An ideal gas expands according to the law `P^(2) V ` = constant . The internal energy of the gas

Options

A. Increases continuously
B. Decreases continuously
C. Remain constant
D. First increases and then decreases

Answer

To solve the problem of how the internal energy of an ideal gas changes when it expands according to the law \( P^2 V = \text{constant} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the given law**: The gas expands according to the law \( P^2 V = \text{constant} \). This implies that as the volume \( V \) changes, the pressure \( P \) also changes in such a way that the product \( P^2 V \) remains constant.

2. **Rearranging the equation**: We can rewrite the equation as:
\[ PV^{1/2} = \text{constant} \]
This is valid because taking the square root of a constant gives another constant.

3. **Identify the type of process**: The equation \( PV^{1/2} = \text{constant} \) indicates that this is a polytropic process. In a polytropic process, the relationship between pressure and volume is defined by \( PV^n = \text{constant} \) where \( n \) is the polytropic index. Here, \( n = \frac{1}{2} \).

4. **Work done in the polytropic process**: The work done \( W \) during a polytropic process is given by the formula:
\[ W = \frac{nR \Delta T}{1 - n} \]
For our case, substituting \( n = \frac{1}{2} \):
\[ W = \frac{2R \Delta T}{1 - \frac{1}{2}} = 2R \Delta T \]

5. **Analyze the change in temperature**: Since the gas is expanding, the work done \( W \) is positive. For an ideal gas, the work done on the gas is related to the change in internal energy and the change in temperature. As the gas expands, it absorbs heat, which leads to an increase in temperature (\( \Delta T > 0 \)).

6. **Relate internal energy to temperature**: The change in internal energy \( \Delta U \) for an ideal gas is given by:
\[ \Delta U = nC_V \Delta T \]
where \( C_V \) is the molar heat capacity at constant volume. Since \( \Delta T \) is positive, \( \Delta U \) will also be positive.

7. **Conclusion**: Therefore, as the gas expands according to the given law, the internal energy of the gas increases continuously.

### Final Answer:
The internal energy of the gas increases continuously.