# Peng-Robinson Equation of State

The Peng-Robinson (PR) equation of state (EOS) is a cubic EOS which was initially developed by Peng and Robinson in 19761 and was modified in 1978 to enhance the phase behavior predictions of the model2. The PR EOS is a common EOS used in many areas of petroleum and chemical engineering.

## PR EOS Structure

The PR EOS model is defined by

where $$a$$ and $$b$$ are the EOS model parameters. The pure component parameters ($$a$$ and $$b$$) can be calculated by

In equations \eqref{eq:a_pr}, \eqref{eq:b_pr} and \eqref{eq:alpha} the coefficients $$\Omega_a$$ and $$\Omega_b$$ , and the value of $$\alpha(T)$$ can be calculated by

## Component Properties

The component properties are composed of the critical pressure ($$p_c$$), critical temperature ($$T_c$$), accentric factor ($$\omega$$) and molecular weight ($$MW$$). Another important component property for cubic EOS models are the volume shift factors descried by Peneloux in 19823. The volume shifts were introduced as a correction term to the phase behaviour predictions of the molar volume predictions.

## Mixing Rules

Note

The main article for cubic EOS models describes the mixing rules with some more detail.

The original mixing rule developed by van der Waals is a quadratic mixing rule. The set of equations describing the mixing are given by

where $$a_{ij}=\sqrt{a_ia_j}$$, $$u_i$$ can be the liquid composition ($$x_i$$), vapor composition ($$y_i$$) or the total composition ($$z_i$$) and $$a_i$$ and $$b_i$$ are the component EOS parameters for component $$i$$.

Modifications to the $$a$$ and $$b$$ mixing rules were developed to enhance the performance of the EOS models. The most common modified mixing rules are given by

where $$k_{ij}$$ is often referred to as the binary interaction parameter (BIP) or sometimes referred to as the binary interaction coefficient (BIC).

1. D. Y. Peng and D. B. Robinson. A new two-constant equation of state. Industrial & Engineering Chemistry Fundamentals, 15:59–64, 1976.

2. D. B. Robinson and D. Y. Peng. The characterization of the heptanes and heavier fractions for the GPA Peng-Robinson programs. Gas processors association, 1978.

3. A. Péneloux, E. Rauzy, and R. Fréze. A consistent correction for redlich-kwong-soave volumes. Fluid phase equilibria, 8:7–23, 1982. doi:https://doi.org/10.1016/0378-3812$82$80002-2