For an ideal diatomic gas:

Options

A. `C_(P)=7/2R`
B. `C_(P)=3/2R`
C. `C_(P)=5/2R`
D. `C_(v)=3/2R`

Answer

To solve the problem regarding the molar heat capacities of an ideal diatomic gas, we will follow these steps:

### Step-by-Step Solution:

1. **Identify Degrees of Freedom**:
For a diatomic gas, the degrees of freedom (f) can be calculated. A diatomic molecule has:
- 3 translational degrees of freedom (movement in x, y, and z directions)
- 2 rotational degrees of freedom (rotation about two axes)
Therefore, the total degrees of freedom for a diatomic gas is:
\[ f = 3 + 2 = 5 \]

2. **Calculate \( C_V \)**:
The molar heat capacity at constant volume (\( C_V \)) can be calculated using the formula:
\[ C_V = \frac{f}{2} R \]
Substituting the value of \( f \):
\[ C_V = \frac{5}{2} R = \frac{5R}{2} \]

3. **Calculate \( C_P \)**:
The molar heat capacity at constant pressure (\( C_P \)) is related to \( C_V \) by the equation:
\[ C_P = C_V + R \]
Substituting the value of \( C_V \):
\[ C_P = \frac{5R}{2} + R \]
To combine these terms, convert \( R \) to have a common denominator:
\[ C_P = \frac{5R}{2} + \frac{2R}{2} = \frac{7R}{2} \]

4. **Final Answer**:
Therefore, the value of \( C_P \) for an ideal diatomic gas is:
\[ C_P = \frac{7R}{2} \]

### Conclusion:
The correct answer is \( C_P = \frac{7R}{2} \).

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