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Molar specific heat at constant volume: $C_v = \frac{f}{2}R$, where $f$ is the number of active degrees of freedom. Molar specific heat at constant pressure: $C_p = C_v + R = \frac{f+2}{2}R$ (Mayer's relation). The extra $R$ represents the work done during isobaric expansion.

For monatomic: $C_v = 12.5$ J/(mol-K), $C_p = 20.8$ J/(mol-K). For diatomic (moderate T): $C_v = 20.8$ J/(mol-K), $C_p = 29.1$ J/(mol-K). These predicted values match experiment beautifully for noble and diatomic gases, validating the equipartition theorem.

The ratio $\gamma = C_p/C_v$ determines adiabatic behavior ($PV^\gamma = \text{const}$) and the speed of sound ($v = \sqrt{\gamma RT/M}$). JEE frequently asks for $\gamma$ of mixtures: calculate $C_{v,\text{mix}}$ and $C_{p,\text{mix}}$ using mole-fraction weighting, then take the ratio. A common trap: $\gamma_{\text{mix}}$ is NOT the mole-fraction average of individual $\gamma$ values. The correct approach always goes through $C_v$ and $C_p$ separately.