[figure] A piston of mass M is hung from a massless spring whose restoring force law goes as F = -kx3, 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 L0 to L1, the total energy delivered by the filament is: (Assume spring to be in its natural length before heating)
- nRT ( L1L0 ) + Mg(L1 - L0) + 3k4 (L14 - L04 )
- nRT ( L1L0 ) + Mg(L1 - L0) + k4 (L14 - L04 )
- nRT ( L12L02 ) + Mg2(L1 - L0) + k4 (L14 - L04 )
- 3nRT ( L1L0 ) + 2Mg(L1 - L0) + k3 (L13 - L03 )
Answer
(2) nRT ( L1L0 ) + Mg(L1 - L0) + k4 (L14 - L04 )
Solution
The energy delivered by the filament is taken as the sum of three parts: the isothermal work on the gas, the rise in potential energy of the piston and the energy stored in the spring.
Gas (isothermal, V = A0L, A0 = cross-section of the chamber):
Wgas = nRT V1V0 = nRT ( L1L0 )
Piston raised through (L1 - L0): Ug = Mg(L1 - L0)
Spring: restoring force of magnitude kx3, with x read as the piston height (as in the options):
Us = L0L1 kx3 dx = k4 (L14 - L04 )
E = nRT ( L1L0 ) + Mg(L1 - L0) + k4 (L14 - L04 )
Answer: (2)