For one mole of an ideal gas, which of these statements must be true?
(a) U and H each depends only on temperature
(b) Compressibility factor z is not equal to 1
(c) C P, m – C V, m = R
(d) dU = C V dT for any process

Solution

Chemistry ma’am · 3 steps

One fact settles this: for an ideal gas, U depends only on T. Test every statement against it.

1. Start from U. Ideal gas molecules do not attract or repel each other, so the internal energy U does not change with volume or pressure. It depends only on T. Since PV = RT for one mole, H = U + PV = U + RT also depends only on T.

   Statement	True or false	Reason
   (a) U and H depend only on T	True	U = U(T) and H = U + RT
   (b) z ≠ 1	False	z = PV/RT = 1 exactly for an ideal gas
   (c) C_(P,m) - C_(V,m) = R	True	Differentiate H = U + RT with respect to T
   (d) dU = C_V dT for any process	True	U depends only on T, so the path does not matter

   C_(P,m) = dHₘ/dT = dUₘ/dT + R = C_(V,m) + R

2. Why (d) holds. C_V is measured at constant volume, but for an ideal gas U changes only when T changes. Any process with the same dT gives the same dU, so dU = C_V dT holds even when the volume changes.
3. So. Statements (a), (c) and (d) are true and (b) is false. The answer is option (a).

The idea to keep. For an ideal gas, U and H are functions of T alone. That is why dU = C_V dT and dH = C_P dT work for every process, not just constant volume or constant pressure.

Answer from the NTA final answer key. Solution by PrepWiser.