history of individual sciences. These courses are designed primarily for science majors who usually have no sense of history when it comes to their own discipline. Many interested non-majors, however, take these courses and often learn more science from them than they do from the freshman course in the subject. Our mathematics department requires our history of mathematics course of its B.A. candidates. Clearly the historical approach is not the only solution to Professor DeSieno's search for unity. I n some cases it has been tried and found wanting. However, our increasing number of courses, greater student enrollment, and overwhelmingly favorable feedback indicate that at GVSC, at least, this is a workable and pleasant solution.
Polywoter and Analytical Chemistry: A Lesson for the Future
To the Editor: To all of us who feel and express concern about the role of chemistry and its several branches in the world of the '70's and beyond, the October, 1971 issue of THISJOURNAL should prove to he highly instructive. Presented here mas a fascinating juxtaposition of the proceedings of the recent AAAS Symposium dealing with "Chemistry and Social Concern" and an account of the astounding history of "polywater." The paramount concern, social or other, of the chemist is the reliability of his observations which are, after all, the foundation of his science. I fervently hope that someday an architect will include in bas-relief or mural decoration of a university chemistry building a representation of Anteus, one of Hercules' many antagonists, whose strength derived from contact with the earth, in order to constantly remind us that chemistry is as strong as its experimental foundation, as the reliability of measurements. The elaborately inflated structure of the theory and lore of polywater is a highly dramatic but not isolated example of relegation of analytical chemistry to a minor role in the design and execution of chemical research. The solution of the polywater question was finally obtained by application of proper analysis although the techniques for the purpose were available long before the question was raised. This point deserves general attention a t a time when a large number of major universities are planning to eliminate analytical chemistry from their curricula entirely, "assimilate" it, or have already done so. Analytical chemistry is a discipline whose applications are easily recognized as "socially relevant," and important as a service branch. That it is equally vital in the proper conduct of all experimental research seems to he harder to grasp by some of our brethren before their "feet are yanked out from under them." Analytical chemistry, the science involving carefully executed measurements of all varieties related to composition and identification and careful evaluation of their reliability, rightfully deserves an
important place in the training program of both undergraduate and graduate chemists. Without some exposure to this discipline, non-chemists may well consider much of chemistry a series of (hopefully) delightful, interesting, hut not totally explicable assertions.
Thermodynamics and Pollution
To the Editor: It is unfortunate that a contribution to the huge pile of articles incorrectly relating thermodynamics to social or philosophical issues should appear in THIS JOURNAL [49, 18 (1972)l. It is not a consequence of the Second Law that "regardless of what is done to cure pollution things are bound to get worse." Even accepting that the entire material universe is a closed thermodynamic system which is by no means evident,' the Second Law has no bearing on the social and political decisions that can he taken and acted upon to minimize the problems of pollution. These include decisions to terminate many of our current practices but also in general terms to develop more efficient new technologies (and the Second Lav can he used to discuss efficiency unambiguously) that will key into the overall biological ecosystem. In particular we need to develop low-temperature technology based on electrochemistry for the short run and on the truly exciting promise of the technology of superconductors2 in the long run. This last application still awaits the development of materials that become superconducting above the ordinary boiling point of liquid hydrogen (20°K). CHARLES E.-HECHT
1 T . a ~.. n.. n. r,r 1... ~-~ a w -n T.IFSHITZ. , E.. "Statistical Phvsics." " , Ad~~, diaon-Wesley, Reading, Mass., 1958, p. 29. 2 MCCLINTOCK, M., "Cryogenics," Reinhold, New York, 1964. CEEGTER. P., Reports on Prog. in Phys.,30, 561 (1967). MATT H I A ~ B.; , PhysicsToday,23 (August, 1971). -.2
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To the Editor: Professor Hecht has in a way asked for depollution of my article "Pollution and Thermodynamics" (or was it "Thermodynamics Polluted?"). The pompous solemnity of thermodynamics seems to have been desecrated whimsically and facetiously. However, if my memory is correct, Professor Don If.Yost of Cal Tech warned his students to stop doing research when it xasn't fun. And, as an undergraduate student of Professor Ward V. Evans, I was dclighted by his amusing and homespun applications of the laws of physical chemistry to d o scribe the comfort of ducks in icy water, the priority of freezing of adjacent hot- and cold-water pipes in an abandoned house, gas-air explosions, keeping apples in Volume 49, Number 6, June 1972
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icy water, where to fish in warm (polluted?) rivers, and so on and on and on. My gem stems, at least indirectly, from a remarkable restatement of the laws of thermodynamics by a genius unknown to me (0)Things could be better or worse (1)You can't win. (2) You can't even break even. (3) Stop trying theimpossible.
It is amazing how the authentic spirit of impossibility shines forth in these simple statements. Two hints that the stretched laws of thermodynamics in my article on the laws of pollutodynamics are not quite as serious as the fate of the universe are (1) At theend, a suggestedreversdof timeand eventsis solemnly labeled "TheFuture." (2) There is a liberal sprinkling of weasel words everywhere.
A second aspect of the article is the difficulty of defining pollution. To a wild animal, an intelligently directed arrowhead is rather serious pollution of the atmosphere. I wonder what the ultimate pollution was for the dinosaurs. Indeed, much of the difficulty in defining pollution is that pollution is a relative term. On first impulse one might suspect that B.O. is pollution, but a reasonable case can be made that B.O. is quite natural while deodorants are unnatural and; therefore, pollutants. Professor Hecht's letter deserves careful rereading. Besides agreeing that activity and pollution are closely related, he offers definite ways of minimizing the toll of the second law. Instead of doom and gloom, he presents hope together with the suggestion that the criterion of success be efficiency as calculated by thermodynamics. I do regret any hint that the rate of decline in quality of life is not serious and cannot be changed. Everyone can do something about the rate . . . except reduce it to zero if the laws of thermodynamics are valid locally for the next few generations. Thus I should like to thank Professor Hecht for filling in some blanks, and I should like to join him in noting that pollution is not a laughing matter. WILLIAM F. SHEEHAN
Integration of the 2p, A 0
T o the Editor:
The definition of the size and shape of an orbital is a contour surface of constant IJ.1that encloses some fraction (say 90%) of the electron probability density. In an article in THIS JOURNAL [OGRYZLO, E. A., AND PORTER, G. B., 40, 256 (1963)l the multiple integral giving the fraction of enclosed probability density for a 2p, A 0 was evaluated approximately by a rather tedious graphical integration method. I wish to point out that
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this integral can be evaluated analytically, making the graphical integration unnecessary. The 2p, orbital has the form 11. = K/w-'/re-+ cas B (1) where the orbital exponent k equals Z/2a, with a the Bohr rhdius and Z the effectivenuclear charge. If we integrate 1$12 over the orbital volume, doing first the 9 integral, then the r integral, and finally the .$ integral, then the fraction f of the probability density enclosed by a 2p, A 0 is
where s, t, and B(r) are as shown in the figure. The
factor 2 is needed to include the probability density in the lower lobe of the orbital. Evaluation of the angular integrals gives
But from eqn. (1) we have cos B(r) = $a'/Wv/'ei'/r, where #is the chosen value of the wave function. Substitution then gives an r integral which is readily evaluated to give as the final result
where c s $~'/'k-'/* = se-X' = te-". The values of s and t are found by setting 0 eqn. (1) to get
=
0 in
Eqn. (3) is then solved for the two values of r that satisfy it; this can be done numerically by trial and error, or graphically by plotting k''a?r-"'re-k' (which isathe value of # on the z axis) versus r and reading off the two values of r that correspond to the chosen value of $. The results of Ogryzlo and Porter agree closely with those obtained from the exact expression (2).
IRA N. LEVINE BROOKLYN COLLEGE of CUNY BROOKLYN, NEWYORK11210
