For a diatomic gas, which options is/are correct?

Options

A. `gamma = 1.40`
B. `C_(P) = (7R)/(2)`
C. `C_(v) = (5R)/(2)`
D. `gamma = 1.67`

Answer

To determine the correct options for a diatomic gas, we will analyze the properties of diatomic gases, specifically focusing on the specific heat capacities (Cp and Cv) and the specific heat ratio (gamma).

### Step-by-Step Solution:

1. **Understanding Gamma (γ)**:
- For a diatomic gas, the specific heat ratio (gamma) is defined as:
\[ \gamma = \frac{C_p}{C_v} \]
- The degrees of freedom (f) for a diatomic gas is 5 (3 translational + 2 rotational).
- The formula to calculate gamma in terms of degrees of freedom is:
\[ \gamma = 1 + \frac{2}{f} \]
- Substituting \(f = 5\):
\[ \gamma = 1 + \frac{2}{5} = 1 + 0.4 = 1.4 \]
- Therefore, the first option stating that \(\gamma = 1.40\) is **correct**.

2. **Calculating Cv**:
- Using Mayer's formula:
\[ C_p - C_v = R \]
- We can express \(C_p\) in terms of \(C_v\):
\[ C_p = C_v + R \]
- From the relationship of gamma:
\[ \gamma = \frac{C_p}{C_v} = \frac{C_v + R}{C_v} \]
- Substituting \(\gamma = 1.4\):
\[ 1.4 = \frac{C_v + R}{C_v} \]
- Rearranging gives:
\[ 1.4C_v = C_v + R \implies 0.4C_v = R \implies C_v = \frac{R}{0.4} = \frac{5R}{2} \]

3. **Calculating Cp**:
- Now substituting \(C_v\) back into the equation for \(C_p\):
\[ C_p = C_v + R = \frac{5R}{2} + R = \frac{5R}{2} + \frac{2R}{2} = \frac{7R}{2} \]
- Therefore, \(C_p = \frac{7R}{2}\) is also **correct**.

4. **Conclusion**:
- The first option (γ = 1.40) is correct.
- The second option (C_p = 7R/2) is correct.
- The third option (C_v = 5R/2) is also correct.
- The statement that γ = 1.67 is **incorrect**.

### Final Answer:
- The correct options are:
- γ = 1.40
- C_p = 7R/2
- C_v = 5R/2