#### 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:

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

Peng-Robinson EOS:

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

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.