unsatisfactory for the description of forces between permanent and induced dipoles hut acceptable for description of the mixing of relatively nonpolar molecules. Since dispersion forces are always attractive, the negative square root is used. In this treatment, therefore, the total interaction paw,,) is always rameter of Engstrom, w = w,, - 1/2(w,, positive. With an acceptable method of calculating the various energetir interactions required for the program, we can proceed to experiments. The table contains a short list of interaction parameters calculated from puhlished data.
+
Student Explorations We asked our students to look a t a variety of effects on a polystyrene chain in solution. Students conducted simulations while systematically varying the polymer chain length, solvent, and temperature (as reflected in the dimensions of the interactions Darameters.. eiven in units of kT). Our microcomputers were not equipped with the floating-point coDrocessor recommended bv Ennstrom. so our students limited their investigations tochains of 10 monomer units or less. The leneth of time for the calculations was less than an hour for most simulations, and students were able to conduct several calculations a t one time by using several computers each in our microcomputer laboratory. We also found it useful to coordinate the investigative efforts of different groups of students. Each group examined a different experimental variable, and the results from all the students were combined and made available to the entire class.
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Conclusions The Monte Carlo simulation by Engstrom and Lindherg is a meaningful addition to a polymer solution theory or polymer laboratory course. The program illustrates several important concepts of polymer shape and provides an introduction to statistical methods for modeling polymer systems. I t is clear that the atudents benefited from the opportunit y to conduct Monte Carlo simulation experiments. Manv siudenta are unfamiliar with statistical cafculations, even if they are familiar with the solution theories. The students gained additional insight into the approximations of the theory and the Monte Carlo method by watching as the computer explored different polymer conformations on the CRT screen. This paper addressed some ambiguities in the original description of the Monte Carlo program by Engstrom and provides a rational basis for deriving energetic parameters for use as input to the program. Our students found the results of the modeling to be more meaningful when they were able to derive the energetic parameters used as input to the program from experimental data representing real chemical systems.
Acknowledgment The author wishes to thank Paul Rasmussen for helpful discussions and to acknowledge suggestions from Sven Engstrom. Literature Cited 1. EngstrDm, s.;Lindher g,M. J. Charn.Educ. 1988,65,973. inSolution; Krieger: 2. For s moreeomploted,~rivation,~Morawetz.H.Moeromolaculas Malshar. FL, 1983: pp 39-12. 3. Brsndrup, J.; Immorgut, H.. Eda. Polymer Handbook Wiley: New York. 1975. I. Van Larr. J.J.: Lurenz, R. 2. Anore.. Al&ern. Chem. 19'25. I(6.442 5. Hildebrand, J. H.: Scott. R. L. The SoluUlily of None bcfrolytes; American Chemical Society Monograph No. 17; Reinhold: NewYork:Ch apfen 7 . 8 . 6. Handbook o/ Chemistry and Physics; Wesst, R. C., Ed.; CRC. Boca Raton, FL, 1980: 6lst ed.
622
Journal of Chemical Education
Altltude and Bolllng Point To the Editor: Boyd Earl [1990,67,45] has done a great service to teachers of thermodynamics. In presenting his formula relating the boiling temperature of a liquid to elevation, he has combined in a useful and interesting context three thermodynamic relations, a t least two of which are of great importance: the (isothermal) barometric formula, the relation for the adiabatic expansion of an ideal gas, and the ClausiusClapeyron equation. While this in itself is remarkable, in addition the derivation is straightforward and elegant, and, as the author points out, it provides an instructive illustration of the use of approximations. Beyond all this is the surprising conclusion of the derivation: in fair approximation, a t least up to 10 km (misprint in the table), the decrease in the boiling temperature of water from its value a t sea level is directly proportional to the elevation. In fact, this lowering is given by where h is the elevation in kilometers, which places the boiling point on the top of Mt. Everest a t 70 "C. I am so moved by this article that I wish to make the following modest contribution. Since there are 3.28 ft in a meter, ATb (in K) is numerically equal to the elevation in thousands of feet to within the accuracy allowed by the approximations involved. Unfortunately, neither scientist nor lay person will he a t ease with this mix of units. However, everyone should be impressed that in OF and miles There is a 10 "Fd r o in ~ the boiline tem~eratureofwater for every mile of elevation, a t least u p to 6mi. In fact the value 10 provides closer agreement With the more accurate formula ( ~ a r l ' seq 4) t h k d o e s the approximate one (eq 6) in this range of elevations. I look forward to using this in my thermodynamics classes as a means of instilling life into the barometric formula, as a practical application of the dreaded (by students) adiabatic ideal gas relations, and as an extension beyond the usual applications of the Clausius-Clapeyron equation. Donald Peterson California State Universiiy Hayward. C A 94542
Paper Model of a Cuboctahedron To the Editor: The articles by Shukichi Yamana, "An Easily Constructed Cuboctahedron Model" (1985, 62, 1088) and "An Easily Constructed Model of a Coordination Polyhedron that Represents the Cubic ClosestPacked Structure"(1987,64,1040) describe nicely the methods of getting the paper models from a sealed, empty envelope. I have followed both the methods in making the models. The models formed finally, however, are identical. The same polyhedron, if seen from different angles, can look as is shown in the f i r e s of the respective papers. I would like to know whether the author and other readers are getting similar experience.
Davld S. Allan
V. D. Kelker
Northwestern University Evanston, IL 60208
Unlvenity of Poona Pune 411 007, India
To the Editor:
I t is a great pleasure for the author to know that some readers are interested in the author's articles and in testing them to express their frank opinions. Now the author would like to answer the auestion sent in from the reader. I t is clear ---in a crystallograpiy texthook that a cuboctahedron can be observed in a cubic closest packed structure. From this viewpoint, the observation of the reader is correct, and i t is selfevident truth. However, this does not lessen the valuesof the two articles previously puhlished by the author (J.Chem. Educ. 1985.62.1088 and 1987,64,1040). I t is needless to say that the authoks methods of folding envelopes are useful f i r various reasons, one of which is to train thinking ahility of a pupil, in addition to using the model ohtained in class (Yamana, S. 8th International Conference on Chemical Education, Tokyo, 1985 and 31st IUPAC '87, Sofia, 1987). This is applied to the present case, as follows: (A) Although these two methods produce the same shape (euhoctahedron), the processes of them are quite different from each other. (B) The model produced by the first article is not the same as the one by the second article in their magnitude. (The first method produces a slightly Larger model than by the second method.) (C) The author thinks it is proper to treat the cubic closest packed structure together with the hexagonal closest packed structure in class. This leads to the conclusion that the second article should he read together with the third one (J. Chem. Educ. 1987,64,1033), whose processes are quite similar to those of the second article and the scales of models obtained in the second and third articles are the same. In this case, the first article is powerless. (D) From these standpoints,the second method can he considered to he a modified method of the first one. On the other hand, since 1968, the author has invented and published methods of making ahout 30 kinds of polyhedra from a (or two) business envelo~e(s).Among them, three methods bf producing three kindsof polyhedra (tetrahedron. octahedron. and twin ~entahedralconessharing a face) have been revised and rep"hlished. The author always thinks that even if a method is devised and published, it should be revised for its better use, if possible. Shuklchl Yamana Depsrtrnent of Kyoushoku Education Kinki University Kowakae. Higashi Osaka 577. Japan
Side view of the inverted tube.
where
and
are the principal radii of curvature a t the point P, and a,R, and x denote the surface tension of the water, the radius of the tuhe and the vertical displacement AP (or QB), respectively. Equations 2 and 3 are obtained by assuming that the vertical cross sections of the surface of the water through AB, and through P perpendicular to AB, are arcs of circles. T o determine the maximum allowed diameter of the tuhe to prevent water from flowing out, it is necessary to substitute eq 2 and eq 3 into eq 1and set
where p is the density of the water andg the gravity constant. T o achieve this, Sasaki assumes, quite rightly, that x
