Domain: Thermodynamics

Determine the Gibbs free energy change associated with heating and compressing hydrogen gas from an initial state of \(\mathrm{H_2(g)}\) at 298 K and 1 atm to a final state of \(\mathrm{H_2(g)}\) at 323 K and 2 atm. Treat the gas as ideal throughout the process and assume that its molar heat capacity at constant pressure remains fixed at \(C_{p,m} = 28.87\ \mathrm{J\,K^{-1}\,mol^{-1}}\) over the temperature range considered. The standard molar entropy of \(\mathrm{H_2(g)}\) at 298 K is \(S_{298\text{K}} = 130.58\ \mathrm{J\,K^{-1}\,mol^{-1}}\). Analyze the problem by accounting for both the temperature-dependent entropy contribution and the effect of pressure change on the Gibbs free energy, and compute the total \(\Delta_{r}G\) for the process.

Grading rubric:
criterion: Correctly invokes the Maxwell relation $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$ to analytically derive the pressure-dependent entropy term.
weight: 10
criterion: Correctly explicitly states the partial derivative $(\partial H/\partial P)_T = 0$ for an ideal gas to rigorously justify why enthalpy is pressure-independent.
weight: 10
criterion: Correctly derives the specific algebraic expansion $\Delta G = \Delta H - S_1(T_2 - T_1) - T_2 \Delta S_{process}$.
weight: 20
criterion: Correctly formulates the entropy integrals with explicit limits $T_1 \to T_2$ and $P_1 \to P_2$ before substituting the logarithmic solutions.
weight: 10
criterion: Correctly isolates and calculates the "process entropy correction" energy term $T_2 \Delta S_{process}$ as exactly $-1110.15\ \mathrm{J\ mol^{-1}}$.
weight: 20
criterion: Correctly explicitly identifies the thermodynamic path as a reversible process to validate the substitution of $TdS$ for $dq$.
weight: 20
criterion: Correctly reports the final answer as $-1432.6\ \mathrm{J\ mol^{-1}}$.
weight: 10

Answer:
-1432.6 J mol^-1