Derive an expression for the pressure of a gas.

The molecules of gas are in random motion. So,  they exert pressure on the walls of the container and strike with the walls of the container. Therefore, they exert thrust or force on the wall.  
The thrust exerted by molecules per unit area of the wall of the container is called pressure. 
 
Consider a cubical vessel of side l  is enclosing an ideal gas. 
Let m be the mass of gas molecules. 
Let v₁ be the velocity of one of the gas molecules. 
If v_1x , v_1y, v_1z are the components of v₁. 
Then, 
 
Let this molecule approaches the face towards L. 
Before collision with the walls, momentum is given by, 
 
Since molecule collides with face L due to X-component of motion, therefore, due to collision only X-component of momentum will change. 
Since collision is elastic, therefore momentum after collision is given by, 
 
Due to collision, the change in momentum of molecule is, 

Let n be the number of molecules per unit volume.
Since the molecules approach towards face L with velocity v_1x, therefore, the molecules lying in the cylinder of length v_1x and area of cross-section A may collide with face L on area in one second. 
Since molecular motion is random, therefore, it is logical to suppose only half of the molecules that lie in cylinder will strike the area A in one second. 


According to Newton’s third law of motion, to every action, there is an equal and opposite reaction.
Therefore, gas molecules exert an equal and opposite force on area A of face L as experienced by gas molecules. 
Force on area A of face L,  
Therefore pressure on face L is, 
                   
Similarly, 
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