 Solution

Use $c_p,c_V$ for the heat capacities of the entire $n$-mole sample. For a stable simple compressible gas, a Maxwell relation giveswhere ${\alpha }_V=V^{-1}(\partial V/\partial T{)}_p$ and ${\kappa }_T=-V^{-1}(\partial V/\partial p{)}_T>0$. This heat-capacity difference and mechanical stability explains the sign without assuming the internal energy depends only on temperature. It is strictly positive for an ordinary expanding gas; the unrestricted word “always” needs the qualification that equality is possible when thermal expansion vanishes.

For an ideal gas, thermodynamic consistency givesConsequently $U=U(T)$, while $H=U+pV=U+nRT$. Differentiating gives $c_p-c_V=nR$: at fixed pressure extra supplied heat permits expansion work in addition to the internal-energy increase.

Differentiate the supplied entropy at fixed $n$:Thus $U=5nRT/2$, consistently recovering $pV=nRT$, andThe coefficient in this entropy corresponds to five active quadratic thermal degrees of freedom per molecule. It should not be replaced by the monatomic value. Reference scales needed to make logarithms dimensionless are absorbed into the entropy constant and do not affect these derivatives.

With open windows the house equilibrates mechanically at approximately the same atmospheric pressure before and after heating. It is an open system: its mole number is not fixed. The isobaric ideal-gas internal-energy density isAt a fixed house volume, $n_{300}/n_{273}=273/300=0.91$: fewer molecules occupy the house, each with greater mean energy. Heating energy can enter while enthalpy leaves with escaping air. Treating the room as a sealed fixed-$n$ gas would give the wrong energy-density comparison.