Chemistry Everyday for Everyone
Quantity Calculus: Unambiguous Designation of Units in Graphs and Tables Mary Anne White Department of Chemistry, Dalhousie University, Halifax, NS B3H 4J3, Canada
We have all seen tables and graphs that we cannot decipher because of their ambiguous representations of physical quantities. For example, in the CRC Handbook on my bookshelf, the parameter that quantifies the ease of demagnetization of a material, maximum energy product, is listed for various magnetic materials with the column heading in the table “Maximum energy product, (BH)max (Gauss-Oersteds × 10᎑6)”; Alnico’s value is listed as 1.70 (1). Does this mean that (BH)max = 1.70 × 10᎑6 Gauss-Oersteds, or does (BH)max = 1.70 × 106 Gauss-Oersteds? If we are not familiar with typical values, a factor of 1012 mistake could result! Similar ambiguities can arise from interpretation of graphical representations of data. There is a simple method to avoid this ambiguity. Known as quantity calculus, this method treats each physical quantity as the product of a numerical value and a unit: physical quantity = numerical value × unit
(1)
and the use of this method allows unambiguous designations in, for example, tables and graphs. Quantity calculus has been known for more than a century. In 1924, it was said that the “system is very little used in schools,” because “[i]t is, in fact, not well known”; nevertheless, “teachers usually adopt it as soon as they hear of it” (2). Thirty-four years later, Guggenheim expressed similar sentiments in this Journal: “The advantages of this notation have been emphasized…. This notation has been used by some of the greatest theoretical physicists, in particular Planck and Sommerfeld. Its use is spreading, but surprisingly slowly. Most people who understand the notation like it and use it” (3). Little has changed with respect to this situation: quantity calculus is still very useful in unambiguous description of units and physical quantities, and although it is underutilized, those who know it use it with favorable results. Quantity calculus is especially useful in handling units, and units are a very important teaching/learning tool in physical science (4, 5). The quantity-calculus view of units has been shown to avoid ambiguities in dimensional analysis, especially in distinguishing the amount of a substance from its mass (6 ), for conversion of units (7, 8), and as a basis for teaching stoichiometry (9). The use of quantity calculus in tabular and graphical presentations—the emphasis of the present paper—is rather
widespread in physical chemistry textbooks, but it is rarely used in other areas of chemistry, despite the fact that its use is recommended by the International Union of Pure and Applied Chemistry (10) and ISO (11), and it has been recognized internationally since at least 1969 (12). Ambiguities in algebraic formulas in textbooks that could be clarified through the use of quantity calculus have been described (13). The lack of standards in graphical presentation and the potential ambiguity of writing graphical scale markers as pure numbers have been pointed out (14 ). Quantity calculus has been described as “especially clear and tidy for labelling the axes of a graph” (3), and graphical presentations using this method have been described briefly for a few simple cases (3, 15). Although the utility of quantity calculus in teaching has been presented previously (16 ), the fact that it is not widely used indicates that the target audience has not yet been reached. In this paper, examples are given showing the use of quantity calculus for the unambiguous presentation of tables and graphs of data. The hope is that these examples will convince chemical educators of the merits of the quantity calculus approach and provide a basis for the teaching of this method. Quantity Calculus: A Method To Handle Physical Quantities From eq 1, the standard enthalpy of formation of water (∆ f H °), for example, is given by ∆ f H ° = ᎑285.9 kJ mol᎑1
(2)
where the numerical value is ᎑285.9 and the unit is kJ mol᎑1. An expression equivalent to eq 2 is ∆ f H °/(kJ mol᎑1) = ᎑285.9
(3)
where the physical quantity, the numerical value and the unit have been treated algebraically. Equation 3 can be generalized to: physical quantity/unit = numerical value
(4)
as it is often the numerical values that are required for tables and graphs. Equation 4 indicates the appropriate expression for equality of numerical values with physical quantities and units. This is the methodology is known as quantity calculus.
Editor’s Note: The quantity-calculus convention for labeling tables and graphs that is described in this article has been adopted by JCE for all papers and is described in our Guidelines for Submission of Manuscripts (on pages 646–648 in this issue and at http://jchemed.chem.wisc.edu/journal/authors/). Please follow the conventions described here when manuscripts are being prepared for publication. It is especially important to follow the convention for graphs. If the JCE editorial staff cannot correct the labeling of axes, we may need to return incorrectly labeled graphs for redrawing.
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Chemistry Everyday for Everyone
Use of Quantity Calculus Let us consider the use of quantity calculus in the handling of three data sets of varying degrees of complexity.
Simple Data Set The following simple data set of time (t) and temperature (T ) values t = 0.0 s, T = 298.6 K t = 10.0 s, T = 302.1 K t = 20.0 s, T = 305.7 K t = 30.0 s, T = 310.2 K can be manipulated by quantity calculus to give the numerical values (to be presented in a table and in a graph) as the physical quantity divided by the unit (see eq 4), as in Table 1. Similarly, temperature can be unambiguously plotted as a function of time with appropriate labeling of the axes, as shown in Figure 1. Note that the numerical values in the table and plotted on the graph are the physical quantities divided by the units (eq 4). Data Set with Use of Orders of Magnitude If you have a set of numerical values such as the following set of thermal expansion data, α (= V ᎑1 [∂V /∂T ]p , where V is volume and p is pressure): T = 300 K, α = 2.1 × 10᎑5 K᎑1 T = 310 K, α = 2.3 × 10᎑5 K᎑1 T = 320 K, α = 2.5 × 10᎑5 K᎑1 T = 330 K, α = 2.7 × 10᎑5 K᎑1 and you wish to express them unambiguously in tabular form, treatment of each physical quantity by quantity calculus leads to Table 2. The representation in this table certainly is preferable to the ambiguous column heading of “α (105 K᎑1)” or even the unambiguous but tedious alternative of including the order of magnitude in the table—stating, for example, 2.1 × 10᎑5 for the first value of α in the table. Using the method of quantity calculus, thermal expansion as a function of temperature can be plotted with unambiguous axis labeling, as in Figure 2. Further Manipulation of Data In chemistry, one of the most common manipulations of data is taking the logarithm of a physical quantity; again quantity calculus allows unambiguous expression of this information (17, 18). Another manipulation is taking the reciprocal, and both are used to determine the activation energy from reaction rate constants (k) determined at different temperatures, as in the following data set: k = 7.8 L mol᎑1 s᎑1, T = 673 K k = 11 L mol᎑1 s᎑1, T = 683 K k = 14 L mol᎑1 s᎑1, T = 693 K k = 18 L mol᎑1 s᎑1, T = 703 K k = 24 L mol᎑1 s᎑1, T = 713 K In order to determine the activation energy from a plot of ln k vs T ᎑1, we must first obtain the natural logarithms of the rate constants and the reciprocals of the temperatures. It is not possible to take the logarithm of a unit, so we must first divide the unit from k. For k = 7.8 L mol᎑1 s᎑1 this gives ln (k /(L mol᎑1 s᎑1)) = ln (7.8) = 2.05
608
(5)
Table 1. Simple Data Set Represented Using Quantity Calculus t/s
T/K
0.0
298.6
10.0
302.1
20.0
305.7
30.0
310.2
Figure 1. Illustration of the principle of quantity calculus to label graph axes for a simple data set (see Table 1). The graph represents temperature (T ) as a function of time (t ).
Table 2. Data Set Incorporating Orders of Magnitude in the Units Represented Using Quantity Calculus T/K
α / (10 ᎑5 K ᎑1)
300
2.1
310
2.3
320
2.5
330
2.7
Figure 2. Illustration of the principle of quantity calculus to label graph axes for a data set incorporating orders of magnitude in the units (see Table 2). The graph represents thermal expansion (α) as a function of temperature (T ).
Journal of Chemical Education • Vol. 75 No. 5 May 1998 • JChemEd.chem.wisc.edu
Chemistry Everyday for Everyone Table 3. Data Set Derived by Manipulation of Primary Data Represented Using Quantity Calculus ln [ k /( L mol ᎑1 s ᎑1)]
T ᎑1 / (10 ᎑3 K ᎑1)
2.05
1.49
2.40
1.46
2.64
1.44
2.89
1.42
3.18
1.40
of units both makes measurement possible and determines the numerical value. The word “calculus” in “quantity calculus” is used according to the definition “method or process of reasoning by computation of symbols” (21), not the present common usage, as an abbreviation of “calculus of variations”, which is strictly related to rates of change. The calculations invoked by the term “quantity calculus” are what we might now call algebraic manipulations of physical quantities. A thorough discussion of the history of quantity calculus and the concepts of quantity and units has been given by de Boer (20). Conclusion Quantity calculus—that is, the manipulation of physical quantities especially by eq 1—has been considered internationally and recommended for use in the physical sciences. This simple method can be used by students, teachers, and researchers to unambiguously report physical quantities, especially in tables and graphs. Acknowledgments
Figure 3. Illustration of the principle of quantity calculus to label graph axes for a data set derived from primary data (see Table 3). The graph shown represents the natural logarithm of the rate constant (k ) as a function of reciprocal temperature (T ᎑1).
I thank Professor P. W. M. Jacobs for introducing me to the concepts of quantity calculus in an undergraduate physical chemistry class in 1972–73. Thanks are also expressed to P. Bessonette for helpful comments on this manuscript. Literature Cited
Following this methodology, the derived data set, expressed in ln k and T ᎑1, is ln [k /(L mol᎑1 s᎑1)] = 2.05; T ᎑1 = 1.49 × 10᎑3 K᎑1 ln [k /(L mol᎑1 s᎑1)] = 2.40; T ᎑1 = 1.46 × 10᎑3 K᎑1 ln [k /(L mol᎑1 s᎑1)] = 2.64; T ᎑1 = 1.44 × 10᎑3 K᎑1 ln [k /(L mol᎑1 s᎑1)] = 2.89; T ᎑1 = 1.42 × 10᎑3 K᎑1 ln [k /(L mol᎑1 s᎑1)] = 3.18; T ᎑1 = 1.40 × 10᎑3 K᎑1 which can be expressed unambiguously as in Table 3. The Arrhenius plot, with the axes labeled unambiguously through quantity calculus, is shown in Figure 3. IUPAC (10) allows other choices for expression of units: for example, the abscissa in Figure 3 could be equivalently labeled “103 K/T ” or “k K /T ” or “103(T / K)᎑1”, with the same information conveyed. Brief History of Quantity Calculus The introduction of quantity calculus has been variously attributed to Professor William Stroud of Leeds University (in his work in the 1880s to eliminate errors in unit manipulation of dynamics quantities [19]) and to James Clerk Maxwell in 1863 (20). Difficulties in definitions of electromagnetic quantities and units led to Maxwell’s main point that the physical quantity is independent of the choice of units; however, the choice
1. Handbook of Chemistry and Physics, 52nd ed.; Weast, R. C., Ed.; CRC: Cleveland, 1971; p E-104. 2. Godfrey, C. Math. Gaz. 1924, 12, 104–105. 3. Guggenheim, E. A. J. Chem. Educ. 1958, 35, 606–607. 4. Norris, A. C. J. Chem. Educ. 1971, 48, 797–800. 5. Wadlinger, R. L. J. Chem. Educ. 1983, 60, 942–945. 6. Davies, W. G.; Moore, J. W.; Collins, R. W. J. Chem. Educ. 1976, 53, 681–682. 7. Lodge, A. Nature 1888, 281–283. 8. Wood, J. A. Educ. Chem. 1993, 30, 52–53. 9. Packer, J. E. Educ. Chem. 1988, 25, 92–95. 10. Mills, I.; Cvitaˇs, T.; Homann, K.; Kallay, N.; Kuchitsu, K. Quantities, Units and Symbols in Physical Chemistry, 2nd ed.; Blackwell: Oxford, 1993; p 3. 11. ISO Standards Handbook 2, Units of Measurement; International Standards Organization: Geneva, 1979. 12. McGlashan, M. L. Pure Appl. Chem. 1970, 21, 1–44. 13. Davies, W. G.; Moore, J. W. J. Chem. Educ. 1980, 57, 303–306. 14. Le Vent, S. Educ. Chem. 1987, 24, 14–15. 15. Norris, A. C. J. Chem. Educ. 1971, 48, 797–800. 16. McGlashan, M. L. Physicochemical Quantities and Units; Monographs for Teachers No 15; Royal Institute of Chemistry: London, 1971. 17. Henry, A. J. Educ. Chem. 1967, 4, 81–86. 18. Molneux, P. J. Chem. Educ. 1991, 68, 467–469. 19. Henderson, J. B. Math. Gaz. 1924, 12, 99–104. 20. de Boer, J. Metrologia 1994/95, 32, 405–429. 21. Webster’s Third New International Dictionary of the English Language. Unabridged. Merriam: Springfield, MA, 1966; p 315.
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