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

Step 1: Use Mayer's relation.
For an ideal gas, $C_p-C_v=R$.

Step 2: Bring in the ratio.
Since $\gamma=\frac{C_p}{C_v}$, we have $C_p=\gamma C_v$. Substitute: $\gamma C_v-C_v=R$.

Step 3: Solve for $C_v$.
$C_v(\gamma-1)=R$, so $C_v=\frac{R}{\gamma-1}$. \[ \boxed{\frac{R}{\gamma-1}} \]