Class 11 Physics · Chapter 12 NotesKinetic Theory

Matter is made of atoms and molecules. In solids, atoms are closely packed and vibrate about fixed positions; in liquids they are nearly as close but can move about, which lets a liquid flow. In gases the molecules are much farther apart and interact only during collisions. The ideal gas equation connects pressure P, volume V and absolute temperature T for µ moles of gas: PV = µRT = k_B N T, where N is the number of molecules, R = 8.314 J mol⁻¹ K⁻¹ and k_B = 1.38 × 10⁻²³ J K⁻¹. A gas that obeys this equation exactly at all pressures and temperatures is an ideal gas. Real gases follow it closely only at low pressures and high temperatures, where molecules are far apart and interactions are negligible. Avogadro's hypothesis states that equal volumes of all gases at the same temperature and pressure contain the same number of molecules; the Avogadro number is 6.02 × 10²³ per mole.

• PV = µRT = k_B N T
• R = 8.314 J mol⁻¹ K⁻¹, k_B = 1.38 × 10⁻²³ J K⁻¹
• Avogadro number N_A = 6.02 × 10²³
• Molar volume of an ideal gas at STP is 22.4 litres

For a fixed amount of gas at constant temperature, pressure varies inversely with volume: PV = constant. This is Boyle's law. At constant pressure, volume is proportional to absolute temperature, which is Charles' law. For a mixture of non-interacting ideal gases, the total pressure equals the sum of the partial pressures of the components, where the partial pressure of each gas is the pressure it would exert alone in the same volume at the same temperature. This is Dalton's law of partial pressures. These laws follow naturally from the ideal gas equation and are obeyed well by real gases at low pressures and high temperatures.

• Boyle's lawPV = constant at fixed T
• Charles' lawV ∝ T at fixed P
• Dalton's lawtotal pressure = sum of partial pressures

The kinetic theory assumes that a gas consists of a very large number of molecules in incessant random motion, that the molecules are much smaller than the distance between them, that intermolecular forces act only during collisions, and that all collisions are perfectly elastic. Consider a molecule of mass m with velocity component v_x hitting a wall of area A. On elastic collision its x-component reverses, so the momentum transferred to the wall is 2mv_x. In a short time Δt, only molecules within a distance v_x Δt of the wall can hit it, and on average half of them move towards the wall. Summing over all molecules and using the fact that the gas is isotropic gives P = (1/3) n m v̄², where n is the number density and v̄² is the mean of the squared speed. The shape of the container does not affect this result.

• P = (1/3) n m v̄²
• n = number of molecules per unit volume
• Collisions are assumed elastic; total kinetic energy and momentum are conserved

Writing the pressure expression in terms of the number of molecules N and comparing it with the ideal gas equation gives PV = (2/3) E, where E is the total translational kinetic energy of the molecules. This leads to E = (3/2) k_B N T, so the average translational kinetic energy of a molecule is (1/2) m v̄² = (3/2) k_B T. Thus the absolute temperature of a gas is a measure of the average kinetic energy of its molecules and is independent of the nature of the gas. The root mean square speed is v_rms = √(v̄²) = √(3k_B T/m) = √(3RT/M), where M is the molar mass. At a given temperature, lighter molecules have greater rms speed. For nitrogen at 300 K, v_rms is about 516 m s⁻¹, comparable to the speed of sound in air.

• Average kinetic energy per molecule = (3/2) k_B T
• v_rms = √(3k_B T/m) = √(3RT/M)
• Temperature depends only on average kinetic energy, not on the gas

The number of independent ways a molecule can move or store energy is called its degrees of freedom. A molecule free to move in space has three translational degrees of freedom, each contributing a term containing the square of a velocity component. A diatomic molecule can also rotate about two independent axes perpendicular to its bond, adding two rotational degrees of freedom. A vibrational mode contributes two squared terms, one kinetic and one potential. The law of equipartition of energy states that in thermal equilibrium at absolute temperature T, the total energy is distributed equally among all modes, each quadratic mode having an average energy of (1/2) k_B T. So each translational or rotational degree of freedom contributes (1/2) k_B T, while each vibrational mode contributes 2 × (1/2) k_B T = k_B T.

• Monatomic gas3 translational degrees of freedom
• Rigid diatomic gas3 translational + 2 rotational = 5 degrees of freedom
• Each vibrational mode contributes k_B T (two squared terms)

Using the law of equipartition, the internal energy of one mole of a monatomic gas is U = (3/2) RT, giving C_v = (3/2) R, C_p = (5/2) R and γ = 5/3. For a rigid diatomic gas, U = (5/2) RT, so C_v = (5/2) R, C_p = (7/2) R and γ = 7/5. If vibration is included, a diatomic gas has U = (7/2) RT, C_v = (7/2) R, C_p = (9/2) R and γ = 9/7. For a polyatomic gas with f vibrational modes, C_v = (3 + f) R and C_p = (4 + f) R. In all cases C_p − C_v = R. For a solid of N atoms vibrating in three dimensions, the total energy is 3RT per mole, so the molar specific heat is about 3R, which agrees with experiment at ordinary temperatures for many solids.

• MonatomicC_v = (3/2)R, C_p = (5/2)R, γ = 5/3
• Rigid diatomicC_v = (5/2)R, C_p = (7/2)R, γ = 7/5
• C_p − C_v = R for any ideal gas
• Molar specific heat of a solid ≈ 3R

Gas molecules move with high speeds but do not travel far in a straight line because they constantly collide with one another. The average distance a molecule travels between two successive collisions is called the mean free path, denoted by l. Treating molecules as spheres of diameter d and considering the relative motion of molecules, the mean free path is l = 1/(√2 n π d²), where n is the number density. The average time between collisions is τ = l/⟨v⟩. For air at STP, with n ≈ 2.7 × 10²⁵ m⁻³ and d ≈ 2 × 10⁻¹⁰ m, the mean free path is about 2.9 × 10⁻⁷ m, roughly 1500 times the molecular diameter. Because the mean free path is large, gases diffuse slowly and must be kept in closed containers.

• l = 1/(√2 n π d²)
• τ = l/⟨v⟩ is the average time between collisions
• Mean free path decreases as number density or molecular size increases