to be elastic. We can derive an expression for the pressure of a gas based on the kinetic theory. We begin with the idea that molecules of a gas are in incessant random motion, colliding against one another and with the walls of the container. All collisions between molecules among themselves or between molecules and the walls are elastic.
This implies that total kinetic energy is conserved. The total momentum is conserved as usual. . .
Pressure of an Ideal Gas Consider a gas enclosed in a cube of side l. Take the axes to be parallel to the sides of the cube, as shown in Fig. . .
A molecule with velocity ( v x , v y , v z ) hits the planar wall parallel to yz - plane of area A (= l ). Since the collision is elastic, the molecule rebounds with the same velocity; its y and z components of velocity do not change in the collision but the x -component reverses sign. That is, the velocity after collision is (- v x , v y , v z ) . The change in momentum of the molecule is: – mv x – ( mv x ) = – mv x .
By the principle of conservation of momentum, the momentum imparted to the wall in the collision = mv x . the wall. Thus, the number of molecules with velocity ( v x , v y , v z ) hitting the wall in time ∆ t is ½ A v x ∆ t n, where n is the number of molecules per unit volume. The total momentum transferred to the wall by these molecules in time ∆ t is: Q = ( mv x ) (½ n A v x ∆ t ) ( .
) The force on the wall is the rate of momentum transfer Q / ∆ t and pressure is force per unit area : P = Q /( A ∆ t ) = n m v x ( . ) Actually, all molecules in a gas do not have the