Kinetic Theory of Gases – JEE Main Physics Formula Sheet & Class 11 Notes | Ideal Gas Law, KE per Molecule, RMS Speed, Degrees of Freedom, Cp Cv Gamma & All Formulas

Ideal Gas Laws — PV=nRT, Boyle, Charles, Gay-Lussac, Dalton

Why the Ideal Gas Equation and Its Special Cases Are the Foundation of All Kinetic Theory JEE Main Questions

Ideal Gas Equation (from Aakash PDF — Kinetic Theory of Gases):

An ideal gas is one in which: (1) molecules have negligible volume compared to container volume; (2) intermolecular forces are zero (no attraction or repulsion); (3) all collisions are perfectly elastic; (4) molecules are in random motion.

PV = nRT

where P = pressure (Pa), V = volume (m³), n = number of moles, R = universal gas constant = 8.314 J/mol·K, T = absolute temperature (K). T(K) = T(°C) + 273.

In terms of total number of molecules N: PV = NkT where k = Boltzmann constant = R/Nₐ = 1.38×10⁻²³ J/K. Nₐ = 6.022×10²³ /mol (Avogadro's number).

Number density (molecules per unit volume): n_density = N/V = P/kT

Special Gas Laws (from Aakash PDF — Kinetic Theory of Gases JEE Main):

Boyle's Law (constant T, n): P₁V₁ = P₂V₂ (isothermal process)

Charles's Law (constant P, n): V₁/T₁ = V₂/T₂ (isobaric process)

Gay-Lussac's Law (constant V, n): P₁/T₁ = P₂/T₂ (isochoric process)

Avogadro's Law (constant P, T): V₁/n₁ = V₂/n₂ (equal volumes → equal molecules)

At STP (Standard Temperature and Pressure): T₀ = 273 K = 0°C, P₀ = 10⁵ Pa (or 1 atm). Molar volume at STP = 22.4 L/mol.

Dalton's Law of Partial Pressures (from Aakash PDF — Kinetic Theory of Gases JEE Main):

For a mixture of non-reacting gases in a container:

P_total = P₁ + P₂ + P₃ + … = ΣPᵢ

where Pᵢ = partial pressure of gas i = nᵢRT/V. Total pressure = sum of partial pressures each gas would exert if alone.

Mole fraction: xᵢ = nᵢ/n_total; Pᵢ = xᵢ × P_total.

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Kinetic Theory — Pressure, KE per Molecule, Temperature-KE Relationship

Why Kinetic Pressure P=⅓ρv² and Average KE=(3/2)kT Are the Core Kinetic Theory JEE Main Derivation Results

Kinetic Theory Assumptions and Pressure Derivation (from Aakash PDF — Kinetic Theory of Gases JEE Main):

Consider N molecules of gas in a cubical container of side L (volume V=L³). Each molecule of mass m moves randomly. Consider one molecule with velocity component vₓ toward the x-wall:

Momentum change per collision with wall = 2mvₓ. Time between successive collisions = 2L/vₓ. Force by one molecule = Δp/Δt = 2mvₓ/(2L/vₓ) = mvₓ²/L.

Total force on wall by N molecules: F = Σ(mvᵢₓ²/L) = mNv²ₓ_avg/L where v²ₓ_avg = average of vₓ².

By isotropy: v²ₓ_avg = v²_avg/3 (since v²=vₓ²+vy²+vz² and symmetry).

Pressure P = F/A = F/L² = mNv²_avg/(3L³) = mNv²_avg/(3V) = (Nm/V)(v²_avg/3) = ρv²_rms/3

P = ⅓ρv²_rms = ⅓(Nm/V)v²_rms

Also written: PV = ⅓Nmv²_rms = ⅓Nm⟨v²⟩

Kinetic Energy and Temperature (from Aakash PDF — Kinetic Theory of Gases JEE Main):

From PV = NkT and PV = ⅓Nmv²_rms = (2/3)N×(½mv²_rms):

NkT = (2/3)N×(½m⟨v²⟩) = (2/3)×(total KE)

→ Total translational KE = (3/2)NkT = (3/2)nRT

Average KE per molecule = ½m⟨v²⟩ = (3/2)kT

Key results from Aakash PDF:

(1) Average KE per molecule = (3/2)kT (depends ONLY on temperature)

(2) KE is independent of pressure, volume, and the nature of the gas

(3) At the same temperature, all ideal gas molecules (regardless of mass) have the SAME average translational KE = (3/2)kT

(4) Temperature is a measure of average kinetic energy: T ∝ KE per molecule

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Molecular Speeds — RMS, Mean, Most Probable — All Formulas and Ratios

Why v_rms, v_mean, v_p Formulas and Their Ratio Are the Most-Tested Kinetic Theory JEE Main Results

Three Molecular Speeds (from Aakash PDF — Kinetic Theory of Gases JEE Main):

1. Root Mean Square (RMS) Speed:

v_rms = √(⟨v²⟩) = √(mean of squares of speeds)

v_rms = √(3RT/M) = √(3kT/m) = √(3P/ρ)

where M = molar mass (kg/mol), m = mass per molecule (kg), ρ = density of gas.

Derivation: from (3/2)kT = ½mv²_rms → v²_rms = 3kT/m = 3RT/M.

2. Mean (Average) Speed:

v_mean = ⟨v⟩ = average of speeds (not average of v²)

v_mean = √(8RT/πM) = √(8kT/πm)

Derivation: from Maxwell-Boltzmann speed distribution integral ∫₀^∞ v × f(v) dv.

3. Most Probable Speed:

v_p = speed at which the Maxwell-Boltzmann distribution f(v) is maximum

v_p = √(2RT/M) = √(2kT/m)

Derivation: from df(v)/dv = 0.

Ratio of Three Speeds (from Aakash PDF — Kinetic Theory of Gases JEE Main):

v_p : v_mean : v_rms = √2 : √(8/π) : √3

= √2 : √(8/π) : √3

= 1 : √(4/π) : √(3/2)

= 1 : 1.128 : 1.225

v_rms > v_mean > v_p (always)

Exact relations: v_rms = √(3/2) × v_p; v_mean = √(4/π) × v_p = (2/√π)×v_p

v_rms/v_mean = √(3π/8); v_mean/v_p = √(4/π)×(1/√2) = 2/√(2π) = √(2/π).

Effect of Temperature on Speeds (from Aakash PDF — Kinetic Theory of Gases JEE Main):

All three speeds ∝ √T. If T doubles: all speeds increase by factor √2.

Effect of Molar Mass: All three speeds ∝ 1/√M. Lighter molecules move faster. H₂ molecules are 4× faster than O₂ (√(32/2)=4). Download the Free PDF for Kinetic Theory of Gases for all speed formula examples for JEE Main.

Degrees of Freedom and Equipartition Theorem

Why Degrees of Freedom and Equipartition Are the Conceptual Core of Kinetic Theory JEE Main Questions

Degrees of Freedom (from Aakash PDF — Kinetic Theory of Gases JEE Main):

Degrees of freedom (f) = number of independent ways a molecule can store kinetic energy = number of independent coordinates needed to specify the complete configuration of the molecule.

Monatomic gas (He, Ar, Ne — single atom): 3 translational degrees of freedom (motion along x, y, z).

f = 3 for monatomic gas.

Diatomic gas (H₂, N₂, O₂, CO — two atoms):

At room temperature: 3 translational + 2 rotational (about two axes perpendicular to the bond axis; rotation about bond axis negligible for point atoms) = f = 5

At high temperature: 3 translational + 2 rotational + 2 vibrational (KE + PE) = f = 7

Triatomic linear molecule (CO₂, CS₂ — three atoms in a line):

3 translational + 2 rotational + 2×2 vibrational modes = but at moderate T: f = 7 (translational + rotational only). At very high T: f = more.

Triatomic non-linear molecule (H₂O, SO₂ — bent shape):

3 translational + 3 rotational = f = 6 (at room temperature)

Summary (from Aakash PDF — Kinetic Theory of Gases JEE Main):

Monatomic: f=3. Diatomic (room T): f=5. Diatomic (high T): f=7. Linear triatomic: f=7 (approx). Non-linear triatomic: f=6.

Equipartition Theorem (from Aakash PDF — Kinetic Theory of Gases JEE Main):

In thermal equilibrium, the average energy associated with EACH degree of freedom is (1/2)kT per molecule (or (1/2)RT per mole).

Energy per degree of freedom per molecule = (1/2)kT

Total average energy per molecule: E = (f/2)kT

Total internal energy of n moles: U = (f/2)nRT = (f/2)NkT

For monatomic (f=3): U = (3/2)nRT ✓ (matches translational KE result)

For diatomic (f=5): U = (5/2)nRT (includes rotational energy)

Download the Free PDF for Kinetic Theory of Gases for all degrees of freedom examples for JEE Main.

Specific Heats Cv, Cp, Gamma — Complete Table for All Gas Types

Why Cv=(f/2)R, Cp=Cv+R, and γ=Cp/Cv=1+2/f Are the Most Directly Tested Kinetic Theory JEE Main Formulas

Molar Heat Capacities from Equipartition (from Aakash PDF — Kinetic Theory of Gases JEE Main):

For n moles of gas with f degrees of freedom: U = (f/2)nRT

Change in internal energy: dU = (f/2)nR dT

At constant volume (no work done by gas): dU = nCv dT

→ Cv = (f/2)R

Mayer's Relation: Cp – Cv = R (for any ideal gas)

Derivation: At constant pressure, dQ = dU + PdV = nCvdT + nRdT = n(Cv+R)dT → Cp = Cv+R.

→ Cp = (f/2 + 1)R = ((f+2)/2)R

Ratio γ = Cp/Cv = ((f+2)/2)R / ((f/2)R) = (f+2)/f = 1 + 2/f

γ = 1 + 2/f

Complete Cv, Cp, γ Table (from Aakash PDF — Kinetic Theory of Gases JEE Main):

Monatomic gas (f=3): He, Ar, Ne, Kr, Xe, Rn

Cv = 3R/2 = 1.5R; Cp = 5R/2 = 2.5R; γ = 5/3 ≈ 1.67

Diatomic gas at room temperature (f=5): H₂, N₂, O₂, CO, HCl

Cv = 5R/2 = 2.5R; Cp = 7R/2 = 3.5R; γ = 7/5 = 1.4

Diatomic gas at high temperature (f=7): H₂ above ~1000K

Cv = 7R/2 = 3.5R; Cp = 9R/2 = 4.5R; γ = 9/7 ≈ 1.29

Triatomic linear molecule (f=7): CO₂, CS₂, N₂O

Cv = 7R/2; Cp = 9R/2; γ = 9/7 ≈ 1.29

Triatomic non-linear molecule (f=6): H₂O, SO₂, H₂S

Cv = 3R = 6R/2; Cp = 4R; γ = 4/3 ≈ 1.33

Key Relations (from Aakash PDF — Kinetic Theory of Gases):

Cv = R/(γ–1); Cp = γR/(γ–1). Also: R = Cp–Cv.

For mixture of gases: Cv_mix = ΣnᵢCvᵢ/Σnᵢ; Cp_mix = ΣnᵢCpᵢ/Σnᵢ; γ_mix = Cp_mix/Cv_mix.

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Mean Free Path, Collision Frequency, and van der Waals Equation

Why Mean Free Path and van der Waals Corrections Are the Advanced Kinetic Theory JEE Main Formulas

Mean Free Path (from Aakash PDF — Kinetic Theory of Gases JEE Main):

The mean free path λ is the average distance a molecule travels between successive collisions.

Derivation: A molecule of diameter d moving with average speed v_mean collides with all molecules within a cylinder of cross-section πd² and length v_mean·t in time t. If n = number density = N/V = P/kT, collision rate = √2·πd²·n·v_mean (√2 accounts for relative motion).

λ = 1/(√2·π·d²·n)

In terms of pressure and temperature: substituting n = P/kT:

λ = kT/(√2·π·d²·P)

Key dependencies: λ ∝ T (at constant P); λ ∝ 1/P (at constant T); λ ∝ 1/d²; λ ∝ 1/n.

At STP: λ ≈ 100 nm for air molecules; molecular diameter d ≈ 10⁻¹⁰ m.

Collision Frequency (from Aakash PDF — Kinetic Theory of Gases JEE Main):

Z = √2·π·d²·n·v_mean (number of collisions per unit time per molecule)

Relation: λ = v_mean/Z.

van der Waals Equation for Real Gases (from Aakash PDF — Kinetic Theory of Gases JEE Main):

Real gases deviate from ideal behaviour because: (1) intermolecular forces are non-zero; (2) molecules have finite volume. van der Waals modified the ideal gas equation:

(P + an²/V²)(V – nb) = nRT

Correction terms:

(a) Pressure correction +an²/V²: accounts for intermolecular attractive forces. At the wall, a molecule is pulled back by surrounding molecules → reduced pressure. Corrected pressure = P + an²/V². Constant a represents intermolecular attraction strength.

(b) Volume correction –nb: accounts for finite volume of molecules. Effective free volume = V – nb. Constant b = 4 × (volume of one molecule) × Nₐ (excluded volume per mole).

Critical Constants (from Aakash PDF — Kinetic Theory of Gases):

At the critical point (inflection point on P-V isotherm), the gas-liquid distinction vanishes:

Critical temperature: Tc = 8a/27Rb

Critical pressure: Pc = a/27b²

Critical volume: Vc = 3nb (for n moles)

Above Tc, a gas cannot be liquefied regardless of pressure applied. Below Tc, compression can liquefy the gas. Download the Free PDF for Kinetic Theory of Gases for all mean free path and van der Waals examples for JEE Main.