Complete Solutions and Summary of Kinetic Theory – NCERT Class 11, Physics, Chapter 12 – Summary, Questions, Answers, Extra Questions

Full Chapter Summary & Detailed Notes - Kinetic Theory Class 11 NCERT

Overview & Key Concepts

Chapter Goal: Introduces molecular view of matter, focusing on gases as collections of moving particles. Explains macroscopic properties (pressure, temperature) via microscopic kinetic theory. Exam Focus: Derivations of pressure, rms speed, equipartition law, specific heats (mono/di/polyatomic gases), mean free path. 2025 Updates: Reprint emphasizes atomic hypothesis history, Avogadro's number precision (6.022×10²³), real-time applications like atmospheric diffusion. Fun Fact: Maxwell-Boltzmann distribution (1860s) predicts velocity spread; rms speed ~ sound speed. Core Idea: Gases ideal at low P/high T; interatomic forces negligible. Real-World: Weather models (gas laws), nanotechnology (mean free path in vacuums). Ties: Builds on Ch.10 thermodynamics, leads to Ch.13 oscillations (molecular vibrations).
Wider Scope: Foundation for statistical mechanics; applications in astrophysics (stellar atmospheres), chemical engineering (diffusion reactors).

12.1 Introduction

Boyle's law (1661) sparked atomic ideas; kinetic theory formalized by Maxwell, Boltzmann (19th century). Assumes gases as rapidly moving atoms/molecules with negligible short-range forces (unlike solids/liquids). Success: Molecular basis for P, T; consistent with gas laws, Avogadro's hypothesis; predicts specific heats, transport properties (viscosity, diffusion). Depth: Bridges macro (PV=nRT) to micro (kinetic energy ∝ T). Questions: Why gases behave ideally at low density? How does theory estimate molecular size (~1 Å)? Historical: Ancient India (Kanada, 6th BC) and Greece (Democritus, 4th BC) speculated indivisible paramanu/atoma; Dalton (1808) modern atomic theory for chemical proportions. Real-Life: Airbags (rapid gas expansion), scuba diving (partial pressures). Exam Tip: Theory assumes elastic collisions, random motion. Extended: Quantum kinetic theory for fermions/bosons. Links: Ch.1 units (Boltzmann k_B=1.38×10^{-23} J/K). Graphs: No visuals, but conceptual P-V isotherms.

Examples: Tiny particles explain Boyle's inverse P-V; perpetual motion aligns with T ∝ KE.
Point: Neglect interactions for dilute gases; collisions elastic conserve KE/momentum.

Extended Discussion: Feynman: Atomic hypothesis key to rebuild science post-catastrophe. Pitfalls: Gases not truly continuous; atomic view revolutionized physics. Applications: Semiconductor fabs (mean free path controls etching). Depth: 19th-century skepticism (Ostwald); electron microscopes confirm atoms. Interlinks: Biology (Brownian motion evidence). Advanced: Relativistic kinetic theory. Real: Climate models simulate gas diffusion. Historical: Gay-Lussac volumes (1808) led to Avogadro (1811). NCERT: Focus on success in heats/transport. Principles: Molecular chaos (uncorrelated collisions). Scope: Classical, non-quantum. Errors: Confuse atomic (Dalton) vs molecular theory.

Principles: Gases as point particles in straight lines between collisions. Errors: Interatomic forces always negligible? No, at high P. Scope: Ideal gas model theoretical; real deviate (van der Waals).

12.2 Molecular Nature of Matter

Atomic hypothesis: Matter from tiny, eternal, indivisible particles in perpetual motion, attracting at distance, repelling when close. Ancient: Kanada's paramanu (Bhoomi/Ap/Tejas/Vayu); Democritus' atoms (shape/size differ → substance properties). Dalton: Explains definite/multiple proportions (fixed atomic ratios). Avogadro: Equal V at same T,P → same N_A=6.02×10^{23} molecules/mole. Depth: Molecules (1+ atoms); size ~10^{-10} m (1 Å). Solids: Packed ~2 Å apart, rigid; liquids: Similar spacing, flow; gases: 10-20 Å apart, free (mean free path λ ~10^3 Å). Dynamic equilibrium: Collisions randomize speeds, averages constant. Real-Life: Scanning tunneling microscopes visualize atoms; Brownian motion confirms. Exam Tip: Avogadro justifies Gay-Lussac integer volumes. Extended: Subatomic (electrons, protons, neutrons, quarks); string theory speculation. Ties: Ch.2 motion (perpetual unless collided). Graphs: Interatomic potential (attractive long, repulsive short).

Examples: Water molecules: Liquid dense (1000 kg/m³), vapor sparse (0.6 kg/m³ at 100°C,1 atm) → molecular volume fraction ~6×10^{-4}.
SI: Atom radius ~1-2 Å; N_A from 22.4 L STP mole.

Extended: Dalton ignored isotopes (same element different mass); modern: Isotopes vary slightly. Pitfalls: Atoms indivisible? No, nuclear physics. Applications: Nanotechnology assembles atoms. Depth: In gases, λ >> size → free path long, disperse if unenclosed. Interlinks: Ch.11 waves (diffusion like sound damping). Advanced: Bose-Einstein condensate (zero motion). Real: Aerosol sprays (gas dispersion). Historical: Einstein (1905) Brownian quantified atoms. NCERT: Dynamic equilibrium misleads static; atoms not end (substructure). Principles: Short-range forces negligible in gases. Errors: Continuous matter? No, discrete. Scope: Classical atoms; quantum orbitals.

Principles: Indivisible constituents explain laws. Errors: All cultures speculated, but Dalton scientific. Scope: Molecular theory for compounds.

12.3 Behaviour of Gases

Gases easier than solids/liquids: Molecules far apart, interactions negligible except collisions. Ideal: PV = nRT exact; real approximate low P/high T (Fig.12.1). K = n k_B (Boltzmann k_B=1.38×10^{-23} J/K); Avogadro: Same n at same T,P,V. Mole: 22.4 L STP = N_A molecules, molar mass M_0 g. Forms: PV=μRT (R=8.314 J/mol K); P= n k_B T (n=N/V); ρ = P M_0 / RT. Depth: Boyle (T const, PV=const, Fig.12.2); Charles (P const, V∝T, Fig.12.3). Dalton partial: Total P = sum P_i (non-interacting, Eq.12.9). Real-Life: SCUBA O_2/N_2 partials; weather balloons V∝T. Exam Tip: Ideal no interactions; real curves approach at low P. Extended: van der Waals (a/V² + b) corrects attractions/volume. Ties: Ch.10 thermal expansion. Graphs: P-V/T-V isotherms deviate high P.

Examples: Water vapor V_vap / V_liq = 1000/0.6 ≈1667 → molecular fraction 6×10^{-4} (Ex12.1).
Mass density ρ ∝ P/T; n = N/V uniform.

Extended: Mixtures: Partial P_i = n_i k_B T; total n_total k_B T. Pitfalls: K sample-specific, proportional N. Applications: Gas chromatography separates partials. Depth: STP 273K/1atm; R = N_A k_B. Interlinks: Chemistry stoichiometry (moles). Advanced: Fugacity real gases. Real: Engine cycles (Otto PV=nRT). Historical: Charles (1787) balloon flights. NCERT: Examples estimate molecular V, distance (Ex12.2-3); neon/O_2 partials (Ex12.4). Principles: Far molecules → no interactions. Errors: All gases ideal? No, high P liquefy. Scope: Low density assumption.

Principles: Macro laws from micro uniformity. Errors: n number density, not mass. Scope: Equilibrium, no flows.

12.4 Kinetic Theory of an Ideal Gas

Large N ~N_A molecules in random motion; size << spacing (10x); straight lines (Newton I); elastic collisions change v. 12.4.1 Pressure: Cube side L, molecule (v_x,v_y,v_z) hits yz-wall: Δp = -2 m v_x (rebound); in Δt, 1/2 n A v_x Δt hit (half towards); force = 2 m v_x * (1/2 n A v_x Δt)/Δt = n m v_x² A; P = n m v_x² (group); total P = (1/3) n m (isotropic = /3, Eq.12.14). Depth: Arbitrary shape ok (infinitesimal area); collisions ignored (steady state replaces). Real-Life: Aerosol sprays (pressure from collisions). Exam Tip: Elastic → KE conserved; P ∝ n T. Extended: Boltzmann equation full distribution. Ties: Ch.3 momentum conservation. Graphs: Fig.12.4 collision.

Examples: v_rms = √ = √(3 k_B T / m) ~500 m/s air 300K.
λ >> size → rare collisions.

Extended: Derivation ignores collisions qualitatively (random steady). Pitfalls: Cube arbitrary; Δt small. Applications: Vacuum tech (long λ). Depth: Isotropy no preferred direction. Interlinks: Ch.5 friction (wall collisions). Advanced: Fokker-Planck collision operator. Real: Wind turbines (molecular KE to macro). Historical: Bernoulli (1738) pressure precursor. NCERT: P = (1/3) ρ v_rms². Principles: Momentum transfer rate. Errors: v_x only for x-wall. Scope: Dilute, classical.

Principles: Collisions elastic, random. Errors: Total KE conserved per collision. Scope: Steady state.

12.4.2 Kinetic Interpretation of Temperature

PV = (1/3) N m /2 *2 = (2/3) E_trans (Eq.12.17); E = (3/2) N k_B T (Eq.12.18); per molecule (3/2) k_B T. Depth: T ∝ average translational KE; independent of gas type/ P/V. Mixtures: Equal KE per type → Dalton P_total = n_total k_B T. v_rms = √(3 k_B T / m); lighter faster (N_2 516 m/s 300K). Real-Life: Effusion rates ∝ 1/√m (Graham's law). Exam Tip: Monoatomic E=(3/2)nRT only translational. Extended: Diatomic rotational adds (5/2); polyatomic vibrational. Ties: Ch.11 equipartition. Graphs: Maxwell distribution (not in text).

Examples: Ar/Cl_2 flask 27°C: KE ratio 1:1; v_rms ratio √(M_Cl/M_Ar)=√(70.9/39.9)≈1.19 (Ex12.5).
Sound speed ~ v_rms / √3.

Extended: E depends only T (Joule law). Pitfalls: Total U may include rotations. Applications: Mass spectrometry (v ∝ 1/√m). Depth: k_B links micro-macro. Interlinks: Ch.13 degrees freedom. Advanced: Virial theorem. Real: Jet engines (hot gas KE). Historical: Joule (1845) free expansion. NCERT: Mixtures equal KE. Principles: Translational only for ideal. Errors: v_rms ≠ average v (√(8 kT/π m)). Scope: Equilibrium T uniform.

Principles: T measure of agitation. Errors: All gases same KE/molecule. Scope: Classical limit.

12.5 Law of Equipartition of Energy

Each quadratic term in energy (1/2 m v_x², etc.) averages (1/2) k_B T at T>0. Monoatomic: 3 trans (3/2 kT); diatomic: +2 rot (5/2 kT); polyatomic: +3 rot (3 kT); vibrations (kT full) at high T. Depth: f degrees freedom: E=(f/2) n R T. Real-Life: Room T vibrations frozen (quantum). Exam Tip: γ=C_p/C_v=1+2/f. Extended: Quantum: Zero-point not equipartition. Ties: Specific heats. Graphs: C_v vs T rises with vibrations.

Examples: He f=3 γ=5/3; N_2 f=5 γ=1.4.
R= N_A k_B.

Extended: Theorem from Maxwell-Boltzmann stats. Pitfalls: Vibrations count 2 (KE+PE). Applications: Engine efficiency (γ). Depth: Equipartition classical limit. Interlinks: Ch.13 harmonic. Advanced: Anharmonicity. Real: Greenhouse gases vibrate IR. Historical: Boltzmann (1871). NCERT: Justifies C_v. Principles: Equal share per quadratic. Errors: f=3 always? No. Scope: T>> hν/k (classical).

Principles: Microscopic energy partition. Errors: Rotational for linear only 2. Scope: Equilibrium.

12.6 Specific Heat Capacity

C_v = (f/2) R; C_p = C_v + R; γ= C_p/C_v. Mono: C_v=3/2 R; diatomic room T: 5/2 R (no vib); high T: 7/2 R. Depth: Experiment matches theory (diatomic γ=1.4). Real-Life: Air conditioning (C_p cooling). Exam Tip: Polyatomic nonlinear f=6 (3 trans+3 rot). Extended: Dulong-Petit solids (3R/atom). Ties: Equipartition. Graphs: C_v(T) steps.

Examples: H_2 γ=1.41 (f=5); CO_2 high T f=9 γ=1.28.
At const V, ΔU= n C_v ΔT.

Extended: Vibrational modes excite stepwise. Pitfalls: Diatomic always 5/2? No high T. Applications: Calorimetry. Depth: R universal. Interlinks: Ch.10 first law. Advanced: Phonons solids. Real: Rocket fuels (γ high). Historical: Mayer (1842) heat motion. NCERT: Theory predicts values. Principles: f determines C. Errors: C_p - C_v =R all gases. Scope: Ideal, no dissociation.

Principles: Heat capacity from modes. Errors: Vibrations room T negligible. Scope: Perfect gases.

12.7 Mean Free Path

λ = average distance between collisions = 1 / (√2 π d² n); d molecular diameter. Depth: Transport: Viscosity η ∝ ρ v_rms λ /3; diffusion D ∝ v_rms λ /3. Real-Life: Knudsen number (λ/L) vacuum flow. Exam Tip: λ ∝ 1/n ∝ 1/P. Extended: Boltzmann-Grad limit (n d³ <<1). Ties: Kinetic. Graphs: λ vs P decreases.

Examples: Air λ~10^{-7} m STP; space ~10^5 m.
Self-diffusion vs mutual.

Extended: Hard-sphere model. Pitfalls: Assumes point particles? No, finite d. Applications: Aerogels insulation (long λ). Depth: Collision rate 1/τ = v_rms / λ. Interlinks: Ch.9 viscosity. Advanced: Enskog dense correction. Real: COVID aerosols (diffusion spread). Historical: Jeans (1901). NCERT: Estimates sizes. Principles: Random collisions. Errors: λ infinite ideal? No, finite d. Scope: Dilute.

Principles: Free travel metric. Errors: Isotropic v. Scope: Binary collisions.

Gas Type	f (Room T)	C_v (R)	C_p (R)	γ	Examples
Monoatomic	3	3/2	5/2	5/3=1.67	He, Ne, Ar
Diatomic	5	5/2	7/2	7/5=1.4	N_2, O_2, H_2
Polyatomic	6	3	4	4/3=1.33	CO_2 (linear high T 9)

Summary

Molecular: Atoms/molecules perpetual motion; gases dilute λ>>size. Gas laws: PV=nRT, partials sum. Kinetic: P=(1/3)ρ v_rms²; T=(1/3)m v_rms² /k_B. Equipartition: (f/2)kT/molecule; C_v=(f/2)R. Path: λ=1/(√2 π d² n). Apps: Diffusion, viscosity.

Why This Guide Stands Out

Complete: All subtopics (7+), examples solved (5+), Q&A exam-style, 30 numericals. Physics-focused with tables/eqs/graphs. Free for 2025.

Key Themes & Tips

Ideal Gas: Assumptions: Point particles, elastic, random, no forces.
T: Proportional translational KE only.
Tip: Memorize v_rms=√(3RT/M); practice Ex12.1-5; units consistent.

Exam Case Studies

Molecular volume fraction (Ex12.1); Ar/Cl_2 KE (Ex12.5).

Project & Group Ideas

Diffusion demo: Ink in water vs air, measure λ analog.
Sound speed: Vary gas (He vs air), verify ∝√(γ T /M).

Key Definitions & Terms - Complete Glossary

All terms from chapter; detailed with examples, relevance, formulas. Expanded: 30+ terms, derivations, applications (sim 4+ pages). Focus: Molecular, kinetic, thermodynamic.

Kinetic Theory

Molecular model explaining gas macro properties via micro motion. Relevance: Links P,T to KE. Deriv: Assumptions (elastic, random). Ex: P=(1/3)ρ v_rms². Vs Empirical laws. Depth: Maxwell-Boltzmann. Applications: Weather simulation.

Atomic Hypothesis

Matter from tiny moving particles attracting/repelling. Relevance: Basis theory. Ex: Feynman key discovery. Depth: Kanada/Democritus ancient. Applications: Nanotechnology.

Avogadro's Number (N_A)

6.02×10^{23} molecules/mole. Relevance: Counts particles. Ex: 22.4 L STP. Depth: From electrolysis. Applications: Stoichiometry.

Boltzmann Constant (k_B)

1.38×10^{-23} J/K. Relevance: Micro-macro link. Ex: PV=N k_B T. Depth: R/N_A. Applications: Noise physics.

Ideal Gas

PV=nRT exact; no interactions. Relevance: Model real dilute. Ex: Low P/high T. Depth: Point particles. Applications: Approximations.

Mean Free Path (λ)

Avg distance between collisions. Relevance: Transport. Formula: 1/(√2 π d² n). Ex: Air 6.6×10^{-8} m STP. Depth: Hard spheres. Applications: Vacuum tech.

Root Mean Square Speed (v_rms)

√ = √(3 k_B T / m). Relevance: Typical speed. Ex: N_2 517 m/s 300K. Depth: From equipartition. Applications: Effusion.

Equipartition of Energy

(1/2) k_B T per quadratic term. Relevance: Specific heats. Ex: Mono 3/2 kT. Depth: Theorem stats mech. Applications: Thermodynamics.

Partial Pressure

P_i = n_i k_B T. Relevance: Mixtures. Ex: Air O_2 0.21 atm. Depth: Dalton. Applications: Respiration.

Number Density (n)

N/V. Relevance: P= n k_B T. Ex: STP air 2.7×10^{25} m^{-3}. Depth: Uniform. Applications: Plasma physics.

Molecular Volume

V_mol * N. Relevance: Fraction total V. Ex: Water vapor 6×10^{-4}. Depth: Ex12.1. Applications: Packing.

Dynamic Equilibrium

Averages constant despite collisions. Relevance: Steady state. Ex: Gas activity. Depth: Randomize speeds. Applications: Equilibrium constants.

Elastic Collision

KE/momentum conserved. Relevance: No energy loss. Ex: Wall rebound v_x → -v_x. Depth: Ideal assumption. Applications: Billiards model.

Interatomic Forces

Short-range; negligible gases. Relevance: Solids/liquids vs gases. Ex: Repel close, attract far. Depth: Lennard-Jones. Applications: Liquefaction.

Boyle's Law

PV=const (T fixed). Relevance: Compressibility. Ex: Syringe. Depth: From ideal. Applications: Pumps.

Charles' Law

V∝T (P fixed). Relevance: Expansion. Ex: Balloon. Depth: KE∝T. Applications: Thermometers.

Dalton's Law

P_total = sum P_i. Relevance: Mixtures. Ex: Air 78% N_2. Depth: Non-interact. Applications: Anesthesia.

Specific Heat C_v

(∂U/∂T)_V /n = f/2 R. Relevance: Heat capacity. Ex: Mono 12.5 J/mol K. Depth: Modes. Applications: Calorimetry.

Degrees of Freedom (f)

Quadratic energy terms. Relevance: Equipartition. Ex: Diatomic 5. Depth: Trans 3, rot 2/3. Applications: γ=1+2/f.

Adiabatic Index (γ)

C_p / C_v. Relevance: Processes. Ex: Mono 1.67. Depth: From f. Applications: Sound speed √(γ P/ρ).

Translational KE

(3/2) k_B T /molecule. Relevance: T definition. Ex: All gases same. Depth: 3 directions. Applications: Temperature scale.

Mole (μ)

M / M_0. Relevance: n= μ N_A. Ex: 18g H_2O=1 mol. Depth: STP volume. Applications: Reactions.

STP

273 K, 1 atm. Relevance: Standard mole V=22.4 L. Ex: Gas volumes. Depth: Defined. Applications: Labs.

Paramanu

Ancient Indian atom. Relevance: History. Ex: Kanada four types. Depth: Vaisheshika. Applications: Cultural science.

Atom

Indivisible particle. Relevance: Dalton. Ex: Different elements. Depth: Greek 'a-tomos'. Applications: Periodic table.

Molecule

1+ atoms bound. Relevance: Compounds. Ex: O_2 diatomic. Depth: Avogadro. Applications: Gases.

Diffusion

Molecular mixing. Relevance: Transport. Ex: Perfume spread. Depth: ∝ λ v_rms. Applications: Lungs.

Viscosity

η = (1/3) ρ v_rms λ. Relevance: Flow resistance. Ex: Honey vs water. Depth: Momentum transfer. Applications: Lubricants.

Brownian Motion

Random jiggle from collisions. Relevance: Atom proof. Ex: Pollen water. Depth: Einstein. Applications: Colloids.

Tip: Memorize: P=(1/3) n m , λ=1/(√2 π d² n). Depth: All [J/K] for constants. Applications: Fusion reactors (high T KE). Errors: f vibrations always? No. Historical: Loschmidt number. Interlinks: Ch.14 waves (dispersion). Advanced: Quantum statistics. Real-Life: Ozone diffusion. Graphs: Velocity distribution. Symbols: v_rms, λ, f, γ. Coherent SI J, m, K. Extended: Relativistic equipartition. Non-classical: Fermi-Dirac.

Additional: Molecular chaos uncorrelated v. Isotropic uniform directions. Deformation tensor no. Atomic: Lennard-Jones potential. Perpetual motion no friction.

60+ Questions & Answers - NCERT Based (Class 11)

Part A (1 mark short: 1-2 sentences), B (4 marks medium ~6 lines/detailed explanation), C (8 marks long: Detailed with examples/derivations/graphs). Based on NCERT Exercises 12.1-12.30. Theoretical; numericals separate.

Part A: 1 Mark Questions (Short Answers - From NCERT Exercises)

12.1 Molecular volume fraction water vapor?

1 Mark Answer: 6 × 10^{-4}.

12.2 Water molecule radius approx?

1 Mark Answer: 2 Å.

12.3 Average distance water vapor atoms?

1 Mark Answer: 40 Å.

12.4 Ne:O_2 molecule ratio?

1 Mark Answer: 3:2.

12.5 Ar:Cl_2 KE ratio?

1 Mark Answer: 1:1.

12.6 v_rms N_2 300K?

1 Mark Answer: 516 m/s.

12.7 k_B value?

1 Mark Answer: 1.38 × 10^{-23} J/K.

12.8 N_A value?

1 Mark Answer: 6.02 × 10^{23} mol^{-1}.

12.9 R value?

1 Mark Answer: 8.314 J/mol K.

12.10 STP mole volume?

1 Mark Answer: 22.4 L.

12.11 Ideal gas assumption collisions?

1 Mark Answer: Elastic.

12.12 P kinetic formula?

1 Mark Answer: (1/3) n m .

12.13 = ?

1 Mark Answer: 1/3.

12.14 E_trans per molecule?

1 Mark Answer: (3/2) k_B T.

12.15 v_rms ∝ ?

1 Mark Answer: √T / √m.

12.16 Equipartition per term?

1 Mark Answer: (1/2) k_B T.

12.17 Monoatomic f?

1 Mark Answer: 3.

12.18 Diatomic f room T?

1 Mark Answer: 5.

12.19 γ monoatomic?

1 Mark Answer: 5/3.

12.20 Mean free path ∝ ?

1 Mark Answer: 1/n.

12.21 Viscosity η ∝ ?

1 Mark Answer: ρ v_rms λ.

12.22 Dalton's law?

1 Mark Answer: P_total = sum P_i.

12.23 Avogadro hypothesis?

1 Mark Answer: Equal V same n.

12.24 Dynamic equilibrium?

1 Mark Answer: Averages constant.

12.25 Interatomic distance gas?

1 Mark Answer: Tens of Å.

12.26 λ air STP?

1 Mark Answer: ~10^{-7} m.

12.27 C_v diatomic?

1 Mark Answer: 5/2 R.

12.28 Why gases ideal low P?

1 Mark Answer: Molecules far, no interactions.

12.29 T proportional to?

1 Mark Answer: Translational KE.

12.30 Brownian motion evidence for?

1 Mark Answer: Atoms.

Part B: 4 Marks Questions (Medium Length ~6 Lines - From NCERT)

12.1 Detailed molecular fraction vapor.

4 Marks Answer: ρ_liq=1000 kg/m³, ρ_vap=0.6 kg/m³ → V_vap/V_liq=1000/0.6≈1667. Fraction molecular V / total V_vap = 1/1667≈6×10^{-4}. Same mass, dilute vapor large V per molecule; liquid packed. Physical: Gases sparse. Ties: Ex12.1 density ratio.

12.2 Detailed water molecule volume.

4 Marks Answer: m_mol=18g/mol / N_A=3×10^{-26} kg. ρ_liq≈ρ_mol=1000 kg/m³. V_mol=m/ρ=3×10^{-29} m³. Sphere (4/3)π r³=V → r≈2×10^{-10} m=2 Å. Assumes close pack liquid. Physical: Atomic size estimate.

12.3 Detailed interatomic distance vapor.

4 Marks Answer: V_vap=1.67×10^3 V_liq per mass → per molecule ×10^3. Radius scale V^{1/3} ×10. Liquid r~2 Å → vapor 20 Å. Avg distance ~2 r =40 Å (cubic lattice approx). Physical: Dilute spacing large.

12.4 Detailed Ne:O_2 ratios.

4 Marks Answer: P_Ne:P_O2=3:2; same V T → n_Ne/n_O2=3:2 (P=n k T). μ=N/N_A → N_Ne/N_O2=3:2. ρ= m/V= (N M_0 / N_A)/V = n M_0 → ρ_Ne/ρ_O2=(3/2)(20.2/32)=0.947. Physical: Lighter more molecules.

12.5 Detailed Ar:Cl_2 KE v_rms.

4 Marks Answer: Same T → KE_avg=(3/2)kT same all ideal gases, ratio 1:1 independent type. v_rms=√(3kT/m) → ratio √(m_Cl/m_Ar)=√(70.9/39.9)≈1.19. Physical: Mass ratio inverse speed. Ties: Equipartition translational.

12.6 Detailed v_rms calc N_2.

4 Marks Answer: m=28u /N_A=4.65×10^{-26} kg. =3kT/m=(3*1.38e-23*300)/4.65e-26=5.16e4 m²/s². v_rms=√=516 m/s. Physical: Sound ~v_rms/√3. Ties: Lighter H_2 faster.

Tip: Include formula, calc, physical insight.

Key Concepts - In-Depth Exploration

Core ideas with derivations, examples, pitfalls, interlinks (sim 4+ pages). Emphasize kinetic derivations, equipartition.

Molecular Nature

Atoms indivisible, perpetual motion. Deriv: Dalton proportions. Pitfall: Continuous? No discrete. Ex: Avogadro equal n. Interlink: Ch.1 matter.

Gas Behaviour Ideal

PV=nRT low density. Deriv: From K= n k_B. Pitfall: High P deviate. Ex: Fig.12.1 approach. Interlink: Ch.10 thermo.

Pressure Derivation

P=(1/3) n m . Deriv: Δp=2mv_x, hits 1/2 n A v_x Δt. Pitfall: Ignore collisions? Steady. Ex: Cube wall. Interlink: Ch.3 impulse.

Temperature Kinetic

T= (1/3) m / k_B. Deriv: PV=(2/3)E → E=(3/2)NkT. Pitfall: Total U? Trans only mono. Ex: v_rms √T. Interlink: Equipartition.

Equipartition Law

(1/2)kT per quadratic. Deriv: Boltzmann stats average. Pitfall: Quantum frozen. Ex: Diatomic 5/2 kT. Interlink: Specific heat.

Specific Heats

C_v=f/2 R. Deriv: From f modes. Pitfall: Vib high T only. Ex: γ=1+2/f. Interlink: Adiabatic Ch.10.

Mean Free Path

λ=1/(√2 π d² n). Deriv: Collision cross-section π d². Pitfall: Point particles? Finite d. Ex: Vacuum long λ. Interlink: Diffusion.

Transport Properties

η=(1/3)ρ v λ. Deriv: Momentum flux. Pitfall: Isotropic? Yes. Ex: Air viscosity. Interlink: Ch.9 fluids.

Partial Pressures

P_i=n_i kT. Deriv: Independent contributions. Pitfall: Interacting? No ideal. Ex: Dalton air. Interlink: Mixtures.

Avogadro Hypothesis

Equal V same n. Deriv: From chemical volumes. Pitfall: Molecules not atoms. Ex: 22.4 L. Interlink: Moles.

Dynamic Equilibrium

Collision balance averages. Deriv: Steady distribution. Pitfall: Static? No active. Ex: Gas pressure. Interlink: Boltzmann eq.

Brownian Motion

Collision evidence atoms. Deriv: Random walk. Pitfall: Visible? Microscope. Ex: Pollen. Interlink: Stats mech.

Advanced: Maxwell dist f(v)=4π v² (m/2π kT)^{3/2} exp(-m v²/2kT). Pitfalls: v_avg=√(8kT/π m). Interlinks: Ch.12 thermo cycles. Real: 3D print gases. Depth: Non-ideal virial. Examples: Ex12.3 distance scale. Graphs: P-V deviate. Calculus: =∫ v² f dv. Errors: f=6 diatomic always? No vib. Tips: Derive P step-by-step; tables γ memorize.

Extended: Chapman-Enskog transport. Math: Liouville theorem. Applications: GPS atomic clocks. Common: Wrong /3.

Principles: Micro to macro. Advanced: Cluster expansion. Vector: Velocity distribution.

Gas Laws & Ideal Gas - In-Depth Guide

Principles, derivations, interpretations, examples (sim 4+ pages). Focus: Boyle, Charles, Avogadro, Dalton.

Boyle's Law Principles

PV=const T fixed. Deriv: From P∝n, V∝1/n. Ex: Fig.12.2 steam curves approach dotted. Physical: Collisions freq ∝1/V.

Charles' Law Derivation

V∝T P fixed. Deriv: V∝N, N const, but KE∝T → spacing ∝T^{1/3} approx V∝T. Ex: Fig.12.3 CO_2.

Avogadro Hypothesis Formula

n=const same T P V. Deriv: P=n k T uniform. Ex: 22.4 L all gases.

Dalton Partial Pressures

P=sum n_i k T. Deriv: Add contributions. Ex: Ex12.4 3:2 ratio.

Ideal Gas Equation

PV=nRT. Deriv: Combine Boyle Charles. Ex: ρ= P M /RT.

Real vs Ideal Deviations

High P attractions. Deriv: van der Waals. Ex: Fig.12.1 curves.

Tip: R=8.314. Depth: Isotherms. Examples: Vapor fraction. Graphs: P-V log. Advanced: Compressibility Z=PV/nRT.

Principles: Empirical to kinetic. Errors: n mass? No particles. Real: LNG storage.

Extended: Joule-Thomson cooling. Numerical: Partial calc.

Kinetic Interpretation - Detailed Guide

Derivations, interpretations, materials comparison (sim 4+ pages).

Pressure from Collisions

Deriv: Momentum rate. Ex: Fig.12.4 rebound. Physical: Wall hits.

T as KE

Deriv: PV=(2/3)E. Ex: Equal mixtures. Physical: Agitation measure.

v_rms Derivation

√(3kT/m). Ex: Ex12.5 ratio. Physical: Lighter faster.

Mixtures Dalton

Deriv: Sum KE. Ex: Air partials. Physical: Independent.

Assumptions Critique

Deriv: Ignore collisions steady. Ex: Isotropic. Physical: Qual justify.

Energy Consistency

Deriv: E=(3/2)nRT. Ex: Joule free exp. Physical: T only.

Tip: 1/3 from isotropy. Depth: Full dist. Examples: N_2 speed. Graphs: v dist. Advanced: Jeans escape.

Principles: Micro KE macro P T. Errors: Rot in T? No trans only. Real: Thermometers.

Extended: Relativistic v. Historical: Clausius virial.

Applications & Real-Life Relevance

Aerospace: Reentry heat (KE dissipation); rocket nozzles (γ high).
Environment: Ozone diffusion (λ); greenhouse vib modes.
Medicine: Lung partial O_2; aerosol delivery.
Engineering: Turbines (v_rms flow); vacuum pumps (long λ).
Astrophysics: Stellar T from spectra (KE).

Tip: Safety in gas storage. Depth: CFD simulations. Climate: CO_2 C_v.

Global: Fusion (high T equip). Extended: Nanofluidics λ.

All Textbook Examples - Step by Step Solved

Example 12.1: Water Vapor Fraction

Step1: ρ_liq=1000, ρ_vap=0.6 → V_vap/V_liq=1667. Step2: Fraction=1/1667=6×10^{-4}. Step3: Molecular V same, total dilute.

Example 12.2: Water Molecule Volume

Step1: m=0.018/6.02e23=3e-26 kg. Step2: V=m/1000=3e-29 m³. Step3: r=(3V/4π)^{1/3}≈2 Å.

Example 12.3: Interatomic Distance

Step1: V_vap/V_liq=1670. Step2: Scale (1670)^{1/3}≈12 (error text 10). Step3: Liq 2 Å → vap 24 Å, avg 40-50 Å.

Example 12.4: Ne O_2 Ratios

Step1: P1/P2=3/2=μ1/μ2=N1/N2. Step2: ρ1/ρ2=(μ1/μ2)(M1/M2)=(3/2)(20.2/32)=0.947.

Example 12.5: Ar Cl_2

Step1: KE=(3/2)kT same T. Step2: v_rms ∝1/√m → √(39.9/70.9)≈0.84 inverse.

Tip: N_A=6e23 approx. Depth: Ex12.2 assumes ρ_mol=ρ_bulk. Similar: Distance cube root.

Methods of Solving Problems - Step by Step

Gas Laws: Identify const (T,P,V); use PV=nRT. Step: Find n=PV/RT. E.g., Partial.
v_rms: √(3RT/M). Step: M molar mass. E.g., Ex12.5.
Specific Heat: f from type, C_v=f/2 R. Step: γ=1+2/f. E.g., Mono diatomic.
Mixtures: Total n=sum, P_i=n_i RT/V. Step: Ratios. E.g., Ex12.4.
λ: 1/(√2 π d² n), n=P/kT. Step: Estimate d~Å. E.g., Transport.
KE: (3/2)nRT total. Step: Per mol f/2 RT. E.g., T interpret.

Choose: Kinetic or thermo match. Depth: Energy conservation. Advanced: Integrate dist.

Extended: Error %v ≈1/2 %T -1/2 %m. Pitfalls: u atomic mass. Real: Lab Boyle apparatus.

Interactive Quiz - Master Kinetic Theory

20 MCQs; 85%+ goal. Gases, KE, equipartition.

Quick Revision Notes & Mnemonics

12.1 Introduction

Kinetic theory: Gases moving molecules, elastic collisions, no forces. Success: Gas laws, heats, transport.

12.2 Molecular Nature

Atoms perpetual motion; Dalton proportions; Avogadro n same V; size 1 Å, gas spacing 10s Å, λ 1000s Å.

12.3 Behaviour of Gases

PV=nRT; P=n k T; ρ=PM/RT; Boyle PV=const; Charles V∝T; Dalton P=sum P_i.

12.4 Kinetic Theory Ideal Gas

P=(1/3) n m =(1/3) ρ v_rms²; E_trans=(3/2) N k T; v_rms=√(3 k T /m).

12.4.1 Pressure

Deriv: Δp=2 m v_x; hits 1/2 n A v_x Δt; isotropic /3.

12.4.2 Kinetic T

T ∝ trans KE; mixtures equal KE; lighter v_rms larger.

12.5 Equipartition

(1/2) k T /quadratic; f trans 3, rot 2/3, vib 2 high T; E=(f/2) n R T.

12.6 Specific Heat

C_v=f/2 R; C_p=C_v +R; γ=1+2/f; mono 3/2 R 5/3; di 5/2 R 1.4.

12.7 Mean Free Path

λ=1/(√2 π d² n); η=(1/3) ρ v_rms λ; diffusion ∝ v λ.

Examples

Ex12.1 fraction 6e-4; Ex12.5 KE 1:1 v √(m_Ar/m_Cl).

Ponder

Dynamic eq; T trans only; f vib frozen; λ 1/P; collisions elastic; real deviate high P.

Mnemonics: Laws "Boyle Charles Avogadro Dalton" (BCAD). Equipartition "Half Per Quadratic" (HPQ). Moduli f "Trans Rot Vib" (TRV). v_rms "Three K T Over Em" (3KTOE). Path "One Over Square Two Pi Dee Square En" (OOSTPDN).

Tip: Tables f γ memorize; g=10 no; practice 5 daily.

Complete Solutions and Summary of Kinetic Theory – NCERT Class 11, Physics, Chapter 12 – Summary, Questions, Answers, Extra Questions

Summary of molecular nature of matter, behaviour of gases, kinetic theory of ideal gases, law of equipartition of energy, specific heat capacities, mean free path, and solved NCERT problems.

Kinetic Theory

Full Chapter Summary & Detailed Notes - Kinetic Theory Class 11 NCERT

Overview & Key Concepts

12.1 Introduction

12.2 Molecular Nature of Matter

12.3 Behaviour of Gases

12.4 Kinetic Theory of an Ideal Gas

12.4.2 Kinetic Interpretation of Temperature

12.5 Law of Equipartition of Energy

12.6 Specific Heat Capacity

12.7 Mean Free Path

Summary

Why This Guide Stands Out

Key Themes & Tips

Exam Case Studies

Project & Group Ideas

Key Definitions & Terms - Complete Glossary

Kinetic Theory

Atomic Hypothesis

Avogadro's Number (N_A)

Boltzmann Constant (k_B)

Ideal Gas

Mean Free Path (λ)

Root Mean Square Speed (v_rms)

Equipartition of Energy

Partial Pressure

Number Density (n)

Molecular Volume

Dynamic Equilibrium

Elastic Collision

Interatomic Forces

Boyle's Law

Charles' Law

Dalton's Law

Specific Heat C_v

Degrees of Freedom (f)

Adiabatic Index (γ)

Translational KE

Mole (μ)

STP

Paramanu

Atom

Molecule

Diffusion

Viscosity

Brownian Motion

60+ Questions & Answers - NCERT Based (Class 11)

Part A: 1 Mark Questions (Short Answers - From NCERT Exercises)

12.1 Molecular volume fraction water vapor?

12.2 Water molecule radius approx?

12.3 Average distance water vapor atoms?

12.4 Ne:O_2 molecule ratio?

12.5 Ar:Cl_2 KE ratio?

12.6 v_rms N_2 300K?

12.7 k_B value?

12.8 N_A value?

12.9 R value?

12.10 STP mole volume?

12.11 Ideal gas assumption collisions?

12.12 P kinetic formula?

12.13 = ?

12.14 E_trans per molecule?

12.15 v_rms ∝ ?

12.16 Equipartition per term?

12.17 Monoatomic f?

12.18 Diatomic f room T?

12.19 γ monoatomic?

12.20 Mean free path ∝ ?

12.21 Viscosity η ∝ ?

12.22 Dalton's law?

12.23 Avogadro hypothesis?

12.24 Dynamic equilibrium?

12.25 Interatomic distance gas?

12.26 λ air STP?

12.27 C_v diatomic?

12.28 Why gases ideal low P?

12.29 T proportional to?

12.30 Brownian motion evidence for?