`C_(V)andC_(P)` denote the molar specific heat capacities of a gas at constant volume and constant pressure, respectively. Then

Options

A. `C_(P)-C_(V)` is larger for a diatomic ideal gas than for a monoatomic ideal gas
B. `C_(P)+C_(V)` is larger for a diatomic ideal gas than for a monoatomic ideal gas
C. `C_(P)//C_(V)` is larger for a diatomic ideal gas than for a monoatomic ideal gas
D. `C_(P).C_(V)` is larger for a diatomic ideal gas than for a monoatomic ideal gas

Explanation

To solve the problem regarding the molar specific heat capacities \( C_V \) and \( C_P \) for monatomic and diatomic ideal gases, we will follow these steps:

### Step 1: Understand the Definitions
- \( C_V \) is the molar specific heat capacity at constant volume.
- \( C_P \) is the molar specific heat capacity at constant pressure.

### Step 2: Recall the Values of \( C_V \) and \( C_P \)
- For a **monatomic ideal gas**:
- \( C_V = \frac{3}{2}R \)
- \( C_P = \frac{5}{2}R \)

- For a **diatomic ideal gas**:
- \( C_V = \frac{5}{2}R \)
- \( C_P = \frac{7}{2}R \)

### Step 3: Calculate \( C_P - C_V \)
- For **monatomic gas**:
\[ C_P - C_V = \frac{5}{2}R - \frac{3}{2}R = R \]

- For **diatomic gas**:
\[ C_P - C_V = \frac{7}{2}R - \frac{5}{2}R = R \]

### Step 4: Calculate \( C_P + C_V \)
- For **monatomic gas**:
\[ C_P + C_V = \frac{5}{2}R + \frac{3}{2}R = 4R \]

- For **diatomic gas**:
\[ C_P + C_V = \frac{7}{2}R + \frac{5}{2}R = 6R \]

### Step 5: Calculate \( \frac{C_P}{C_V} \)
- For **monatomic gas**:
\[ \frac{C_P}{C_V} = \frac{\frac{5}{2}R}{\frac{3}{2}R} = \frac{5}{3} \approx 1.67 \]

- For **diatomic gas**:
\[ \frac{C_P}{C_V} = \frac{\frac{7}{2}R}{\frac{5}{2}R} = \frac{7}{5} = 1.4 \]

### Step 6: Calculate \( C_P \times C_V \)
- For **monatomic gas**:
\[ C_P \times C_V = \left(\frac{5}{2}R\right) \times \left(\frac{3}{2}R\right) = \frac{15}{4}R^2 \]

- For **diatomic gas**:
\[ C_P \times C_V = \left(\frac{7}{2}R\right) \times \left(\frac{5}{2}R\right) = \frac{35}{4}R^2 \]

### Step 7: Compare the Results
- **For \( C_P - C_V \)**: Both gases yield the same result \( R \).
- **For \( C_P + C_V \)**: Diatomic gas has a larger value (6R) compared to monatomic gas (4R).
- **For \( \frac{C_P}{C_V} \)**: Monatomic gas has a larger ratio (1.67) compared to diatomic gas (1.4).
- **For \( C_P \times C_V \)**: Diatomic gas has a larger product \( \frac{35}{4}R^2 \) compared to monatomic gas \( \frac{15}{4}R^2 \).

### Conclusion
From the calculations, we can conclude that:
1. \( C_P - C_V \) is the same for both gases.
2. \( C_P + C_V \) is larger for diatomic gas.
3. \( \frac{C_P}{C_V} \) is larger for monatomic gas.
4. \( C_P \times C_V \) is larger for diatomic gas.