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

${\beta }_T=-\frac{1}{V}{\left(\frac{\partial V}{\partial P}\right)}_T=-\frac{1}{V}\frac{{\partial }^{2}G}{\partial P^2}$

• Adiabatic compressibility

${\beta }_S=-\frac{1}{V}{\left(\frac{\partial V}{\partial P}\right)}_S=-\frac{1}{V}\frac{{\partial }^{2}H}{\partial P^2}$

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

• Specific heat at constant pressure

$c_P=\frac{T}{N}{\left(\frac{\partial S}{\partial T}\right)}_P=-\frac{T}{N}\frac{{\partial }^{2}G}{\partial T^2}$

• Specific heat at constant volume

$c_V=\frac{T}{N}{\left(\frac{\partial S}{\partial T}\right)}_V=-\frac{T}{N}\frac{{\partial }^{2}A}{\partial T^2}$

• Coefficient of thermal expansion

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

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:

$c_P=c_V+\frac{TV{\alpha }^2}{N{\beta }_T}$

${\beta }_T={\beta }_S+\frac{TV{\alpha }^2}{N{c}_P}$

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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