A sample of 1.0 mol of a monoatomic ideal gas is taken through a cyclic process of expansion and compression as shown in Fig. 6.1. What will be the value of $\Delta H$ for the cycle as a whole? ![Fig. 6.1 — cyclic process on a pressure-volume plot: from point 1 the gas expands along a curve to point 2, is compressed at constant pressure from 2 to 3, and returns at constant volume from 3 to 1](/figures/thermodynamics/cyclic-process-pv-diagram-ideal-gas-exemplar-fig-6-1.svg)

A sample of 1.0 mol of a monoatomic ideal gas is taken through a cyclic process of expansion and compression a

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\Delta H

Fig. 6.1 — cyclic process on a pressure-volume plot: from point 1 the gas expands along a curve to point 2, is compressed at constant pressure from 2 to 3, and returns at constant volume from 3 to 1 — Figure 6.1

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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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Enthalpy is a state function, so for any cyclic process that returns the system to its initial state, the change in enthalpy must be zero. $\Delta H = 0$ for the complete cycle.

Why enthalpy change vanishes in a cycle

Enthalpy $H$ is a state function, meaning its value depends only on the current state of the system (pressure, temperature, composition) and not on the path taken to reach that state. This is the central insight.

When a system undergoes a cyclic process, it traces some path through different states but ultimately returns to exactly where it started: same pressure, same temperature, same volume, same everything. Because $H$ depends only on the state, and the initial and final states are identical, the change in enthalpy around the complete cycle must be:

\Delta H_{\text{cycle}} = H_{\text{final}} - H_{\text{initial}} = H_{\text{initial}} - H_{\text{initial}} = 0

This reasoning applies to any state function—internal energy $U$ , Gibbs free energy $G$ , entropy $S$ (for a reversible cycle), etc.

Step-by-step reasoning

Identify what changes during the cycle.

The gas expands and compresses through various intermediate states. At different points in the cycle, $P$ , $V$ , and $T$ all vary. The enthalpy at each intermediate point is different.
Recognize the defining property of a cycle.

The process is cyclic: the gas returns to its initial state. Every thermodynamic property that defines the state—pressure, volume, temperature, and therefore enthalpy—has the same value at the end as at the beginning.
Apply the state-function property.

For a state function, only the endpoints matter:

\Delta H = \int_{i}^{f} dH = H_f - H_i

Since $H_f = H_i$ for a cycle, we have $\Delta H = 0$ .

Check independence from path details. …

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Q.A sample of 1.0 mol of a monoatomic ideal gas is taken through a cyclic process of expansion and compression as shown in Fig. 6.1. What will be the value of $\Delta H$ for the cycle as a whole?

Concept understanding — State Function

What is a State Function? The Intuition

The Precise Statement

Common State Functions in Chemistry & Physics

Common Path Functions (the opposite)

The Mathematical Signature

A Concrete Example: Internal Energy

Concept: State Function Property of Enthalpy

Why enthalpy change vanishes in a cycle

Step-by-step reasoning

Q.A sample of 1.0 mol of a monoatomic ideal gas is taken through a cyclic process of expansion and compression as shown in Fig. 6.1. What will be the value of ΔH\Delta HΔH for the cycle as a whole?

Why enthalpy change vanishes in a cycle

Step-by-step reasoning

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Q.A sample of 1.0 mol of a monoatomic ideal gas is taken through a cyclic process of expansion and compression as shown in Fig. 6.1. What will be the value of $\Delta H$ for the cycle as a whole?

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