Material properties (thermodynamics)

• Compressibility (or its inverse, the bulk modulus)

• Isothermal compressibility

${\displaystyle {\kappa }_T=-\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }P}\right)}_T=-\frac{1}{V}\frac{{\mathrm{\partial }}^{2}G}{\mathrm{\partial }{P}^2}}$

• Adiabatic compressibility

${\displaystyle {\kappa }_S=-\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }P}\right)}_S=-\frac{1}{V}\frac{{\mathrm{\partial }}^{2}H}{\mathrm{\partial }{P}^2}}$

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

• Specific heat at constant pressure

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

• Specific heat at constant volume

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

• Coefficient of thermal expansion

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

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 ${\displaystyle {\kappa }_T}$, the specific heat at constant pressure ${\displaystyle c_P}$, and the coefficient of thermal expansion ${\displaystyle \alpha }$.

For example, the following equations are true:

${\displaystyle c_P=c_V+\frac{TV{\alpha }^2}{N{\kappa }_T}}$

${\displaystyle {\kappa }_T={\kappa }_S+\frac{TV{\alpha }^2}{N{c}_P}}$

The three "standard" properties are in fact the three possible second derivatives of the Gibbs free energy with respect to temperature and pressure. Moreover, considering derivatives such as ${\displaystyle \frac{{\mathrm{\partial }}^{3}G}{\mathrm{\partial }P\mathrm{\partial }{T}^2}}$ and the related Schwartz relations, shows that the properties triplet is not independent. In fact, one property function can be given as an expression of the two others, up to a reference state value.

The second principle of thermodynamics has implications on the sign of some thermodynamic properties such isothermal compressibility.