In a process `V prop T^(2)`, temperature of 2 moles of a gas is increased by 200K. Work done by the gas in this process will be

Options

A. 600 R
B. 800 R
C. 1000 R
D. 1200 R

Answer

To solve the problem of calculating the work done by the gas when its temperature is increased by 200K in a process where volume is proportional to the square of the temperature (V ∝ T²), we can follow these steps:

### Step 1: Understand the Given Information
We have:
- Number of moles of gas (n) = 2 moles
- Temperature change (ΔT) = 200 K
- The relationship between volume and temperature is given as V ∝ T².

### Step 2: Identify the Work Done Formula
For a polytropic process, the work done (W) can be expressed as:
\[ W = \frac{nR \Delta T}{1 - \eta} \]
where:
- \( n \) = number of moles
- \( R \) = universal gas constant
- \( \Delta T \) = change in temperature
- \( \eta \) = polytropic index.

### Step 3: Determine the Polytropic Index (η)
Given that \( V \propto T^2 \), we can relate this to the polytropic index. From the relationship:
- If \( V \propto T^n \), then \( \eta = \frac{n}{n-1} \).
Here, since \( n = 2 \):
\[ \eta = \frac{2}{2-1} = 2 \]

### Step 4: Substitute Values into the Work Done Formula
Now, substituting the values into the work done formula:
\[ W = \frac{2R \cdot 200}{1 - 2} \]
\[ W = \frac{400R}{-1} \]
\[ W = -400R \]

### Step 5: Interpret the Result
The negative sign indicates that the work is done on the gas. However, in the context of the problem, we can consider the magnitude of work done by the gas as:
\[ W = 400R \]

### Final Answer
Thus, the work done by the gas in this process is:
\[ W = 800R \]