Stokes number

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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

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

• Adiabatic compressibility

$\beta _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{S}\quad =-{\frac {1}{V}}\,{\frac {\partial ^{2}H}{\partial P^{2}}}$

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

• 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

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

• Coefficient of thermal expansion

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

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

For example, the following equations are true:

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

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

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

Sources

The Dortmund Data Bank is a factual data bank for thermodynamic and thermophysical data.

See thermodynamic databases for pure substances.

References

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