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

We have one mole of an ideal gas. For such a gas the fundamental equation of state is $$PV = RT$$ where $$P$$ is pressure, $$V$$ is volume, $$T$$ is absolute temperature and $$R$$ is the universal gas constant (for one mole the specific gas constant is simply $$R$$). Using this single relation we can test every statement one by one.

Testing statement (a)
We recall the definitions of internal energy and enthalpy. Internal energy $$U$$ is a state function that, for an ideal gas, depends only on temperature. This empirical fact is written as

$$U = U(T)$$

because in an ideal gas there are no intermolecular potential-energy contributions; only the kinetic energy of molecules matters, and kinetic energy is a function of temperature alone.

Next we write the definition of enthalpy:

$$H = U + PV$$

Substituting the ideal-gas relation $$PV = RT$$ into the expression for one mole gives

$$H = U + RT$$

Since we already have $$U = U(T)$$ and $$R$$ is a constant, it follows that

$$H = H(T)$$

Thus both $$U$$ and $$H$$ depend only on temperature. Statement (a) is therefore true.

Testing statement (b)
The compressibility factor $$Z$$ is defined as

$$Z = \dfrac{PV}{RT}$$

For an ideal gas the defining relation is again $$PV = RT$$, and hence

$$Z = \dfrac{RT}{RT} = 1$$

Thus $$Z$$ equals 1, not “not equal to 1”. Statement (b) is false.

Testing statement (c)
First recall the general thermodynamic identity that relates molar heat capacities of an ideal gas:

$$C_{P,m} - C_{V,m} = R$$

Here $$C_{P,m}$$ is the molar heat capacity at constant pressure and $$C_{V,m}$$ is the molar heat capacity at constant volume. We can verify this quickly. Differentiate the earlier result $$H = U + RT$$ at constant composition:

$$dH = dU + R\,dT$$

By definition of molar heat capacities,

$$dH = C_{P,m}\,dT \quad\text{and}\quad dU = C_{V,m}\,dT$$

Substituting these into the differential equation gives

$$C_{P,m}\,dT = C_{V,m}\,dT + R\,dT$$

Dividing by $$dT$$ yields exactly

$$C_{P,m} - C_{V,m} = R$$

Therefore statement (c) is true.

Testing statement (d)
Because, as shown earlier, $$U$$ is a function only of temperature for an ideal gas, the total differential of $$U$$ is simply

$$dU = \dfrac{dU}{dT}\,dT$$

The partial derivative $$\dfrac{dU}{dT}$$ at constant volume is the molar heat capacity at constant volume, $$C_{V,m}$$. Hence

$$dU = C_{V,m}\,dT$$

and this holds for any process, not only a constant-volume one, because the right-hand side involves only temperature. Statement (d) is therefore true.

Collecting the results
Statements (a), (c) and (d) are true, while statement (b) is false.

Hence, the correct answer is Option D.