Kinetic Theory of Gases
We want to express the pressure on the wall in terms of the force of impact of molecules hitting the walls of the container. Let’s begin with a single molecule in a cubic container with side L.
The diagram below represents a molecule moving in the xy plane and bouncing off a wall in the yz plane. In a perfectly elastic collision the x component of the velocity would be reversed, that is, go from vx to -vx. Thus every molecule that strikes the wall has a change in momentum of 2mvx perpendicular to the wall. The component mvy parallel to the wall does not change.
The force applied to the wall is given by the change in momentum per unit time. The molecule will travel the length of the cube, collide with the opposite wall and return to the first wall, so the time between impacts with the selected wall is given by twice the length of a side of the cube divided by the velocity in the x direction (2L/vx). The force exerted on the wall by the single molecule is then
where m is the mass of the molecule, vx is its velocity in the x direction and L is the length of a cube side. The pressure, or force per unit area, becomes mvx2/L3. In three dimensions a molecule will have components in the x, y and z directions. The total velocity is the sum of these three components.
The motion in all directions is equally probable so vx2 = vy2 = vz2. Thus v2 = 1/3vx2. The pressure exerted by the one molecule is then given by
where L3 is the volume per molecule. In terms of moles of gas we have
where V is volume and M is the molecular weight. We need to express the velocity as the average velocity when dealing with many molecules.
Now we know that the direction of motion in the gas is equally probable in all directions. If it were not, the body of gas as a whole would move. We know that the component of velocity in the x direction will vary over a wide range (depending on the temperature). So we want the average velocity in the x direction. The velocity in the +x direction is as probable as in the -x direction. Thus the average velocity in the x direction is zero. If we average the square of the velocity in the x direction we have a non-zero value. So we use the square-root of the average square of the velocity. This is called the root-mean-square.