$A$ piston of mass $M$ is hung from a massless spring whose restoring force law is given by $F = -kx^3$,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 the lower part of the chamber,so that the piston moves up from a height $L_0$ to $L_1$. Calculate the total energy delivered by the filament. (Assume the spring to be in its natural length before heating.)

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