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:


 * 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 $$\beta_T$$, 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.