A gas is expanded to double its volume by two different processes. One is isobaric and the other is isothermal. Let `W_(1)` and `W_(2)` be the respective work done, then find `W_(1)` and `W_(2)`

Options

A. A) `W_(2)=W_(1)ln(2)`
B. B) `W_(2)=(W_(1))/("ln"(2))`
C. C) `W_(2)=(W_(1))/(2)`
D. D) Data is insufficient

Answer

To solve the problem of finding the work done in two different processes (isobaric and isothermal) when a gas is expanded to double its volume, we can follow these steps:

### Step 1: Understand the Processes
1. **Isobaric Process**: In an isobaric process, the pressure remains constant. The work done \( W_1 \) during an isobaric expansion can be calculated using the formula:
\[ W_1 = P \Delta V \]
where \( \Delta V \) is the change in volume.

2. **Isothermal Process**: In an isothermal process, the temperature remains constant. The work done \( W_2 \) during an isothermal expansion can be calculated using the formula:
\[ W_2 = nRT \ln\left(\frac{V_f}{V_i}\right) \]
where \( n \) is the number of moles, \( R \) is the gas constant, \( V_f \) is the final volume, and \( V_i \) is the initial volume.

### Step 2: Calculate Work Done in Isobaric Process
1. Let the initial volume \( V_i = V_1 \) and the final volume \( V_f = V_2 = 2V_1 \).
2. The change in volume \( \Delta V = V_f - V_i = 2V_1 - V_1 = V_1 \).
3. Thus, the work done in the isobaric process is:
\[ W_1 = P \cdot V_1 \]

### Step 3: Calculate Work Done in Isothermal Process
1. For the isothermal process, using the same initial and final volumes:
\[ W_2 = nRT \ln\left(\frac{V_f}{V_i}\right) = nRT \ln\left(\frac{2V_1}{V_1}\right) = nRT \ln(2) \]

### Summary of Results
- The work done in the isobaric process is:
\[ W_1 = P \cdot V_1 \]
- The work done in the isothermal process is:
\[ W_2 = nRT \ln(2) \]

### Step 4: Relationship Between \( W_1 \) and \( W_2 \)
From the derived equations, we can see that:
\[ W_2 = W_1 \cdot \ln(2) \quad \text{(if we express \( P \) in terms of \( nRT/V_1 \))} \]

### Final Answer
- \( W_1 = P \cdot V_1 \)
- \( W_2 = nRT \ln(2) \)