Question

The work done to increase the volume of 2 moles of an ideal gas from \( V \) to \( 2V \) at a constant temperature \( T \) is \( W \). The work to be done to increase the volume of 2 moles of the same gas from \( 2V \) to \( 4V \) at the same constant temperature \( T \) is:

• A. \( 0.5W \)
• B. \( W \)
• C. \( 2W \)
• D. \( 4W \)

Hint:

For isothermal processes, the work ratio follows: \[ W = nRT \ln \frac{V_f}{V_i} \] where work doubles if volume doubles proportionally in logarithmic scaling.

Answer 1

The Correct Option is C

Step 1: Work Done in Isothermal Expansion For an ideal gas undergoing isothermal expansion, work is given by: \[ W = nRT \ln \frac{V_f}{V_i} \] where: - \( n = 2 \) moles, - \( R \) is the universal gas constant, - \( T \) is the absolute temperature, - \( V_i \) and \( V_f \) are the initial and final volumes. Step 2: Expressing Work Ratio Given: \[ W = 2RT \ln \frac{2V}{V} = 2RT \ln 2 \] For the second expansion \( 2V \to 4V \): \[ W' = 2RT \ln \frac{4V}{2V} = 2RT \ln 2 \] Thus: \[ W' = 2W \] Conclusion Thus, the correct answer is: \[ 2W \]