A ** state function** is a mathematical function that describes the thermodynamic state of a system at equilibrium.

The ideal gas equation is a state function because it describes the state of a gas at equilibrium with a specific set of values: volume, temperature and pressure. In other words, a state function for a system at equilibrium relates one or more input thermodynamic properties to an output thermodynamic property.

Since a particular state of a system at equilibrium is characterised by a set of numbers, the output value of a state function for that state is independent on the path (i.e. process) taken to reach that value.

###### Question

Show that the output of the state function is independent on the path taken to reach it.

###### Answer

The total differential of is

The second cross partial derivatives of is

and

Eq19 refers to the path where the function *V* is changed with respect *T* at constant *p*, followed by a change with respect to *p* at constant *T*, whereas eq20 refers to another path where the function *V* is changed with respect to *p* at constant *T*, followed by a change with respect to *T* at constant *p*. If eq19 is equal to eq20, the change of *V* is independent of the path taken. Substituting in eq19 and eq20 gives for both equations. Therefore, the change of *V* is path-independent and the differential given by eq18 is called an ** exact differential**.

Finally, if the output value of the function of pressure and the output value of the function of volume are path-independent, then the output value of the product of the two functions or the output value of the sum of the two functions are also path-independent.