Question

The ratio of the specific heats \( \frac{C_p}{C_v} = \gamma \) in terms of degrees of freedom 'n' is given by

• A. \( \left( 1 + \frac{n}{2} \right) \)
• B. \( \left( 1 + \frac{1}{n} \right) \)
• C. \( \left( 1 + \frac{n}{3} \right) \)
• D. \( \left( 1 + \frac{2}{n} \right) \)

Hint:

For monatomic gases, \( \gamma \) is 5/3, and for diatomic gases, \( \gamma \) is 7/5, as these values correspond to the number of degrees of freedom for these gases.

Answer 1

The Correct Option is D

Step 1: Understanding the relation between \( C_p \) and \( C_v \).
The ratio \( \frac{C_p}{C_v} = \gamma \), where \( C_p \) is the specific heat at constant pressure and \( C_v \) is the specific heat at constant volume. This ratio is related to the degrees of freedom 'n' of the gas particles. For ideal gases, the ratio is dependent on the number of translational, rotational, and vibrational degrees of freedom.
Step 2: Formula application.
The general expression for \( \frac{C_p}{C_v} \) in terms of the degrees of freedom 'n' is derived from the kinetic theory of gases. The correct formula is \( \left( 1 + \frac{2}{n} \right) \), which accounts for the translational and rotational degrees of freedom.
Step 3: Conclusion.
The correct answer is \( \left( 1 + \frac{2}{n} \right) \), as it matches the known formula for the ratio of specific heats in terms of degrees of freedom.