Question

$\gamma_1$ is the ratio of the specific heat capacities of the rigid diatomic gas at constant pressure to that at constant volume and $\gamma_2$ is the corresponding value for a non-rigid diatomic gas molecule with an additional vibrational mode, then the ratio of $\gamma_1$ to $\gamma_2$ is

• A. 49 : 45
• B. 7 : 5
• C. 9 : 7
• D. 7 : 9
• E. 49 : 30

Hint:

Remember that each vibrational mode adds 2 to the degrees of freedom because it involves both potential and kinetic energy terms in the Hamiltonian.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The ratio of specific heat capacities, denoted by $\gamma$, is given by the formula $\gamma = 1 + \frac{2}{f}$, where $f$ represents the number of degrees of freedom of the gas molecule.

Step 2: Detailed Explanation:
1. For a rigid diatomic gas:
It has 3 translational and 2 rotational degrees of freedom.
Total degrees of freedom, $f_1 = 5$.
\[ \gamma_1 = 1 + \frac{2}{5} = \frac{7}{5} \]
2. For a non-rigid diatomic gas with an additional vibrational mode:
A vibrational mode consists of two degrees of freedom (one for kinetic energy and one for potential energy).
Total degrees of freedom, $f_2 = 3 \text{ (trans)} + 2 \text{ (rot)} + 2 \text{ (vib)} = 7$.
\[ \gamma_2 = 1 + \frac{2}{7} = \frac{9}{7} \]
3. Calculate the ratio $\gamma_1 : \gamma_2$:
\[ \frac{\gamma_1}{\gamma_2} = \frac{7/5}{9/7} = \frac{7}{5} \times \frac{7}{9} = \frac{49}{45} \]

Step 3: Final Answer:
The ratio of $\gamma_1$ to $\gamma_2$ is 49 : 45.