Kinetic Theory

🎯 Key Points

• P = (1/3)ρv_rms²; average KE per molecule = (3/2)k_BT - temperature IS a direct measure of average translational KE
• Degrees of freedom: monoatomic f=3, diatomic f=5 (moderate T), polyatomic f=6 (+2 per vibration mode)
• Equipartition: each degree of freedom contributes ½k_BT; total energy/molecule = (f/2)k_BT
• Cv=(f/2)R, Cp=Cv+R, γ=Cp/Cv - γ DECREASES as atomicity/degrees of freedom increases (monoatomic γ=5/3 highest)
• Speed order: v_p < v_avg < v_rms (most probable < average < root-mean-square), all ∝ √(T/M)
• Mean free path λ ∝ 1/(n·d²) - increases as pressure/density drops, decreases as molecular size increases

Postulates of Kinetic Theory of Gases

• A gas consists of a very large number of identical molecules in random motion, separated by distances much larger than their own size
• Molecules exert negligible force on each other except during collisions (ideal gas assumption)
• Collisions between molecules, and with the walls of the container, are perfectly elastic, and of negligible duration
• Between collisions, molecules move in straight lines with constant velocity (Newtons laws apply)
• The molecules obey Maxwell-Boltzmann distribution of speeds; the density and distribution of molecules is assumed uniform

Pressure of an Ideal Gas from Kinetic Theory

• Kinetic theory derivation gives: P = (1/3)·ρ·v_rms², where ρ is density of gas and v_rms is the root mean square speed
• Equivalently: PV = (1/3)Nm·v_rms², where N is number of molecules and m is mass of each molecule
• This can be written as PV = (2/3)E, where E is the total translational kinetic energy of all the gas molecules
• This kinetic theory result combined with the ideal gas equation PV = nRT leads directly to the interpretation of temperature

Kinetic Interpretation of Temperature

• Average kinetic energy per molecule = (3/2)k_B·T, where k_B is the Boltzmann constant (k_B = 1.38 × 10⁻²³ J/K)
• This shows absolute temperature is a direct measure of the average translational kinetic energy of the molecules of a gas
• At absolute zero (T = 0 K), translational kinetic energy of molecules becomes zero (classically), which is why 0 K is the lowest possible temperature
• RMS speed: v_rms = √(3k_BT/m) = √(3RT/M), where M is the molar mass

Degrees of Freedom

• Degrees of freedom = number of independent ways a molecule can possess energy (independent coordinates needed to specify its position/configuration)
• Monatomic gas (e.g. He, Ar): 3 translational degrees of freedom only, f = 3
• Diatomic gas (e.g. O₂, N₂) at moderate temperature: 3 translational + 2 rotational, f = 5 (vibrational modes ignored at moderate T)
• Polyatomic (nonlinear) gas: 3 translational + 3 rotational + vibrational modes, f = 6 plus 2 per vibrational mode

Law of Equipartition of Energy

• In thermal equilibrium, energy is distributed equally among all the degrees of freedom, and each quadratic degree of freedom contributes (1/2)k_BT of average energy per molecule
• Total average energy per molecule = (f/2)k_BT, where f is the total number of degrees of freedom
• This is the basis used to compute the molar specific heats of gases from their molecular structure

Specific Heat Capacities of Gases

• Monatomic gas (f=3): C_v = (3/2)R, C_p = (5/2)R, so γ = C_p/C_v = 5/3 ≈ 1.67
• Diatomic gas (f=5, no vibration): C_v = (5/2)R, C_p = (7/2)R, so γ = 7/5 = 1.4
• Diatomic gas with vibration (f=7): C_v = (7/2)R, C_p = (9/2)R, γ = 9/7 ≈ 1.29
• Triatomic nonlinear/polyatomic gas (f=6, no vibration): C_v = 3R, C_p = 4R, γ = 4/3 ≈ 1.33
• γ decreases as the number of degrees of freedom (atomicity) increases

Mean Free Path

• Mean free path (λ) is the average distance a molecule travels between two successive collisions
• λ = 1/(√2·n·π·d²), where n is number density of molecules and d is molecular diameter
• Mean free path increases if density decreases (lower pressure) and decreases as molecular size/density increases
• This concept explains the basis of diffusion, viscosity, and thermal conductivity of gases

Maxwell Speed Distribution

• Molecules in a gas do not all move at the same speed, the Maxwell-Boltzmann distribution gives the fraction of molecules with speeds in a given range
• Root mean square speed: v_rms = √(3RT/M) (largest of the three characteristic speeds)
• Average speed: v_avg = √(8RT/πM) (mean of the speed distribution)
• Most probable speed: v_p = √(2RT/M) (speed at the peak of the distribution curve)
• Relation: v_p< v_avg< v_rms, in the approximate ratio √2 : √(8/π) : √3

Maxwell−Boltzmann speed distribution: at a given temperature, lighter molecules have a broader distribution and a higher most-probable speed than heavier ones. Image: Kadykianus, CC BY-SA 4.0, via Wikimedia Commons.

Molecular Nature of Matter

• Matter is made of atoms and molecules; the atomic hypothesis (Dalton) explains the laws of chemical combination and the macroscopic behaviour of gases
• Gay-Lussac's law of combining volumes together with Avogadro's hypothesis (equal volumes of all gases at the same temperature and pressure contain equal numbers of molecules) established the molecular picture
• Intermolecular forces are weakly attractive at large separations and strongly repulsive at short range; the equilibrium separation gives matter its stability and its resistance to compression
• Molecules are tightly bound in solids (fixed positions), loosely bound and mobile in liquids, and nearly free in gases - kinetic theory applies most cleanly to gases where interactions are negligible except during collisions

Behaviour of Gases: Gas Laws and the Ideal Gas Equation

• Boyle's law: at constant temperature, PV = constant, so P ∝ 1/V
• Charles's law: at constant pressure, V/T = constant, so V ∝ T (T in kelvin)
• Gay-Lussac's (pressure) law: at constant volume, P/T = constant
• Avogadro's law: at the same P and T, equal volumes of all gases contain equal numbers of molecules
• Combining these gives the ideal gas equation: PV = nRT = Nk_BT, where n is the number of moles, N the number of molecules, R = 8.314 J/(mol·K) the universal gas constant, and k_B = R/N_A the Boltzmann constant
• A real gas behaves like an ideal gas at low pressure and high temperature, where molecular separations are large and intermolecular forces negligible

Avogadro's Number and the Mole

• One mole of any substance contains Avogadro's number N_A = 6.022 × 10²³ particles
• One mole of any ideal gas occupies 22.4 litres at STP (0°C and 1 atm)
• The Boltzmann constant k_B = R/N_A = 1.38 × 10⁻²³ J/K links per-mole quantities to per-molecule quantities
• In terms of number density n = N/V, the ideal gas law per molecule reads P = n·k_B·T

Specific Heat Capacity of Solids and Water

• In a solid each atom has 3 vibrational modes, each carrying both kinetic and potential energy, giving 6 quadratic degrees of freedom; equipartition then gives internal energy U = 3RT per mole and molar specific heat C = 3R ≈ 25 J/(mol·K)
• This is the Dulong-Petit law, obeyed well by most solids at ordinary temperatures (it fails at low temperature, where quantum effects reduce C below 3R)
• For water, treating each of its 3 atoms as contributing 3R gives molar specific heat ≈ 9R ≈ 75 J/(mol·K), in good agreement with experiment
• These predictions from the law of equipartition of energy are strong evidence for the kinetic-molecular model of matter

🚀 JEE Advanced Edge

Mixture of gases - effective γ and molar mass: For a mixture of n₁ moles of gas 1 and n₂ moles of gas 2, the effective Cv is the mole-weighted average: Cv(mix) = (n₁Cv₁+n₂Cv₂)/(n₁+n₂), and similarly for Cp - letting you find the mixture's effective γ even when the two gases have different atomicities (e.g., a He + O₂ mixture).

Vibrational degrees of freedom at high temperature: At very high temperatures, diatomic molecules gain 2 additional vibrational degrees of freedom (f=7 instead of 5), lowering γ from 7/5 toward 9/7 - a subtle point examiners use to test whether students just memorise f=5 for ALL diatomic gases regardless of temperature.

Worked problem: Find the ratio of RMS speeds of hydrogen (M=2) and oxygen (M=32) molecules at the same temperature. Approach: v_rms ∝ 1/√M (same T, R cancel), so v_rms(H₂)/v_rms(O₂) = √(M_O₂/M_H₂) = √(32/2) = √16 = 4 - hydrogen molecules move 4× faster on average at the same temperature.

Ideal Gas Assumptions and Real Gas Deviations

• An ideal gas obeys PV = nRT exactly at all pressures and temperatures.
• Real gases obey it closely only at low pressure and high temperature, when molecules are far apart.
• Deviations arise because real molecules have finite size and exert intermolecular attractive forces.
• At high pressure the finite molecular volume matters; near condensation the attractions matter.
• The van der Waals equation corrects the ideal gas law for these two effects.

Types of Molecular Speeds

• The rms speed is v_rms = square root of (3RT/M), the square root of the mean of the squared speeds.
• The average (mean) speed is v_avg = square root of (8RT/(pi M)).
• The most probable speed is v_mp = square root of (2RT/M), the peak of the Maxwell distribution.
• Their ratio is v_mp : v_avg : v_rms = 1 : 1.13 : 1.22, so v_rms is greater than v_avg is greater than v_mp.
• All three speeds are proportional to the square root of temperature and inversely to the square root of molar mass.

Dalton's Law of Partial Pressures

• In a mixture of non-reacting ideal gases, each gas exerts the pressure it would if it alone occupied the container.
• The total pressure equals the sum of the partial pressures: P = P₁ plus P₂ plus P₃ and so on.
• This follows from kinetic theory because molecules of different gases contribute independently to the wall pressure.
• For a mixture, P = (n₁ plus n₂ plus ...) R T / V, treating all molecules together.
• The partial pressure of a component is proportional to its mole fraction in the mixture.

Brownian Motion and Diffusion

• Brownian motion is the ceaseless random zig-zag motion of tiny particles suspended in a fluid.
• It is caused by unbalanced collisions of fast-moving fluid molecules with the suspended particle.
• The motion is more vigorous for smaller particles, lower fluid viscosity and higher temperature.
• It gives direct visual evidence for the molecular nature of matter and molecular motion.
• Diffusion, the gradual mixing of gases, is likewise explained by molecular motion and is slowed by a short mean free path.