The Fundamental Equation:




These are the starting point for EOS:

Now take the derivative of the above with respect to other independent variables:






If we differentiate w.r.t: S,etc. we get:

There are similar arguments for D H, D G, D A, electromagnetism, gravity, etc.:

Eg: --- Given:



What other relationships do we have to work from?


If we recognize that each property such as P, T, V, etc. can be used in our n + 2 postulate. It can be shown via "Legendre Trasformations" that a derivative can replace these values in our description:

These lead to the rest of the thermodynamic identities and a few definitions:

There are 3 Legendre transforms of the fundamental equation:

2 first order transforms: (one variable changed)

1 second order transform:







Or in differential form:

Now we have the ability to define the remainder of our identities:


From previous page: =T =V

To summarize:


One final transform:


Additional useful relationships:

Recall that previously we defined:

dividing expression 2) by dT and evaluating at constant P yields:



If we divide 2) by dP, evaluate at Const. T: etc.

Other similarly derived physical constants include :

  1. Volume expansivity

  2. Isothermal compressibility:


We can se our definitions:

and from:

Replace b into equation 5):