provocative opinion Ideas of Equality and Ratio Mathematical Basics for Chemistry and the Fallacy of Unitary Conversion Ei-lchiro Ochiai Juniata College, Huntingdon, PA 16652 New pedagogical approaches to mathematics, science, and chemistry have been the topic of numerous journals and conferences. "Discovery Chemistry" is a good example (I). The problem areas in science education, particularly two in chemistry, are (A) (a-1)qualitative reasoning/(a-2)qualitative factual operation, and (B)(b-1) quantitative reasoning1 (b-2)quantitative operation.
Problem (a-2) is "understanding, remembering, and correlating facts and concepts". An old Chinese proverb that says much about this prbblem is Tell me, I will forget; Show me, I may remember; Involve me, and I will understand. "Discovery or investigative chemistry" attempts to involve students in their own learning. The other issue is "correlating facts and concepts". There is, in this country, a tendency that I coin as "trivia-game syndrome". Information is given (by instructors) or perceived (by students) as fragmentary facts. One fact has nothing to do with an' other, or so it is perceived. The fact that all our knowledge is divided into the so-called "disci~lines"does not h e l ~ to reduce this tendency. However, even in a limlted discipline like chemistw. correlation amone subdisciolines and facts does not seemto be emphasized. In realit;many concepts are related to each other. Problem (a-1) is "inductive and deductive operations" and is supposed to be addressed by a pedagogy Such as investigative and discovery chemistry. This aspect hax been and is the strongest asset in the education of this country. It is the major emphasis in many social science and humanities courses. It seems to have been rediscovered only recently in the natural and physical sciences. While the qualitative aspects can be made "fun and exciting" or is being made understandable (perhaps to a limited extent) by tbese recent developmeks, th&quantitative aspects (b-lh-2) are being perceived a s "drudgery" by many (2).And "drudgery" is equated to "evil" in this country. Not much study has been done on this aspect. Only the idea that use of a computer will magically eliminate the major portion of the drudgery of quantitative operation seems to be accepted without question. However, the main problem, at least in entrv-level chemistry courses, is not the complicated mathematical operations that require computers. Most problems in entm-level chemistry require on?; arithmetic,;( -, x and 1)and H little algebra. As pointed out often (31, students can solve a problem, if it is couched in a mathematical form. An analogous chemistry problem apparently confuses students Here are two dlffcrent problems. One is a "lan&wageproblem" and
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Journal of Chemical Education
the other is a "translation pmblem". The language issue is that students are baffled by the unfamiliar technical words used in the question and are thus Drevented from thinkine through thebuestion logically. ~ 6 is sreal and importag but belongs to the qualitative issues mentioned earlier. In most cases, however, it is a phony issue as far as quantitative problem solving is concerned. The real issue is the "translation problem". That is, students cannot translate a chemical question into a mathematical problem. This is the so-called "word problem". The word problem appears to be deemphasized or neglected a t K-12 levels. Because of lack of training in this respect at K-12 levels, entry-level college courses have to teach problem solving of this kind. Facing the difficulties in solving quantitative problems, students tend to resort to "algorithmic methods" (4). Instead of trying to understand the question and solving it logically, they fmd a pattern in the question and then apply an algorithmic method. They "find a formula and plug numbers into it". A typical example is
c,v, = c,v,
(1)
Students try to apply this equation to almost any problem involving calculation of concentrations. Another example is dimensional analysis (or factor-label method, in general). Often this, too, is abused. But the textbooks emphasize the dimensional analysis. I suspect that textbook authors believe that such an algorithm would reduce the agony of students. Students oRen ask: "Is this number to be divided by that number or multiplied?" A suggestion that "dimensional analysis" be applied to the question to figure out which (division or multiplication) is right would on the surface help students answer the question, but this does not necessarily foster understanding. These two are typical examples of what Bodner calls "algorithmic solution" (4). There is essentially only one basic mathematical relationship among quantities in chemical problems. That is the ratio of the quantities in question. This ratio could take a value of "one" or could take a certain definite value (e.g., 1to 2 or 1/21, In the former case the two quantities are the same in the value. These relationships amear in most of basic chemical calculations in many differLnt disguises. To solve a quantitative chemical aurstion is olten to find this relationship. If such a relationship can be identified in a question, it can be solved without resorting to any specific algorithm. This can be called "unalgorithmic method" (4). The mathematical equality (sign of " = ") means that the quantity (or some measure) on the right-hand side of the equal sign is the same as that on the left-hand side. Equation (1)reflects the fact that the quantity of a substance (solute) does not change upon dilution. That quantity in this particular case happens to becalculated by quantity=
C(concentration) x V(vo1ume) (assuming that the concentration is given in terms of quantity/volume). In other words, eq (1)is but a simple example of equality. Students need to be instructed to think about the relationship (in this case what is equal to what), not blindly to use an equation (i.e., algorithmic method). Take an equation: 1mile = 1.6 km (2) as an example. The implication of this equation is that the distance represented by 1mile is the same as that represented by 1 6 km. Then mathematically, and 1.6km/l mile = 1
(4)
hold true. It also is true mathematically that multiplying by 1(or any constant value for that matter) both sides of an equation would not change the equality. Therefore, one can multiply the left-hand side of an equation by 1 and the right-hand side by either 1mildl.6 k m or 1.6 kmll mile and can assume that the equality still holds. Thus, one can do, e.g., n miles x 1=n miles x (1.6 kmll mile) = 1.6n km (5) Here, the idea of dimensional analysis is employed to get the proper conversion factor. The equality still holds; i.e., the distances represented by n miles and that of 1.6n k m are the same. This idea is extended to stoichiometric calculations. Let us take the following chemical equation as an example:
Textbooks tell you that since 2 rnol of Mn2+react with 5 rnol of S20a2-an equation 2 mol Mn2t 5 mols20a2 (7) holds. Obviously, 2 rnol of Mn2+is not equal to 5 rnol of S20a2-.However, textbooks do not tell you why the leehand side is equal to the right-hand side nor what kind of quantity actually is represented in this equation. There is a "hidden agenda", to which we will return briefly. Now let's try an exercise of stoichiometriccalculation. If one assumes eq (I), then 2 ma1 ~ 8 1 mol 5 szos2=1 (8) mol hIn2+=1
Therefore, by rearranging eq (11)one obtains, e.g.,
(13)
This is exactly the opposite of the wrrect relationship (11).Textbook authors certainly do not recommend this kind of operation, but they do not caution against it, either. This kind of confusion can be seen in other situations, as well. For example, for a question such as: "express the fact that there are 14 students for each faculty member in this college, by an algebraic equation representing the number of students by S and that of faculty by F', students often give a wrong answer: F= 14S.Obviously, they think that one faculty member is equivalent to 14 students, so lF=14S. Admittedly, their idea of "equivalent" is vague (see below), but the idea is not too much different from the idea expressed in eq 7. This relationship, however, gives a correct answer, when it is used as the basis of unitary wnversion. That is, lFI14S = 1or l4SIlF = 1 and apply it to such a question as "How many faculty are there in this wllege if the total student number is 1680?" The answer is 1680 S x (1Fl14 S) = 120 F
(10)
You get what you need from this calculation. That is, how many moles of SzOs2would react with n moles of MI?+. Let us call this kind of calculation "unitary conversion method". It works. Let us now do the same eltercise using the concept of "ratio" (let's call this "ratio method"). Equation (6) indicates that the ratio of number of moles of Mn2+to that of S2OS2-is 2 to 5 in this reaction. The relationship can be expressed by: No. of moles of ~ n ~ + / Nofo moles . of S2oS2=W5
No. of moles of Mn2+Ii%.of moles 820s2 = 512
may be interpreted on the basis of, as it were, "instructional unit (i.~.)".That is, each student is one "i.u!', and one faculty provides service for 14 'S.U.". Hence, the IeRhand side = 1 F x (14 i.u.llF)= 14 i.u.
n mol of Mn2+x 1 = n mol of iVn2+x (5 m o l ~ ~ 0 , m ~mo/l 2Mn2?
~~02-
The wntent of this equation is the same as that of eq 10. Of wurse, either method should work, and they do. However, the "unitary conversion method" can be quite misleading, for two reasons. One is that the meaning of an equality such as 7 is not obvious. The second is that the idea can be mistakenly abused. The first (i.e., the hidden agenda) is in fact related to the idea of "reaction chemical equivalents", which the chemical community of this country has banished from general chemistry. Equation 7 implies that the reaction chemical equivalents on the left-hand side is equal to that on the right-hand side for the chemical reaction in question 6. That is, the left-hand side means that 2 rnol Mn2+would require 2 x 5 rnol of "e" (electron)to be removed in order to be oxidized to 2Mn04 (i.e., in this case 1rnol of Mn2+is 5 equivalents). whereas, the righthand side indicates that 5 rnol S2OaL would accept 5 x 2 rnol of "e"; so 2 x 5e = 5 x 2e. Textbook authors do not tell us that this is the trick, but they simply say that 2 rnol of Mn2+and 5 mol S20a2are equivalent in this reaction. This could be misused, too. For example, eq 7 can mathematically lead to:
(9)
are mathematically true. Therefore,
= ((32)x n) mol of
(12)
120 faculty members. It should be noted that in this calculation F and S do not represent the numbers of faculty and students. They are symbols representing those two different categories of entities. The meaning of lF= 14s
and 5
No. of moles s ~ o , (32) ~ =x No. of moles of Mn2+
(11)
and the right- hand side = 14s x (1i.u.llS)= 14 i.u
An alternative interpretation is to assume that the wrong equation represents an empirical formula of the wllege. That is, the wllege can be represented by a chemical formula: (FSI4)n.Then if 14n = 1680, n= 120. Of course, we should not go this length to justify a wrong answer. However, the "unitary conversion method" does actually require this kind of justification. A less confusing way to solve this kind of problem is to use the "ratio method". In this casethe ratio of number of Volume 70 Number 1 January 1993
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students to that of faculty (actually "student/faculty ratio" is commonly used in the real life) is 14 to 1;i.e., SIF= l a 1
or
FIS = 1/14
The question above can be answered as F= (1114)x S
thus, F= (V14)x 1680 = 120
Why have textbooks adopted the "unitary conversion method", when simpler and more basic ideas of equality and ratio suffice? Perhaps the idea of unitary conversion (or factor-label method) was introduced in order to algorithmize the stoichiometric calculations. I would argue that the "unitary conversion method", if necessary, should be applied only to the cases of conversion of units, where the equality is obvious and straightforward. Indeed the "unitam conversion method" is unuecessarv for this kind ofcalcuiation, too. For example, eq 1.5)can beobtained simply by multiplying both sides by n [a constant); i.e., since 1 mile = 1.6 km 1 mile x n (= n miles) = 1.6 km x n= (1.6n km)
and it also can be changed to n !un=(1/1.6)n miles. The "unitary conversion (factor-label) method" is actually cumbersome, and I don't think that we gain anything from it. The only advantage it might have is to help students keep track of a calculation in terms of units (dimensionor label). However, there are still a lot of pitfalls in this aspect, too. For example, from "bomb calorimeter" experiments, a student obtained the following results: weight of sample (naphthalene) = 0.7895 g, MI (calculated from AT and the heat capacity of the system, after correction for the iron wlre portlon~= -29.49 kJ: AE2 I heat of combustion of naphthalene Der mole, = -154 kJImol He was then to estimate ~~~----- ~~AH val
