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14.5: 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 Δf will be only a function of the value of “f” at the final and initial states,

Contact your instructor if you are unable to see or interpret this graphic.(14.13)

Since f=f(x,y,z), we can mathematically relate the total differential change (df) to the partial derivatives Contact your instructor if you are unable to see or interpret this graphic.,Contact your instructor if you are unable to see or interpret this graphic., and Contact your instructor if you are unable to see or interpret this graphic. of the function, as follows:

Contact your instructor if you are unable to see or interpret this graphic.(14.14)
where, in general:

Contact your instructor if you are unable to see or interpret this graphic. the change of f with respect to x, while y and z are unchanged.

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

Contact your instructor if you are unable to see or interpret this graphic.(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:

Contact your instructor if you are unable to see or interpret this graphic.(14.16)

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

Contact your instructor if you are unable to see or interpret this graphic.(14.17)

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

Contact your instructor if you are unable to see or interpret this graphic.(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 Ψ(x,y,z),

Contact your instructor if you are unable to see or interpret this graphic.(14.19a)

where:

Contact your instructor if you are unable to see or interpret this graphic.(14.19b)
Contact your instructor if you are unable to see or interpret this graphic.(14.19c)
Contact your instructor if you are unable to see or interpret this graphic.(14.19d)

we say that dψ is an exact differential and consequently Ψ(x,y,z) a state function if all the following conditions are satisfied:

Contact your instructor if you are unable to see or interpret this graphic.(14.20a)
Contact your instructor if you are unable to see or interpret this graphic.(14.20b)
Contact your instructor if you are unable to see or interpret this graphic.(14.20c)

Equations (14.20) are called the exactness condition.

Contributors

  • Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.