In the Classroom
Teaching Theoretical Physical Chemistry: Density Functional Theory and the Taylor Expansion of the van der Waals Free Energy
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Brian G. Moore School of Science, Penn State Erie—The Behrend College, Erie, PA 16563 John D. Mize Department of Chemistry, Eastern Washington University, Cheney, WA 99004
It can be very difficult to involve undergraduates in research on theoretical topics. The problem of getting “up to speed” in a particular area can be extremely difficult. The approach that we have taken is to try to find a nontrivial example demonstration of the theoretical technique that allows students to analyze the problem completely. This example should be something that builds on material the student is familiar with in course work, while at the same time being novel, so that the result is not known in advance. This concept of using a “bridging’’ problem can be attempted in general for any area of theoretical research. The details will be different, depending on the student and the area of research. One particular case is discussed in detail in our article available in JCE Online. The area of research was the statistical mechanical version of density functional (DF) theory. The system analyzed was the free energy derived from the van der Waals equation of state. First we set up the non-DF problem of locating the spinodal line and liquid–gas coexistence conditions, using dimensionless variables. The variables (such as P, T, density) were made dimensionless in a manner that does not assume the existence of a critical point. This first part of the problem has known solutions. We find that it is important to start with a calculation with known results so the students can check the results. W Supplementary materials for this article are available in JCE Online at http://jchemed.chem.wisc.edu/Journal/Issues/1998/Jul/ abs858.html.
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The second part was to construct a DF approximation to the van der Waals free energy. One must isolate the nonideal part of the free energy and make a Taylor expansion of this function. This procedure illustrates several of the steps taken in a real DF research problem. Furthermore, this problem had never been done before so the results were not known in advance. The results were then analyzed for various levels of the truncation of the expansion. At second-order truncation there is no solution—one can show that no critical point exists. At third order only a nonphysical solution is found. At fourth order a solution is found with qualitatively the same features as the exact answer, although the coexistence curve is no longer symmetrical as in the exact van der Waals case. At a fifthorder truncation, one can see that the solution is converging on the exact answer. In the case described here, a more ambitious project was originally proposed. However, initial work showed that it would not be possible for students to do more than a very small piece of the project in the time available. (The student was committed to intensive work on this project, but only had a 10-week quarter available to do research.) The “bridging” problem was instead worked out and in large part finished during the 10-week period. Aside from showing an example of an approach to theoretical research involving an undergraduate, the full article available in JCE Online may also be beneficial to nonspecialists interested in an introduction to the area of density functional theory.
Journal of Chemical Education • Vol. 75 No. 7 July 1998 • JChemEd.chem.wisc.edu
