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# Material properties (thermodynamics)

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:

• Isothermal compressibility $\beta_T=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T \quad = -\frac{1}{V}\,\frac{\partial^2 G}{\partial P^2}$ $\beta_S=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_S \quad = -\frac{1}{V}\,\frac{\partial^2 H}{\partial P^2}$
• 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 $c_V=\frac{T}{N}\left(\frac{\partial S}{\partial T}\right)_V \quad = -\frac{T}{N}\,\frac{\partial^2 A}{\partial T^2}$ $\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P \quad = \frac{1}{V}\,\frac{\partial^2 G}{\partial P\partial T}$

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 βT, the specific heat at constant pressure cP, 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.

## References

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