`C_v and C_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 then for a monatomic ideal gas
B. `C_p + C_v` is larger for a diatomic ideal gas than for a monatomic ideal gas
C. `C_P//C_v` is larger for a diatomic ideal gas than for a monatomic ideal gas
D. `C_p C_v` is larger for a diatomic ideal gas than for a monatomic ideal gas

Explanation

To solve the question regarding the molar specific heat capacities \( C_v \) and \( C_p \) of gases, we need to analyze the statements provided and derive the necessary relationships. Here’s a step-by-step solution:

### Step 1: Understanding \( C_v \) and \( C_p \)
- \( C_v \) is the molar specific heat capacity at constant volume.
- \( C_p \) is the molar specific heat capacity at constant pressure.
- For any ideal gas, the relationship between \( C_p \) and \( C_v \) is given by:
\[ C_p - C_v = R \]
where \( R \) is the universal gas constant.

### Step 2: Analyzing the first statement
- The first statement claims that \( C_p - C_v \) is greater for diatomic gases than for monatomic gases.
- Since \( C_p - C_v = R \) is a constant for all ideal gases, this statement is **false**.

### Step 3: Analyzing the second statement
- The second statement claims that \( C_p + C_v \) is greater for diatomic gases than for monatomic gases.
- We know:
\[ C_v = \frac{f}{2} R \]
where \( f \) is the degrees of freedom.
- For monatomic gases, \( f = 3 \) (translational motion only), so:
\[ C_v = \frac{3}{2} R \]
and thus:
\[ C_p = C_v + R = \frac{3}{2} R + R = \frac{5}{2} R \]
- For diatomic gases, \( f = 5 \) (3 translational + 2 rotational), so:
\[ C_v = \frac{5}{2} R \]
and thus:
\[ C_p = C_v + R = \frac{5}{2} R + R = \frac{7}{2} R \]
- Therefore:
\[ C_p + C_v = \frac{5}{2} R + \frac{7}{2} R = 6R \quad \text{(for diatomic)} \]
\[ C_p + C_v = \frac{3}{2} R + \frac{5}{2} R = 4R \quad \text{(for monatomic)} \]
- This statement is **true**.

### Step 4: Analyzing the third statement
- The third statement claims that \( \frac{C_p}{C_v} \) is greater for monatomic gases than for diatomic gases.
- We know:
\[ \frac{C_p}{C_v} = 1 + \frac{R}{C_v} \]
- For monatomic gases:
\[ \frac{C_p}{C_v} = 1 + \frac{R}{\frac{3}{2}R} = 1 + \frac{2}{3} = \frac{5}{3} \]
- For diatomic gases:
\[ \frac{C_p}{C_v} = 1 + \frac{R}{\frac{5}{2}R} = 1 + \frac{2}{5} = \frac{7}{5} \]
- Since \( \frac{5}{3} > \frac{7}{5} \), this statement is **false**.

### Step 5: Analyzing the fourth statement
- The fourth statement claims that \( C_p \cdot C_v \) is greater for diatomic gases than for monatomic gases.
- We calculate:
\[ C_p \cdot C_v \text{ for monatomic} = \left(\frac{5}{2} R\right) \left(\frac{3}{2} R\right) = \frac{15}{4} R^2 \]
\[ C_p \cdot C_v \text{ for diatomic} = \left(\frac{7}{2} R\right) \left(\frac{5}{2} R\right) = \frac{35}{4} R^2 \]
- Since \( \frac{35}{4} R^2 > \frac{15}{4} R^2 \), this statement is **true**.

### Conclusion
The correct statements are:
- Statement 2: \( C_p + C_v \) is greater for diatomic gases than for monatomic gases.
- Statement 4: \( C_p \cdot C_v \) is greater for diatomic gases than for monatomic gases.