For one mole of an ideal gas, which of these statements must be true? (a) Internal energy (U) and enthalpy (H) each…

🧠 Four key properties of one mole of ideal gas — know which always hold For an ideal gas: $U$ and $H$ are functions of $T$ only; $Z = 1$ always; Mayer's relation $C_P - C_V = R$ holds; $dU = C_v\,dT$ for any process.

🗺️ Checking each statement (a) $U$ and $H$ each depend on temperature — True. Both are state functions of $T$ only for ideal gas. ✓

(b) $Z \neq 1$ — False. By definition, an ideal gas obeys $PV = nRT$, so $Z = PV/nRT = 1$ always. ✗

(c) $C_{P,m} - C_{V,m} = R$ — True. Mayer's relation for any ideal gas. ✓

(d) $dU = C_v\,dT$ for any process — True. Since $U$ depends only on $T$ for an ideal gas, this holds regardless of the process type — isochoric, isobaric, isothermal, or adiabatic. ✓

Statements (a), (c), (d) are correct → option (d).

⚠️ Trap: Statement (d) fools many students who think $dU = C_v dT$ applies only to isochoric processes. For an ideal gas, it holds for all processes.

$\boxed{\text{Answer: (d)}}$