Peng-Robinson Equation of State

$p = (R*T)/(V_m-b ) - (a*alpha)/(V_m^2+2*b*V_m - b^2)$

$(T) "Temperature"$

$(T_c)"Critical Temperature"$

$(V_m)"Molar Volume"$

$(p_(sat))"Saturated Vapor Pressure"$

$(p_c)"Critical Pressure"$

The Math / Science

The Peng-Robinson equation describes the state of the gas under given conditions, relating pressure, temperature and volume of the constituent matter. The Peng-Robinson equation of state has the basic form:

$p = (R*T)/(V_m - b) - (a*alpha)/ (V_m^2 + 2 * b * V_m - b^2)$

Variables a, b, and $alpha$ are further described by:

$a = (0.457235 * R^2 * T_c^2)/p_c$ , where R is the universal gas constant (8.3144626181532 J/(K·mol)) and T_c and p_c are defined in the Inputs above.

$b = (0.077796 * R * T_c) / p_c$

$alpha = (1 + kappa * (1 - T_r^(1/2) ) )^2$ , where $T_r = T/T_c$

Further $kappa$ in the definition of $alpha$ is defined as:

$kappa = 0.37464 + 1.54226* omega - 0.26992* omega^2$ , where $omega$ is the acentric factor which is a function of the saturated vapor pressure and the critical pressure

ω = $p^(sat) / p_c$

The Peng-Robinson equation of state (named after D.-Y. Peng and D. B. Robinson) can model some liquids as well as real gases.

It is easy to see in the first term of the Peng-Robinson equation that this state equation had its origins in the Ideal gas law: pV = nRT, since $V/n$ gives you molar volume. To apply the Ideal gas law to real gases, correction terms have been included that are composed of empirically derived offsets.

The Peng-Robinson equation was specifically deigned to meet the following criteria¹:

The parameters should be expressible in terms of the critical properties and the acentric factor.
The model should provide reasonable accuracy near the critical point, particularly for calculations of the compressibility factor and liquid density.
The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature pressure and composition.
The equation should be applicable to all calculations of all fluid properties in natural gas processes.

Usage

This equation is popular for us with natural gas systems found in petroleum industries. Equations of state are useful in describing the physical properties of fluids, solids, and even the interior of stars.

Chemistry Calculators

R - Gas Constant: 8.3144626181532 J/(K⋅mol)
Boyle's Law Calculator: P₁ • V₁ = P₂ • V₂
Charles Law Calculator: V₁• T₂ = V₂ • T₁
Combined Gas Law Calculator: P•V / T= k
Gay-Lussac Law: T₁•P₂ =T₂•P₁
Ideal Gas Law: P•V = n•R•T
Bragg's Law: n·λ = 2d·sinθ
Hess' Law: ΔH⁰_rxn=ΔH⁰_a+ΔH⁰_b+ΔH⁰_c+ΔH⁰_d
Internal Energy: ΔU = q + ω
Activation Energy: E_a = (R*T₁⋅T₂)/(T₁ - T₂) ⋅ ln(k₁/k₂)
Arrhenius Equation: k = Ae^E_a/(RT)
Clausius-Clapeyron Equation: ln(P₂/P₁) = (ΔH_vap)/R * (1/T₁ - 1/T₂)
Compressibility Factor: Z = (p*V_m)/(R*T)
Peng-Robinson Equation of State: p = (R*T)/(V_m - b) - (a*α)/(V_m² + 2*b*V_m - b²)
Reduced Specific Volume: v_r = v/(R* T_cr/ P_c)
Van't Hoff Equation: ΔH⁰ = R * ( -ln(K₂/K₁))/ (1/T₁ - 1/T₂)

References

^ Peng-Robinson Equation