An ideal gas expands in such a way that `PV^2 = `constant throughout the process.

Options

A. This expansion is not possible without heating
B. This expansion is not possible without cooling
C. Internal energy remains constant in this expansion
D. Internal energy increases in this expansion

Explanation

To solve the problem, we need to analyze the relationship between pressure (P), volume (V), and temperature (T) of an ideal gas under the condition that \( PV^2 = \text{constant} \).

### Step-by-Step Solution:

1. **Understand the Given Condition**:
We are given that \( PV^2 = \text{constant} \). This implies that as the gas expands or contracts, the product of pressure and the square of volume remains constant.

2. **Relate Pressure to Temperature**:
Using the ideal gas law, we know:
\[ PV = nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

3. **Express \( P \) in Terms of \( V \) and \( T \)**:
From the ideal gas law, we can express \( P \) as:
\[ P = \frac{nRT}{V} \]

4. **Substitute \( P \) into the Given Condition**:
Substitute \( P \) into the equation \( PV^2 = \text{constant} \):
\[ \left(\frac{nRT}{V}\right)V^2 = \text{constant} \]
Simplifying this gives:
\[ nRTV = \text{constant} \]

5. **Analyze the Relationship Between Volume and Temperature**:
From \( nRTV = \text{constant} \), we can rearrange it to find:
\[ TV = \frac{\text{constant}}{nR} \]
This shows that \( T \) is inversely proportional to \( V \):
\[ T \propto \frac{1}{V} \]

6. **Determine the Effect of Expansion on Temperature**:
As the gas expands (i.e., \( V \) increases), the temperature \( T \) must decrease to keep the product \( TV \) constant. Therefore, during expansion, the temperature decreases.

7. **Internal Energy Consideration**:
For an ideal gas, the internal energy \( U \) depends only on temperature. Since we have established that the temperature decreases during expansion, the internal energy also decreases.

8. **Conclusion**:
- During the expansion of the gas, the temperature decreases.
- The internal energy of the gas also decreases.

### Final Answer:
The correct option is that the gas cools down during expansion, and the internal energy decreases.