Chapter 12: Kinetic Theory | Summary

Gases at low pressures and high temperatures satisfy the ideal gas equation PV = µRT, where µ is the number of moles and R = 8.314 J mol⁻¹ K⁻¹ is the universal gas constant. This can also be written as PV = kB NT, where kB = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant and N is the number of molecules.

Boyle's law (PV = constant at fixed T), Charles' law (V ∝ T at fixed P), and Dalton's law of partial pressures (total pressure equals the sum of partial pressures of constituent gases) all follow directly from the ideal gas equation. Real gases deviate from ideal behaviour at high pressures and low temperatures, as molecular interactions become significant.

Kinetic theory models a gas as a large number of molecules in incessant random motion. The molecules collide elastically with each other and with the walls of the container, conserving total kinetic energy and momentum. The pressure on a wall is derived as P = (1/3) nm v̄², where n is the number density, m is the molecular mass, and v̄² is the mean squared speed.

Combining this with the ideal gas equation yields the kinetic interpretation of temperature: the average kinetic energy per molecule (½ m v̄²) equals (3/2) kB T. Temperature is therefore a measure of the average translational kinetic energy of molecules, independent of the nature of the gas. The root mean square speed is vrms = √(3kBT/m), showing that lighter molecules move faster at the same temperature.

The law of equipartition of energy states that in thermal equilibrium at temperature T, the total energy is equally distributed among all possible energy modes, with each mode contributing an average energy of ½ kBT. Each translational and rotational degree of freedom contributes ½ kBT, while each vibrational mode contributes kBT because it has both kinetic and potential energy components.

A monatomic gas molecule has 3 translational degrees of freedom. A diatomic molecule treated as a rigid rotator has 3 translational and 2 rotational degrees of freedom (total 5). If vibration is included, each vibrational mode adds 2 quadratic terms, contributing an additional kBT to the energy. Polyatomic molecules have 3 translational and 3 rotational degrees of freedom plus f vibrational modes.

Using the law of equipartition of energy, the molar specific heats of gases can be derived. For monatomic gases: Cv = (3/2)R, Cp = (5/2)R, and γ = 5/3. For diatomic gases (rigid): Cv = (5/2)R, Cp = (7/2)R, and γ = 7/5. For diatomic gases with vibration: Cv = (7/2)R, Cp = (9/2)R, and γ = 9/7. For polyatomic gases with f vibrational modes: Cv = (3 + f)R and Cp = (4 + f)R.

The relation Cp − Cv = R holds for all ideal gases regardless of their atomicity. The predicted values agree well with experimental values for several gases. For solids, each atom vibrating in three dimensions contributes energy 3kBT per atom, giving a molar specific heat C ≈ 3R (Dulong-Petit law), which agrees with experiments at ordinary temperatures except for carbon.

The mean free path l is the average distance a molecule travels between two successive collisions. For molecules of diameter d and number density n, the exact expression accounting for the relative motion of all molecules is l = 1/(√2 π n d²). The time between collisions is τ = 1/(nπ⟨v⟩d²).

For air molecules at STP, with d = 2 × 10⁻¹⁰ m and n = 2.7 × 10²⁵ m⁻³, the mean free path is approximately 2.9 × 10⁻⁷ m (about 1500 molecular diameters) and the collision time τ ≈ 6.1 × 10⁻¹⁰ s. The mean free path depends inversely on number density and molecular size, explaining why gas molecules cannot be confined without a container and why diffusion is slow despite high molecular speeds.

Kinetic theory provides a unifying microscopic framework that connects the observable thermodynamic properties of gases—pressure, temperature, and specific heat—to molecular-level quantities such as speed, mass, and degrees of freedom. Its most profound result is that temperature is a direct measure of the average translational kinetic energy of molecules, an insight that bridges the macroscopic and microscopic worlds through the Boltzmann constant.

From a board examination perspective, this chapter is highly important as it involves both conceptual understanding and numerical problem-solving. Key areas include derivation of pressure from kinetic theory, kinetic interpretation of temperature, application of the equipartition law to calculate Cv, Cp, and γ for different gases, and calculation of mean free path. Students should be comfortable with the expressions vrms = √(3kBT/m), l = 1/(√2 πnd²), and the specific heat formulae for mono-, di-, and polyatomic gases.