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Step-by-Step Explanation
Step 1: Analyze the statements for an ideal gas
(a) For an ideal gas, both internal energy ($U$) and enthalpy ($H$) depend
only on temperature. This is a well-known property of an ideal gas,
because interactions between gas molecules are assumed to be negligible,
making $U$ and $H$ independent of pressure or volume and purely
functions of temperature.
Step 2: Examine the compressibility factor for an ideal gas
(b) The compressibility factor $Z$ is defined as
$Z = \frac{pV}{nRT}$. For an ideal gas, by definition,
$Z = 1$. Therefore, the statement that $Z \neq 1$ is incorrect
for an ideal gas.
Step 3: Relationship between $C_{P,m}$ and $C_{V,m}$
(c) For one mole of an ideal gas, the molar heat capacities
at constant pressure $C_{P,m}$ and constant volume $C_{V,m}$
differ by the universal gas constant $R$, i.e.,
$C_{P,m} - C_{V,m} = R$.
This is a fundamental result derived from thermodynamics
for an ideal gas.
Step 4: Change in internal energy for any process
(d) The internal energy change of an ideal gas is given by
$dU = C_{V}\,dT$ for any process. This means that the internal
energy of an ideal gas depends only on temperature change, not on
how that temperature change was brought about (path independence).
Step 5: Conclude the correct statements
From the above explanations:
Statement (a) is true.
Statement (b) is false because for an ideal gas, $Z = 1$.
Statement (c) is true.
Statement (d) is true.
Therefore, the correct statements are (a), (c), and (d).
