# EoS: Equations of State for Liquids and Solids

### Liquids and Solids

The specific volume (or density) of liquids and solids are essentially constant (except for the liquid near the critical point), therefore they are often called "incompressible".

Nevertheless, there is, in fact, a small dependence of the specific volume on temperature and pressure:

##### Definition:

The coefficient of thermal expansion is a measure of the dependence of a liquid/solid's specific volume on temperature and is given as
$\beta \equiv \frac{1}{v} \left (\frac{\partial v}{\partial T} \right )_P$

##### Definition:

The isothermal compressibility is a measure of the dependence of a liquid/solid's specific volume on pressure and is given as
$\kappa \equiv -\frac{1}{v} \left (\frac{\partial v}{\partial P} \right )_T$

These quantities can be used in a simple equation of state in the following way:

$v = v_o\left [1+\beta (T-T_o)-\kappa(P-P_o)\right ]$

##### Definition:

Alternatively, the liquid specific volume at saturation conditions is given by the Rackett Equation:

$v^{l,sat}=\frac{RT_c}{P_c}(0.29056-0.08775\omega)^{[1+(1-T_r)^{2/7}]}$

##### Outcome

Apply the Rackett equation, the thermal expansion coefficient, and the isothermal compressibility to find v for liquids and solids