### 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