JEE Physics — Kinetic Theory

Chapter Overview & Weightage

Kinetic Theory of Gases is a short, formula-heavy chapter that overlaps with thermodynamics. JEE Main asks 1-2 questions almost every year, usually on rms speed, mean free path, or degrees of freedom.

Typical JEE weightage: $1-2$ questions in JEE Main, occasional combined-with-thermo question in JEE Advanced.

Year	JEE Main Qs	JEE Advanced Qs
2020	2	0
2021	1	1
2022	2	0
2023	2	1
2024	2	0

Key Concepts You Must Know

Postulates of kinetic theory (random motion, elastic collisions, point particles, no inter-molecular forces)
Pressure-velocity relation: $P = \tfrac{1}{3}\rho \langle v^2\rangle$
Three speeds: rms, average, most probable
Degrees of freedom for monatomic, diatomic, polyatomic gases
Equipartition theorem: $\tfrac{1}{2}k_B T$ per degree of freedom
Specific heats $C_V$ and $C_P$ , ratio $\gamma = C_P/C_V$
Mayer’s relation: $C_P - C_V = R$ (per mole)
Mean free path: $\lambda = 1/(\sqrt{2}\pi d^2 n)$

Important Formulas

$v_{\text{rms}} = \sqrt{3RT/M}, \quad v_{\text{avg}} = \sqrt{8RT/(\pi M)}, \quad v_{\text{mp}} = \sqrt{2RT/M}$

Ratio: $v_{\text{rms}} : v_{\text{avg}} : v_{\text{mp}} = \sqrt{3} : \sqrt{8/\pi} : \sqrt{2}$ .

Monatomic: $U = \tfrac{3}{2}nRT$ , $C_V = \tfrac{3}{2}R$ , $C_P = \tfrac{5}{2}R$ , $\gamma = 5/3$

Diatomic (rigid): $U = \tfrac{5}{2}nRT$ , $C_V = \tfrac{5}{2}R$ , $C_P = \tfrac{7}{2}R$ , $\gamma = 7/5$

Diatomic (with vibration): $C_V = \tfrac{7}{2}R$ , $C_P = \tfrac{9}{2}R$ , $\gamma = 9/7$

$\lambda = \frac{k_B T}{\sqrt{2}\pi d^2 P}$

where $d$ is molecular diameter and $P$ is pressure.

Solved Previous Year Questions

PYQ 1 (JEE Main 2023)

The rms speed of a gas at $300$ K is $500$ m/s. At what temperature will it be doubled?

$v_{\text{rms}} \propto \sqrt{T}$ . Doubling speed → quadrupling $T$ .

$T_2 = 4 \times 300 = 1200$ K.

PYQ 2 (JEE Main 2024)

The ratio $C_P/C_V$ for a gas is $1.4$ . Identify the gas type.

$\gamma = 1.4 = 7/5$ . This is a rigid diatomic gas (5 DOF: 3 translational + 2 rotational, no vibration).

PYQ 3 (JEE Advanced 2021)

A mixture of $1$ mole helium ( $\gamma_1 = 5/3$ ) and $2$ moles oxygen ( $\gamma_2 = 7/5$ ). Find the effective $\gamma$ .

For a mixture: $C_V = (n_1 C_{V_1} + n_2 C_{V_2})/(n_1+n_2)$ .

$C_{V_1} = 3R/2$ , $C_{V_2} = 5R/2$ .

$C_V = (1 \cdot 3R/2 + 2 \cdot 5R/2)/3 = (3R/2 + 5R)/3 = (3R/2 + 10R/2)/3 = (13R/2)/3 = 13R/6$ .

$C_P = C_V + R = 13R/6 + 6R/6 = 19R/6$ .

$\gamma = C_P/C_V = 19/13 \approx 1.46$ .

Difficulty Distribution

Difficulty	% of JEE Qs	Typical type
Easy	$35\%$	Direct rms/average/mp speed plug-ins
Medium	$50\%$	DOF, $C_V/C_P$ for mixtures
Hard	$15\%$	Mean free path with pressure-temperature variation

Expert Strategy

Memorise the three-speed formulas in the form $v = \sqrt{kRT/M}$ with $k = 3, 8/\pi, 2$ . Three numbers, three speeds, no confusion.

For DOF questions, count translational ( $3$ ) + rotational ( $2$ for diatomic, $3$ for non-linear polyatomic) + vibrational ( $2$ per mode, only if temperature is high enough). NCERT usually ignores vibrational unless explicitly asked.

For mixtures, treat the gas constant $R$ as universal but $C_V$ as the mole-weighted average. Same for internal energy: $U_{\text{mix}} = n_1 U_1 + n_2 U_2$ .

Common Traps

Using $T$ in Celsius instead of Kelvin. Every kinetic-theory formula uses absolute temperature. $T = 27^\circ$ C means $T = 300$ K.

Forgetting that molar mass must be in kg/mol for SI calculations. $M$ (N $_2$ ) = $0.028$ kg/mol, not $28$ .

Saying $\gamma$ for a triatomic non-linear molecule is $7/5$ . It’s $4/3$ (since $C_V = 3R$ , $C_P = 4R$ from $6$ DOF: 3 trans + 3 rot).