Kinetic Theory

Matter is made of very small particles called molecules, an idea that goes back to Kanada and Democritus and was placed on a firm footing by Dalton's atomic theory (1808) and Avogadro's law (1811). Avogadro's law says that equal volumes of all gases at the same temperature and pressure contain equal numbers of molecules, which lets us define a mole: one mole of any substance contains Avogadro's number N_A = 6.022 × 10²³ molecules and its mass equals the molecular mass in grams.

Robert Brown's discovery in 1827 of the continuous, zig-zag motion of pollen grains suspended in water, later called Brownian motion, gave the first direct visual evidence that molecules of the fluid are themselves in ceaseless random motion and keep colliding with the suspended particles. Einstein's theory of Brownian motion (1905), verified experimentally by Perrin, firmly established the molecular picture of matter and let scientists estimate the size of a molecule.

An exam favourite: the exact law of force between molecules is not known, but it is repulsive at very short distances (so molecules don't collapse into each other), weakly attractive at intermediate distances (this attraction is what condenses a gas into a liquid or solid), and negligible beyond a few molecular diameters — which is why a gas exerts pressure only through collisions with the walls of its container.

At low pressures and high temperatures, well away from their liquefaction point, all gases obey two simple empirical laws. Boyle's law: at constant temperature, PV = constant for a fixed mass of gas. Charles' law: at constant pressure, V/T = constant, i.e. V ∝ T. Combining these with Avogadro's law (equal volumes of gas at the same P and T contain equal numbers of moles) gives the ideal gas equation, PV = μRT, where μ is the number of moles and R is the universal gas constant, the same for every gas: R = 8.314 J mol⁻¹ K⁻¹.

If the gas has N molecules, μ = N/N_A, so the equation can also be written PV = Nk_BT, where k_B = R/N_A = 1.38 × 10⁻²³ J K⁻¹ is Boltzmann's constant. A gas that obeys PV = μRT exactly at all pressures and temperatures is called an ideal or perfect gas; real gases approach this behaviour only in the limit of low density.

On a P–V diagram, curves of constant T (isotherms of an ideal gas) are rectangular hyperbolas, since PV is constant along each one. A very common graph-based question gives PV/T or PV/μT on the y-axis against P on the x-axis for a real gas and asks the student to identify that this ratio stays constant (equal to R) only for an ideal gas, and deviates from R as pressure rises for a real gas.

Kinetic theory explains the macroscopic behaviour of gases (pressure, temperature) in terms of the microscopic motion of their molecules, on these assumptions:

• A gas consists of a very large number of identical molecules in random motion, obeying Newton's laws.
• The size of a molecule is negligible compared to the average distance between molecules (a typical gas is mostly empty space).
• Molecules exert no force on each other, or on the walls, except during a collision.
• Collisions between molecules, and between a molecule and the wall, are perfectly elastic, and the time of a collision is negligible compared to the time between collisions.
• Between collisions, molecules move in straight lines with uniform velocity (they obey Newton's first law).
• The number density of molecules is uniform throughout the gas (no preferred position or direction).

These assumptions describe an ideal gas. Real gases depart from this picture at high pressure (molecular size is no longer negligible) and low temperature (intermolecular attraction becomes significant, eventually causing liquefaction) — this is exactly why real gases deviate from PV = μRT under those conditions.

This is the single most important derivation in the chapter and is asked almost every year as a 5-mark long answer. Consider a cube of side L containing N molecules of an ideal gas, each of mass m. Take one molecule moving with velocity components (v_x, v_y, v_z) and follow its collisions with the wall perpendicular to the x-axis.

On hitting this wall elastically, the molecule's x-component of velocity reverses (v_x → −v_x) while v_y and v_z are unchanged. So the change in its momentum is 2mv_x, all delivered to the wall. The molecule travels a distance 2L between two successive collisions with this same wall, so it strikes the wall once every 2L/v_x seconds. The average force this one molecule exerts on the wall is therefore (2mv_x)/(2L/v_x) = mv_x²/L.

Adding the contributions of all N molecules, the total force on the wall is (m/L)Σv_x² = (m/L)·N⟨v_x²⟩, where ⟨v_x²⟩ is the mean of v_x² over all molecules. Pressure is force divided by area L², so P = Nm⟨v_x²⟩/L³ = Nm⟨v_x²⟩/V.

Because molecular motion is random, there is no preferred direction, so ⟨v_x²⟩ = ⟨v_y²⟩ = ⟨v_z²⟩ = (1/3)⟨v²⟩, where ⟨v²⟩ = ⟨v_x²⟩ + ⟨v_y²⟩ + ⟨v_z²⟩ is the mean square speed. Substituting, P = (1/3)(Nm/V)⟨v²⟩ = (1/3)ρ⟨v²⟩, since Nm/V is the density ρ of the gas. This can also be written PV = (1/3)Nm⟨v²⟩.

Comparing PV = (1/3)Nm⟨v²⟩ with the ideal gas equation PV = Nk_BT gives (1/3)m⟨v²⟩ = k_BT, so the average translational kinetic energy per molecule is (1/2)m⟨v²⟩ = (3/2)k_BT. This is the kinetic interpretation of temperature: the absolute temperature of a gas is a direct measure of the average translational kinetic energy of its molecules, and it does not depend on the mass of the molecule or on the pressure/volume, only on T. It is also why at absolute zero, molecular motion (and hence pressure) should vanish for an ideal gas.

For a mole of gas, total translational kinetic energy is (3/2)RT, and this leads directly to the root-mean-square speed, v_rms = √⟨v²⟩ = √(3k_BT/m) = √(3RT/M), where M is the molar mass. A frequently tested fact: at a given temperature, lighter molecules have a higher rms speed than heavier ones (v_rms ∝ 1/√M), which is why hydrogen escapes Earth's atmosphere more easily than oxygen or nitrogen.

Two related conclusions worth memorising: (i) at the same temperature, different gases have the same average kinetic energy per molecule, but different rms speeds because their masses differ; (ii) for a mixture of gases in thermal equilibrium, each component has the same mean kinetic energy (3/2)k_BT, and Dalton's law of partial pressures follows directly from adding the pressure each species would exert independently.

A molecule's energy can be split among independent 'degrees of freedom' — the independent ways it can store energy, such as translation along x, y, z, or rotation about an axis, or vibration. The law of equipartition of energy states that in thermal equilibrium, the total energy is distributed equally among all the degrees of freedom, and the energy associated with each degree of freedom (whether kinetic from translation/rotation, or the kinetic-plus-potential pair from vibration) equals (1/2)k_BT per molecule.

A monatomic gas molecule (like He or Ar) has only 3 translational degrees of freedom (f = 3), so its average energy is 3 × (1/2)k_BT = (3/2)k_BT, matching the kinetic-theory result exactly. A diatomic molecule (like O₂ or N₂) has 3 translational plus 2 rotational degrees of freedom (rotation about the bond axis is not counted, since the moment of inertia there is negligible), so f = 5 if treated as rigid, giving average energy (5/2)k_BT. At high temperature, vibration along the bond also becomes active; a vibrational mode contributes 2 degrees of freedom (one kinetic, one potential), so f rises to 7 and average energy to (7/2)k_BT.

A polyatomic, non-linear molecule (like H₂O or NH₃) has 3 translational + 3 rotational degrees of freedom, plus 2 for every vibrational mode present; ignoring vibration, f = 6 and average energy is 3k_BT. This law is the bridge between the microscopic picture and the specific heats worked out in the next section.

Using the law of equipartition, the internal energy of μ moles of gas with f degrees of freedom per molecule is U = μ(f/2)RT. Since C_V = (1/μ)(dU/dT) at constant volume, C_V = (f/2)R, and using Mayer's relation C_P − C_V = R (which follows from PV = μRT), C_P = (f/2 + 1)R. The ratio γ = C_P/C_V then follows for each type of gas.

For a monatomic gas, f = 3: C_V = (3/2)R, C_P = (5/2)R, γ = 5/3 ≈ 1.67. For a rigid diatomic gas, f = 5: C_V = (5/2)R, C_P = (7/2)R, γ = 7/5 = 1.4. For a diatomic gas with vibration active, f = 7: C_V = (7/2)R, C_P = (9/2)R, γ = 9/7. For a non-linear polyatomic gas, f = 6: C_V = 3R, C_P = 4R, γ = 4/3.

The same reasoning extends to solids. Each atom in a solid vibrates about its mean position and has 3 kinetic plus 3 potential degrees of freedom (f = 6), so its average energy is 3k_BT, and for one mole, U = 3RT, giving a molar specific heat C_V = 3R ≈ 25 J mol⁻¹K⁻¹, independent of the solid — this is the law of Dulong and Petit, verified for most solids at ordinary temperature (it fails at low temperature, where quantum effects take over). For water, treating each of the 3 atoms in the H₂O molecule as an independent vibrating unit gives 3 × 6 = 18 degrees of freedom per molecule, so C_V = 9R ≈ 75 J mol⁻¹K⁻¹, which matches the observed molar specific heat of water closely.

A gas molecule does not travel far before it collides with another molecule; the average distance it covers between two successive collisions is called the mean free path, l. To estimate it, model each molecule as a sphere of diameter d. A molecule moving with average speed v sweeps out, in time Δt, a cylindrical tube of cross-section πd² and length vΔt; it collides with any molecule whose centre lies inside this tube. If n is the number density of molecules, the number of collisions in time Δt is n(πd²vΔt), so the mean free path (distance travelled divided by number of collisions) works out to l = 1/(nπd²), before accounting for the fact that the other molecules are also moving.

Correcting for the relative motion of all the molecules (not just one moving through stationary targets) brings in a factor of √2, giving the standard result l = 1/(√2 nπd²). Since PV = Nk_BT gives n = N/V = P/k_BT, the mean free path can also be written l = k_BT/(√2 πd²P), showing that l increases with temperature and decreases with pressure and with molecular size — compressing a gas or cooling it (which lowers the volume it occupies at fixed N) packs molecules closer and shortens the mean free path.

Mean free path explains diffusion: a gas molecule does not travel to a distant point in a straight line, it takes a random, zig-zag path made of many short free flights, which is why the smell of a gas released in one corner of a room takes noticeable time to reach the other corner even though molecular speeds are hundreds of m/s.