Q.(a) Derive the expression of pressure exerted by the gas molecules on the walls of the container. OR

Considering molecular collisions with a container wall and summing the momentum transferred gives P = (1/3) rho c-bar^2 for an ideal gas.

This question offers a choice between (a) deriving the pressure exerted by gas molecules on the container walls, and (b) deriving Newton's formula for the speed of sound with Laplace's correction; part (a) is answered here.

Consider N molecules, each of mass m, of an ideal gas enclosed in a cubical container of side L (volume V = L^3). Let the gas molecules move randomly with different velocities.

Step 1, momentum change on collision: consider one molecule moving with velocity component c(x) along the x-direction, striking the wall perpendicular to the x-axis. Since the collision is elastic, it rebounds with velocity -c(x). The change in momentum of the molecule is

Delta p = m c(x) - (-m c(x)) = 2 m c(x)

Step 2, time between successive collisions with the same wall: the molecule must travel a distance 2L (to the opposite wall and back) between successive collisions with this wall, so the time interval is

Delta t = 2L / c(x)

Step 3, force from one molecule: the rate of change of momentum (= force exerted by the wall on the molecule, and by Newton's third law, the force exerted by the molecule on the wall) is

f = Delta p / Delta t = 2 m c(x) / (2L/c(x)) = m c(x)^2 / L

Step 4, total force from all N molecules: summing over all N molecules (each with its own x-component of velocity) and using the average value of c(x)^2:

F = (m/L) N x average(c(x)^2)

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