---
title: "Test for property; state and path function"
subject: "Fundamental of Thermodynamics & Heat Transfer"
chapter: 1
key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:thermodynamic-property-test-state-path"
source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/thermodynamic-property-test-state-path
site: BCT Engineering
---

# Test for property; state and path function

Fundamental of Thermodynamics & Heat Transfer · Chapter 1: Introduction

- Asked in: 2073 Magh, 4 marks
- Video: [Path and Point Function](https://www.youtube.com/watch?v=9CjPDXujjBY) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/thermodynamic-property-test-state-path
- Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-1

## Question

Explain how will you find out whether a given variable is a thermodynamic property or not. Also define state function and path function.

## Short answer

A quantity is a property if its cyclic integral is zero and change depends only on endpoints (exact differential). State functions (e.g., internal energy) are path-independent; path functions (heat, work) are inexact and path-dependent.

## Answer

#### Test for a thermodynamic property
A quantity is a thermodynamic property only if:
1. It has a definite value at each equilibrium state of the system.
2. Its change between two states depends only on the end states, not on the path. Mathematically its differential is exact:
   ∮ dX = 0 for any cycle, and ∫₁² dX = X₂ − X₁.
3. For X = f(x, y), the mixed derivatives are equal: ∂²X/∂x∂y = ∂²X/∂y∂x (Euler's test).

If the cyclic integral is not zero or the value depends on the process followed, it is not a property. Example: pressure, volume, T are properties; heat and work are not.

#### State function (point function)
A quantity whose value depends only on the state of the system, not on how the state was reached. Its differential is exact (dU, dH, dP, dV). Examples: internal energy, enthalpy, entropy.

#### Path function
A quantity whose value depends on the path (process) followed between two states. Its differential is inexact (δQ, δW). The area under a process curve on a P-V diagram (work) changes with the path. Examples: heat and work.

```text
 P |1
   | \  path A: W_A
   |  \_____
   |   path B: W_B   (W_A != W_B)
   |        \__2
   +--------------- V
```

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which test shows a quantity X is a property?
   A. Its value depends on the process followed
   B. Its cyclic integral is nonzero
   C. Its cyclic integral is zero and its differential is exact
   D. It crosses the boundary during a process
   Answer: C. A property has an exact differential, so the cyclic integral of dX is zero and change depends only on end states. (Key point: Test for property)

2. Which pair are path functions?
   A. Heat and work
   B. Internal energy and enthalpy
   C. Pressure and volume
   D. Entropy and temperature
   Answer: A. Heat and work depend on the path and have inexact differentials (δQ, δW). (Key point: Path function)

3. Why does work change with the path between the same two states?
   A. Because pressure is a path function
   B. Because the area under the P-V curve differs for different paths
   C. Because the end states change
   D. Because work is a state function
   Answer: B. Work is the area under the process curve, which varies with the path. (Key point: Work on P-V diagram)

4. What is a state function (point function)?
   A. A quantity that depends on the path with inexact differential
   B. A quantity present only at the boundary
   C. A quantity depending only on the state, with exact differential
   D. A quantity that is always zero over a cycle of heat
   Answer: C. State functions like U, H, P, V depend only on state and have exact differentials. (Key point: State function)

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