In a process, the pressure of an ideal gas is proportional to square of the volume of the gas. If the temperature of the gas increases in this process, then work done by this gas

Options

A. (a) is positive
B. (b) is negative
C. (c) is zero
D. (d) may be positive or negative

Explanation

To solve the problem step by step, we will analyze the relationship between pressure, volume, and temperature for an ideal gas, and then determine the work done by the gas.

### Step 1: Understand the relationship given in the problem
The problem states that the pressure (P) of an ideal gas is proportional to the square of its volume (V). This can be expressed mathematically as:
\[ P \propto V^2 \]
This implies that:
\[ P = k V^2 \]
where \( k \) is a proportionality constant.

### Step 2: Use the Ideal Gas Law
The ideal gas law states:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = temperature

### Step 3: Substitute the expression for pressure into the ideal gas law
Substituting \( P = k V^2 \) into the ideal gas law gives:
\[ k V^2 V = nRT \]
This simplifies to:
\[ k V^3 = nRT \]

### Step 4: Relate temperature to volume
From the equation \( k V^3 = nRT \), we can express temperature as:
\[ T = \frac{k V^3}{nR} \]
This shows that temperature is directly proportional to the cube of the volume:
\[ T \propto V^3 \]

### Step 5: Analyze the effect of increasing temperature
If the temperature \( T \) increases, it follows from the relationship \( T \propto V^3 \) that the volume \( V \) must also increase.

### Step 6: Determine the work done by the gas
The work done \( W \) by an expanding gas is given by the formula:
\[ W = \int P \, dV \]
Since we have established that \( P = k V^2 \), we can substitute this into the work formula:
\[ W = \int k V^2 \, dV \]
This integral will yield a positive value since the volume is increasing.

### Conclusion
Since the volume of the gas is increasing as the temperature increases, the work done by the gas is positive.

### Final Answer:
The work done by the gas is positive.

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