A ** path function** is a mathematical function that describes thermodynamic processes that are involved in a change of equilibrium states. Unlike a state function, whose output is independent of the path taken to reach it, the output of a path function is path-dependent.

An example of a path function is the work done by an ideal gas. Consider an ideal gas in a piston-cylinder device immersed in a water bath (see diagram below).

It is evident from the above PV diagram that there are three different paths for work done in bringing the system from one equilibrium state to another equilibrium state :

Amongst the three paths, A to B to C requires the greatest work, while A to D to C requires the least amount of work. This is because the system is expanding reversibly against a higher constant pressure throughout the process from A to B at , versus that from D to C at . The path from A to C requires intermediate work done as the system is expanding reversibly against a decreasing pressure from to . Relative work done by the three paths can be inspected visually by estimating the areas under the respective PV curves. The precise values are calculated using eq6 and eq7. Since work done has values that are dependent on the path taken from one thermodynamic state to another, it is a path function.

Lastly, the differential form of the reversible isobaric expansion of an ideal gas is

We have used the symbol to emphasise that is a path function but the symbol is also acceptable. Eq23 is called an ** inexact differential** because its integral is not path independent. Even though work has the same units (Joules) as energy, it is misleading to say that work is a form of energy, which if it is, will be described by a thermodynamic function that is path-independent. Work is rather, a process of energy transfer between a system and its surroundings.

###### Question

Show that is an inexact differential, while is an exact differential.

###### Answer

Comparing the first equation to the general form of a differential equation , we have and . We need to show that for an inexact differential:

and

For , its second cross partial derivatives are and . Therefore is an exact differential.