot k\cdot T$ (ideal gas law), we find: Number of a molecule collisions during the time $\Delta T$ = $N=\frac{P}{k\cdot T}\pi d^2 \sqrt2 \lt v\gt \Delta T$ Collision frequency of a molecule = $\frac{N}{\Delta T}$ $=$ $\frac{P}{k\cdot T}\pi d^2 \sqrt2 \lt v\gt $

Collision frequency of a molecule of a gaseous substance: $z=\frac{P}{k\cdot T}\pi d^2 \sqrt2 \lt v\gt$ where $P$ is pressure $T$ is Kelvin temperature $k$ is the Boltzmann constant $d$ is the diameter of the molecule $\lt v\gt$ is the average velocity →   with: $\lt v\gt=\sqrt{\frac{8RT}{\pi M}}$