Class 11 · Chapter 13

Kinetic Theory of Gases

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Kinetic Theory of Gases Formula Sheet

7 formulas across 3 topics in Kinetic Theory of Gases.

1 min read

Updated 2026-07-04 · v1.0.1

Ideal Gas Equation & Gas Laws

One equation, PV = nRT, contains Boyle, Charles and Gay-Lussac as special cases — fix one variable and read off the rest.

PV = nRT = NkT

Ideal gas equation

EasyAsked very oftenJEE MainNEETMHT-CETBoards

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
n	number of moles	mol	[N]
N	number of molecules	dimensionless	[M⁰L⁰T⁰]
k	Boltzmann constant (R/Nₐ)k = 1.38 × 10⁻²³ J/K	J/K	[ML²T⁻²Θ⁻¹]
R	universal gas constant	J/(mol·K)	[ML²T⁻²Θ⁻¹N⁻¹]

Valid when

Boyle: PV = const at fixed T; Charles: V ∝ T at fixed P; Gay-Lussac: P ∝ T at fixed V
Density form: PM = ρRT (M = molar mass)

Common mistakes

Two connected containers share PRESSURE, not temperature — conserve total moles n₁ + n₂ when gas redistributes

Read the concept →

Worth remembering

Real gases approach ideal behaviour at LOW pressure and HIGH temperature — far from liquefaction

Kinetic Pressure & Molecular Speeds

Pressure is molecular bombardment. From one formula, P = ⅓ρv̄², every speed and energy result follows.

P = ⅓ ρ v(rms)² = ⅓ (mN/V) v(rms)²

Pressure of an ideal gas (kinetic theory)

MediumAsked very oftenJEE MainNEETMHT-CETBoards

Momentum transfer of molecules bouncing off the walls, averaged over random directions

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
ρ	density of the gas	kg/m³	[ML⁻³]
v(rms)	root-mean-square speed	m/s	[LT⁻¹]
m	mass of ONE molecule	kg	[M]

Valid when

Point molecules, elastic collisions, no intermolecular forces — the ideal gas postulates

Common mistakes

Kinetic energy density link: P = ⅔ × (translational KE per unit volume)

Read the concept →

v(rms) = √(3RT/M) = √(3kT/m) = √(3P/ρ)

RMS speed

EasyAsked very oftenJEE MainJEE AdvNEETMHT-CETBoards

Equating ⅓ρv(rms)² with P from the ideal gas law

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
M	molar mass (in kg/mol!)	kg/mol	[MN⁻¹]
T	absolute temperature	K	[Θ]

Valid when

v(rms) ∝ √T and ∝ 1/√M — lighter and hotter means faster

Common mistakes

Using M in g/mol with R = 8.314 gives answers off by √1000 — convert to kg/mol
Speed ORDER at the same T: v(rms) > v(avg) > v(mp), in ratio √3 : √(8/π) : √2

Read the concept →

v(avg) = √(8RT/πM); v(mp) = √(2RT/M)

Average and most-probable speeds

MediumAsked oftenJEE MainNEETMHT-CET

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v(avg)	mean speed of the Maxwell distribution	m/s	[LT⁻¹]
v(mp)	most probable speed (peak of the distribution)	m/s	[LT⁻¹]

Valid when

Maxwell-Boltzmann speed distribution

Read the concept →

Worth remembering

At the same temperature all gases share the same average translational KE — but NOT the same speeds (lighter gas moves faster)

Energy, Degrees of Freedom & Mean Free Path

Equipartition hands each quadratic degree of freedom ½kT. Count the degrees and every heat capacity in the chapter writes itself.

KE(trans) = (3/2) kT per molecule = (3/2) RT per mole

Average translational kinetic energy

EasyAsked very oftenJEE MainNEETMHT-CETBoards

½m v(rms)² with v(rms)² = 3kT/m

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
k	Boltzmann constant	J/K	[ML²T⁻²Θ⁻¹]
T	absolute temperature	K	[Θ]

Valid when

Depends ONLY on temperature — identical for every gas
Translational part is (3/2)kT even for polyatomic molecules

Common mistakes

Doubling the CELSIUS temperature does not double KE — kelvin only

Read the concept →

U = (f/2) nRT; Cᵥ = (f/2)R; γ = 1 + 2/f

Equipartition: energy and heat capacity from degrees of freedom

MediumAsked very oftenJEE MainJEE AdvNEETMHT-CETBoards

Each quadratic degree of freedom carries ½kT on average

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
f	degrees of freedommonatomic 3; diatomic 5 (7 with vibration at high T); nonlinear polyatomic 6	dimensionless	[M⁰L⁰T⁰]

Valid when

Vibrational modes freeze out at ordinary temperatures for diatomic gases

Common mistakes

Mixture of gases: Cᵥ(mix) = (n₁Cᵥ₁ + n₂Cᵥ₂)/(n₁ + n₂) — mole-weighted, and γ(mix) comes from Cₚ(mix)/Cᵥ(mix), never by averaging γ values

Read the concept →

λ = 1/(√2 π d² n(v)) = kT/(√2 π d² P)

Mean free path

MediumAsked sometimesJEE MainNEET

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
λ	average distance between collisions	m	[L]
d	molecular diameter	m	[L]
n(v)	number density N/V	m⁻³	[L⁻³]

Valid when

λ ∝ T at fixed pressure; λ ∝ 1/P at fixed temperature

Read the concept →

Worth remembering

Absolute zero in kinetic language: the temperature at which molecular translational KE would vanish

NEET loves speed-ratio and KE-per-molecule one-liners; JEE Main's favourite is the gas-mixture γ — always via Cᵥ, never by averaging γ.

Stuck on a concept in Kinetic Theory of Gases?

Message Ajay Sir directly on WhatsApp for doubt support on this chapter.

Ask a doubt

Previous chapter12. Thermodynamics Next chapter 14. Oscillations

Back to Class 11 chapters