G. Olof Larson Ferris State College Big Rapids, Michigan 49307
Chemical Exponentialism for Beginners
There is a great need for finding easier ways to present mathematical concepts essential to problem-solving in chemistry. Handling of numbers is the real stumbling block for many .beginners, and efforts to persuade students to develop logical solution set-ups (based on unit labeling, balanced equations, mole concepts, "chemical mainstreets," etc.) are often frustrated by the barrier encountered in "working out" the numerical answers. It is suggested, therefore, that essential math operations. needed for chemical calculations. should be reviewed or taught in a way that will give the students new insight for understanding and handling of numbers, so as to reduce (hopefully eliminate) this barrier to solving chemical problems. The approach outlined below assumes a latent ability (and willingness) on the part of the student to recall rudimentary operations in algebra. Following an introduction or review of operations with whole powers of 10, e.g., lo3 X lo3 = lo6, 103/102 = lo1, 102/103 = lo-', and 102/1Q2= lo0 --= 1, it is reasonable to point out, even to beginning students, that numbers between 1 and 10 can be identiJied with a series of fractional powers of 10. This is most conveniently illustrated as a linear exponential scale (similar to the C and D scales on a slide rule (Fig. I)). Identity u
rather, on the basis of "m-s" values, they become m01e~~le~/rno1e or "thinkable" quantities of about g/a.m.u. To reinforce the concept of intrinsic "m-s" values of numbers between 1 and 10 a simple exercise in constructing an exponential scale is useful. Starting with the relative positions (i.e., the identity 2.0 = "m-s" values) of the other cardinal numbers can be estimated quite well. For example and Also, since we estimate that 9 Therefore
--
(between
and 10l.O).
and 6
=
3 X 2 =
100.48
X
100.30
=
100.78
Moreover To fill in 2.5
=
10/4
= 10~.00/100.60 =
100.40
and Figure 1.
A linear exponential scale.
of the exponents with common logarithms might be mentioned, but without emphasis. Rather, at this point in the discussion, a concept of relative "multiplying strengths" of numbers from 1 to 10 is suggested, as the basis for understanding the uneven distribution on the scale. = 4%= 3.16. That is, 3.16 is For example, half as "strong" as 10 for multiplying purposes, in the same way that 10 (or 10l.O) has one half the multiplying strength ("m-s") of 100 (or 102.0). And, just as it takes 3 tens (i.e., lo3) to equal the "m-s" of 1000, the "m-s" of the number 2 (or is one third as great as the "m-s" of the number 8 (or 10G.90). The fact that 1.0 (10O.O) has no "multiplying strength" is interesting, in this context, just as it is important to stress the uniformly relative nature of the "m-s" value for all numbers, regardless of reference base (e.g., in this case the number 10). Further, very large numbers (like Avogadro's number) and very small numbers (e.g., g/a.m.u.) need not be pointed out as "unfathomably" large or small, but
1.4 w
42
w I O 0 . I 5 (or 1.4 w 10/7 w
100.15)
etc. (e.g., 10/8 and other reciprocals). I n addition to promoting a new awareness of numbers, the above exercise also encourages critical comparisons of actual versus estimated values, affording a natural opportunity to teach scale reading, which is a hurdle for some students as they learn to use the slide rule. So far, in this discussion, there has been a deliberate avoidance of the terms "logarithm," "mantissa," "characteristic," and "antilogarithm." Rather, the emphasis is on identifying numbers with their exponential counterparts, for example, 2.0 = 20 = 2 X 10 E 100.30 X 101.00 = 101.30, 200 = 102.30, etc. I t is more instructive to say "2.5 is 10°.40"than to say ''the log of 2.5 is 0.40." Also, it is just as easy to write the as it is exponential identity of a number (2.0 = to write its representative log notation (log 2.0 = 0.30). Of course, as familiarity develops with practice, either notation generally simplifies to the writing of representative exponents (or mantissas) only. But emphasis on the exponential identity of numbers in the Jirst exposure serves as a natural introduction to the slide Volume 47, Number 10, October 1970
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693
rule and to the later ~ationaluse of log tables, when greater precision is required. With such a background in "exponentialism" the "log scale" on a slide rule is easily recognized as an exponential scale, to be compared directly with one of the "number" scales. Also, the D scale on the typical linear slide rule can be described as a passive "logdistance" ruler, upon which the companion C or C I scales operate to measure out appropriate log-distances right or left, for multiplication or division. Furthermore, with a strong consciousness of the exponential counterparts of numbers, the student is in a position to think of whole number fractions, such as 303/273 and 740/760, as having definite "m-s" values. These values are, of course, expressed in terms of log d-ferences and, on the slide rule, he should identify the corresponding log distances between the two numbers of each fraction. He will then also be able to see that the common "divide before multiplying" operational sequence (using the C and D scales) is simply a means of measuring the appropriate log distance ("m-s" value of the fraction) to the left or right on the D scale, in accord with the expected effect of, for example, increasing the temperature or decreasing the pressure on a volume of gas. I n this connection, it is also instructive to use a compass (or divider) to illustrate the relative "multiplying strength" of a fraction, or to carry out complete gas law calculations, e.g., 12.5 1 X 303/273 X 740/760, by measuring "m-s" values of fractions with a paper straight edge and pencil on a linear D scale alone, analogous to the operation of a binary (circular) slide rule. A further lesson from use of the exponential scale and the slide rule is the concept of precision. Assuming that the data justifies the effort, answers are uniformly readable on a 10-in. "D scale," to 1-2 parts per thousand, with careful settings, while the number of significant figures varies from 4 on the left to 3 on the right (with the transition from estimated 1/10 or 1/5 to 1/2 divisions corresponding roughly with this limit of precision). This contrasts with the usual rule of rounding off numbers and retaining significant figures, and the problems encountered for such set-ups as 965 X 1.250 = 1206. Practice in rewriting all numbers of a correctly formulated problem set-up in exponential form (i.e., identifying the fractional as well as the whole power of 10 components exponentially) not only helps the student develop a facility for estimating the numerical answers, but it also prepares him for a real understanding of p H and pK notation, as related to concepts of chemical equilibrium. For example, in solving for the p H of a 0.1 M solution of a weak acid HA (K, = 5 X 10-lo), after writing the equation and related equilibrium expression, K,,
694
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Journal of Chemical Education
Figure 2. To convert from pH to [H+], position upper arrowhead (left) a t correct pH (whole number). Find [H+] as number corresponding to decimol pH value times the power of 1 0 indicated b y lower (right) arrowhead. For [H+] to pH conversion, reverse above procedure. When starting with a negative p H value, rewrite as positive decimal and larger negative 0.401, in order to determine whole number (e.g., p H = - 0 . 6 0 = - 1 right setting. Similarly, when converting a large [H +] (>1 ) to pH, b e sure to a d d the decimal (+) and whole number (-1 components algebraically. When the decimal p H value is zero, e.g., pH = 6.00, ignore the decimal converter and read the relationship directly on the lower scales, since [H+] = 10 10' = lo4.
+
x
and substituting the equation-derived concentrations into (or for) the bracketed symbols, the student is in a position to solve successively for X2 and X (= [H+]), and then finally convert to pH. The calculation becomes simpler if he first converts the set-up to full exponential form. Since
and I n this case the numerical value of [H+] was not needed, except "enroute," and solving problems via "exponential identities" is analogous to travelling underground on the subway: There is no need to check "ground level" a t each station. I n fact, it can increase the cost of the trip. Graphical illustration of decimal p H values versus [H+]'s is possible with a modified log scale design (Fig. 2), which can be moved into various positions on an unit pH versus [H+] scale, to analyze the number relationships. The big task in teaching chemical calculations is to persuade the student to ask "What is given?" (analogous to a geographical "Where am I?") and "What is asked for?," i.e., what units are implied in the answer (like "Where am I going?"). When all the factors are arranged to give the proper (unit labeled) answer, there remains the arithmetic. I t has to be done, but should be facilitated by any reasonable means. A mastery of chemical "exponentialism" by the beginner should be well worth the time spent.
