uent vectors. Each such vector, therefore, represents a correctly balanced chemical equation. If r = 1,that is, if only one reaction is possible, all that remains to do is to multiply the vector c. by an appropriate numerical factor to clear the fractions. If r > 1the orthonormality of the reaction vectors c m , + ...,c, is a much stronger constraint then the chemical condition that the reactions only need to be independent. We can form linear combinations of the reaction vectors that result in simpler expressions taken here to imply that each reaction contains a t least one species that does not participate in any other reaction. Other chemical criteria might be imposed. Independence is still guaranteed and clearing fractions proceeds as in a single reaction. Literature Cited 1. Bmmkrg, J P Phyarid Ckmistry, 2nd. ed.; AUyn and B a n : NeuYorh 1984, m any physics1chemistry textbrnk.
D.
2. Gmkell, R. Introduction to Metollugirnl ThormodyMmics. 2nd. ed.; Hemianhpre 1981: n436 --,3. BlaWey. G. R. J Chpm Educ. 1982.69, 728. 4. Smith W R.;Missen. R. Chemical Reoetion Epuilibrim AMly"s; W h y : New ""-7, .ow 6. Missen. R.W.: R.J. Chem. Edue. 1089.66.211: erratum J Chpm Edue 1989,'66, 53k. ' ~~~~
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W . Smith.W.
. .
.
6. Albertv. R.A.PhvsMl Chomiatri, 7th ed.: Wilw: NeuYmk. 1987.
The Pointer Function- An Invaluable Calculational Tool Henry Freiser University of Arizona Tucson. AZ 85721
Figure 1. Use of the pointer fundion to find the real roots of a sixthorder function of x.
Readers of this Journal need no convincing of the convenience resulting from conducting chemical calculations with the aid of computer spreadsheets. It is the purpose of this article to describe a simplified method of solving even the most complex algebraic problems by means of a simple function, called the pointer finction, whose graphical display over the entire range of interest of the variable of interest gives all possible real roots in that range ( I ) . Thus, while solutions of equations of higher order than quadratic can be simplified by using the method of successive appmximations with the help of QuattroPro (Borland Inc.) (or by means of special preprogrammed techniques such as Eureka), the use of the pointer function represents a significant improvement in simplicity while serving to focus on the basic mathematical relationships involved. The use of this function has the added advantage of providing a "global overlay" on the graphical representation of the problem under consideration.
logarithms of negative values are not possible, using the logarithm of the absolute value of FG):
The Pointer Functlon The principle of the pointer function can be stated very simply: If a problem can be reduced to an algebraic expression of one indenendent variable !OH. ..DM.. x.. etc.)... called FG), which has dne or more real soLtions (roots) in a given range of interest for the variable, these can be found very simply and accurately without sacrificing any rigor. Values of F(x) a r e obtained, with the help of the spreadsheet, for a series of values of the independent variable, x, over the entire range of interest. By definition, the roots are those values of x a t which FW is zero. One could simply graph F(x) versus x and locate the points where the curve crosses the x-axis (i.e., where x = 0). With some functions, however, the appmach to zero may be so gradual that the exact location is difficult to find. This difficulty is obviated by using the logarithmic expression and, since
Consider first how the pointer function is used in solving a general algebraic problem involving a higher order equation.
+ lo4)) @MG(@ABS(F(x) This function, called apointer function (you can see why in Figure 11, is a convenient way to recognize rwts. (The reason for adding 10" to F(x) is to avoid the awkwardness resulting when a value of x results in a F(x) = 0 exactly. Without the lo", this pointer value would read "ERR.".) Applications of the Pointer Function The usefulness of this approach is not restricted to the tmes of ~roblemsencountered in analvtical chemistrv. or A n to those in chemistry in general. ~ k a m ~ lin e sphy&s, engineering, and mathematics are equally useful. Algebraic Applications
'Note: In this article three menu commands will be listed for each step where approriate. First, in bald, will be the QPro command, followed by curly brackets containing the Quattro and Lotus 123 commands,respectively. What are the real roots of F(x) in the interval 0
