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
${\displaystyle \kappa _{T}=-{\frac {1}{V))\left({\frac {\partial V}{\partial P))\right)_{T}\quad =-{\frac {1}{V))\,{\frac {\partial ^{2}G}{\partial P^{2))))$
• Adiabatic compressibility
${\displaystyle \kappa _{S}=-{\frac {1}{V))\left({\frac {\partial V}{\partial P))\right)_{S}\quad =-{\frac {1}{V))\,{\frac {\partial ^{2}H}{\partial P^{2))))$
• Specific heat at constant pressure
${\displaystyle c_{P}={\frac {T}{N))\left({\frac {\partial S}{\partial T))\right)_{P}\quad =-{\frac {T}{N))\,{\frac {\partial ^{2}G}{\partial T^{2))))$
• Specific heat at constant volume
${\displaystyle c_{V}={\frac {T}{N))\left({\frac {\partial S}{\partial T))\right)_{V}\quad =-{\frac {T}{N))\,{\frac {\partial ^{2}A}{\partial T^{2))))$
${\displaystyle \alpha ={\frac {1}{V))\left({\frac {\partial V}{\partial T))\right)_{P}\quad ={\frac {1}{V))\,{\frac {\partial ^{2}G}{\partial P\partial T))}$

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 ${\displaystyle \kappa _{T))$, the specific heat at constant pressure ${\displaystyle c_{P))$, and the coefficient of thermal expansion ${\displaystyle \alpha }$.

For example, the following equations are true:

${\displaystyle c_{P}=c_{V}+{\frac {TV\alpha ^{2)){N\kappa _{T))))$
${\displaystyle \kappa _{T}=\kappa _{S}+{\frac {TV\alpha ^{2)){Nc_{P))))$

The three "standard" properties are in fact the three possible second derivatives of the Gibbs free energy with respect to temperature and pressure. Moreover, considering derivatives such as ${\displaystyle {\frac {\partial ^{3}G}{\partial P\partial T^{2))))$ 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.[1]

The second principle of thermodynamics has implications on the sign of some thermodynamic properties such isothermal compressibility.[1][2]

## References

1. ^ a b S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States", Link to Archiv e-print Link to Hal e-print
2. ^ Israel, R. (1979). Convexity in the Theory of Lattice Gases. Princeton, New Jersey: Princeton University Press. doi:10.2307/j.ctt13x1c8g
• Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8.