#heat capacity ratio

Specific Heat Capacity and Latent Heat

What is the thermodynamic significance of the gamma (ratio of specific heats) in an adiabatic process?

**Latent heat** energy needed for phase change without temperature change, overcomes intermolecular forces, e.g., heating ice at 0°C to water at 0°C requires 334 kJ/kg, then heating water to 100°C requires c ΔT, then vaporization 2260 kJ/kg, illustrating two types of heat. γ = (C_p)/(C_v) determines the steepness of the P-V curve in an adiabatic process ( P V^γ = constant ), reflecting how internal energy changes with volume, influenced by the gas’s degrees of freedom. Using first law ΔU = Q - W, W = ∫ P dV, isobaric W = P ΔV, isothermal W = n R T ln(V₂/V₁), adiabatic P V^γ =

Ref: NCERT > Physics Book > Thermodynamics > Specific Heat Capacity and Latent Heat

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Second Law Heat Engines and Kelvin-Planck

A gas is compressed adiabatically from 15 L to 5 L , increasing its pressure from 3 atm to 12 atm . What is gamma ?

**Second law Kelvin-Planck statement** no process possible whose sole result is absorption of heat from reservoir and complete conversion to work, heat engine must have at least two reservoirs hot and cold, efficiency η = W/Q_h =1 - Q_c/Q_h

Ref: NCERT > Physics Book > Thermodynamics > Second Law Heat Engines and Kelvin-Planck

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Second Law Heat Engines and Kelvin-Planck

A gas undergoes an adiabatic expansion from 30 L to 90 L , reducing its pressure from 9 atm to 1 atm . What is the value

**Carnot engine** reversible engine operating between T_h and T_c has maximum efficiency η =1 - T_c/T_h, T in kelvin, e.g., T_h=400 K T_c=300 K η=0.25, real engines less due to irreversibilities, second law defines direction of spontaneous processes and entropy increase. For adiabatic: P₁ V₁^γ = P₂ V₂^γ . 9 × 30^γ = 1 × 90^γ . 9 = ((90)/(30))^γ ⇒ 9 = 3^γ . 3^γ = 3² ⇒ γ = 2 . Using first law ΔU = Q - W, W = ∫ P dV, isobaric W = P ΔV, isothermal W = n R T ln(V₂/V₁), adiabatic P V^γ = const and η

Ref: NCERT > Physics Book > Thermodynamics > Second Law Heat Engines and Kelvin-Planck

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Degrees of Freedom and Molar Specific Heat

What is the ratio of C_p to C_v for a gas with 3 translational and 2 rotational degrees of freedom?

**Specific heat relation** C_p - C_v = R for ideal gas per mole, Mayer's relation, due to work done at constant pressure, degrees of freedom include translational, rotational, vibrational, each quadratic term contributes ½ R to C_v. Degrees of freedom = 3 + 2 = 5.C_v = (5)/(2) R, C_p = C_v + R = (5)/(2) R + R = (7)/(2) R.γ = (C_p)/(C_v) = (7)/(2) R(5)/(2) R = (7)/(5) = 1.4. Substituting values gives 1.4, 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 > Degrees of Freedom and Molar Specific Heat

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