In the Classroom
Improving Conceptions in Analytical Chemistry: ciVi = cfVf Margarita Rodríguez-López Department of Chemistry, Pontifical Catholic University of Puerto Rico, Ponce, PR 00731 Arnaldo Carrasquillo, Jr.* Department of Chemistry, University of Puerto Rico, Mayaguez, PR 00681-9019; *
[email protected] The analytical chemistry skills required to support the development and improvement of chemical processes have been described recently (1, 2). The misconception addressed herein has been broadly disseminated in recent quantitative analytical chemistry educational literature (3, 4). It may be generalized as the failure to recognize and to account analytically for changes in substance density. Chemists who provide analytical support to manufacturing processes are especially prone to this type of systematic error if volumetric analytical methods are used to ascertain mass-based measures of composition. One of the analytical challenges posed by process chemistry is the common occurrence of changes in the physical state of process-streams, for example, gas phase-to-liquid phase transitions and vice versa. Changes in the stream composition of chemical process are commonplace and are usually accompanied by concomitant changes in process-stream density. Measurements of chemical composition that are based on the concept of mass fraction (w)—weight兾weight percent, ppm, ppb, and so forth—are frequently used for describing the mass balance within chemical processes. Mass fraction measures remain constant in the absence of chemical changes, even if changes in physical state or changes in volume take place. Such mass-based units are deemed convenient because they vary only with changes in chemical composition. Within the context of the analytical laboratory, volumetric units of concentration such as molarity, normality, and formality are convenient and sometimes preferable. One reason for the preference is that analytical methods and technology tend to rely heavily on volumetric measurements and concepts. Examples range from the application of Beer’s law, to volumetric dilution procedures, volumetric titrations as well as volumetric injection of samples into GC and HPLC instrumentation. It should come as no surprise that a sound understanding of dimensional analysis is a needed prerequisite to adequately perform and report quantitative chemical analyses in support of chemical processes.
IUPAC Convention and the Volumetric Dilution Equation Equation 2 shows the volumetric dilution equation using IUPAC recommended symbols, where cx represents the amount concentration with SI units of moles meter᎑3 and Vx represents the mixture volume with SI units of meter3: c iVi = c f Vf
(2)
Mathematical expressions similar to eq 2 can be found in many analytical chemistry textbooks. However, most textbooks do not make use of the IUPAC-recommended definitions and symbols for describing chemical and physical quantities (6). For example, although IUPAC defines concentration as a “group of four quantities characterizing the composition of a mixture with respect to the volume of the mixture (mass, amount, volume and number concentration)” (7), most textbooks use the term concentration in a broader sense. Molal and mass fraction-based measures of composition—percent, ppm, ppb, and so forth—are included. Because most textbooks use the term concentration in such a broad sense, misconceptions can take place when equations similar to eq 2 are memorized by learners and recalled for future use. Teaching and Learning Misconceptions Related to the Volumetric Dilution Equation Equation 3 will be used in this manuscript to describe a common misconception derived from eqs 1 and 2:
The Volumetric Dilution Equation Mathematical expressions similar to eq 1 are traditionally employed for teaching dimensional analysis in volumetric dilution problems: MiVi = MfVf
(1)
The mathematical symbols Mx and Vx have been used in some textbooks to represent, respectively, the amount concentration and the total volume of the initial or final solutions. In a typical textbook, these quantities will have units of moles liter᎑1 and liter, respectively. In that instance, the product, MxVx, represents the absolute amount of solute and carries www.JCE.DivCHED.org
units of moles. Equation 1 is used to represent a chemical reality that is assumed to hold true during any dilution, that is, the initial amount of solute (MiVi ) is equal to the final amount of solute (MfVf ). The mathematical equality described by eq 1 is therefore rooted to the chemical concept of molar amount and to the chemical principle of mass conservation. Excellent descriptions of eq 1 are available in most general chemistry textbooks (5).
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C iV i = Cf Vf
(3)
The misconception consists in believing that Cx represents any measure of composition or of “concentration”. In fact, this overgeneralization (believing that Cx can represent any measure of composition) is explicitly or implicitly proposed in at least two contemporary analytical chemistry textbooks (3, 4). Learners exposed to such information form the misconception that “you can use any units for concentration and volume as long as you use the same units on both sides”. The contextual suggestion is that beyond volumetric units of concentration, mass-based measures of composition may also be used by direct substitution into eq 3.
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In the Classroom
The resultant misconception can be demonstrated by the following problem: What is the final concentration if 1.00 mL of a 14.1% (wt/wt) aqueous CuSO4 solution is diluted to a final volume of 50.0 milliliters? Answer: 0.282% (Please note that the answer is incorrect.)
The ensuing discussion demonstrates when and why this calculation and the conceptions leading to it are not correct. Mass-based units of “concentration” are used as a cautionary example. Experimentally, it can be easily demonstrated that dilution of one milliliter of a 14.1% aqueous CuSO4 solution to a final volume of 50.0 mL produces a more concentrated solution than would be predicted from the direct application of eq 3, that is, a more accurate answer to the question above is 0.33%. A quantitative description of this experimental observation follows. Equation 4 is a mass-based analogue of eq 1 and may be used to teach how to perform gravimetric dilutions w im i = w f m f
(4)
where wi represents the initial mass-fraction or “mass-based concentration” of a solute in a solution, that is, mass of solute per mass of solution; mi represents the total mass of the initial concentrated solution, that is, the mass of solution that will be subject to dilution; wf is the final mass-fraction or “mass-based concentration” obtained after dilution; and mf is the total mass of the dilute solution to be prepared. Unit analysis reveals that in fact, the mathematical symbols in eq 4 convey a chemical reality, that is, the initial mass of solute (wimi) is equal to the final mass (wfmf ) of solute. This can safely be assumed to hold true during any dilution. The mathematical equality in eq 4 is, as before, ensured by the principle of mass conservation (i.e., the mass of solute remains constant during the dilution):
mass solute mass conc soln
mass × conc soln =
mass solute mass dil soln
mass × dil soln (4a)
Equation 4 may readily be converted to a volumetric expression (eq 6) by substituting the definition of solution density (ρsoln) into eq 4: ρsoln = msoln兾Vsoln
(5)
w iρ iV i = w f ρ f V f
(6)
It should be noticed that when the densities of the initial and final solution are the same (ρi = ρf) eq 6 is simplified to eq 6b: w iV i = w f V f
(6b)
Comparison of eqs 6 and 3 reveals that eq 3 may be employed analytically using mass-based units of “concentration” such as percent, ppm, ppb, and so forth, only if the initial density of the concentrated solution and the final density of the dilute solution (ρi and ρf ) are identical. It is well known
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that changes in solution composition are usually accompanied by concomitant changes in solution density. Hence, eqs 3 and 6b are seldom analytically applicable when mass-based units are involved. On the other hand, eq 6 is analytically accurate and may be adapted to be used with any of the massbased units mentioned (i.e., percent, ppm, ppb, and ppt). In our initial example, if the initial and final densities of the CuSO4 solutions where ρi = 1.16 and ρf = 0.999, eq 6 could then be used to obtain the corrected answer 0.33%. In fact, the systematic percentage relative error introduced by use of eqs 3 or 6b when solution composition is expressed with mass-based units may be estimated (᎑13.9%) through the use of eq 7: Percent Relative Error = [(ρf兾ρi) − 1]100
(7)
The misconception described above, that is, overgeneralization of eqs 1 or 2 and the indiscriminate application of eqs 3 and 6b, find their origin in learning and teaching strategies that are over reliant in one aspect of dimensional analysis, that is, unit analysis or “unit cancellation”. Although it is an important skill in the arsenal of future chemists, the use of unit analysis or “unit cancellation” strategies should perhaps be deemphasized in lieu of promoting a more integrated combination of chemical, physical, and mathematical concepts for the solution of quantitative analytical chemistry problems. A more consistent use of IUPAC recommended symbols and definitions would also help ensure less ambiguous scientific communication. Acknowledgments MRL and ACJr. wish to acknowledge (i) Pedro Gancedo (Du Pont Iberica, S.A.) for stimulating discussions on the subject, (ii) Milka Hernáiz for her assistance (undergraduate Chemical Engineering program at UPRM), and (iii) José Cortes (UPRM) for his encouragement. Special thanks go to Paulino Tuñon (UNIOVI) and A. J. Miranda Ordieres (UNIOVI) for their educational devotion. Literature Cited 1. Mabrouk, P. A. J. Chem. Educ. 1998, 75, 527–528. 2. Chauvel, J. P.; Henslee, W.; Melton, L. Anal. Chem. 2002, 74, 381A–384A. 3. Harris, D. C. Quantitative Chemical Analysis, 6th ed.; W. H. Freeman: New York, 2003; p 21. Harris D. C. Quantitative Chemical Analysis, 5th ed.; W. H. Freeman: New York, 1999; p 17. 4. Harvey, D. Modern Analytical Chemistry; McGraw Hill: New York, 2000; pp 31–32. 5. Ebbing, D. D.; Gammon, S. D. General Chemistry, 6th ed.; Houghton Mifflin: New York, 1999; pp 161–162. 6. CRC Handbook of Chemistry and Physics, 82nd ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2001. 7. IUPAC. Compendium of Chemical Terminology, 2nd ed.; McNaught, Alan D., Wilkinson, Andrew, Compiled; Blackwell Science: London, 1997. http://www.iupac.org (accessed May 2005).
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