Kinetic Theory

2.1 Core concepts

Kinetic theory explains the macroscopic properties of gases — pressure, temperature, specific heat, viscosity, diffusion — by treating a gas as a collection of huge numbers of tiny atoms or molecules in rapid, random motion. The inter-atomic forces between gas molecules are negligible compared to those in solids and liquids; they are short-ranged and ignorable except during the brief instants of collision (NCERT §12.1, p. 244). The theory was developed in the nineteenth century by Maxwell, Boltzmann and others, but its roots run back to John Dalton's atomic hypothesis at the start of the 1800s.

Richard Feynman called the atomic hypothesis the single most important sentence in physics: "all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another" (NCERT §12.2, p. 244). Dalton's atomic theory explained the laws of definite and multiple proportions in chemistry; Gay-Lussac's law of combining gas volumes (in small whole-number ratios when gases react) and Avogadro's law (equal volumes of any gas at the same temperature and pressure contain the same number of molecules) made the molecular picture quantitative (NCERT §12.2, p. 245).

The scale of atoms is set by the Bohr radius: an atom is roughly 1 Å = 10⁻¹⁰ m across. In solids and liquids the inter-atomic spacing is about 2 Å — atoms are essentially touching. In gases at ordinary conditions the inter-atomic distance is tens of ångströms and the mean free path between collisions is thousands of ångströms (NCERT §12.2, p. 245). This is why a gas can be compressed easily, a liquid much less, and a solid hardly at all.

The ideal gas equation (NCERT §12.3, p. 246–247). At low pressures and temperatures well above their liquefaction point, real gases approximately satisfy a single combined law. For a fixed amount of gas, PV/T = constant. Including the number of molecules, the constant becomes Nk_B where k_B is Boltzmann's constant, the same for every gas — exactly what Avogadro's hypothesis predicts. Equivalently, working with moles, PV = μRT, where μ = N/N_A is the number of moles, N_A = 6.02 × 10²³ is the Avogadro number, and R = N_A k_B = 8.314 J mol⁻¹ K⁻¹ is the universal gas constant. Three other useful forms follow: PV = k_B NT, P = k_B nT (n = number density), and P = ρRT/M₀ (ρ = mass density, M₀ = molar mass). A gas that satisfies PV = μRT exactly at all P and T is an ideal gas; real gases deviate but approach ideal behaviour at low P and high T (NCERT Fig. 12.1, p. 246). Boyle's law (PV = const at fixed T), Charles' law (V ∝ T at fixed P) and Dalton's law of partial pressures (P_total = P₁ + P₂ + … for a non-reacting mixture of ideal gases) all fall out as special cases (NCERT §12.3, p. 247).

The molar volume of any gas at STP (0 °C, 1 atm) is 22.4 L, and the mass of 22.4 L equals the gas's molecular weight in grams — i.e., one mole. This single number underwrites almost every numerical problem in stoichiometry.

Kinetic-theory derivation of pressure (NCERT §12.4.1, p. 248–249). Consider a gas in a cube of side l. A molecule of mass m moving with x-velocity v_x bounces elastically off the wall normal to the x-axis; its x-momentum changes by 2mv_x. The number of molecules striking unit area of that wall per unit time is ½ n v_x (factor of ½ because only molecules with v_x > 0 move toward the wall). The total momentum delivered per unit area per unit time is the pressure:

P = n m v̄_x² (where v̄_x² is averaged over molecules).

By isotropy v̄_x² = v̄_y² = v̄_z² = (1/3) v̄², so

P = (1/3) n m v̄² (NCERT Eq. 12.14, p. 249).

This is the central result of kinetic theory. Two things are remarkable about it. First, it relates a macroscopic quantity (pressure) to a microscopic one (mean-square molecular speed). Second, multiplying through by V gives PV = (2/3) N × (½ m v̄²) = (2/3) E, where E is the total translational kinetic energy of all the molecules.

Kinetic interpretation of temperature (NCERT §12.4.2, p. 250). Comparing PV = (2/3) E with PV = N k_B T gives

(½ m v̄²) = (3/2) k_B T (NCERT Eq. 12.19, p. 250).

The average translational kinetic energy of a molecule depends only on the absolute temperature T — not on the nature of the gas, the pressure or the volume. This is the deep content of "temperature": it measures the random kinetic energy of microscopic motion.

From the same relation the root-mean-square (rms) speed of a molecule is

v_rms = √(v̄²) = √(3 k_B T/m) = √(3 RT/M₀) = √(3 P/ρ) (NCERT p. 250 & Summary p. 256).

For nitrogen (M₀ = 28 g/mol) at 300 K, v_rms ≈ 516 m s⁻¹, comparable to the speed of sound in air — which is no coincidence, since sound waves are pressure disturbances carried by molecular motion. At the same T, lighter molecules have larger v_rms; that is why hydrogen and helium escape Earth's upper atmosphere but heavier gases like nitrogen and oxygen are retained.

A direct consequence is the kinetic-theory derivation of Dalton's law (NCERT Eq. 12.21, p. 250). For a mixture of non-reacting gases at the same T, each species contributes a partial pressure n_i k_B T independent of the others, and the total pressure is the sum.

Degrees of freedom and equipartition (NCERT §12.5, p. 252–253). A point particle free in three-dimensional space needs three numbers (v_x, v_y, v_z) to specify its instantaneous state; we say it has 3 translational degrees of freedom (DOF). A particle constrained to a plane has 2 DOF; on a line, 1. Each translational DOF contributes a quadratic term ½ m v_i² to the kinetic energy.

A diatomic molecule (O₂, N₂) is like a dumb-bell; in addition to translating it can rotate about two axes perpendicular to its interatomic line, giving 2 rotational DOF. Rotation about the interatomic line itself is ignored — the moment of inertia is negligibly small. So a rigid diatomic has 3 trans + 2 rot = 5 DOF. If the bond can also vibrate, the vibration contributes both a kinetic term ½ m(dy/dt)² and a potential term ½ ky², i.e., two quadratic terms per vibrational mode.

The law of equipartition of energy (Maxwell) says that in thermal equilibrium at absolute temperature T, every quadratic term in the molecular energy expression carries an average energy of ½ k_B T. So each translational and rotational DOF contributes ½ k_B T; each vibrational mode contributes 2 × ½ k_B T = k_B T (NCERT §12.5, p. 253).

Specific heats (NCERT §12.6, p. 253–254).

• Monatomic gas (He, Ar, Ne): 3 trans DOF. Internal energy per mole U = (3/2) RT. So C_v = (3/2) R, C_p = (5/2) R, γ = C_p/C_v = 5/3 ≈ 1.67 (Table 12.1).
• Rigid diatomic gas (O₂, N₂ at moderate T): 3 trans + 2 rot = 5 DOF. U = (5/2) RT, so C_v = (5/2) R, C_p = (7/2) R, γ = 7/5 = 1.40.
• Non-rigid diatomic (one vibrational mode also excited): U = (7/2) RT, so C_v = (7/2) R, C_p = (9/2) R, γ = 9/7 ≈ 1.29 (Eq. 12.35).
• Polyatomic (3 trans + 3 rot + f vibrational modes): C_v = (3 + f) R, C_p = (4 + f) R, γ = (4 + f)/(3 + f) (Eq. 12.36). For any ideal gas the Mayer relation C_p − C_v = R holds. The experimental measurements in Table 12.2 agree well with these predictions for monatomic and rigid-diatomic gases at moderate temperatures. Solids (NCERT §12.6.4, p. 254). Each atom in a solid vibrates about a fixed equilibrium position in 3 dimensions; each dimension has 2 quadratic terms (KE + PE of harmonic motion), so each atom has 6 quadratic terms, contributing 3 k_B T to the energy. For one mole this gives U = 3 RT and the molar specific heat C = 3R ≈ 24.9 J mol⁻¹ K⁻¹ — the empirical Dulong–Petit law (Eq. 12.37). Carbon as diamond is a notable exception at room temperature because its vibration frequencies are so high that quantum freezing-out reduces the effective DOF. Mean free path (NCERT §12.7, p. 255). Treating molecules as hard spheres of diameter d, two molecules collide whenever their centres come within distance d. In time Δt a molecule of mean speed ⟨v⟩ sweeps out a cylinder of volume π d² ⟨v⟩ Δt. If we naively imagine all other molecules at rest, the average distance between collisions is l = 1/(n π d²). Accounting for the fact that all molecules are moving (so the relevant speed is the relative speed √2 ⟨v⟩), the correct mean free path is λ = 1/(√2 n π d²) (NCERT Eq. 12.40, p. 255). For air at STP n ≈ 2.7 × 10²⁵ m⁻³, d ≈ 2 × 10⁻¹⁰ m gives τ ≈ 6 × 10⁻¹⁰ s and λ ≈ 2.9 × 10⁻⁷ m ≈ 1500 d. That is why a gas, despite molecular speeds of hundreds of metres per second, diffuses across a room only in minutes — each molecule executes a random walk of step size ~λ. Because n ∝ P/T (ideal gas), the mean free path scales as λ ∝ T/P (NCERT Example 12.9, p. 255) — it grows when the gas is rarefied (low P) or heated (high T), and shrinks under compression.

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 12.1 (p. 246): PV/μT vs P plot — real gases approach the constant ideal-gas value at low P and high T; deviations grow at high P or near liquefaction.
• Fig. 12.2 (p. 247): Experimental P–V isotherms (steam) compared with Boyle's-law curves — verifies PV = const at high T.
• Fig. 12.3 (p. 247): V–T plot (CO₂) showing V ∝ T at fixed pressure (Charles' law).
• Fig. 12.4 (p. 249): Elastic collision of a gas molecule with a cube wall — the geometry that yields P = (1/3) n m v̄².
• Fig. 12.5 (p. 251): Gas molecules diffusing through a porous wall — basis of isotope enrichment (²³⁵U vs ²³⁸U via UF₆ diffusion).
• Fig. 12.6 (p. 252): The two independent rotational axes of a diatomic molecule, perpendicular to the inter-atomic axis.
• Fig. 12.7 (p. 255): Volume π d² ⟨v⟩ Δt swept by a molecule of diameter d — geometric basis for the mean-free-path estimate.
• Tables 12.1–12.2 (p. 254): Predicted vs measured C_v, C_p, C_p − C_v and γ for monatomic, diatomic and triatomic gases — direct memorisation table.
• Table 12.3 (p. 254): Specific heat capacity of solids ≈ 3R (Dulong–Petit), with carbon (diamond) as the quantum-frozen exception.

2.4 Common confusions / NTA trap points

• Pressure formula uses number density n (molecules per unit volume), not total N or moles; P = (1/3) n m v̄² and P = k_B nT (§12.3–12.4).
• v̄² uses absolute temperature T (kelvin), and m is the mass of a single molecule; v_rms = √(3 RT/M₀) uses molar mass M₀, while √(3 k_B T/m) uses single-molecule mass. NTA often swaps these and offers a wrong-by-N_A distractor.
• Average translational KE per molecule is (3/2) k_B T, but for a rigid diatomic molecule the total average energy is (5/2) k_B T. Confusing "translational KE" with "total internal energy" is a classic trap.
• A vibrational mode contributes two quadratic terms (KE + PE) ⇒ k_B T per vibration, not ½ k_B T (Point to Ponder 3, p. 257).
• For a rigid diatomic γ = 7/5 = 1.40; if vibration is included γ = 9/7 ≈ 1.29. Do not mix the two unless the question explicitly mentions vibration.
• ⟨v²⟩ ≠ ⟨v⟩² in general — the average of a squared quantity is not the square of the average (Point to Ponder 5, p. 257); mean speed and rms speed are different.
• Mean free path uses √2 in the denominator: λ = 1/(√2 n π d²). The naive (target-at-rest) estimate 1/(nπd²) is the wrong distractor.
• Because PV = k_B NT ⇒ n ∝ P/T, the mean free path λ ∝ T/P — increases with T at fixed P, decreases with P at fixed T.
• For helium (monatomic) C_v = (3/2) R ≈ 12.5 J mol⁻¹ K⁻¹; for any ideal gas C_p − C_v = R, independent of atomicity.
• Boyle's law is isothermal (T constant); Charles' law is isobaric (P constant). Mixing the two yields wrong answers.
• Dalton's law applies only to non-reacting gases.
• Carbon at room temperature does not obey Dulong–Petit because of quantum freezing of high-frequency modes.

2.5 Key formulas table