Question

The ratio of specific heat at constant pressure to specific heat at constant volume (\( \gamma \)) for a gas is \( \gamma = \left( 1 + \frac{2}{f} \right) \) where \( f \) is the number of degrees of freedom of a molecule of a gas. The ratio of \( \gamma_{r} \) for rigid diatomic to \( \gamma_{m} \) for monatomic is

• A. \( \frac{14}{23} \)
• B. \( \frac{25}{21} \)
• C. \( \frac{21}{25} \)
• D. \( \frac{23}{14} \)

Hint:

The ratio of specific heats \( \gamma \) for a gas depends on the number of degrees of freedom, with monatomic molecules having fewer degrees of freedom than diatomic molecules.

Answer 1

The Correct Option is C

Step 1: Understanding the specific heat ratio.
For a rigid diatomic molecule, the number of degrees of freedom is \( f = 5 \), and for a monatomic molecule, \( f = 3 \). The ratio of specific heats is given by: \[ \gamma = 1 + \frac{2}{f} \] Step 2: Finding \( \gamma \) for rigid diatomic and monatomic molecules.
For a rigid diatomic molecule: \[ \gamma_{r} = 1 + \frac{2}{5} = \frac{7}{5} \] For a monatomic molecule: \[ \gamma_{m} = 1 + \frac{2}{3} = \frac{5}{3} \] Step 3: Finding the ratio of \( \gamma_{r} \) to \( \gamma_{m} \).
The ratio is: \[ \frac{\gamma_{r}}{\gamma_{m}} = \frac{\frac{7}{5}}{\frac{5}{3}} = \frac{21}{25} \] Step 4: Conclusion.
The correct answer is (C), \( \frac{21}{25} \).