The Benedict–Webb–Rubin equation (BWR), named after Manson Benedict, G. B. Webb, and L. C. Rubin, is an equation of state used in fluid dynamics. Working at the research laboratory of the M. W. Kellogg Company, the three researchers rearranged the Beattie–Bridgeman equation of state and increased the number of experimentally determined constants to eight.^[1][2]

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

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, 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, 47 (9): 449–454.