Given that f(x,y,z) is any state function that characterizes the system and (x,y,z) is a set of independent variable properties of that system, we know that any change f will be only a function of the value of “f” at the final and initial states,
Since f=f(x,y,z), we can mathematically relate the total differential change (df) to the partial derivatives , , and of the function, as follows:
where, in general:
the change of f with respect to x, while y and z are unchanged.
If we want to come up with the total change, , of a property (we want to go from 14.14. to 14.13), we integrate the expression in (14.14) to get:
Let us visualize this with an example. For a system of constant composition, its thermodynamic state is completely defined when two properties of the system are fixed. Let us say we have a pure component at a fixed pressure (P) and temperature (T). Hence, all other thermodynamic properties, for example, enthalpy (H), are fixed as well. Since H is only a function of P and T, we write:
and hence, applying 6.2, any differential change in enthalpy can be computed as:
The total change in enthalpy of the pure-component system becomes:
Now we are ready to spell out the exactness condition, which is the mathematical condition for a function to be a state function. The fact of the matter is, that for a function to be a state function — i.e., its integrated path shown in (14.15) is only a function of the end states, as shown in (14.13) — its total differential must be exact. In other words, if the total differential shown in (14.14) is exact, then f(x,y,z) is a state function. How do we know if a total differential is exact or not?
Given a function (x,y,z),
we say that
is an exact differential and consequently
a state function if all the following conditions are satisfied:
Equations (14.20) are called the exactness condition.