Kinetic Theory of Gases formulas for JEE Main & Advanced

20 formula entries for Kinetic Theory of Gases, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Ideal Gas Equation

P V = n R T

When to use. Use for any problem involving state variables of an ideal gas, especially in thermodynamic processes.

Trap. Do not forget to use temperature in Kelvin, not Celsius.

Avogadro's Law

V \propto n

at constant

T

and

P

When to use. Use when comparing volumes or number of moles of different gases under same conditions.

Trap. Do not apply this to real gases without corrections.

Pressure of an Ideal Gas (Kinetic Theory)

P = \frac{1}{3} \frac{N m {\overset{ˉ}{v}}^{2}}{V} = \frac{1}{3} ρ {\overset{ˉ}{v}}^{2}

When to use. Use to relate macroscopic pressure to microscopic molecular properties (mass, speed, density).

Trap. Do not confuse ${\overset{ˉ}{v}}^{2}$ (mean square speed) with $\overset{ˉ}{v}$ (mean speed); they are different.

Root Mean Square Speed

v_{rms} = \sqrt{\frac{3 R T}{M}} = \sqrt{\frac{3 k T}{m}}

When to use. Use to calculate molecular speeds from temperature and molar mass, or to compare gases.

Trap. Do not confuse $M$ (molar mass in kg/mol) with $m$ (mass per molecule).

Average Kinetic Energy per Molecule

\overset{ˉ}{K} = \frac{3}{2} k T

When to use. Use to relate temperature to molecular energy, or to compare energies of different gases.

Trap. Do not forget this is per molecule; for per mole, multiply by $N_{A}$ : $\frac{3}{2} R T$ .

Total Internal Energy of an Ideal Gas

U = \frac{f}{2} n R T

When to use. Use to find internal energy change in thermodynamic processes (isothermal, adiabatic).

Trap. Do not forget $f$ is degrees of freedom: 3 for monatomic, 5 for diatomic (at moderate T), etc.

Molar Specific Heats of Ideal Gases

C_{V} = \frac{f}{2} R

C_{P} = (\frac{f}{2} + 1) R

γ = \frac{C_{P}}{C_{V}} = 1 + \frac{2}{f}

When to use. Use to compute heat capacity or adiabatic index for thermodynamic processes.

Trap. Do not forget that $C_{P} - C_{V} = R$ for ideal gases.

Maxwell-Boltzmann Speed Distribution

f (v) = 4 π N {(\frac{m}{2 π k T})}^{3 / 2} v^{2} e^{- m v^{2} / (2 k T)}

When to use. Use to understand molecular speed distributions or to derive average speeds.

Trap. Do not confuse $f (v)$ with probability; it is the number of molecules with speed in $d v$ .

Most Probable, Mean, and RMS Speeds

v_{p} = \sqrt{\frac{2 k T}{m}} = \sqrt{\frac{2 R T}{M}}

\overset{ˉ}{v} = \sqrt{\frac{8 k T}{π m}} = \sqrt{\frac{8 R T}{π M}}

v_{rms} = \sqrt{\frac{3 k T}{m}} = \sqrt{\frac{3 R T}{M}}

When to use. Use to compare molecular speeds or to compute average kinetic energy.

Trap. Do not confuse the three speeds; they have different constants: $v_{p} < \overset{ˉ}{v} < v_{rms}$ .

Using the Kinetic Theory to Derive Gas Laws

P V = N k T = n R T

When to use. Use when asked to derive ideal gas equation from microscopic theory.

Trap. Do not skip steps; show that $\frac{1}{2} m {\overset{ˉ}{v}}^{2} = \frac{3}{2} k T$ .

Solving Problems with Gas Mixtures

P_{total} = P_{1} + P_{2} + \dots

P_{i} = \frac{n_{i} R T}{V}

When to use. Use when dealing with multiple gases in the same container.

Trap. Do not forget that each gas occupies the entire volume.

Calculating Number Density and Mean Free Path

n = \frac{N}{V} = \frac{P}{k T}

λ = \frac{1}{\sqrt{2} π d^{2} n}

When to use. Use to find how far a molecule travels between collisions, or to relate pressure to density.

Trap. Do not confuse $n$ (number density) with moles; use context.

Mean Free Path

λ = \frac{1}{\sqrt{2} π d^{2} n}

When to use. Use to understand transport phenomena (diffusion, viscosity) or collision rates.

Trap. Do not forget that $λ$ is inversely proportional to pressure and density.

Law of Equipartition of Energy

\overset{ˉ}{E} = \frac{f}{2} k T

per molecule,

U = \frac{f}{2} n R T

When to use. Use to compute internal energy or specific heats of polyatomic gases.

Trap. Do not forget that only quadratic degrees count (e.g., translational, rotational, vibrational).

Adiabatic Process for Ideal Gas

P V^{γ} = constant

T V^{γ - 1} = constant

T^{γ} P^{1 - γ} = constant

When to use. Use for adiabatic compression/expansion problems (e.g., in engines).

Trap. Do not forget that $γ = C_{P} / C_{V}$ depends on degrees of freedom.

Isothermal Process for Ideal Gas

P_{1} V_{1} = P_{2} V_{2}

When to use. Use for processes at constant temperature (e.g., slow expansion).

Trap. Do not forget that internal energy change is zero for ideal gas in isothermal process.

Dalton's Law of Partial Pressures

P_{total} = \sum P_{i}

When to use. Use for gas mixtures to find individual gas contributions.

Trap. Do not apply to reacting gases or at very high pressures.

Using the Kinetic Theory to Compare Gases

\overset{ˉ}{K} = \frac{3}{2} k T

v_{rms} \propto \frac{1}{\sqrt{M}}

When to use. Use to compare molecular speeds or energies of different gases at same temperature.

Trap. Do not assume same RMS speed; lighter molecules move faster.

Relation Between Pressure and Kinetic Energy

P = \frac{2}{3} \frac{N}{V} \overset{ˉ}{K} = \frac{2}{3} n \overset{ˉ}{K}

(where

n

is number density)

When to use. Use to connect macroscopic pressure to microscopic energy.

Trap. Do not confuse $\overset{ˉ}{K}$ (per molecule) with total kinetic energy.

Temperature as a Measure of Kinetic Energy

\overset{ˉ}{K} = \frac{3}{2} k T

When to use. Use to interpret temperature microscopically.

Trap. Do not forget that this applies only to translational motion; rotational/vibrational energy may differ.

Values to remember

$R = 8.314 J/mol \cdot K$ Ideal Gas Equation
$R = 0.0821 L \cdot atm/mol \cdot K$ Ideal Gas Equation
$N_{A} = 6.022 \times 10^{23} {mol}^{- 1}$ Avogadro's Law
$k = 1.38 \times 10^{- 23} J/K$ Root Mean Square Speed
$f = 3$ (monatomic) Total Internal Energy of an Ideal Gas
$f = 5$ (diatomic, rigid) Total Internal Energy of an Ideal Gas
$f = 7$ (diatomic, with vibration) Total Internal Energy of an Ideal Gas
$f = 5$ (diatomic, no vibration) Degrees of Freedom
$γ = 5 / 3$ (monatomic) Molar Specific Heats of Ideal Gases
$γ = 7 / 5$ (diatomic, rigid) Molar Specific Heats of Ideal Gases
$γ = 4 / 3$ (polyatomic, rigid) Molar Specific Heats of Ideal Gases
$d \approx 10^{- 10} m$ (typical molecular diameter) Calculating Number Density and Mean Free Path
$γ = 7 / 5$ (diatomic) Adiabatic Process for Ideal Gas

Read the full chapter: Kinetic Theory of Gases for JEE, with what JEE asks and the common traps.