Kinetic Theory of GasesJEE Main

Molecular speeds, degrees of freedom, γ — interactive Physics simulation for IIT-JEE.

Concept

Gas pressure is molecular bombardment. Temperature measures average translational KE — the same $\tfrac32 k_BT$ for every gas, so lighter molecules must fly faster. Energy spreads equally over all quadratic degrees of freedom ( $\tfrac12 k_BT$ each), which fixes the molar heats and $\gamma$ for mono-, di- and polyatomic gases.

Key formula

P = \frac{1}{3}\frac{mN}{V}\overline{v^2}, \quad v_{rms} = \sqrt{\frac{3RT}{M}}, \quad C_v = \frac{f}{2}R, \quad \gamma = 1 + \frac{2}{f}

Derivation

A molecule bouncing between walls transfers momentum $2mv_x$ per hit; summing over N molecules gives $PV = \tfrac13 Nm\overline{v^2} = \tfrac23 N\cdot\overline{KE}_{trans}$ . Comparing with $PV = Nk_BT$ yields $\overline{KE} = \tfrac32 k_BT$ .

Equipartition: each quadratic term in the energy gets $\tfrac12 k_BT$ . Monoatomic f=3, diatomic f=5 (2 rotations), polyatomic f=6.

Speed ladder: $v_{mp} : v_{mean} : v_{rms} = \sqrt2 : \sqrt{8/\pi} : \sqrt3$ .

Scenarios to explore

Kinetic Theory of Gases — Molecular speeds, degrees of freedom, Cv, Cp and γ.

Real-world applications

Why H₂ escapes planetary atmospheres (v_rms vs escape speed).
Speed of sound = √(γRT/M) — directly uses γ from here.
Gas thermometry and Maxwell distribution experiments.

JEE exam tips

v_mp < v_mean < v_rms always — ratio √2 : √(8/π) : √3 ≈ 1.41 : 1.60 : 1.73.
γ: monoatomic 5/3, diatomic 7/5, polyatomic 4/3 — memorize the trio.
Mixture Cv = (n₁Cv₁ + n₂Cv₂)/(n₁+n₂) — mole-weighted.

Common mistakes

Using molar mass in grams — SI needs kg/mol.
Same v_rms for all gases at a temperature — only KE is equal, speed differs by mass.
Vibrational modes at room temperature — usually frozen out; f = 5 for diatomic, not 7.

Exam traps to avoid

Doubling T raises v_rms by √2 only.
Average VELOCITY vector is zero; average SPEED is not.