# 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
${\displaystyle \beta _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}\quad =-{\frac {1}{V}}\,{\frac {\partial ^{2}G}{\partial P^{2}}}}$
${\displaystyle \beta _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{S}\quad =-{\frac {1}{V}}\,{\frac {\partial ^{2}H}{\partial P^{2}}}}$
• Specific heat at constant pressure
${\displaystyle c_{P}={\frac {T}{N}}\left({\frac {\partial S}{\partial T}}\right)_{P}\quad =-{\frac {T}{N}}\,{\frac {\partial ^{2}G}{\partial T^{2}}}}$
• 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 ${\displaystyle \beta _{T}}$, the specific heat at constant pressure ${\displaystyle c_{P}}$, and the coefficient of thermal expansion ${\displaystyle \alpha }$.

For example, the following equations are true:

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

The three "standard" properties are in fact the three possible second derivatives of the Gibbs free energy with respect to temperature and pressure.