Calculator
How to use this calculator
- Select a model (PR, SRK, Virial, or Ideal).
- Enter temperature and pressure with your preferred units.
- For PR/SRK, provide Tc, Pc, and ω for the component.
- Choose vapor or liquid root when multiple Z values exist.
- Press Submit to view φ, ln(φ), Z, and fugacity f = φP.
Formula used
- Fugacity for a pure fluid: f = φ · P
- Reported outputs: φ, ln(φ), and Z
- ln(φ) ≈ (B·P)/(R·T)
- Z ≈ 1 + (B·P)/(R·T)
- Best suited for lower densities and moderate pressures.
- A = aP/(R²T²), B = bP/(RT)
- Cubic Z equation solved for real roots.
- Vapor root = largest Z, liquid root = smallest Z.
- ln(φ) = Z − 1 − ln(Z−B) − A/(2√2B)·ln((Z+(1+√2)B)/(Z+(1−√2)B))
- A = aP/(R²T²), B = bP/(RT)
- Cubic Z equation solved for real roots.
- ln(φ) = Z − 1 − ln(Z−B) − (A/B)·ln(1 + B/Z)
- Inputs: Tc, Pc, and ω define the temperature correction.
Example data table
| Component | T (K) | P (bar) | Method | Phase | Z | φ | f (bar) |
|---|---|---|---|---|---|---|---|
| Methane (CH4) | 300.00 | 50.00 | PR | Vapor | 0.901770 | 0.901320 | 45.0660 |
| Carbon dioxide (CO2) | 320.00 | 80.00 | PR | Vapor | 0.556023 | 0.678969 | 54.3175 |
| Nitrogen (N2) | 298.15 | 100.00 | SRK | Vapor | 1.017443 | 1.005895 | 100.5895 |
Calculation history
| Timestamp | Method | Phase | T (K) | P (Pa) | Z | φ | ln(φ) | f (Pa) |
|---|---|---|---|---|---|---|---|---|
| No calculations saved yet. | ||||||||
Practical notes
- At high pressure or near the critical region, φ can deviate strongly from 1.
- For some conditions, cubic EOS yield multiple real Z roots; phase selection matters.
- Virial results depend on B(T); use values consistent with your temperature.
- For mixtures, additional mixing rules and binary parameters are required.
Non‑ideal gas behavior at elevated pressure
Fugacity coefficients quantify departures from ideal-gas behavior. At 1 bar, many gases have φ near 1.000, but deviations grow with density. Around 50 bar, light hydrocarbons often show φ about 0.85–1.15, and near-critical states can move further. At 300 K and 100 bar, dense gases may show Z far below 1. This calculator compares models at the same T and P with consistent unit conversions quickly.
Reading the outputs: Z, φ, ln(φ), and f
Z captures volumetric nonideality through PV = ZRT. φ links chemical potential to pressure for a pure fluid using f = φP. ln(φ) is included because EOS formulas compute it directly and it is convenient in derivations. Example: φ = 0.90 at 80 bar gives f = 72 bar, showing lower effective pressure. When φ exceeds 1, repulsive forces dominate and f becomes greater than P.
Choosing a model for the pressure range
Ideal gas assumes φ = 1 and suits low-pressure screening. The second-virial method uses ln(φ) ≈ BP/(RT) and is reliable when |BP/(RT)| is small. Cubic equations such as PR and SRK extend to higher pressures by solving for Z and then ln(φ). If multiple real roots occur, vapor uses the largest Z and liquid uses the smallest Z. Select phase root with care.
Property data quality and sensitivity checks
PR and SRK require Tc, Pc, and acentric factor ω to shape the attraction term. Property errors matter most at high pressure. Confirm Tc is in kelvin and Pc uses the intended unit; a 10% Pc shift changes A and B and can move Z and φ. If results seem extreme, compare PR versus SRK and inspect how ln(φ) changes with P.
Using exports and history for reporting
Traceable outputs reduce rework. The history stores up to 50 rows, so you can sweep pressure (10, 25, 50, 75, 100 bar) at fixed temperature and observe φ trends. Export the current result or history to CSV for analysis, or to PDF for sharing. Recording method, phase root, T(K), P(Pa), Z, φ, ln(φ), and f(Pa) supports audits. For comparisons, keep the same method when generating multi-point curves across a dataset.
FAQs
What is a fugacity coefficient?
It is the ratio f/P for a pure fluid, adjusting pressure to an effective value that matches real‑fluid chemical potential. φ = 1 is ideal; deviations indicate intermolecular attraction or repulsion.
When should I choose vapor versus liquid root?
For cubic EOS, multiple Z roots can appear. Use vapor (largest Z) for gases and superheated states. Use liquid (smallest Z) for condensed phases or when modeling subcooled liquids at the same T and P.
Why do PR and SRK require Tc, Pc, and ω?
Those properties set reduced temperature and scale EOS parameters. ω tunes the temperature dependence to match vapor-pressure behavior, improving φ predictions over a wider range than a simple corresponding-states fit.
Can I use the virial option at very high pressure?
Virial ln(φ) ≈ BP/(RT) is a low-density approximation. If |BP/(RT)| is not small, errors grow quickly. For high pressures or near-critical regions, prefer PR or SRK and compare results.
Why can φ be greater than 1?
At high density, repulsive effects and excluded volume can dominate, making ln(φ) positive and φ > 1. This means the effective pressure f = φP exceeds the actual pressure, common for compressed gases.
How should I report units in exports?
Use SI in reports whenever possible: T in K and P in Pa. The calculator converts inputs to SI internally and exports consistent values for φ, Z, ln(φ), and f to support comparisons.