Thermodynamics of Van der Waals Mixtures; Symbolic and
numerical computations, visualizations using Maple
A.H.M. Levelt, University of Nijmegen, The Netherlands
Friday, May 3, 2002, at U. of Waterloo.
Abstract:
The physicst J.D. van der Waals is best known for his equation of
state for fluids (1873), which incorporates the liquid-vapor
transition and criticality. In 1890, he proposed a generalization
to fluid mixtures, the first time the complex phase behavior of
mixtures could be modeled. Special cases of the Van der Waals
binary mixtures equation were studied by the mathematician D.J.
Korteweg (1891), he of the Korteweg-de Vries equation, and by J.J.
van Laar (1903-1906). In both cases, huge symbolic computations
were performed with paper and pencil.
Only around 1970, when electronic computers became widely
available, was the study resumed by the chemist R.L. Scott at UCLA.
He and his student Van Konynenburg studied the general case
extensively, formulating the starting equations using paper and
pencil, and performing all further calculations numerically, while
plotting the results using a Calcomp plotter. The availability of
these "global" results provides a framework for understanding the
older work. The calculation of phase separations in multicomponent
mixtures by means of equations of state is now widespread throughout
the chemical process industry.
My attention was drawn to the work of Korteweg and Van Laar by the
physicist Paul Meijer (CUA, Washington, DC). In particular, he
noticed several exact results in Van Laar's work, which he and I then
confirmed, using Maple. He also alerted me to the differential
geometry applied to thermodynamic surfaces by Korteweg, and the
resulting complex graphs. Presently, my sister, J.M.H. Levelt
Sengers (NIST, Gaithersburg, MD) and I, are trying to understand
Korteweg's work from a modern physical and mathematical point of view.
In my talk, I give a limited overview of this field, and discuss
some mathematical details. But the emphasis will be on my
applications of Maple to the required symbolic computations, as well
as to the visualization of surfaces in 3-D, of which examples will be
given in the talk. No previous knowledge of thermodynamics is
required; the necessary minimum will be explained.
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