VLE Activity Coefficient: Wilson Equation

The Wilson Equation is a parameter-based technique used to estimate vapor-liquid equilibrium activity coefficients. (Note that Cranium and Synapse assign different activity coefficients to each type of phase releationship, i.e., VLE, LLE, SLE, SVE.) In addition to parameter values for mixture components, the technique also requires values for the each component's liquid density.

The estimation code contains four main steps:

  1. the binary interaction parameter keyword is formed by adding a prefix to designate the pressure (code lines 012 through 015)
  2. the critical temperature is retrieved or estimated (code lines 017 through 020)
  3. the critical pressure is retrieved or estimated (code lines 022 through 028 - note the technique requires the critical pressure be in the non-standard units of atmospheres)
  4. these values are inserted into the technique's final calculations (code lines 045 through 054)


  1. Robert C. Reid, John M. Prausnitz and Bruce E. Poling. "The Properties of Gases and Liquids." McGraw-Hill Book Company. New York, New York, USA. Edition 4, 1987.
Example Calculation

Being an equation-based technique, the Lee + Kesler Relation requires values for input physical properties. Specifically, property values are required for the normal boiling point, critical temperature, and critical pressure.

For example, values for the required properties of 3-isopropyl-6-methylene-1-cyclohexene are given in the table to the right. Note that each of these values was estimated by a group contribution technique.

Inserting these property values into the model shown at the top of the page yields an estimate of

Example Calculation
Boiling Point1431.76K
Critical Temperature2630.74K
Critical Pressure327.54atm
Numerator --- -0.8876 ---
Denominator --- -2.5143 ---
Estimate --- 0.3530 ---
1) Estimated by the Stein + Brown Method
2) Estimated by the Joback Method
3) Estimated by the Lydersen Method
Evaluation Summary

The figure to the right shows good agreement between estimates and data values. The two outliers, located in the graph's top-middle section, are o-terphenyl and m-terphenyl, were excluded from the calculation of the summary statistics. Note that as an equation-oriented technique, estimation errors may be caused by errors in the input physical properties.

This evaluation was performed on January 31, 2023 using Cranium, Professional Edition version 5.0.

Summary Statistics
# Observations314- - -
Avg Abs % Error12.677%
Max Abs % Error930.2%
Min Abs % Error0.070346%
Avg Abs Error0.028767- - -
Max Abs Error0.63288- - -
Min Abs Error0.000068148- - -
Avg Error0.014307- - -
Max Error0.63288- - -
Min Error-0.47644- - -
Example Estimation Errors
ChemicalData [K]Estimates [K]% Error
Ethyl acetate0.360.384.152
Accuracy Code

Our evaluation determined that the Lee + Kesler estimation technique should not be used for perfluorocarbons. The technique's applicability rule shows this exclusion on lines 012 through 025. To summarize:, the code checks if there are only two types of elements in the chemical's molecular structure and if these two elements are carbon and fluorine.

  1. the code compiles the elements contained within the chemical's molecular structure (code lines 008 and 009)
  2. the code checks if there are only two types of element (code line 010)
  3. the positions of 'Fluorine' and 'Carbon' in the elements array are determined (code lines 012 through 014)
  4. if both fluorine and carbon are present in the array, i.e., their positions in the array are not equal to -1, then the chemical is a perfluorocabon and the technique is not applicable (code lines 016 through 026)

For non-perfluorocarbons, the average absolute error is 0.0288.

Online Calculation

The following calculator demonstrates the use of the Lee + Kesler estimation method. Enter values for the input properties in the table to the right and then press the 'Estimate' 'button. Property estimates are generated by an online version of Cranium, Web Server Edition. The Web Server Edition of Cranium enables you to distribute physical property values and estimates via the internet. Please contact us if you would would like to learn more about how you organization can utilized the Web Server Edition of Cranium.

Calculation: enter values for the required physical properties in the units given in the table below. Then press the 'Estimate Property Value' button. A request will be sent to an online Web Server Edition of Cranium to perform the estimation. (Note that if you do not know values for the required input properties, draw your chemical's molecular structure in the editor to the right and press the editor's Estimate button. Cranium will estimate the required input property values.)

Acentric Factor Estimation
Boiling PointK
Critical TemperatureK
Critical Pressureatm
Estimated Acentric Factor
- - -
- - -
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How It Works

The physical properties entered above are added to a property estimation request which is sent to an instance of our Cranium, Web Server Edition software product running on a Microsoft Azure virtual machine. Cranium processes the request generating an estimated physical property value. This resulting value is then sent back to this webpage for display.

Click here to learn more about how you can use our Cranium Web Server to distribute your company's physical property data, estimates, and knowledge throughout your organization or contact us for further details.