The Unitless Angle of Sin
Bean Counter's Caution To the Editor:
The recent article. "The 'Bean Lab'. A S i m ~ l eIntroduction to ~quilihrium: by ~ickinsona n d ~ r h a r i(1) t is a useful "ex~erimental"wav to exDose students to a micmsco~ic picture of equilihrim". (A mire detailed discussion of ;he same idea was published earlier in this Journal by Gunther Harsch in a n article entitled, "Kinetics and Mechanism-A Games Approach" (2) .) It should he made clear, however, t h a t the Bean Lab method is valid only for illustrating equilibria in chemical reactions where the microscopic mechanism of the reaction is a n elementary mechanism, which is identical to the stoichiometric equation for the reaction. That is, in the reaction
discussed in the article, the authors implicitly assume that the mechanism consists of a bimolecular collision steo between a n R and a P to yield 2W's and a B, plus a trikolecular collision step between 2W's and a B to yield an R and a P. If the reaction is actually governed by some other mechanism, then the details of how the Bean Lab is played need to follow the mechanism and not the stoichiometric equation. Literature Cited 1. Diekinson, P.D.: E r h d t , W. J. Chem. Educ. 1991,68.990. 2. Harsch, G.J Chem Educ 1384.61, lm9.
To the Editor:
Recently erruneoui statements about "units" of angles ameared in this Journal I1). That the anele. .. . 8. in functions such as sln 8, is 3 u r f l t l e ~argument ~ ir seen vla elementitry results of'tri~onometrvand calculus such as ?in' 9 + ~06'8 = 1,(sin 8) 18'approaches unity as 8 + 0 and d (sin 8) I dB = cos 0. The first shows that the trigonometric functions, such as the sine, are unitless (a point not in dispute): they are simply ratios. The latter show that sin 8 and 9 have the same units and so 9 must he unitless. These functions can be expanded as infinite series (whose use so conveniently yields the computed values of the functions), e.g.,
which also demands that 8 be unitless because sin 0 is and because only a unitless quantity to different powers could be summed. Other series show that n is likewise unitless. The great self-consistency in correct science, while pmviding us with the liberty to choose convenient expressions, is constraining as well. Unless all connecting relationships are consistent, we can be sure of a n error. Mathematical constraints such as sign, parity, unitlessness, etc., are powerful. Thus they force us to ask if some seeming inconsistency lies instead in our own incomplete understanding.
Leslle J. Schwartz
St. John Fisher College Rochester, NY 14618
Literature Cited 1. Wadhnge~R.J. Chpm. Edue 1991.68.708
A. H. Kalantar
The Mercury Beating Heart To the Editor:
The recent article by Avnir (I) atributes to Carl Adolf Paalzow (1858)the discovery of the mercury beating heart; however, in Berzelius's Chemistry he describes this phenomenon and says that i t was obsemed the first time by Ermann and studied in detail by Herschel's son and P f e . Berzelius describes the oscilating mercury when it is covered with a liquid and two electrodes connected to one electric battery are then introduced into that liquid. He says "the movement is very rapid in strong acid and no movement when the liquid is basic, although the electropositive metals presence can produce pulsating movement of mercury". Berzelius attributes to Runge the orginal description of what happens when the mercury is covered with a solution of salt, a crystal of blue vitriol is put over the metal, and the metal is then touched with iron or zinc. Literature Cited 1. A&,
D.J. Chem. Edue. 1983,66,211.
2. Berzelius. Chemlslry; Spaniah edition of 1845:Vol.I, Eleetmchemistry, p. 95.
M. and M'T. Martin Sanchez E.U. Pablo Montesino Santisirna Trinidad 37 28010 Madrid, Spain 1042
Journal of Chemical Education
University of Alberta Edmonton, Canada, T6G 2G2
Answer to Punle on Page 1039