Thermodynamic Consistency Test Based on Differential Residuals

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Consider the following four binary mixtures: (1) acetone and methanol at 58 °C, (2) MEK (methyl ethyl ketone, or butanone) and toluene at 50 °C, (3) chloroform and 1,4-dioxane at 50 °C, and (4) diethyl ketone and -hexane at 65 °C. For each mixture, we provide the isothermal vapor-liquid equilibrium (VLE) diagram and the equilibrium curve. We also plot the following quantities (known as differential residuals):




The method presented here is called the differential thermodynamic consistency test. It tests whether the experimental VLE data is consistent or not [1]. These differentials should lie with for the data to be thermodynamically consistent.

The theoretical values of , , and are calculated using the two-parameter Margules equation:


However, the approach can be extended to any activity coefficients model.

For one of the mixtures (acetone and methanol at 58 °C), you can vary the values of the experimental data to add either white noise or a systematic error. The noise level is restricted to a magnitude equal to 0.05. Clearly, when there is no noise and no error at all, the data is perfectly consistent (i.e., all differential residuals are equal to zero).

For the diethyl ketone and -hexane mixture, the experimental data is clearly inconsistent.

For (1) the chloroform and 1,4-dioxane and (2) the MEK and toluene mixtures, the experimental data can be considered as thermodynamically consistent.


Contributed by: Housam Binous, Ahmed Bellagi, and Ali Kh. Al-Matar (May 2015)
Open content licensed under CC BY-NC-SA




[1] J. M. Smith, H. C. Van Ness, and M. M. Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., New York: McGraw-Hill, 2005.

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