Identification of Thermodynamic State Functions

To determine which of the given quantities are state functions, let us recall the thermodynamic definitions:

1. State Function:
   A state function is a property of a thermodynamic system whose value depends only on the current state of the system (specified by state variables such as pressure $P$, volume $V$, temperature $T$, and amount of substance $n$) and is completely independent of the path or process taken to reach that state. Mathematically, for a state function $X$, the change $\Delta X$ between state $1$ and state $2$ depends only on the initial and final states:
   $\Delta X = X_2 - X_1$
   and its cyclic integral is zero:
   $\oint dX = 0$

2. Path Function:
   A path function is a thermodynamic quantity whose value depends on the specific path followed by the system during a transformation from one state to another. Heat ($q$) and work ($w$) are classic examples of path functions.

Now, let us analyze each of the given options:

• (A) Internal energy ($U$):
  Internal energy is the total energy contained within a thermodynamic system. By the First Law of Thermodynamics, $dU = \delta q + \delta w$. Although heat and work are path functions, their sum $dU$ is an exact differential. Thus, internal energy is a fundamental extensive state function.

• (B) Irreversible expansion work ($w_{\text{irrev}}$):
  Work done during an irreversible expansion depends on the external pressure $P_{\text{ext}}$ and the specific mechanism of expansion ($w = -\int P_{\text{ext}} dV$). Since work depends on the process/path followed, it is a path function, not a state function.

• (C) Reversible expansion work ($w_{\text{rev}}$):
  Although reversible work represents the maximum (or minimum) work along a reversible path, it is still a form of work and depends specifically on the path chosen between the two states (e.g., reversible isothermal path vs. reversible adiabatic path yield different amounts of work). Hence, it is a path function, not a state function.

• (D) Molar enthalpy ($H_{\text{m}}$):
  Enthalpy is defined as:
  $H = U + PV$
  Since $U$, $P$, and $V$ are all state functions, enthalpy $H$ is also a state function. Molar enthalpy is defined as:
  $H_{\text{m}} = \frac{H}{n}$
  Since both enthalpy $H$ and the number of moles $n$ depend only on the state of the system, molar enthalpy is an intensive state function.

Therefore, the state functions are:

• (A) Internal energy
• (D) Molar enthalpy

Correct Options: A, D