Question

The molar specific heats of an ideal gas at constant pressure and volume are denoted by \( C_p \) and \( C_v \), respectively. If \( \gamma = \dfrac{C_p}{C_v} \) and \( R \) is the universal gas constant, then \( C_v \) is equal to:

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

Hint:

The ratio of specific heats \( \gamma \) is the ratio of \( C_p \) to \( C_v \), and \( C_v \) can be calculated using the ideal gas constant \( R \) and \( \gamma \).

Answer 1

The Correct Option is A

Step 1: The relationship between the specific heats at constant pressure and constant volume is given by: \[ \gamma = \dfrac{C_p}{C_v}. \] Step 2: Using the gas constant \( R \) and the ideal gas equation, we can solve for \( C_v \): \[ C_v = \dfrac{R}{\gamma - 1}. \]
Final Answer: \[ \boxed{\dfrac{R}{\gamma - 1}} \]