Consider a gas enclosed in a cube. Let the axes be parallel to the sides of the cube (Fig.).
A molecule having velocity (Vx ,Vy,Vz) hits the planar wall parallel to γz plane of area A(l2). As the collision is elastic, the molecule rebounds with the same velocity, y and z components of velocity do not change in the collision but the sign of x-component is reversed. Then, the velocity of collision is (-Vx, Vy, Vz).
The change in momentum of molecule,
= -mVx – (mVx) = -2 mVx …(1)
According to the principle of conservation of momentum. The momentum imported to wall (in collision) = 2mVx. Consider a small time interval ∆ t, a molecule with velocity Vx will hit the wall when it is within the volume of AVx∆t and can hit the wall in time ∆t. The number of molecules with velocity (Vx, Vy, Vz) hitting the wall in time ∆t is AVx∆ t n where n is the number of molecules per unit volume.
The total momentum transferred to wall by these molecules in time ∆t is :
There is a distribution in velocities. So, equation (3) stands for pressure due to the group of molecules having velocity Vx in x-direction and n is the number density of these molecules.
If the gas is isotropic, i.e., the velocity of molecules in the vessel has no preferred direction.
here, 1/2 Mv-2 is the total kinetic energy of the gas.
If v = 1
\(\therefore P= \frac{2}{3}\)