Material properties (thermodynamics)

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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

${\displaystyle \kappa _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}\quad =-{\frac {1}{V}}\,{\frac {\partial ^{2}G}{\partial P^{2}}}}$

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

• 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

${\displaystyle c_{V}={\frac {T}{N}}\left({\frac {\partial S}{\partial T}}\right)_{V}\quad =-{\frac {T}{N}}\,{\frac {\partial ^{2}A}{\partial T^{2}}}}$

• Coefficient of thermal expansion

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

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]