Question

The molar specific heat of an ideal gas at constant pressure and constant volume is '\(C_P\)' and '\(C_V\)' respectively. If '\(R\)' is a universal gas constant and the ratio of '\(C_P\)' to '\(C_V\)' is \(\gamma\), then '\(C_P\)' is equal to

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

Hint:

Use two key relations: \(\gamma = C_P/C_V\) and \(C_P - C_V = R\).

Answer 1

The Correct Option is C

Concept:
For an ideal gas: \[ \gamma = \frac{C_P}{C_V}, \quad C_P - C_V = R \] Step 1: Express \(C_P\) in terms of \(C_V\). \[ C_P = \gamma C_V \]
Step 2: Substitute in Mayer's relation. \[ C_P - C_V = R \] \[ \gamma C_V - C_V = R \] \[ C_V(\gamma - 1) = R \]
Step 3: Find \(C_V\). \[ C_V = \frac{R}{\gamma - 1} \]
Step 4: Find \(C_P\). \[ C_P = \gamma C_V = \gamma \cdot \frac{R}{\gamma - 1} \] \[ C_P = \frac{R\gamma}{\gamma - 1} \]
Step 5: Conclusion. \[ C_P = \frac{R\gamma}{\gamma - 1} \]