Kinetic Theory: Pressure and Temperature from Collisions

The Microscopic Picture

Imagine a box filled with a vast number of tiny, hard spheres bouncing around randomly. These spheres represent gas molecules. They move in straight lines until they hit a wall or another molecule, at which point they bounce elastically. There are no attractive forces between them, and their own volume is negligible compared to the box. This is the model of an ideal gas.

Pressure, in this picture, is not a mysterious fluid property but simply the result of countless tiny impacts. Each time a molecule hits a wall, it pushes on it. The sum of all these pushes, averaged over time and area, is the pressure. Temperature, conversely, is not a separate substance but a statistical property of the motion itself. Faster molecules mean higher temperature. This lecture connects these two intuitive ideas to the precise equation \( PV = Nk_B T \).

Pressure from Momentum Transfer

Consider a single molecule of mass \( m \) moving with velocity component \( v_x \) towards a wall perpendicular to the x-axis. When it collides elastically with the wall, its x-component of velocity reverses from \( v_x \) to \( -v_x \). The change in momentum of the molecule is \( \Delta p_x = -m v_x - m v_x = -2m v_x \). By Newton's third law, the wall receives an impulse of \( +2m v_x \).

To find the force, we need the rate of these collisions. If the molecule travels a distance \( 2L \) (where \( L \) is the length of the box in the x-direction) between successive hits on the same wall, the time interval is \( \Delta t = 2L / v_x \). The average force exerted by this single molecule is the impulse divided by the time interval: \( F_{1,x} = \frac{2m v_x}{2L/v_x} = \frac{m v_x^2}{L} \).

For \( N \) molecules, the total force on the wall is the sum of the forces from all molecules: \( F_{\text{total}} = \sum_{i=1}^N \frac{m v_{i,x}^2}{L} \). Pressure is force per unit area. The area of the wall is \( A = L_y L_z \). Since the volume \( V = L L_y L_z \), we have \( L = V / A \). Substituting this into the force expression and dividing by area \( A \):

We define the mean square velocity component as \( \langle v_x^2 \rangle = \frac{1}{N} \sum_{i=1}^N v_{i,x}^2 \). Thus, the pressure is:

Isotropy and the Mean Square Speed

The gas is in equilibrium and the container is cubic (or the motion is isotropic), meaning there is no preferred direction. Therefore, the average squared velocity components in the x, y, and z directions must be equal: \( \langle v_x^2 \rangle = \langle v_y^2 \rangle = \langle v_z^2 \rangle \).

The total mean square speed is the sum of these components: \( \langle v^2 \rangle = \langle v_x^2 \rangle + \langle v_y^2 \rangle + \langle v_z^2 \rangle = 3 \langle v_x^2 \rangle \). Consequently, \( \langle v_x^2 \rangle = \frac{1}{3} \langle v^2 \rangle \). Substituting this back into the pressure equation:

Rearranging gives the fundamental kinetic relation:

This equation connects macroscopic variables \( P \) and \( V \) to microscopic quantities \( N \), \( m \), and \( \langle v^2 \rangle \). It shows that pressure is proportional to the density of molecules and their average kinetic energy.

Connecting to Temperature

We now compare this result with the empirical Ideal Gas Law, \( PV = N k_B T \), where \( k_B \) is the Boltzmann constant and \( T \) is the absolute temperature. Equating the two expressions for \( PV \):

Cancelling \( N \) and rearranging for the kinetic energy term:

The term \( \frac{1}{2} m \langle v^2 \rangle \) is the average translational kinetic energy per molecule, denoted \( \langle K_{\text{trans}} \rangle \). Thus, we arrive at the crucial result:

This equation defines the physical meaning of absolute temperature: it is directly proportional to the average translational kinetic energy of the gas molecules. It is independent of the mass of the molecules, meaning that in a mixture of gases at the same temperature, all molecules have the same average kinetic energy, though their average speeds will differ.