Consider, two ideal diatomic gases A and B at some temperature T. Molecules of the gas A are rigid, and have a mass m. Molecules of the gas B have an additional vibrations mode, and have a mass `m/4`. The ratio of the specific heats (`C_(V)^(A)` and `C_(V)^(B)`) of gas A and B, respectively is:

Options

A. `7:9`
B. `3:5`
C. `5:9`
D. `5:7`

Answer

To find the ratio of the specific heats \( C_V^A \) and \( C_V^B \) for the two ideal diatomic gases A and B, we will follow these steps:

### Step 1: Identify the degrees of freedom for gas A
Gas A is a rigid diatomic gas. For diatomic gases, the degrees of freedom can be calculated as follows:
- Translational degrees of freedom: 3 (movement in x, y, z directions)
- Rotational degrees of freedom: 2 (rotation about two axes)

Since gas A is rigid, it does not have vibrational degrees of freedom. Thus, the total degrees of freedom \( F_A \) for gas A is:
\[ F_A = 3 + 2 = 5 \]

### Step 2: Identify the degrees of freedom for gas B
Gas B is also a diatomic gas but has an additional vibrational mode. Therefore, its degrees of freedom are:
- Translational degrees of freedom: 3
- Rotational degrees of freedom: 2
- Vibrational degrees of freedom: 2 (one vibrational mode contributes 2 degrees of freedom)

Thus, the total degrees of freedom \( F_B \) for gas B is:
\[ F_B = 3 + 2 + 2 = 7 \]

### Step 3: Calculate the specific heats \( C_V^A \) and \( C_V^B \)
The specific heat at constant volume \( C_V \) for an ideal gas can be calculated using the formula:
\[ C_V = \frac{1}{2} F R \]
where \( F \) is the degrees of freedom and \( R \) is the universal gas constant.

For gas A:
\[ C_V^A = \frac{1}{2} F_A R = \frac{1}{2} \times 5 R = \frac{5R}{2} \]

For gas B:
\[ C_V^B = \frac{1}{2} F_B R = \frac{1}{2} \times 7 R = \frac{7R}{2} \]

### Step 4: Calculate the ratio \( \frac{C_V^A}{C_V^B} \)
Now we can find the ratio of the specific heats:
\[ \frac{C_V^A}{C_V^B} = \frac{\frac{5R}{2}}{\frac{7R}{2}} = \frac{5R}{7R} = \frac{5}{7} \]

### Conclusion
The ratio of the specific heats \( C_V^A \) and \( C_V^B \) is:
\[ \frac{C_V^A}{C_V^B} = \frac{5}{7} \]