Is Dimensional Analysis the Best We Have to Offer? Sebastian G. Canagaratna Ohio Northern University, Ada, OH 45810 The Two Methods of Calculation There are two distinct ways in which chemistry students in the US are taught to carry out their calculations. One method, which is used quite extensively in the early stages of the introductory courses, is known by three names: the "factor label" method, the "unit conversion" method, or the method of dimensional analysis (DA). The other method, which will be called the method of equations (ME),is used later in the introductory courses to deal with more complex problems such as pH and equilibrium-constant calculations. This method is the traditional method used in physical chemistry, physics, and mathematics. It is also called the method of relations. ME has the following steps.
Identify the quantities involved. Express the relation between quantities with an equation. Substitute for quantities. Solve for the unhowns. There is hardly a single US text at the freshman level that presents material exclusively by ME. Whole books have been written to familiarize the student with DA ( I ) . It would be interesting to know whether this is the method of choice in other countries. However, the popularity of DAis recent, perhaps hegining 3 0 4 0 years ago. Only an occasional voice bas heen raised against i t (2,3). But has this method been an unmixed hlessing to the instructor? In this article I scrutinize the suitability of this method for the role it plays in concept formation and for its applicability in more advanced work. An Example to Compare the Methods To facilitate cornparison consider the following specific example.
Calculate the volume of 10 g of a solution whose density is 0.92 gImL. When using ME, the students first recognizes that volume must be calculated from the mass and that these two quantities are related by the following equation. mass density = volume Thus, volume of solution = mass of solution - 10g = ll mL density of solution 0.92 B mL
A common alternative is to use symbols, yielding expressions such as the following.
When using DA, students focus on the unit in which the answer is to be expressed, and they learn to manipulate the data to obtain this unit. ? mL solution = 10 g solution x
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1mL solution - 11mL ~olution 0.92 g solution -
Journal of Chemical Education
The Use of Relations and Definitions
This example brings out the essential differences hetween the two approaches. With ME there is an explicit recognition of the quantities involved and of the relation between them. When setting up the calculation, students must use explicit knowledge of definitions and relations. The substitution is done with proper units (quantity calculus) so that any error will be detected a t least in the final stage. With DA there is no such need for explicit recognition of quantities and the relations between them. The labelling accompanies the units-not the quantities, as in ME. The labelling serves as a guide in setting up the calculation. As can be seen, there is hardly any difference between the two methods in the mathematical skill required. The difference between the two is summarized below. DA is a units-based approach, while ME is a relations-based approach. Criteria for Evaluation I n evaluating DAit is helpful to be explicit about the criteria to be used.
'Does this method place sufficientemphasis on the basic principles? Does the method enhance retention of basic principles? Is there continuity and consistency of treatment? Does the method prepare the student well for further work in physical chemistry and analytical chemistry? Emphasis on Basic Principles DAis more properly called unit analysis because i t relies heavilv on units to euide students to the solution. The algorith;n of this metKod, stripped to its essentials, is informally summarized below.
Determine the unit of the quantity required. Then string together the available quantities so that all units cancel, except the required one. Rote Memory Versus Understanding and Using Definitions
Most students do proceed in this way when they do not understand the physics of the problem. Some instructors regard it as an advantage that the student can obtain the correct answer just by looking at the units involved-without knowing anything about the physics of the problem! Some texts even give examples of the dimensional analysis approach in which quantities are used long before they are treated in the text. This sends an obvious but misleading message: Don't worry about the physics, just pay attention to the units. It is only fair to add that most of the better texts map a logical path to the solution. Then how do students remember the units? When students remember the units of each quantity without reference to definitions, they are using rote memory. When asked to defme density or molar mass, students often give the answer as units: g/mL or g/mol! Whengiven quantities like 100 glmol, 5 mm2,or 3 lb/ft3, many students are unable to identify the quantities by name, though they are able to work out numerical problems!
It is not accidental that many definitions in the texts are poor or incomplete: DAproper has no need for precise definitions. Since students remember units by reference to an eouation. whv not encouraee them to use the eouation? 6 e math in;olved is only &tiplieation or divisiok Is there anv Dedaeoeic merit in telling students to avoid equations, es"pkialG $they already kniw them? Will the use of ME interfere with a student's understandine? ., We must be sure that students appretiate the physical significance of equations and that they do not work by rote (4,7,. Criticizing the Algorithm
There are two basic difficulties with the DA algorithm. Both concern the ambiguities that arise when students rely on using mere units to set up the calculation. Incorrect selections can be made when students do not understand the physical concepts. Ambiguity in Choosing the Correct Factor
DA never dictates which numerical factor to use when there is choice among factors that all carry a similar unit. Suppose students are given the radius and length of a right circular cylinder, and asked to calculate the volume. To get the units of volume, should they multiply the square of the radius by the length--or the square of the length by the radius. Both procedures will give the same units, with different numerical factors of course. Without understanding the physics of the problem, they can not make a decision. Ambiguity in Choosing the Correct Quantity
Different physical quantities can have the same units. Thus, in thermodynamics, the student encounters heat, work, internal energy, enthalpy, Gibbs free energy, etc., which are all expressed in the SI unit Joule. Given that entropy has units of J/K,without knowing the fundamental relationships, how does the student decide which goes in the numerator-heat or work or something else? Even in the denominator, should the student use temperature or change in temperature? As another example, a quantity with units of g/mL could be a density or a mass concentration. Thus, the guiding principle in DA-that units can be used to set up a calculation-can lead to errors. Stressing Relationships over Quantities
Quantitative chemistry involves relationships between quantities, not relationships between units. Any method of instruction that ignores this or lessens the emphasis must be regarded as pedagogically unsound. It is not surprising that students who have been brought up on DA have difficulty with thermodynamics, where the relationshi~sbetween auantities becomes so imnortant. Also, the lack of focus i n concepts and relationships between auantities in DA does little to encouraee retention. Even in'the best presentations of DA there are pedagogically unsound features. The comments of Tang and Weeny-Kennicutt (see ref 5, foreword) are particularly relevant here. The over-emohasis of this method mav train vou with a mannrad lmc of logwal reasoning, by rushmg towards the final answer and c l n u d ~ n gthe chcrntcal rneanmg of the lnterrnedlate steps. From the popularity of the method, it is reasonable to infer that even the weak student f i d s DA easier to work with than ME. Students like aleorithms that enable them to get the answer in one line of calculation by simply stringing together appropriate units. Although students like an algorithmic approach, all too often they end up ap-
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plying it inappropriately (6).There is no indication that students solve new problems better after receivinginstruction in DA. Enhancing Retention of Basic Principles
Principles and concepts are more easily retained if a network of associations can be built to link them. ME is admirably suited for this. Beginning with definitions and general laws, students can build a whole network of associations between the various quantities. This encourages them to think in terms of quantities, not numbers or units. Mapping the Path To Clarify Relations
A eood scheme of correlation is based on the conce~tof integsive and extensive quantities (7.8).Such a netwdrk of associations is indis~ensablein " m a ~ ~ i the n e Dath" in a multistep problem. Some texts occasionally "map the path" before solving the problem by DA, but a systematic attempt to do this in every problem will help considerably. The solutions manual by Tang and Keeny-Kennicutt (5)is particularly excellent in this regard, oRen giving as many as three different ways of working a problem. While "mapping the pathn is desirable, its effectiveness in DA is questionable because it is a units-based approach, not a relations-based approach. What is a perfectly logical path for instructors, who already know the definitions, becomes a memorized oath to the student. Students will quickly forget any dis'cussion of relations that an instructor mieht use to suodement DA. Most students eenerallv disregard what is not covered on exams; they are interested only in the quickest route to the final answer. In supplementing DA with a map of the path, the need for relationships is clear, but the DAmethod steers the student away from the best way of displaying relationships concisely and precisely. 1f proponents of DA wcrc to be& emphasizing relationships by means of equations, the difference between DA and ME would be minimal. but then there would then be no distinct advantage in uskg DA! The effect of the two methods on learning habits is very different. Students working by ME are stopped dead in their tracks if they do not know the physical relations involved. They are then forced to study these relations. With DA, units are often used as a crutch to ignore and avoid concepts and relationships.
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Continuity and Consistency of Treatment
Few freshman chemistrv texts mention that DA works only with direct (and, by &ension, inverse) proportionalitv relationshi~s.However, not all such relationshi~sare treated by D A . - Glaw ~ ~ rel&onships can be so treaGd, but they hardly ever are. Some texts treat colligative properties by DA, others by ME. In some texts one part of a problem is treated by DA and another part by ME! ME is also used early in the course for temperature conversions, specific gravity, etc. Clearly it can not be argued that DA must be used because the students can not handle the math of the other methods. The relationships encountered in the later parts of the course go beyond direct proportionality, and DA must be abandoned. It seems pedagogicallyunsound to expect a student to use one method in one part of chemistry and another method in other parts of chemistry, or in physics or mathematics. Using Substitution
Instruction in chemistry has over-emphasized the conversion factor method so much that even straightforward Volume 70 Number 1 January 1993
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transformations like the following are done by this method. Convert 0.50 kg to g.
The most direct method is the method ofsubstitution. In the following example, k represents the number 1000.
Preparation for More Advanced Work Besides just using direct and inverse proportionality, most problems in thermodynamics also involve linear, differential, and integral relationships. Furthermore, advanced work abounds in theoretical derivations. Anv method that emphasizes calculation of numerical answers and Days little attention to relationshios between auantities &hardly be regarded as preparation for dealiig with theoretical derivations.
mass = 0.50 kg = 0.50 x 1000 g After all, the principle of substitution-that in any relationship anything can be substituted by it equal-is so fundamental that it is difficult to believe that we should invent tricks to avoid it. What would we think of the following evaluation ofy. Suppose that we know that y = 32
We wish to evaluate y when z = 2. Then z = 2 gives the conversion factor
Thus,
We teach our students that k = 1000, m = lo3, etc., regardless of the units with which they are associated. Our methods of instruction should help reinforce this idea, not obscure it. Again, regardless of units, we have
and so forth. Because a prefixed unit like kg may be regarded ask x g, these unit factors can be used to introduce the prefixes.
DeterminingFurther Information Required
Solvingnumerical problems should not be the only thing we teach our students. Most problems given on exams are "artificial" because the instructor has carefully chosen the required data. If the data is insufficient, can the student proceed with confidence to find out what further information is required? If asked to prepare a 1 molJL solution from a 6 mol/L solution, can the student supply additional information to complete the task? When given the task of calculating the molality of a solute from its concentration, can students determine what further information is required? Correct Symbolization Since the emphasis in DAis on units in numerical problems, symbolization is sorely neglected. Thus, it is not surprising that most definitionsin freshman textbooks are inadequate and incorrectly tied to particular units. To illustrate these points, consider the following problem. 50 mL of a liquid A, which has a mass of 43.8 g, was mixed with 10 mL of another liquid B, which has a mass 9.8 g, to give 60 mL of a homogeneous mixture. What is the density of the solution?
Here we have several masses and several volumes. Astudent who starts with the relation d=- mass
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Students will be able to omit the bracketed steps after some practice. Similarly, given that 1in. = 2.54 cm,5 in3is converted to units of cm3 most simply by direct substitution of 2.54 cm for 1 in. The point is not that the unit factor method must always be avoided. Indeed, this method is at its best and most legitimate in unit conversions, but as shown above there are often quicker routes. Even complicated conversions can be carried out using the "multiply by 1"trick in a slightly different form, using equations of the following form.
For example, mi
ft
in.
cm
5mi=5x~x-x-xm=5x5280x12x2.54x0.01m ft m. cm m
Instead of cancelling units as in DA, we replace the ratios of units like mi/ft by their numerical values, which are obtained by direct substitution or from their relationships. For example, a mile is 5280 times larger than a foot. Familiarity may favor the traditional unit factor method, but the above method is hardly more difficult. Also, it has the pedagogic advantage of forcing the student to think about the relative sues of-the units. 42
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is not paying attention to detail. The Need for ImprovedDefinitions
The texts are partly to blame: The usual definition is incomplete! Apmper definition that pays attention to detail would be the following. density of a sample = mass of the sample volume of the sample In symbols, this would he
where i indicates a mixture. This will alert students that they need the mass and volume of the mixture--not the components. Below are examples of another common definition from two texts. molarity = moles of solute liters of solution molarity =
number of moles of solute number of liters of solution
Note the carelessness with which terms are being used. Is "moles of solute" the same as "number of moles of solute"? Is 'liters of solution"the same as "volume of solution in liters"? A more serious deficiency is that solute is not mentioned on the lee. Aproper definition would be the following.
amount of i concentration of solute i in a solution = volume of solution In symbols, this would be
This definition is not tied to any particular set of units. Encouraging Correct Interpretations
Careless interpretation can thwart our efforts to encourage students to develop the habit of paying attention to detail. For example, the following statement is correct. If the density of a solid is 5.9 g cm3, then the mass of 1cm3 of the solid is 5.9 g. However, it is both careless and just plain wrong to say that density is the mass of 1em3. Density and mass are different quantities and have different units. Similarly, it is wrong to define the molarity as the moles of solute in a liter of solution, as most texts do. It is also wrong to define the molar mass as the mass of 1mol. I found very few freshman texts that defmed the molar mass by an equation or described it precisely!
From the data in the problem and from understanding the relationships, an intermediate step is indicated. It involves the amount of NaC1, which is also an extensive quantity V(soln)+ n(NaC1)+ W(NaC1) (Notice that there are no units!) Because extensive quantities are directly proportional, we get the following equation. mass of NaC1, W(NaCI)= n(NaC1)
V(soln) x ~ ( ~ ~ 1 ~ )
Thus, the starting point is clear: the required quantity. The factors to be used for division and multiplication follow from the above path. Setting up the path requires understandine the various relationshius and definitions. The factors areiatios of extensive quantities, and thus are intensive quantities. Thus, the students must now use their knowledge of definitions of various intensive quantities. Hopefully they will be able to write the following sequence. W(NaC1)= M(NaCl)x dNaCl) x V(soln)
Attention to Detail
If quantities and units are to bewme meaningful t o the students, then our methods should help them appreciate the significance of these wncepts, not blur the distinction between them. It is easy to shrug off this attention to detail as pedantry. h a t t e n t h to detail soon becomes a habit. When this happens understanding suffers. DA does pay some attention to detail by labelling (e.g., 43.8 g Aor 50 mL 6 M HCl solution, ete.) so that only numbers with identical labels wiU be canceled. These "descriptive units" have been discussed in the pages of this J o u r a l (2,3,8). However, this device introduces some inconsistency: 5 g A is different from 10 g B and can never cancel, but what about 5 g A + 10 g B? What hybrid unit does this produce? Introducing labels when calculating specific gravity will produce a curious result. Comparing Slngle-Line Calculations in the TWO Methods
Undoubtedly one of the reasons for the appeal of DA is that it permits quick calculations on only one line without requiring much attention to detail. Thus, the following problem can be worked i n one line by DA. Calculate the mass of sodium chloride in 100 mL of a 1.25 mol/L solution of NaCl. Unfortunately this oRen encourages working by rote, and the better texts (10)break these problems into steps. However, with ME working the solution in one linewhen possible--will be guided entirely by knowledge of concepts and relationships. (This is not recommended until the student has a firm grasp of the fundamentals!) The following outline traces the thought processes that lead to the required solution for the stated problem. Working by Steps in ME
To solve the problem we need the mass of NaC1. Because mass is an extensive quantity, it can be calculated from any other associated extensive quantity, for example, the volume of solution. V(soln)+ W(NaC1)
Conclusion
I hope that this critique of DAwill instigate research and discussion on the relative merits of DA and ME. Is DA so clearly superior to ME that it is worth the time spent? ME can be used for any problem that is usually worked with DE, and it is actually a better method. Also, ME is the only method used in the higher levels of chemistry and in physics and mathematics, whereas DE is adequate for only a part of even the introductory wurses. Overall, ME is a superior method with added benefits for the student: a firmer grasp of relationships and associations, and better preparation for theoretical work. For problems in which both methods can be used, the math involved is only multiplication and division, so the math involved can not be seriously advanced as a reason to favor the use of DA. Can we successfullv use ME in all of ouantitative chemistry? Yes, provided we first prepare the student by inculcating an attention to detail. Students should first gain experie&e in tasks such as the following.
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identifyingquantities and processes in a problem correct symbolization of these quantities explicit identification of laws relevant to the situation We must also test the students on these aspects. Only further research can t d l whether this will be successful, but I believe we should try, starting at the high school level. Literature Cited 1. h b l . h o l d B . PmgmmmdPmblom Soluiwfor Fimt-Yw Chrmutry: noughton W8lin: Boston. 1983. 2. Cardulla, prank J. Chem Educ. 1987.64.519. 3. (a) Kemp, H. R. J Chem. Edm. 1987, 64, 191: 1898,66,272. See also (b) Lythmtt, J. J Chem. Ed-. 1980.67.248: (cl Bmoka, D.W. J. C h . Educ. 1987, 64.53; (dl Navidi, M. H.:Baker, D. J. Chrm. Edue. 1984,61,5ZZ 4. Gold,M. J. Chem. Educ. 1388.65.780. 6. Tang, Y-N.;Keeny-Kenninnin, W. Solutions Manual t o d n n i p n y Cowml Chemistry (by Whitten, Galley,and David; Ssvnders College: NY,1987 6. Frank,D.V; Baher,C.V;Hermn.J.D. J. Cham. Educ. 1984,64,514. 7. cansg-ma, S. G. J. Chem Educ accepted fmpuMiFafion. 8. Dierks, W ;Weninger, J.; H e m n , J. D. J. Chem. Edue. 1985,62,839. 9. Wadlinger, R. J Chem. Educ. 1998, €0.942. 10. Kotr, J. C.; k U , K Chemkfry and Chemiml M i u i t y : Savndera College: NY, 1987.
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