A gas have 3 translation and 2 Rotational degree of freedom. Find `(C_p)/(C_v)`?

Options

A. `5/7`
B. `7/5`
C. `2/5`
D. `5/2`

Answer

To find the ratio \( \frac{C_p}{C_v} \) for a gas with 3 translational and 2 rotational degrees of freedom, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Degrees of Freedom (F)**:
- The gas has 3 translational degrees of freedom and 2 rotational degrees of freedom.
- Therefore, the total degrees of freedom \( F \) is:
\[ F = 3 + 2 = 5 \]

2. **Use the Formula for \( \gamma \)**:
- The ratio \( \frac{C_p}{C_v} \) is denoted by \( \gamma \).
- The formula for \( \gamma \) in terms of the degrees of freedom \( F \) is:
\[ \gamma = 1 + \frac{2}{F} \]

3. **Substitute the Value of F**:
- Now, substitute \( F = 5 \) into the formula:
\[ \gamma = 1 + \frac{2}{5} \]

4. **Calculate \( \gamma \)**:
- Simplifying the expression:
\[ \gamma = 1 + 0.4 = 1.4 \]
- To express this as a fraction:
\[ \gamma = \frac{7}{5} \]

5. **Conclusion**:
- Therefore, the ratio \( \frac{C_p}{C_v} \) is:
\[ \frac{C_p}{C_v} = \frac{7}{5} \]

### Final Answer:
\[ \frac{C_p}{C_v} = \frac{7}{5} \]