#### Equations of State: Screencasts

For each of the screencasts below, we suggest you list the most important points after watching each screencast, as a way to increase retention.

###### Screencast 1:

Introduces the van der Waals equation of state (EOS), which is cubic, and explains its three roots.

###### Screencast 2:

Shows how to determine a saturation pressure from an equation of state.

##### Important Equations:

\[P_r = \frac{P}{P_c}\]

where \(P_r\) is the reduced pressure, \(P\) is pressure, and \(P_c\) is the critical pressure.

\[T_r = \frac{T}{T_c}\]

where \(T_r\) is the reduced temperature, \(T\) is temperature, and \(T_c\) is the critical temperature (must be absolute temperatures).

\[V_r = \frac{V}{V_c}\]

where \(V_r\) is the reduced molar volume, \(V\) is molar volume, and \(V_c\) is the critical molar volume.

\[\omega = -(1 + log_{10} (\frac{P^{sat}}{P_c}))\]

where \(\omega\) is the acentric factor, evaluated at \(P^{sat}\) value for \(T_r\) = 0.7, where \(P^{sat}\) is the saturation pressure.

\[Z = \frac{PV}{RT}\]

where \(Z\) is the compressibility factor, and R is the ideal gas constant.

van der Waals Equation of State (EOS):

\[P = \frac{RT}{V – b} – \frac{a}{V^2} = \frac{\rho RT}{1 – b\rho} – a\rho ^2\]

where V is molar volume, \(\rho\) is molar density and \(a\) and \(b\) are constants. The \(a\) term accounts for attractive interactions between molecules and the \(b\) term accounts for the volume occupied by molecules (replusive term). The \(a\) and \(b\) constants are determined from critical pressures and temperatures:

\[a = \frac{27R^2T^2}{64PC} \hspace{1cm} b = \frac{RT_c}{8P_c}\]

\[P = \frac{\rho RT}{1 – b\rho} – \frac{a\rho ^2}{(1 + 2b\rho – b^2 \rho ^2)}\]

and \(a = a_c \alpha\) , \(b = 0.0779607\frac{RT_c}{P_c}\)

where \(a_c = 0.45723553R^2T_c^2/P_c\) and \(a = [1 + \kappa(1 – T_r^{0.5})]^2\), and \(\kappa = 0.37464 + 1.5422\omega – 0.26992\omega ^2\)

At saturation conditions (vapor-liquid equilibrium):

\[f_{liquid} = f_{vapor}\]

where \(f\) is the fugacity, which can be calculated from the equation of state.