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 1: Understand the relationship between internal energy and temperature
The internal energy \( U \) of an ideal gas is a function of its temperature. The change in internal energy \( \Delta U \) can be expressed as:
\[ \Delta U = n C_v \Delta T \]
where \( n \) is the number of moles, \( C_v \) is the heat capacity at constant volume, and \( \Delta T \) is the change in temperature.

### Step 2: Use the ideal gas law
The ideal gas law states that:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

### Step 3: Relate the given law to the ideal gas law
We are given that \( P^2 V = \text{constant} \). Let's denote this constant as \( k \):
\[ P^2 V = k \]
From the ideal gas law, we can express \( P \) in terms of \( V \) and \( T \):
\[ P = \frac{nRT}{V} \]
Substituting this into the equation \( P^2 V = k \):
\[ \left(\frac{nRT}{V}\right)^2 V = k \]
This simplifies to:
\[ \frac{n^2 R^2 T^2}{V} = k \]
or
\[ n^2 R^2 T^2 = kV \]

### Step 4: Express temperature in terms of volume
Rearranging the above equation gives us:
\[ T^2 = \frac{kV}{n^2 R^2} \]
Taking the square root:
\[ T = \sqrt{\frac{k}{n^2 R^2}} \sqrt{V} \]
Let \( C' = \sqrt{\frac{k}{n^2 R^2}} \), then:
\[ T = C' \sqrt{V} \]

### Step 5: Analyze the relationship between temperature and volume
From the equation \( T = C' \sqrt{V} \), we can see that as the volume \( V \) increases, the temperature \( T \) also increases because \( C' \) is a positive constant.

### Step 6: Relate temperature back to internal energy
Since the internal energy \( U \) is directly related to temperature, and we have established that temperature increases with volume, we conclude that:
\[ \Delta U \propto \Delta T \]
Thus, as the volume increases, the internal energy of the gas also increases.

### Conclusion
The internal energy of the gas increases continuously as it expands according to the law \( P^2 V = \text{constant} \).