# Material properties (thermodynamics)

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$c=$
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$\beta =-$
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vte :
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
       $\kappa _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}\quad =-{\frac {1}{V}}\,{\frac {\partial ^{2}G}{\partial P^{2}}}$
   * Adiabatic compressibility
       $\kappa _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{S}\quad =-{\frac {1}{V}}\,{\frac {\partial ^{2}H}{\partial P^{2}}}$
   * Specific heat at constant pressure
       $c_{P}={\frac {T}{N}}\left({\frac {\partial S}{\partial T}}\right)_{P}\quad =-{\frac {T}{N}}\,{\frac {\partial ^{2}G}{\partial T^{2}}}$
   * Specific heat at constant volume
       $c_{V}={\frac {T}{N}}\left({\frac {\partial S}{\partial T}}\right)_{V}\quad =-{\frac {T}{N}}\,{\frac {\partial ^{2}A}{\partial T^{2}}}$
       $\alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}\quad ={\frac {1}{V}}\,{\frac {\partial ^{2}G}{\partial P\partial T}}$
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 $\kappa _{T}$, the specific heat at constant pressure $c_{P}$, and the coefficient of thermal expansion $\alpha$.
For example, the following equations are true:
    $c_{P}=c_{V}+{\frac {TV\alpha ^{2}}{N\kappa _{T}}}$
    $\kappa _{T}=\kappa _{S}+{\frac {TV\alpha ^{2}}{Nc_{P}}}$
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 ${\frac {\partial ^{3}G}{\partial P\partial T^{2}}}$ 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.
The second principle of thermodynamics has implications on the sign of some thermodynamic properties such isothermal compressibility.