Question

A piston of mass M is hung from a massless spring whose restoring force law goes as F = -kx, where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with 'n' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height $ L_0 $ to $ L_1 $, the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

• A. \( 3nRT \ln \left( \frac{L_1}{L_0} \right) + 2Mg(L_1 - L_0) + \frac{k}{3} (L_1^3 - L_0^3) \)
• B. \( nRT \ln \left( \frac{L_1}{L_0} \right) + \frac{Mg}{2} (L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4) \)
• C. \( nRT \ln \left( \frac{L_1}{L_0} \right) + Mg(L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4) \)
• D. \( nRT \ln \left( \frac{L_1}{L_0} \right) + Mg(L_1 - L_0) + \frac{3k}{4} (L_1^4 - L_0^4) \)

Hint:

Apply the work-energy theorem (WET) to equate the total energy supplied to the change in potential energy and work done by the gas. Remember to consider the work done against gravity and the spring force.

Answer 1

The Correct Option is C

Using WET: Total energy supplied = gravitational potential energy + spring potential energy + work done by gas 

\( Mg(L_1 - L_0) + \int_{0}^{L_1-L_0} kx dx + nRT \ln \left( \frac{L_1}{L_0} \right) + W_{ext} = 0 \) \( \frac{k}{4} [x^4]_{0}^{L_1 - L_0} + Mg(L_1 - L_0) + \int_{0}^{L_1-L_0} kx dx + nRT \ln \left( \frac{L_1}{L_0} \right) + W_{ext} = 0 \) 

\( \frac{k}{4} (L_1^4 - L_0^4) + Mg(L_1 - L_0) + nRT \ln \left( \frac{L_1}{L_0} \right) + W_{ext} = 0 \) \( W_{ext} = \frac{k}{4} (L_1^4 - L_0^4) + Mg(L_1 - L_0) + nRT \ln \left( \frac{L_1}{L_0} \right) \)