• Consider a cube shaped container filled with an ideal gas. We will consider only one molecule. This molecule hits the walls of the container and bounces back
• Let the velocity of the molecule when it is moving be (v_x, v_y,v_z)
• When the molecule bounces back, the velocity will be (-v_x, -v_y,-v_z)
• Change in momentum = p_f – p_i where p_f is the final momentum and p_i is the initial momentum
• p_f – p_i = - mv_x - mv_x = -2mv_x
• This change in momentum is imparted to the wall due to the collision
• Momentum imparted to the wall in collision by one molecule = 2mv_x
  • But as there are many molecules, we have to calculate total momentum imparted to the wall by all of them
  • To calculate the number of molecules that hit the wall
    • Area of wall = A
    • Therefore in time Δt lying within the volume Av_xΔt can hit the wall
    • This means all molecules in time Δt lying within the volume Av_xΔt can hit the wall
    • But on an average half of molecules move towards the wall and half a way from the wall
    • Hence, ½ Av_xΔt will hit the wall
  • The number of molecule that hit the wall = n. So the average number of molecules that hit the wall = n * ½ Av_xΔt
  • Total molecules that hit the wall = ½ nAv_xΔt
  • Therefore the total momentum imparted to the wall = 2 mv_x * ½ Av_xΔt
  • After simplifying Total momentum imparted to the wall = Anv_x²Δt m
  • Force exerted on the wall = Rate of change of momentum = Anv_x² m
  • Pressure on the wall = P = F/A = nmv_x²
  • Therefore p = nmv_x² is true for group of molecules moving with velocity v_x

Note –

• All the molecules inside the gas will not have the same value of velocity
• All will have different velocities
• The above equation therefore, is valid for pressure due to the group of molecules with speed v_x in the x – direction and n stands for the number density of that group of molecules
• Therefore total pressure due to all such groups will be obtained by summing over the contribution due to all molecules P = nm vector(v_x²) where vector(v_x²) is the average of v_x²
• Since gas is isotropic the molecules move randomly which means the velocity of all the molecules can be in any direction
• Therefore vector(v_x²) = vector(v_y²) = vector(v_z²) = v ² Hence, average = 1/3 v ²
• Pressure P = (1/3) nm v² where v² is the average of squared speed