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Thermodynamic equations
	Laws of thermodynamics
	Conjugate variables
	Thermodynamic potential
	Material properties specific heat compressibility thermal expansion
	Maxwell relations
	Bridgman's equations
	Exact differential
	Table of thermodynamic equations
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The thermodynamic properties of materials are intensive thermodynamic parameters which are specific to a given material. Each is directly related to a second order differential of a thermodynamic potential. Examples for a simple 1-component system are:

Compressibility (or its inverse, the bulk modulus)

Isothermal compressibility

$\beta_T=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T \quad = -\frac{1}{V}\,\frac{\partial^2 G}{\partial P^2}$

Adiabatic compressibility

$\beta_S=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_S \quad = -\frac{1}{V}\,\frac{\partial^2 H}{\partial P^2}$

Specific heat (Note - the extensive analog is the heat capacity)

Specific heat at constant pressure

$c_P=\frac{T}{N}\left(\frac{\partial S}{\partial T}\right)_P \quad = -\frac{T}{N}\,\frac{\partial^2 G}{\partial T^2}$

Specific heat at constant volume

$c_V=\frac{T}{N}\left(\frac{\partial S}{\partial T}\right)_V \quad = -\frac{T}{N}\,\frac{\partial^2 A}{\partial T^2}$

Coefficient of thermal expansion

$\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P \quad = \frac{1}{V}\,\frac{\partial^2 G}{\partial P\partial T}$

where P is pressure, V is volume, T is temperature, S is entropy, and N is the number of particles.

For a single component system, only three second derivatives are needed in order to derive all others, and so only three material properties are needed to derive all others. For a single component system, the "standard" three parameters are the isothermal compressibility $β T$ , the specific heat at constant pressure $c P$ , and the coefficient of thermal expansion $α$ .

For example, the following equations are true:

$c_P=c_V+\frac{TV\alpha^2}{N\beta_T}$

$\beta_T=\beta_S+\frac{TV\alpha^2}{Nc_P}$

The three "standard" properties are in fact the three possible second derivatives of the Gibbs free energy with respect to temperature and pressure.

[edit] References

Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd Ed. ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8.

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