Ideal-gas property: for an ideal gas, internal energy (and enthalpy) depend only on temperature. Therefore internal energy is independent of which variables?

Introduction / Context:
In the ideal-gas model, interactions between molecules are neglected, so thermodynamic properties simplify. A key result is that internal energy u (and enthalpy h) are functions of temperature only. This greatly streamlines energy balances for compressors, turbines, and heaters when ideal behavior is acceptable.

Given Data / Assumptions:

• Ideal gas: intermolecular potential energy contributions are negligible.
• State variables: temperature T, pressure P, and volume V are related by PV = nRT.

Concept / Approach:
For an ideal gas, u = u(T) and h = h(T). Consequently, changing pressure or volume at constant temperature does not change internal energy. Any dependence of u on P or V would arise only through T. Therefore, internal energy is independent of both pressure and volume for a given temperature.

Step-by-Step Solution:

Verification / Alternative check:
From statistical mechanics, ideal-gas u reflects translational kinetic energy proportional to T; no configurational energy arises without interactions.

Why Other Options Are Wrong:

• (a) or (b) alone are incomplete; both are independent variables not affecting u at fixed T.
• (d) contradicts the ideal-gas result.

Common Pitfalls:
Extending the conclusion to real gases or to mixtures undergoing reactions or phase changes; for real gases, u can depend on both T and P weakly via interactions, especially at high pressures.

Final Answer:
both (a) and (b)