Kinetic Theory of Gases — NEET Notes

Molecular nature of matter

The first thing kinetic theory asks you to believe is that matter is made of atoms. This idea is so familiar today that it is easy to forget it was a real intellectual achievement: Kanada in ancient India and Democritus in Greece speculated about indivisible particles thousands of years ago, but it was John Dalton, around 200 years ago, who proposed the modern atomic theory to explain the laws of definite and multiple proportions in chemistry. Avogadro's hypothesis — equal volumes of all gases at the same temperature and pressure contain the same number of molecules — combined with Dalton's atoms to explain Gay-Lussac's law of combining volumes. Even at the end of the nineteenth century, however, some famous scientists still did not believe atoms existed.

Today we know that an atom is about 1 Å (10⁻¹⁰ m) across. In solids, atoms sit roughly 2 Å apart, rigidly fixed; in liquids, the separation is similar but the atoms can slide past each other, which is why liquids flow. In gases, the inter-atomic distance is tens of angstroms, and the atoms travel thousands of angstroms — the mean free path — before colliding with another molecule. This is the structural fact that lets kinetic theory ignore inter-molecular forces for gases and treat the molecules as free particles obeying Newton's first law between collisions. The interatomic force itself has long-range attraction and short-range repulsion: atoms attract when a few angstroms apart and repel when squeezed closer.

The static appearance of a gas is misleading. A gas in equilibrium is in dynamic equilibrium — molecules constantly collide, exchange velocities, and re-distribute energy. Only the averages stay constant. This image — many particles, each obeying simple laws, with measurable quantities like pressure and temperature emerging from averages — is the conceptual core of statistical physics.

Behaviour of gases — Boyle, Charles, Avogadro

Gases are easier to describe than solids and liquids because intermolecular forces can be neglected. At low pressures and temperatures well above the liquefaction point, all gases approximately obey one compact relation that brings together a century of experimental work:

PV = nRT

The ideal gas equation — R = 8.314 J mol⁻¹ K⁻¹

Here P is pressure (Pa), V is volume (m³), n is the number of moles, T is the absolute temperature in kelvin, and R is the universal gas constant, 8.314 J mol⁻¹ K⁻¹. Two equivalent forms are useful: PV = Nk_BT (writing N for the number of molecules and k_B = R/N_A for the Boltzmann constant, 1.38 × 10⁻²³ J K⁻¹), and P = ρRT/M (writing ρ for the mass density and M for the molar mass). N_A = 6.022 × 10²³ is Avogadro's number — the number of molecules in 22.4 L of any gas at STP.

Three classical gas laws are special cases of this single equation. Boyle's law (1661) — at constant n and T, PV = constant — was the first quantitative relation discovered for gases. Charles' law — at constant n and P, V ∝ T — says volume grows linearly with absolute temperature. Avogadro's law — at fixed P and T, V ∝ n — follows from the same equation. Kinetic theory does something deeper than collect these laws: it derives all three from first principles, starting only with the assumption that gas molecules move freely between elastic collisions.

Ideal versus real gases

An ideal gas is one that obeys PV = nRT exactly at every pressure and temperature. No real gas is truly ideal. Real gases deviate because they have finite molecular volume and their molecules feel attractive forces at moderate distances — assumptions kinetic theory deliberately discards. NCERT illustrates this with PV/T-vs-P plots that should be horizontal flat lines for an ideal gas but bow downward and then upward for real gases. The deviations shrink as pressure decreases and temperature increases — exactly the regime where molecules are far apart and inter-molecular interactions are negligible.

Kinetic theory of an ideal gas — the assumptions

Kinetic theory builds the gas as a collection of an enormous number of molecules — typically of order N_A — that are in incessant random motion. The whole framework rests on a short list of simplifying assumptions, first laid down clearly by Maxwell in 1860 and used by NCERT and NIOS in the same form.

• A gas consists of a very large number of identical, rigid molecules moving randomly with all possible velocities.
• The molecules are point particles compared with the inter-molecular separation — the volume occupied by the molecules themselves is negligible compared with the volume of the container.
• Inter-molecular forces are negligible except during the brief instant of a collision. Between collisions, molecules travel in straight lines at constant velocity (Newton's first law).
• Collisions of molecules with each other and with the walls of the container are perfectly elastic — kinetic energy and momentum are conserved.
• The time spent in a collision is negligible compared with the time between successive collisions.
• The distribution of molecules is uniform throughout the container, and the motion is isotropic — no preferred direction.

From this list alone, kinetic theory will produce the ideal gas equation, the kinetic interpretation of temperature, the formulas for v_rms, the equipartition of energy, the specific-heat ratios of various gases, and an estimate of the mean free path. The whole edifice rests on Newton's laws applied with averaging.

Pressure of an ideal gas — derivation

Take a gas in a cubical container of side l. Choose the x-axis perpendicular to one wall of area A = l². A molecule of mass m moving with velocity (v_x, v_y, v_z) hits the wall. Because the collision is elastic, only v_x reverses sign — the velocity after collision is (−v_x, v_y, v_z). The change in momentum of the molecule is −2mv_x, so by conservation of momentum the wall receives an impulse of +2mv_x in each collision.

How many molecules hit the wall in a small time Δt? Only those within a distance v_xΔt of the wall can reach it, and of these only half are moving toward the wall (the other half move away). So the number of molecules of the chosen velocity hitting the wall in Δt is ½ (Av_xΔt)·n, where n is the number density (molecules per unit volume). The total momentum transferred is the impulse per molecule times this number — and the force on the wall is the momentum transfer rate, with pressure the force per unit area. Working through and averaging over the velocity distribution gives P = nm⟨v_x²⟩. The gas is isotropic, so ⟨v_x²⟩ = ⟨v_y²⟩ = ⟨v_z²⟩ = ⟨v²⟩/3. The final result is the classic kinetic-theory expression for pressure.

Multiply both sides by V — recalling that nV = N, the total number of molecules in the container — to get PV = ⅓ N m ⟨v²⟩ = (2/3) N · (½ m ⟨v²⟩). The bracketed quantity is the average translational kinetic energy per molecule. Defining the total translational kinetic energy as E gives the compact form PV = (2/3) E. This is the door to a molecular interpretation of temperature.

Kinetic interpretation of temperature

Compare PV = (2/3) E from kinetic theory with PV = Nk_BT from the ideal gas law. Equating the right-hand sides gives the central result of the chapter:

This identification is fundamental. Temperature, a macroscopic thermodynamic variable measured with a thermometer, equals (up to a constant factor of (2/3)k_B) the average translational kinetic energy of a single molecule — a purely molecular quantity. Two different gases in thermal equilibrium have the same kinetic energy per molecule, even if their molecules have very different masses. The Boltzmann constant k_B is the conversion factor between the two languages.

The total translational kinetic energy of one mole of an ideal gas is therefore E = (3/2) RT. For a monatomic gas this is the entire internal energy. For diatomic and polyatomic gases there are additional rotational and vibrational modes that store energy — that is what the law of equipartition handles below.

RMS, average, and most probable speeds

Three different "average" speeds appear in problems on the Maxwell-Boltzmann distribution, and NEET loves to test the difference between them. All three follow from the same distribution but capture different statistics.

The root-mean-square speed v_rms = √⟨v²⟩ is the one that comes directly from the energy formula ½ m⟨v²⟩ = (3/2)k_BT. Solving gives v_rms = √(3k_BT/m) = √(3RT/M), with M the molar mass. For nitrogen at 300 K this evaluates to about 516 m s⁻¹ — comparable to the speed of sound in air. Two consequences are exam-relevant: at the same T, lighter molecules move faster (v_rms ∝ 1/√M); for the same gas, v_rms ∝ √T.

The average speed v_avg is the arithmetic mean of |v| over the distribution, and works out to v_avg = √(8RT/πM). The most probable speed v_mp — the peak of the Maxwell-Boltzmann curve, the speed possessed by the largest fraction of molecules — is v_mp = √(2RT/M). The three differ only in the numerical factor under the square root.

Law of equipartition of energy

Look back at the result ⟨½ m v_x²⟩ = (1/2) k_BT. The energy per component of velocity — not per molecule — is (1/2) k_BT. Maxwell proved that this is a general principle: every independent quadratic term in the energy expression of a molecule has, at thermal equilibrium, an average energy of (1/2) k_BT. This is the law of equipartition of energy.

At absolute temperature T, every quadratic term in the molecule's energy averages (1/2)k_BT.

Law of equipartition — Maxwell & Boltzmann

A quadratic term means any term that involves the square of a coordinate or velocity component — translational kinetic energy (½ m v²), rotational kinetic energy (½ I ω²), or vibrational energy. Each translational degree of freedom contributes one quadratic term (the kinetic part) and so adds (1/2)k_BT to the average energy. Each rotational degree of freedom likewise contributes (1/2)k_BT. But each vibrational mode contributes two quadratic terms — one kinetic (½ m (dy/dt)²) and one potential (½ k y²) — so a vibrational mode adds 2 × (1/2)k_BT = k_BT to the average energy. NCERT lists this as Point to Ponder #3.

Degrees of freedom

The degrees of freedom of a molecule are the number of independent coordinates or motions needed to specify its configuration. A point free to move in space needs three coordinates, so it has three translational degrees of freedom. A rigid dumbbell (a diatomic molecule treated as two atoms joined by a rigid rod) needs three translational coordinates for the centre of mass plus two angles to describe its orientation — two rotational degrees of freedom (the third, rotation about the bond axis, has negligible moment of inertia and is excluded quantum-mechanically). If the bond is not rigid, a vibrational mode adds one more way of carrying energy. A non-linear polyatomic molecule like H₂O has three translational plus three rotational degrees of freedom — six total before any vibrational modes are excited.

Specific heat capacities from kinetic theory

Once you know f for a gas, the specific heat at constant volume follows immediately. The internal energy per mole is U = (f/2) RT, so the molar specific heat at constant volume is C_v = dU/dT = (f/2) R. For any ideal gas, the relation C_p − C_v = R holds (the extra R covers the work done in expanding against constant external pressure as the temperature is raised by one kelvin). So C_p = ((f + 2)/2) R, and the ratio γ = C_p/C_v = (f + 2)/f.

The numerical values for monatomic and rigid-diatomic gases predicted by this scheme — C_v ≈ 12.5 J mol⁻¹ K⁻¹, γ = 1.67 for monatomic; C_v ≈ 20.8 J mol⁻¹ K⁻¹, γ = 1.40 for rigid diatomic; C_v ≈ 24.9 J mol⁻¹ K⁻¹, γ = 1.33 for rigid triatomic — agree closely with measurements at ordinary temperatures, as NCERT Tables 12.1 and 12.2 confirm. Discrepancies for gases like Cl₂ and C₂H₆ are resolved by including vibrational modes — the experimental specific heats are larger than the rigid-rotator prediction because vibrational modes are partly excited at room temperature. The agreement provides striking experimental verification of equipartition.

An aside on solids: the same equipartition logic applied to a solid (where each atom oscillates in three dimensions, contributing 2 × (1/2)k_BT × 3 = 3k_BT to the average energy) gives the molar specific heat of a solid as C = 3R — the Dulong-Petit law, valid at ordinary temperature for most solids (carbon, with its strong bonds, is the famous exception).

Mean free path

Although gas molecules move at speeds of hundreds of metres per second, gas leaking from a cylinder takes minutes to spread across a room. The reason is that molecules collide repeatedly along the way — they cannot travel unhindered. The average distance a molecule covers between two successive collisions is called the mean free path λ.

To estimate λ, treat the molecules as spheres of diameter d. Focus on one molecule moving with average speed ⟨v⟩. It will collide with any other molecule whose centre comes within d. In a time Δt it sweeps a cylinder of volume π d² ⟨v⟩ Δt, and if n is the number density, the number of collisions in that time is n π d² ⟨v⟩ Δt. The rate of collisions is n π d² ⟨v⟩, so the average time between collisions is τ = 1/(n π d² ⟨v⟩) and the mean free path is λ = ⟨v⟩ τ = 1/(n π d²). This treatment assumes all other molecules are at rest; a more careful calculation using relative velocities introduces a factor of √2 in the denominator.

Three features of λ matter for problems. First, λ depends inversely on the number density n — at low pressure (highly evacuated vessels), λ can be as large as the container itself. Second, λ scales as 1/d² — bulkier molecules have shorter free paths. Third, because n = P/k_BT, λ can also be written as k_BT/(√2 π d² P), so at constant temperature λ is inversely proportional to pressure, and at constant pressure λ is directly proportional to temperature. The mean free path is what makes gases gases — a property that lies between the molecular size and the container size, large enough that molecules behave nearly independently between collisions.