the volume as 125 mL. I would appreciate hearing opinions from interested readers. I will report the results of this survey to all who send their opinions and, if theEditor agrees, in a future issue of this Journal. Lowell M. Schwartz
Solvlng Quadratic Equations To the Editor:
Hugus and Hentz, in their discussion ( I ) of solubility product calculations, encounter a quadratic equation of the form
University of Massachursns 8astan. MA 02125
y2+yy+1=0
with y given to six significant digits as -56.8235. Using the quadratic formula in its usual form they obtain m a s 56.7883 to the same significance, and thus the two roots y, and yp are Solving Llmitlng Reagent Problems To the Editor:
The procedure for solving limiting reagent problems described by A. H. Kalantar, J. Chem. Educ. 1985,62,106is an excellent example of problem solving using a logical series of well-defined steps. However, his remarks pertain in^ to each step in the procedure leads to some uncertainty as to the exact meaning of the term, X. Step three states, "Change after X moles react", and step four refers to X as ". . .extent of reaction". In step three, the number of reacting moles is not X hut clearly 3X, 2X, and 8X, respectively. The meaning of the term, "extent of reaction", is not clear. As step three in Kalantar's urocedure indicates. the coefficients of a r~.acrimcan be muiriplied hy any pos&ive number, in this raje, X.and theruuurtun issfill balanced with respect turhe moles of reactants-and products. It seems the essence of his procedure is to find the value of a term, X, so that when multiplied by all of the coefficients (units of moles implied), the limiting number of moles is obtained. Clearly, as the results in step six indicate, the smallest value of X obtained for the three reactants in step five, when substituted in step four gives non-negative results. I suggest that the term X he thought of as simply a reaction coefficient multiulier. The value of this term. when multiplied by each reaction coefficient will yield value equal to the number of moles present of the limiting reagent and values less than the number of moles initially present for the other reagents
a
Dean 0. Skovlln California State University. Northridge Northridge, CA 91330
To the Editor:
Thanks are due D. 0.Skovlin for his clarification and suggestion. That X is merely a "reaction coefficient multiplier", proportional to the number of moles reaction, is especially obvious if one works with homogeneous (ideal) gas phase reactions a t fixed T and V. Then the number of moles of each reactant and oroduct is simnlv nrooortional to that . component's partial pressure or X VIRT. When the urocedure is oresented in class. it is clear that one should begin with an kxamp~einvolving only two reactants and that the balancing coefficient of one of them should be one. For this Journal's readership, I chose the more complicated example (with admittedly imprecise remarks) to emphasize that the change (line 3) is better written with integer (rather than fractional) coefficients. These same integers are then used for halancing, for the change and, in equilibrium problems, for exponents. Fractions also work. but mv.exnerience indicates that the students have far . fewer difficulties when the problem is set up using only integers.
-. - .
A. H. Kalantar The University of Albem Edmonton. Alberta Canada T6G 2G2
472
y , = [-(-56.8235)
+ 56.7883]/2 = 56.8059
y, = [-(-56.8235)
- 56.7883]/2 = 0.0176
and
Journal of Chemical Education
(Using y to only two or even three significant figures, as the data of the problem warrant, one obtains zero for the smaller root.) They note that the smaller root loses significant digits and suggest that it he obtained by a Taylor expansion of the radical. This process would no doubt work, but most general chemistry students would lose the point and, certainly, the chemistry of the discussion. A simpler and pedagogically hetter procedure may he found in ref 2. This paper obtains the easily proved result that the product of the two roots of a quadratic equation is given by the constant term divided by the coefficient of the quadratic term:
or, in the present case
Subtraction losses are thus minimized and y2 = l/yl = 0.0176038 to six significant figures. (In the case y is kept to two fieures. - . the smaller root is 0.018 rather than zero.) In addition to yielding hetter accuracy, this process can be done with fewer kevstrokes and/or temDorarv storages and so is less error-Prone than the direct &e oE the aigehraic formula. Llterature Cited 1. Hugus. ZZ.and Henfr,F.C. J. Chrm.Educ. 1385.62.645 2. I.udwiz. 0. C.. J . Chem. Educ. 1983.60.547.
Ollver G. Ludwlg Villanova University Viiianova, PA 19085
"Elemental Etymology: What's In a Name?" To the Editor:
An error in the article "ElementalEtymology: What's In A Name?" 11985, 62, 787) has been pointed out to me. W. J. Balfour of the University of Victoria has informed me that the Ytterby mentioned in the article is not the Ytterby of ytterbiumlyttrium/terbium/erbium fame. The Ytterby of chemical interest is a small village, not seen on most maps, near the town of Vaxholm on the east side of Sweden. The Ytterby I had referred to was north of Gothenborg on the west side of the country. Balfour has visited Ytterby and says that he found no evidence of the town's historic mine (from which many new elements were eventually discovered), hut he did note that many of Ytterhy's streets were named after elements. I am pleased to be able to correct this error. David W. Ball Rice University Houston. TX 77251
