Feb 152012
 February 15, 2012  Posted by at 11:30 pm Chemical Engineering, Chemistry Tagged with:  Comments Off on Useful Definitions and Identities for Thermodynamic Potentials

This is just a summary of many of the definitions and identities I find useful when working with the basic thermodynamics of a system or process.  It is not meant to be a comprehensive list.

Mathematical Identities

Using the reasoning behind the notation used for derivatives of thermodynamic potentials, there are some useful identities given the following prototypical system

Eq. 1) \displaystyle X=X\left(Y,Z\right)

This equation represents a generic thermodynamic potential of an arbitrary system or process as a function of two implicitly extensive thermodynamic potentials.  Adopting the notation

Eq. 2) \displaystyle\underline{X}=\frac{X}{N}

To denote the relationship between intensive variables, extensive variables, and the size of the system allows for the trivial conversion between system size dependent and system size agnostic thermodynamic relationships.  The total differential for the extensive form of the thermodynamic potential for the prototypical system is given by
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Feb 142012

When I took my process thermodynamics course as an undergraduate student, I was told to use what I had initially considered to be a completely redundant notation for partial derivatives for thermodynamic potentials.  The notation involved wrapping the partial derivative in a set of parentheses and noting which variables were “held constant” as a subscript.

Ex. \displaystyle\left(\frac{\partial P}{\partial V}\right)_{T,\vec{N}}

The derivation leading to this was woefully devoid of the mathematical basis for this apparently redundant notation.  Furthermore, so was the general literature on the subject, most of which consisted of fleeting introductions to their respective application.  As I played with the idea, it became clear why this notation was indeed very necessary.  It was only as I was solving a separate problem on my own that such notation yielded valuable information about the nature of the potentials being described.

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