Question

Starting from the first and second law of thermodynamics, derive the fundamental equation for A in its natural variables. a) Derive the Maxwell relation that s related to this equation. b) Show that for df= gdx + hdy we have an exact equation if:

Answer 1

As we have seen, the fundamental thermodynamic relation implies that the natural variable in which to express are and : .

That means that on purely mathematical grounds, we can write

But comparison with the fundamental thermodynamic relation, which contains the physics, we can make the following identifications:

These (especially the second) are interesting in their own right. But we can go further, by differentiating both sides of the first equation by and of the second by :

Using the fact that the order of differentiation in the second derivation doesn't matter, we see that the right hand sides are equal, and thus so are the left hand sides, giving

By starting with , and , we can get three more relations.

a) In thermodynamics, the Maxwell equations are a set of equations derived by application of Euler's reciprocity relation to the thermodynamic characteristic functions. The Maxwell relations, first derived by James Clerk Maxwell, are the following expressions between partial differential quotients:

First Law of Thermodynamics

The first law of thermodynamics is represented below in its differential form

dU=đq+đw

where

• U is the internal energy of the system,
• q is heat flow of the system, and
• w is the work of the system.

The "đ" symbol represent inexact differentials and indicates that both q and w are path functions. Recall that U is a state function. The first law states that internal energy changes occur only as a result of heat flow and work done.

It is assumed that w refers only to PV work, where

w=−∫pdV

The fundamental thermodynamic equation for internal energy follows directly from the first law and the principle of Clausius:

dU=đq+đw

dS=δqrevT we have

dU=TdS+δw

Since only PV work is performed,

dU=TdS−pdV

The above equation is the fundamental equation for U with natural variables of entropy S and volumeV.