Question #42545

This question tests the synthesis of the First Law of Thermodynamics with classical gravitational self-energy.

The primary trap lies in students either attempting to use $W = nRT \ln(V_f/V_i)$ (which fails due to unknown $T$ and non-constant boundary pressure) or incorrectly invoking the Virial Theorem ($2K + U = 0$) which only applies to natural hydrostatic equilibrium, not a constrained isothermal process.

If a student blindly applies the Virial Theorem, they will incorrectly calculate $\Delta U_{\text{int}} = -\frac{1}{2}\Delta U_g$, leading to Option C.

By recognizing that $\Delta U_{\text{int}} = 0$ for an isothermal ideal gas, students must deduce that the work done by the gas exactly equals the change in gravitational binding energy ($\Delta U_g$), requiring the correct formula for the self-energy of a uniform sphere.