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To analyze the work done during the isothermal expansion of an ideal gas, we start with the fundamental equation for work done in an isothermal process:
### Step 1: Work Done Formula
The work done (W) during an isothermal process is given by the formula:
\[
W = -nRT \ln\left(\frac{V_2}{V_1}\right)
\]
where:
- \( n \) = number of moles of the gas,
- \( R \) = universal gas constant,
- \( T \) = absolute temperature,
- \( V_1 \) = initial volume,
- \( V_2 \) = final volume.
### Step 2: Absolute Value of Work Done
Taking the absolute value of work done, we have:
\[
|W| = nRT \ln\left(\frac{V_2}{V_1}\right)
\]
### Step 3: Expressing Work in Terms of Volumes
We can express this as:
\[
|W| = nRT \left(\ln V_2 - \ln V_1\right)
\]
This can be rewritten using the properties of logarithms:
\[
|W| = nRT \ln V_2 - nRT \ln V_1
\]
### Step 4: Relationship Between Work and Volume
If we denote \( V \) as the final volume \( V_2 \) and keep \( V_1 \) constant, we can simplify the expression:
\[
|W| = nRT \ln V - nRT \ln V_1
\]
### Step 5: Graphical Representation
This equation resembles the equation of a straight line \( y = mx + C \), where:
- \( y \) is \( |W| \),
- \( x \) is \( \ln V \),
- \( m \) (the slope) is \( nRT \),
- \( C \) (the y-intercept) is \( -nRT \ln V_1 \).
### Step 6: Analyzing the Graph
In the context of the graph:
- As the temperature \( T_2 \) is greater than \( T_1 \), the slope \( nRT_2 \) will be greater than \( nRT_1 \).
- The intercept \( -nRT_2 \ln V_1 \) will be less negative than \( -nRT_1 \ln V_1 \).
### Conclusion
From the above analysis, we can conclude:
1. The slope of the line corresponding to \( T_2 \) is steeper than that for \( T_1 \).
2. The intercept for \( T_1 \) is less negative than that for \( T_2 \), indicating that the work done increases with temperature for a given change in volume.
Thus, the correct graphical representation of the dependence of work done on the final volume at different temperatures is accurately depicted in the provided option.
