# Peng-Robinson Equation of State

The Peng-Robinson equation of state, abbreviated as PR EOS, is a widely used mathematical model in thermodynamics and fluid mechanics for describing the behavior of real gases and their deviation from ideal gas behavior. The equation is used to predict the properties of gases and their phase behavior, particularly in the presence of high pressure and non-ideal conditions.

The parameters a and b are empirical constants that depend on the specific gas being studied. They are related to the attractive forces (parameter a) and the excluded volume (parameter b) of the gas molecules. These parameters are crucial for accurately predicting phase behavior and other thermodynamic properties.

The Peng-Robinson equation of state is particularly useful for modeling the behavior of gases and their phase transitions under high pressure and non-ideal conditions, such as in the oil and gas industry for the prediction of phase equilibria in reservoir fluids, as well as in the design of chemical processes and separation units. It's important to note that while the Peng-Robinson equation of state is a significant improvement over the ideal gas law, it is still an empirical model and may have limitations, especially for gases that exhibit highly non-ideal behavior or for systems where interactions are strongly affected by other factors, such as chemical reactions.

## Peng-Robinson Equation of State Formula |
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\(\large{ p = \frac{R \; T}{ V_m \;-\; b } - \frac{ a \; \alpha }{ V_m^2 \;+\; 2\;b \; V_m \;-\; b^2 } }\) | ||

Symbol |
English |
Metric |

\(\large{ p }\) = pressure of gas | \(\large{\frac{lbf}{in^2}}\) | \(\large{Pa}\) |

\(\large{ a }\) = correction for the intermolecular forces | \(\large{dimensionless}\) | |

\(\large{ b }\) = adjusts for the volume occupied by the gas particles | \(\large{in^3}\) | \(\large{mm^3}\) |

\(\large{ \alpha }\) (Greek symbol alpha) = \( \left( 1 + k \; \left( 1 - T_r^{0.5} \right) \right)^2 \) | \(\large{dimensionless}\) | |

\(\large{ k }\) = \( 0.37464 + 1.54226 \;\omega - 0.26922\; \omega^2 \) | \(\large{dimensionless}\) | |

\(\large{ \omega }\) (Greek symbol omega) = acentric factor | \(\large{dimensionless}\) | |

\(\large{ V_m }\) = molar volume of gas \(\left( \frac{V}{n} \right) \) | \(\large{in^3}\) | \(\large{mm^3}\) |

\(\large{ n }\) = number of moles of gas | \(\large{dimensionless}\) | |

\(\large{ R }\) = specific gas constant (gas constant) | \(\large{\frac{ft-lbf}{lbm-R}}\) | \(\large{\frac{J}{kg-K}}\) |

\(\large{ T }\) = temperature of gas | \(\large{R}\) | \(\large{K}\) |

\(\large{ T_c }\) = critical temperature of gas | \(\large{R}\) | \(\large{K}\) |

\(\large{ T_r }\) = \(\frac{T}{T_c}\) | \(\large{R}\) | \(\large{K}\) |