Question

Specific heats of an ideal gas at constant pressure and volume are denoted by $C_p$ and $C_v$ respectively. If $\gamma = \frac{C_p}{C_v}$ and $R$ is the universal gas constant, then $C_v$ is equal to

• A. $\frac{(\gamma - 1)}{(\gamma + 1)}$
• B. $(\gamma - 1)R$
• C. $\frac{R}{\gamma}$
• D. $\frac{R}{\gamma - 1}$

Hint:

Mayer's relation and the adiabatic ratio are foundational to thermodynamics. Combining them always yields the standard forms $$C_v = \frac{R}{\gamma - 1}$$and$$C_p = \frac{\gamma R}{\gamma - 1}$$ . Memorizing these two expressions directly saves massive calculation time on multiple-choice questions!

Answer 1

The Correct Option is D

We are given the ratio of specific heats $\gamma = \frac{C_p}{C_v}$, which can be rewritten as $C_p = \gamma C_v$. According to Mayer's relation for an ideal gas: $$C_p - C_v = R$$ Substituting $C_p = \gamma C_v$ into Mayer's formula: $$\gamma C_v - C_v = R$$ Factoring out $C_v$ from the left-hand side: $$C_v(\gamma - 1) = R \implies C_v = \frac{R}{\gamma - 1}$$
Final Answer:
The specific heat at constant volume $C_v$ is equal to $\frac{R}{\gamma - 1}$, which corresponds to option (D).