A piston of mass M is hung from a massless spring whose restoring force law goes as F=-k x³, 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)

Solution

Physics sir · 4 steps

Isothermal, so all the filament's heat becomes gas work: it lifts the piston and loads the spring.

1. First law. At constant T an ideal gas has Δ U = 0, so Q = W_(gas). The energy from the filament equals the work the gas does as the piston rises.
2. Gas side. With cross-section A, V = AL, so V₁/V₀ = L₁/L₀. Isothermal work is W_(gas) = nRT ln(L₁/L₀).
3. Where it goes. The piston rises slowly, so it gains no kinetic energy. The gas work raises the piston, which takes Mg(L₁ - L₀), and stores energy in the spring, which takes ∫ kx³ dx = k/4x⁴ between the limits. The options write this spring term between the piston positions L₀ and L₁, which gives k/4(L₁⁴ - L₀⁴).

   Q = nRT ln L₁/L₀ + Mg(L₁ - L₀) + k/4(L₁⁴ - L₀⁴)

4. So. The total energy delivered matches option (b). Only (b) has the plain ln(L₁/L₀), the full Mg(L₁ - L₀) and the k/4 that comes from integrating x³.

The idea to keep. Isothermal ideal gas: heat in equals work out, nRT ln(V₂/V₁). For a force F = -kxⁿ, stored energy is kxⁿ⁺¹/(n+1): integrate the force, never differentiate it.

Answer from the NTA final answer key. Solution by PrepWiser.