At ordinary temperatures, the molecules of an ideal gas have only translational and rotational kinetic energies. At high temperatures they may also have vibrational energy.
As a result of this, at higher temperature

Options

A. `C_(V) = (3 R)/(2)` for a monatomic gas
B. `C_(V) gt (3 R)/(2)` for a monatomic gas
C. `C_(V) lt (3 R)/(2)` for a diatomic gas
D. `C_(V) gt (3 R)/(2)` for a diatomic gas

Explanation

To solve the question regarding the kinetic energy of molecules in an ideal gas at different temperatures, we will analyze the degrees of freedom for monoatomic and diatomic gases at low and high temperatures.

### Step-by-Step Solution:

1. **Understanding Degrees of Freedom:**
- Degrees of freedom refer to the number of independent ways in which a system can possess energy. For gases, this includes translational, rotational, and vibrational motion.

2. **Degrees of Freedom at Low Temperature:**
- For a **monoatomic gas** (e.g., helium), the molecules have only translational kinetic energy. Therefore, the degrees of freedom (F) is:
\[ F = 3 \quad \text{(3 translational)} \]
- For a **diatomic gas** (e.g., nitrogen), at low temperatures, the molecules have translational and rotational kinetic energy, leading to:
\[ F = 5 \quad \text{(3 translational + 2 rotational)} \]

3. **Degrees of Freedom at High Temperature:**
- At high temperatures, diatomic gases can also gain vibrational energy. Thus, the degrees of freedom for diatomic gases increases to:
\[ F = 7 \quad \text{(3 translational + 2 rotational + 2 vibrational)} \]
- For monoatomic gases, the degrees of freedom remain the same:
\[ F = 3 \quad \text{(still only translational)} \]

4. **Specific Heat Capacity (C_v):**
- The specific heat capacity at constant volume (C_v) can be calculated using the formula:
\[ C_v = \frac{F}{2} R \]
- For **monoatomic gases**:
\[ C_v = \frac{3}{2} R \quad \text{(remains constant at all temperatures)} \]
- For **diatomic gases**:
- At low temperature:
\[ C_v = \frac{5}{2} R \]
- At high temperature:
\[ C_v = \frac{7}{2} R \]

5. **Conclusion:**
- At high temperatures, the specific heat capacity \(C_v\) for diatomic gases is greater than that for monoatomic gases. Therefore, we can conclude:
- For monoatomic gases, \(C_v\) is always \( \frac{3}{2} R \).
- For diatomic gases, \(C_v\) increases from \( \frac{5}{2} R \) to \( \frac{7}{2} R \) as temperature increases.

### Final Answer:
- At higher temperatures, the specific heat capacity \(C_v\) for diatomic gases is greater than \( \frac{3}{2} R \), confirming that the energy contributions from vibrational modes become significant.