# Material properties (thermodynamics)

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 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 $\beta_T$, the specific heat at constant pressure $c_P$, and the coefficient of thermal expansion $\alpha$.

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.

## Sources

The Dortmund Data Bank is a factual data bank for thermodynamic and thermophysical data.

## References

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