NCERT Solutions for Class 11 Physics Chapter 13 – Kinetic Theory
Chapter 13 of Class 11 Physics, Kinetic Theory, provides a microscopic explanation of the macroscopic behaviour of gases, bridging the gap between atomic theory and classical thermodynamics. The chapter begins with the ideal gas equation (PV = nRT) and the molecular interpretation of pressure — arising from countless random collisions of gas molecules with container walls. Students derive the kinetic expression for pressure (P = ⅓ρ‹v²›) and connect it to temperature, showing that the average kinetic energy of a molecule is directly proportional to absolute temperature. This leads to expressions for RMS speed, mean speed, and most probable speed of gas molecules — all important for JEE and NEET. The chapter introduces degrees of freedom — the number of independent ways a molecule can absorb energy — and the profound Law of Equipartition of Energy, which states that each degree of freedom contributes ½kT to average energy. This explains why different gases have different molar specific heats. The mean free path concept quantifies how far a molecule travels between collisions, connecting microscopic kinetics to macroscopic transport phenomena like viscosity and conductivity. Kinetic Theory is one of the most conceptually rich and frequently tested chapters in Class 11 Physics.
NCERT Solutions PDF – Class 11 Physics Chapter 13 (All Exercises)
The PDF provides complete step-by-step solutions to all NCERT exercises on ideal gas laws, molecular speeds, equipartition, specific heats, and mean free path. Aligned with CBSE curriculum.
Important Formulas – Chapter 13: Kinetic Theory
| Formula | Expression | Description |
|---|---|---|
| Ideal Gas Equation | PV = nRT = NkT | n = moles; R = 8.314 J/mol·K; k = 1.38×10⁻²³ J/K; N = molecules |
| Kinetic Pressure | P = (1/3)ρ‹v²› = (1/3)(mN/V)‹v²› | Derived from molecular collisions with walls |
| RMS Speed | v_rms = √(3RT/M) = √(3kT/m) | M = molar mass; m = molecular mass |
| Mean Speed | v_mean = √(8RT/πM) | Average speed of molecules |
| Most Probable Speed | v_p = √(2RT/M) | Speed corresponding to peak of Maxwell distribution |
| Speed Ratio | v_p : v_mean : v_rms = 1 : 1.128 : 1.225 | √2 : √(8/π) : √3 |
| Average KE per molecule | KE = (3/2)kT | Directly proportional to absolute temperature |
| Equipartition of Energy | Energy per degree of freedom = (1/2)kT | Each quadratic term in energy contributes ½kT |
| Molar specific heat (Cv) | Cv = (f/2)R | f = degrees of freedom; monoatomic f=3, diatomic f=5 |
| Mean Free Path | λ = 1 / (√2 · n · πd²) | n = number density; d = molecular diameter |
| Avogadro's Number | N_A = 6.022 × 10²³ mol⁻¹ | Number of molecules per mole |
Subtopics Explained – Chapter 13: Kinetic Theory
Ideal Gas and Molecular Model
An ideal gas consists of point-mass molecules with no intermolecular forces, undergoing perfectly elastic random collisions. Real gases approach ideal behaviour at high temperatures and low pressures. The ideal gas law PV = nRT combines Boyle's, Charles's, and Avogadro's laws into one equation.
Pressure from Kinetic Theory
Gas pressure is the cumulative effect of billions of molecular collisions per second on container walls. The derivation shows P = (1/3)nmv², where n is number density and v is molecular speed. Connecting this to the ideal gas equation yields the fundamental result: average KE = (3/2)kT.
Molecular Speeds (Maxwell Distribution)
Molecules in a gas don't all move at the same speed. The Maxwell-Boltzmann distribution gives three characteristic speeds. v_rms is the root-mean-square speed (used in KE). v_mean is the arithmetic average. v_p (most probable) is the peak of the distribution. All three increase with temperature as √T.
Degrees of Freedom and Equipartition
Degrees of freedom (f) represent independent modes of energy absorption. Monoatomic gases (like He, Ar) have 3 translational DOF. Diatomic gases (H₂, O₂) have 5 (3 translational + 2 rotational). The Equipartition Theorem assigns ½kT per DOF, explaining why Cv = (3/2)R for monoatomic and (5/2)R for diatomic gases.
Mean Free Path
Mean free path (λ) is the average distance a molecule travels between successive collisions. It is inversely proportional to the number density and the square of molecular diameter. At standard conditions, λ for air ≈ 68 nm, much larger than molecular size (~0.3 nm). λ is important in transport phenomena — viscosity, conductivity, and diffusion all depend on it.
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|---|---|---|
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Quick Reference Table – Degrees of Freedom and Specific Heats
| Gas Type | Examples | Degrees of Freedom (f) | Cv | Cp | γ = Cp/Cv |
|---|---|---|---|---|---|
| Monoatomic | He, Ne, Ar | 3 | (3/2)R | (5/2)R | 5/3 ≈ 1.67 |
| Diatomic (rigid) | H₂, O₂, N₂ | 5 | (5/2)R | (7/2)R | 7/5 = 1.4 |
| Diatomic (with vibration) | H₂ at high T | 7 | (7/2)R | (9/2)R | 9/7 ≈ 1.29 |
| Polyatomic (non-linear) | H₂O, NH₃ | 6 | 3R | 4R | 4/3 ≈ 1.33 |