Benedict–Webb–Rubin equation

${\displaystyle P=\rho RT+\left(B_{0}RT-A_{0}-{\frac {C_{0}}{T^{2}}}\right)\rho ^{2}+\left(bRT-a\right)\rho ^{3}+\alpha a\rho ^{6}+{\frac {c\rho ^{3}}{T^{2}}}\left(1+\gamma \rho ^{2}\right)\exp \left(-\gamma \rho ^{2}\right)}$,

where ${\displaystyle \rho }$ is the molar density.

A modification of the Benedict–Webb–Rubin equation of state by Professor Kenneth E. Starling of the University of Oklahoma:[3]

${\displaystyle P=\rho RT+\left(B_{0}RT-A_{0}-{\frac {C_{0}}{T^{2}}}+{\frac {D_{0}}{T^{3}}}-{\frac {E_{0}}{T^{4}}}\right)\rho ^{2}+\left(bRT-a-{\frac {d}{T}}\right)\rho ^{3}+\alpha \left(a+{\frac {d}{T}}\right)\rho ^{6}+{\frac {c\rho ^{3}}{T^{2}}}\left(1+\gamma \rho ^{2}\right)\exp \left(-\gamma \rho ^{2}\right)}$,

where ${\displaystyle \rho }$ is the molar density. The 11 mixture parameters (${\displaystyle B_{0}}$, ${\displaystyle A_{0}}$, etc.) are calculated using the following relations

${\displaystyle {\begin{aligned}&A_{0}=\sum _{i}\sum _{j}x_{i}x_{j}A_{0i}^{1/2}A_{0j}^{1/2}(1-k_{ij})\\&B_{0}=\sum _{i}x_{i}B_{0i}\\&C_{0}=\sum _{i}\sum _{j}x_{i}x_{j}C_{0i}^{1/2}C_{0j}^{1/2}(1-k_{ij})^{3}\\&D_{0}=\sum _{i}\sum _{j}x_{i}x_{j}D_{0i}^{1/2}D_{0j}^{1/2}(1-k_{ij})^{4}\\&E_{0}=\sum _{i}\sum _{j}x_{i}x_{j}E_{0i}^{1/2}E_{0j}^{1/2}(1-k_{ij})^{5}\\&\alpha =\left[\sum _{i}x_{i}\alpha _{i}^{1/3}\right]^{3}\\&\gamma =\left[\sum _{i}x_{i}\gamma _{i}^{1/2}\right]^{2}\\&a=\left[\sum _{i}x_{i}a_{i}^{1/3}\right]^{3}\\&b=\left[\sum _{i}x_{i}b_{i}^{1/3}\right]^{3}\\&c=\left[\sum _{i}x_{i}c_{i}^{1/3}\right]^{3}\\&d=\left[\sum _{i}x_{i}d_{i}^{1/3}\right]^{3}\end{aligned}}}$

where ${\displaystyle i}$ and ${\displaystyle j}$ are indices for the components, and the summations go over all components. ${\displaystyle B_{0i}}$, ${\displaystyle A_{0i}}$, etc. are the parameters for the pure components for the ${\displaystyle i}$th component, ${\displaystyle x_{i}}$ is the mole fraction of the ${\displaystyle i}$th component, and ${\displaystyle k_{ij}}$ is an interaction parameter.

Values of the various parameters for 15 substances can be found in Starling's Fluid Properties for Light Petroleum Systems..[3]

A further modification of the Benedict–Webb–Rubin equation of state by Jacobsen and Stewart:[4] · [5]

${\displaystyle P=\sum _{n=1}^{9}a_{n}\rho ^{n}+\exp \left(-\gamma \rho ^{2}\right)\sum _{n=10}^{15}a_{n}\rho ^{2n-17}}$

where:

${\displaystyle \gamma =1/\rho _{c}^{2}}$

The mBWR equation subsequently evolved into a 32 term version (Younglove and Ely, 1987) with numerical parameters determined by fitting the equation to empirical data for a reference fluid.[6] Other fluids then are described by using reduced variables for temperature and density.[7]

• Real gas

• Benedict, Manson; Webb, George B.; Rubin, Louis C. (1942), "Mixtures of Methane, Ethane, Propane and n-Butane", Journal of Chemical Physics, 10 (12): 747–758, Bibcode:1942JChPh..10..747B, doi:10.1063/1.1723658, ISSN 0021-9606
• Benedict, Manson; Webb, George B.; Rubin, Louis C. (1951), "An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures. Constants for Twelve Hydrocarbons", Chemical Engineering Progress (CEP), 47 (8): 419–422
• Benedict, Manson; Webb, George B.; Rubin, Louis C. (1951), "An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures Fugacities and Liquid-Vapor Equilibria", Chemical Engineering Progress (CEP), 47 (9): 449–454.