🧪 Real Gas Equations of State
Evaluate compressibility factor Z, molar volume, and departure functions using van der Waals, RK, and PR EOS.
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Thermodynamics
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Jun 2026
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📝 Configuration
Key Equations:
vdW: P = RT/(v−b) − a/v²
RK: P = RT/(v−b) − a/[T0.5v(v+b)]
PR: P = RT/(v−b) − aα/[v(v+b)+b(v−b)]
Z = Pv/(RT)
vdW: P = RT/(v−b) − a/v²
RK: P = RT/(v−b) − a/[T0.5v(v+b)]
PR: P = RT/(v−b) − aα/[v(v+b)+b(v−b)]
Z = Pv/(RT)
📊 Results
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📘 Methodology
Equations of State
Three cubic EOS are implemented: van der Waals (simplest), Redlich-Kwong (temperature-dependent attraction), and Peng-Robinson (most accurate near critical). Each is solved iteratively via Newton-Raphson.
Departure Functions
Departure functions quantify how real gas enthalpy and entropy differ from ideal gas values at the same T and P. They are essential for thermodynamic property calculations in process simulation.
Assumptions
- Single-component pure fluid.
- Vapor-phase root selected (largest v root).
- No mixing rules (single species).
- Constant acentric factor.
📘 Calculation Methodology: Real Gas Cubic Equations of State (SRK & Peng-Robinson)
Mathematical Model & Theory
Cubic equations of state model non-ideal PVT behavior and phase equilibrium by correcting ideal gas pressure for intermolecular attractive forces ($a$) and molecular co-volume ($b$):
$$P = \frac{R T}{v - b} - \frac{a(T)}{v(v + b) + b(v - b)} \quad (\text{Peng-Robinson})$$
$$Z^3 - (1 - B)Z^2 + (A - 2B - 3B^2)Z - (AB - B^2 - B^3) = 0$$
Assumptions
- Cubic EOS parameter tuning using critical properties ($T_c, P_c$) and acentric factor $\omega$.
- Vapor and liquid roots determined from cubic polynomial discriminant.
Academic References
- Peng, D. Y., & Robinson, D. B. (1976): Industrial & Engineering Chemistry Fundamentals.
- Poling, B. E. et al.: The Properties of Gases and Liquids, McGraw-Hill.
Worked Engineering Example
Problem Statement:
Carbon dioxide at $T = 320\text{ K}$, $P = 8.0\text{ MPa}$ ($T_c = 304.1\text{ K}$, $P_c = 7.38\text{ MPa}$, $\omega = 0.224$). Find the compressibility factor $Z$ via Peng-Robinson.
Step-by-step Solution:
1. Reduced parameters: $T_r = 320/304.1 = 1.052$, $P_r = 8.0/7.38 = 1.084$.
2. Evaluating PR coefficients yields $A = 0.442$, $B = 0.0805$.
3. Solving cubic $Z^3 - 0.9195 Z^2 + 0.261 Z - 0.029 = 0$ yields real root $Z \approx 0.628$.
Final Result:
Compressibility factor is $Z = \mathbf{0.628}$ (dense non-ideal fluid behavior).
Carbon dioxide at $T = 320\text{ K}$, $P = 8.0\text{ MPa}$ ($T_c = 304.1\text{ K}$, $P_c = 7.38\text{ MPa}$, $\omega = 0.224$). Find the compressibility factor $Z$ via Peng-Robinson.
Step-by-step Solution:
1. Reduced parameters: $T_r = 320/304.1 = 1.052$, $P_r = 8.0/7.38 = 1.084$.
2. Evaluating PR coefficients yields $A = 0.442$, $B = 0.0805$.
3. Solving cubic $Z^3 - 0.9195 Z^2 + 0.261 Z - 0.029 = 0$ yields real root $Z \approx 0.628$.
Final Result:
Compressibility factor is $Z = \mathbf{0.628}$ (dense non-ideal fluid behavior).