An ideal gas expands in such a manner that its pressure and volume can be related by equation `PV^(3)=constant`. During this process, the gas is

Options

A. heated
B. cooled
C. neither heated nor cooled
D. first heated and then cooled

Answer

To solve the problem, we need to analyze the relationship between pressure (P), volume (V), and temperature (T) of an ideal gas during the expansion process described by the equation \( PV^3 = \text{constant} \).

### Step-by-Step Solution:

1. **Understand the Given Relationship**:
The problem states that the ideal gas expands such that \( PV^3 = \text{constant} \). This means that as the gas expands, the product of pressure and the cube of volume remains constant.

2. **Use the Ideal Gas Law**:
Recall the ideal gas equation:
\[ PV = nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

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

4. **Substitute Pressure into the Given Relationship**:
Substitute the expression for \( P \) into the equation \( PV^3 = \text{constant} \):
\[ \left(\frac{nRT}{V}\right)V^3 = \text{constant} \]
Simplifying this gives:
\[ nRTV^2 = \text{constant} \]

5. **Analyze the Relationship**:
Since \( nR \) is constant (the number of moles and the gas constant do not change), we can write:
\[ TV^2 = \text{constant} \]
This indicates that the product of temperature \( T \) and the square of volume \( V^2 \) remains constant.

6. **Determine the Effect of Volume Change on Temperature**:
If the volume \( V \) is increasing (as the gas expands), then for the product \( TV^2 \) to remain constant, the temperature \( T \) must decrease. This is because if one factor in a product is increasing, the other must decrease to keep the product constant.

7. **Conclusion**:
Since the volume is increasing, the temperature of the gas must be decreasing. Therefore, the gas is being cooled during the expansion process.

### Final Answer:
The gas is **cooled** during the expansion process.