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2 public questions tagged with this topic.

Collision Frequency and Mean Free Path Variation

A solid has a molar specific heat capacity of 24.9 J mol⁻¹ K⁻¹. How many degrees of freedom per atom does it have? (R =

**Collision frequency** Z = √2 n π d² v_avg, n number density, d molecular diameter, v_avg average speed, proportional to n and v_avg, mean free path λ = v_avg/Z =1/(√2 n π d²), inversely proportional to n, so λ ∝1/P at constant T because n ∝ P, collision frequency increases with pressure, λ decreases. C = f × (R)/(2), 24.9 = f × (8.31)/(2).f = (24.9 × 2)/(8.31) ≈ 5.99 ≈ 6. Substituting values gives 6, which matches expected kinetic theory result, confirming mean free path λ = 1/(√2 n π d²), ideal gas law P V = n R T and v_rms = √(3 R T/M) relations.

Ref: NCERT > Physics Book > Behaviour of Perfect Gas and Kinetic Theory > Collision Frequency and Mean Free Path Variation

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RMS Speed and Temperature Dependence

A solid has a molar specific heat capacity of 25 J mol⁻¹ K⁻¹ at room temperature. How many degrees of freedom does each

**RMS speed** v_rms = √(3 R T/M) = √(3 k_B T/m) where M molar mass (kg/mol), m molecular mass (kg), k_B=1.38×10⁻²/³ J/K, R=8.314 J/mol·K, T absolute temperature (K). Proportional to √T and 1/√M, lighter gases faster at same T, e.g., H₂ faster than O₂, temperature increase raises v_rms as √T. For a solid, C = f × (R)/(2) × N_A = f × (R)/(2) × 1 = f × (R)/(2).Given C = 25 J mol⁻¹ K⁻¹, R = 8.31 J mol⁻¹ K⁻¹.25 = f × (8.31)/(2), f = (25 × 2)/(8.31) ≈ 6. Substituting values gives 6, which matches expected kinetic theory result, confirming mean

Ref: NCERT > Physics Book > Behaviour of Perfect Gas and Kinetic Theory > RMS Speed and Temperature Dependence

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