Identify the state functions and path functions out of the following : enthalpy, entropy, heat, temperature, w

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Q. Identify the state functions and path functions out of the following : enthalpy, entropy, heat, temperature, work, free energy.

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Concept understanding — State Function

What is a State Function? The Intuition

Imagine you're standing at the base of a hill. Your height above sea level is a number — say, 100 metres. Now, you walk to the top of the hill. Your height is now 500 metres.

Here's the key question: Does it matter how you got to the top? Did you take the steep path, the winding road, or did you get carried up by a helicopter?

The answer is no. Your height at the top is 500 metres, regardless of the path you took. Height is a state function — it depends only on where you are, not on how you got there.

Now contrast that with distance walked. If you took the winding road, you walked 2 km. If you took the steep path, you walked 500 m. The distance walked depends entirely on the path. That's not a state function — it's a path function.

Note

A state function is a property whose value depends only on the current state of the system, not on the history or path taken to reach that state.

The Precise Statement

In thermodynamics and physics, a state function (or state variable) is any property of a system that is determined entirely by the system's current equilibrium conditions — typically its temperature, pressure, volume, composition, and so on.

If you know the state of the system (say, "1 mole of ideal gas at 300 K and 1 atm"), then every state function has a fixed value. You don't need to know whether the gas was heated slowly, compressed quickly, or cooled then expanded. The value is the same.

Common State Functions in Chemistry & Physics

Property	Symbol	Why it's a state function
Pressure	$P$	Depends only on current $T$ , $V$ , $n$
Volume	$V$	Depends only on current $P$ , $T$ , $n$
Temperature	$T$	A fundamental state variable
Internal Energy	$U$	Depends only on current $P$ , $T$ , composition
Enthalpy	$H$	$H = U + PV$ — a combination of state functions
Entropy	$S$	Depends only on current state
Gibbs Free Energy	$G$	$G = H - TS$ — again, a combination

Common Path Functions (the opposite)

Property	Why it's a path function
Work ( $W$ )	Depends on how you change volume (e.g., fast vs slow)
Heat ( $Q$ )	Depends on how you transfer energy (e.g., conduction vs radiation)

The Mathematical Signature

Here's the crisp, exam-ready way to recognise a state function:

Important

For a state function $f$ , the cyclic integral is zero:

$\oint df = 0$

This means: if you go from state A to state B and back to A by any path, the net change in $f$ is zero. The value of $f$ at A is always the same when you return.

Equivalently, the change in a state function between two states is path-independent:

\Delta f = f_{\text{final}} - f_{\text{initial}}

This is a single number — no integral over a path needed.

A Concrete Example: Internal Energy

Consider a gas in a cylinder. You take it from State 1 ( $T_1 = 300$ K, $P_1 = 1$ atm) to State 2 ( $T_2 = 400$ K, $P_2 = 2$ atm).

Path A: Heat the gas at constant volume, then compress it at constant temperature.
Path B: Compress the gas at constant temperature, then heat it at constant volume.

The work done ( $W$ ) and heat transferred ( $Q$ ) will be different for Path A vs Path B. But the change in internal energy $\Delta U$ will be identical for both paths. That's because $U$ is a state function — it only cares about the starting and ending states. …

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State Functions vs. Path Functions

Concept: A state function depends only on the current state of the system (initial and final conditions), independent of the path taken. A path function depends on the specific process or route followed.

Classification:

The thermodynamic properties divide as follows:

Property	Type	Reason
Enthalpy ( $H$ )	State function	Depends only on $P$ , $T$ , composition
Entropy ( $S$ )	State function	Depends only on state variables
Temperature ( $T$ )	State function	Defines equilibrium state
Free energy ( $G$ or $A$ )	State function	Combination of state functions

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State functions depend only on the current state of the system, not how it got there; path functions depend on the route taken. State functions: enthalpy, entropy, temperature, free energy. Path functions: heat, work.

The Core Idea: Where You Are vs. How You Got There

Thermodynamics distinguishes between two types of quantities. A state function is a property that depends only on the present condition of the system—its pressure, temperature, composition, and so on. If you know the state, you know the value, regardless of the system's history. Think of altitude on a mountain: your elevation depends only on where you stand, not which trail you hiked.

A path function, by contrast, depends on the process itself—the specific route taken between two states. The work you do climbing that mountain depends on whether you took the steep direct path or the gentle winding trail, even though you end at the same elevation.

Mathematically, state functions have exact differentials: when you integrate around any closed loop, the net change is zero. Path functions do not; their integrals depend on the path.

Sorting the Six Quantities

Let's examine each in turn.

Enthalpy, $H$

Defined as $H = U + PV$ , where $U$ is internal energy. Both $U$ , $P$ , and $V$ are properties of the state. Once you specify temperature, pressure, and composition, $H$ is fixed. The enthalpy change $\Delta H$ between two states is the same no matter whether you heat at constant pressure, do work, or follow any other route.

State function.
Entropy, $S$

A measure of disorder or the number of accessible microstates. For a reversible process, $\mathrm{d}S = \frac{\delta q_{\text{rev}}}{T}$ , and the integral $\int \mathrm{d}S$ around a cycle is zero (second law). Entropy is a property of the equilibrium state.

State function.
Heat, $q$

Energy transferred due to a temperature difference. The amount of heat exchanged between two states depends entirely on the process: an adiabatic path transfers zero heat, an isothermal path transfers $q = nRT \ln(V_f/V_i)$ for an ideal gas, and so on. Heat is not a property stored in the system.

Path function.
Temperature, $T$

An intensive property that characterizes thermal equilibrium. Once the state is fixed, so is the temperature. It does not depend on how the system reached that state.

State function.
Work, $w$ …

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Q.Identify the state functions and path functions out of the following : enthalpy, entropy, heat, temperature, work, free energy.

The Core Idea: Where You Are vs. How You Got There

Sorting the Six Quantities

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Q.Identify the state functions and path functions out of the following :
enthalpy, entropy, heat, temperature, work, free energy.

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