Question

Use a simple model: a total of N molecules are contained in a box of side length d. Each molecule has a mass m and a velocity u. By considering the collisions between the molecules and the walls of the box, using the assumptions in kinetic theory of gas derive the equation:

pV = 1⁄3 N m<c^2>

where p is the pressure of the gas, V is its volume and <c^2> is the mean square speed of the gas. 

Answer 1

If the length of the cube, mass of the molecule and velocity are l, m and v respectively,

Momentum in the x-direction $mu_x$

Momentum in the -x-direction $-mu_x$

Change in momentum $2mu_x$

Total time taken - from one end to the other and vice versa $\frac{2l}{u_x}$

Rate of change in momentum $\frac{mu_x^2}{l}$

According to Newton's Second Law, the rate of change of momentum is the force exerted by the molecule on the wall.

Therefore, Force $\frac{mu_x^2}{l}$

Pressure on the wall, $\frac{mu_x^2}{l^3}$

If there are N molecules in the container, $P=\frac{Nmu_x^2}{V}$

Since velocity in each direction is equal,

$$P=\frac{Nmc^2}{3V}$$