Mechanics of Manipulating a Function of State

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 $\Delta$ f will be only a function of the value of “f” at the final and initial states,

$$\Delta f=f_2-f_1=f(x_2,y_2,z_2)-f(x_1,y_1,z_1)$$ (14.13)

Since f=f(x,y,z), we can mathematically relate the total differential change (df) to the partial derivatives $\frac{\partial f}{\partial x}$ ,$\frac{\partial f}{\partial y}$ , and $\frac{\partial f}{\partial z}$ of the function, as follows:

$$df={\left(\frac{\partial f}{\partial x}\right)}_{y,z}dx+{\left(\frac{\partial f}{\partial y}\right)}_{x,z}dy+{\left(\frac{\partial f}{\partial z}\right)}_{x,y}dz$$ (14.14)

where, in general:

${\left(\frac{\partial f}{\partial x}\right)}_{y,z}=$ the change of f with respect to x, while y and z are unchanged.

If we want to come up with the total change, $\Delta f$ , of a property (we want to go from 14.14. to 14.13), we integrate the expression in (14.14) to get:

$$\Delta f=f_2-f_1=\underset{x_1}{\overset{x_2}{\int }}{\left(\frac{\partial f}{\partial x}\right)}_{y,z}dx+\underset{y_{{}_1}}{\overset{y_2}{\int }}{\left(\frac{\partial f}{\partial y}\right)}_{x,z}dy+\underset{z_1}{\overset{z_2}{\int }}{\left(\frac{\partial f}{\partial x}\right)}_{x,y}dz$$ (14.15)

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:

$$H=H(P,T)$$ (14.16)

and hence, applying 6.2, any differential change in enthalpy can be computed as:

$$dH={\left(\frac{\partial H}{\partial P}\right)}_{T}dP+{\left(\frac{\partial H}{\partial T}\right)}_{P}dT$$ (14.17)

The total change in enthalpy of the pure-component system becomes:

$$\Delta H=H_2-H_1=\underset{P_1}{\overset{P_2}{\int }}{\left(\frac{\partial H}{\partial P}\right)}_{T}dP+\underset{T_1}{\overset{T_2}{\int }}{\left(\frac{\partial H}{\partial T}\right)}_{P}dT$$ (14.18)

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 $\Psi$ (x,y,z),

$$d\Psi =M(x,y,z)dx+N(x,y,z)dy+Q(x,y,z)dz$$ (14.19a)

where:

$$M(x,y,z)={\left(\frac{\partial \Psi }{\partial x}\right)}_{y,z}$$ (14.19b)

$$N(x,y,z)={\left(\frac{\partial \Psi }{\partial y}\right)}_{x,z}$$ (14.19c)

$$Q(x,y,z)={\left(\frac{\partial \Psi }{\partial z}\right)}_{x,y}$$ (14.19d)

we say that $d\Psi$ is an exact differential and consequently $\psi (x,y,z)$ a state function if all the following conditions are satisfied:

$${\left(\frac{\partial M}{\partial y}\right)}_{x,z}={\left(\frac{\partial N}{\partial x}\right)}_{y,z}$$ (14.20a)

$${\left(\frac{\partial N}{\partial z}\right)}_{x,y}={\left(\frac{\partial Q}{\partial y}\right)}_{x,z}$$ (14.20b)

$${\left(\frac{\partial M}{\partial z}\right)}_{x,y}={\left(\frac{\partial Q}{\partial x}\right)}_{y,z}$$ (14.20c)

Equations (14.20) are called the exactness condition.