by mass. Gravimetric methods are more accurate and precise in many instances, however. Volumetric methods suffer, because volume-based dimensions for concentration vary with temperature, and because volumetric glassware does not give the precision that balances routinely achieve. Calibrated pumps must be verv to aporoach the . expensive . precision oi balances. The advent of fast, combter-interfaceable balances solves part of the problem of desiming ~ need to stir the automated gravimetric tiirators. B U the analyte as titrant is added introduces complications. It is interesting and easy to carry out semiautomatic gravimetr~ctitrations in the educational laboratory to demonstrate the advantaees (and difficulties, of the method. Mass-based concentration units like molality can be used to allow conversion of mass of added solution to amount of reactant (in moles). Titrant may be taken from a beaker on the interfaced balance Dan bv means of a medicine dropper or pipet and added toihe stirred sample. The droooer is returned to the beaker before it is reweighed. mass of titrant added and pH (or conductivity,etc.1 are recorded after each addition. LIMSport allows direct acquisition of mass, pH and conductivity data, and the spreadsheet format facilitates creation of titration curves and calculation of final results. Equivalence points can be determined easily in LIMSport by two methods: (1) A linear regression ofihe titration data yields a line that intersects the titration curve at the equivalence point (61,or 12) first and second derivatives of the titration data can be calculated easily in a spreadsheet. The first derivative of masswtmuand pH in columns A and B is estimated by coping the formula (B2-Bl)IW-Al) from cell C2 down column C, and the second derivative is estimated by copying the formula (C3-C2)/(A3-A2) from cell D2 down column D. Other methods for estimatine derivatives mav be used (7). One of the advantages of L I M S ~ is O that ~ ~ it is based on Lotus 1-2-3, and thus draws on an extensive literature. For example, more spreadsheet statistics (probably more appropriate for Junior or Senior level courses) are described in a book by Mezei (8)and many calculations appropriate for general chemistry are described in the book by Breneman and Parker (9): The author will supply a wpy of the Lotus template for the gas laws experik&t and-the accompanying handout for no charge. Please submit a blank floppy disk and preaddressed mailer. Literature Cited 1. vit., E. J Chem Edue I" press. 2. Shaner. R. A. Gos Low Aooomlua: Instiblte for Chemiesi Eduestim. Univeaitv of
5. Ylt..E.J.Chom.Educ ISM. 61.803-806
dison, 1990.
7. Orvis, W J. 1-2-3For Scientists and Engineers: S y k San Frandsm, C 4 1987. 8. Mwei, L. M. P m t I c d S p d h e f Statistics & CWW Fitting for Scientists and En@news;P M t i e e H d l : Englelvacd Cliffs. NJ, 1990. 9. Parker. 0.J.: Breneman. G. L. Sorpodrpodhat Chomlstrv:Rwtice-Hall: Endewmd
A Direct Approach for Solving Simultaneous Eauations--Combinina a Basic Understandina oi Equilibria with the Eonvenience and POW& of a Spreadsheet Donald R. Catter, Melissa S. ~ r ~ e ' , and William A. h4attson2 Randolph-Macon Woman's College Lynchburg, VA 24503
The use of modem computer methods for multiequilibria calculations has been described in numerous articles (e.g., 1 4 .., includine articles on the use of s~readsheets(6-9). . . Limitations include divergence, slow convergence, and complex programming a s mentioned by Cobranchi, et al.
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(I). Educators and students need a method that relies primarily on a working knowledge of equilibria; diversions from &e chemical topics to d a t e d and important areas such as computer programming should be done very selectively in chemist* c&ses. The modern spreadsheet can be used in chemical education as a n easily understood, powerful tool for aiding problem solving, even in introductory courses (10, 11).It provides both programming and maohine caoabilities in a more Dalatable format than traIf the few basic guideditibna~co&~uter lines suggested in this paper are followed, the spreadsheet can be a n effective tool for solving complex multiequilibria problems, such a s those encountered in sophomore-level quantitative analysis courses. The easv-to-use iterative method described reinforces chemical concepts. It normally converges rapidly on exact solutions to simultaneous equations without the need for initial steps involving combination/simplificationof the equations, without the need for complex programming or extensive mathematical manipulation, and without the need for extremely accurate initial approximations of equilihrium concentrations. All equilibri"k concentrations are available without the need for additional calculations. Multiple solutions needed for changing condition type problems (c.g., acid-base titrations1 can he obtained using macros to automate the spreadsheet calculations. Titration curves and related plots can then be readily generated the process is more complex and and displayed. time-consuming; it would not be appropriate for a normal class assimmeit This method for solving simultaneous equations is conveniently done on a spreadsheet3, but the pnnciples could be applied using more traditional computer programming.
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Description of the Method The general approach is presented below, followed by a specific example using Figures 1through 4: 1) Define the problem and determine all pertinent equilib-
ria. 2, Gcncrntc as many independent4equilibrium constnnt ex-
prenmms and charge balance and mnRS halanee e x p n s sions as there are unknown concentrations. 3) Calculate initial approximations for equilibrium eoncentrations. (Note:If these initial estimates are poor, the simultaneous eouations mieht diveree rather than converge to exact solutions (i.e., successive values will not approach a single value)): a) First, use an introductory understanding of equilibria and chemical reactions to estimate concentrations for perhaps two easily approximated species. For example, for a reaction favoring products (i.e.,Kc is large) (e.g., titrations and man" acid-base neutralization, . precipi. tation, and complexation reactions), simple stoichiometry will normally provide reasonable estimates of
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'Current address is Lynchburg College, Lynchburg. VA24503. 2Authorto whom correspondence should be addressed. 3QuatroPro. Version 2.0, Boriand: Scotts Valley, CA. %minate dependent equations. For example, do not use Kds if K,'s are being used; use K,'s and &. Frequently charge and mass balance equations will not be independent of one another.
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several concentrations-if A i s mixed with excess B to 10) If the concentrations do not converee. - . see "Sueeestions for form C and D, then B, C, and D could a l l be estimated Problems with the Method" below. knowing the balanced chemical equation and assnmR e f e r r i n g t o t h e detailed example p r o b l e m in F i g u r e s 1 i n g complete reaction. t h r o u g h 4, F i g u r e 1i s a statement o f t h e problem a n d inib) Then, use these estimates and appropriate equilibr i u m constant expressions to estimate the other ret i a l conditions i n c l u d i n g t h e calculation o f approximate maining e a u i l i b r i u m coneentrati&s a n d t h e " g e n e r a t i o n o f t h e si- concentrations. 4) Rank order the approximate concentrations calculated in multaneous equations in an appropriate form. These steps 3) from highest to lowest. a r e m o s t conveniently done o n t h e spreadsheet, but m a y be 5 ) Use the charge balance and mass balance equations to dedone elsewhere. velop expressions for the species having the hzgkst estimated concentrations in step 4 (e.g., use a mass balance equation to salve far the species having the highest concentration in terms of the conI'roblem: Calculate all equilibrium concentrations at the first equivalence centration of the other species). In using point for the titration of 25.0 m L of 0.1 M phosphoric acid these equations containing addition and with 0.1 M sodium hydroxide ( T = 25 Celsius). subtraction convergence i s most oRen successfully achieved by: a) usine the s i m ~ l e s temation first (mass Chemical Reaction: balance') wit; the sbecies having the H3P04 + NaOH o H2P04(-) + Na(+) + H20 largest nnt~cipntrdequilihrium roncmtration, Equilibria: b) using the charge balance expression to H3P04 + H2O o H30(+) + H2P04(-) solve for the most chareed snecies havine H2P04(-) + H20 o H30(+) + HP04(2-) a major concentration! and HP04(2-) + HZ0 o H30(+) + P04(%) c) avoiding: 2H20 o H30(+) + OH(-) the subtraction of two species that are about the Equations (6 unknowns-need 6 equations): same value and Equilibrium Constant Expressions: the calculation of a minor species. 1) Kal = [H30(+)] ' [HZP04(-)I I[H3P04] 6, Develop expressions for the remaining spe2) Ka2 = [H30(+)] ' [HP04(2-)] / [H2P04(-)] cies using the equilibrium constant exprrs3) Ka3 = [H30(+)] ' [P04(3-)] I[HP04(2-)] sions. 4) Kw = [H30(+)] ' [OH(-)] 7) Create a table containing a column for each Charge Balance Expression: species, decreasing in concentration from Sum [+ charge] = Sum [-charge] left ta right. Far the first and second rows, use the expressions developed in step 5 and [Na(+)] + [H30(+)] =[OH(-)] + [H2P04(-)] + 2[HP04(2-)] + 3[P04(3-)] step 6 to solve for the concentration of each 5) 0.05 + [H30(+)] = [OH(-)] + [H2P04(-)] + 2[HPO4(3-)] + 3[P04(3-)] species using the appropriate constants and (0.1 M NaOH diluted to 0.05 M at 1st equivalence point) the previous best estimates Le., the most Mass Balance Expression: current estimatesL6 [H3PO4](dilute initial) = [H3P04](all equil. forms) 8) To obtain calculated converging concentra6) 0.05 M = [H3P04] + [ ~ 2 ~ 0 4 ( - + ) ][HP04(2-)] + [P04(%)] tions to the desired precision, copy the secEquilibrium Constants (from p 110, 817 of Skoog, West, and Holler (12)): ond line of the table a n appropriate number Kb3 = 1.42E-12 Ka1 = 7.11E-03 of times.7 Ka2 = 6.34E-08 Kb2 = 1.59E-07 9) Convergence should signify that the desired Ka3 = 4.20E-13 Kbl = 2.40E-02 exact solutions for the simultaneous equations have been obtained. However, to check Kw = I.0IE-14 the validity of the calculated concentrations, the calculated values should be inInitial Approximations: serted into the original equations. [H2P04(-)] = 5.00E-02 = Titration n n lies to right; K=llKb3=7ElI [HPO4(2-)I = 5.63E-05 = (Ka2 ' [H2P04(-)]) "112 [H30(+)] = 5.63E-05 = (Ka2 ' [H2P04(-)]) "112 since Ka2 > Kb3 5 F ~examp r e, for the charge oa ance expression nvo vlng the phosphate spec es in F f g ~ r 1, e tne eqLa [OH(-)] = 1.79E-10 = Kw I[H30(+)] non w o ~ l d have been solvea lor me [P04(3-11 specles [H3P04] = 3.96E-04 = [H30(+)] ' [H2P04(-)] 1 Kal tf 3'[P04(3-1] haa oeen greater lo or of sim iar magnl[P04(3-)] = 4.20E-13 = [HP04(2-)I ' Ka3 I[H30(+)] tlrae compared to otner possib e cno ces ( e , only [hP04(2-I]. [r130(.)), or [OH(-)] slncethe [H2P04\-1] Dominant Species (decreasing 11's): Minor species: has alreadv been oefmea at t h ~ scant) That was not lH2P04(-)1 P H I the case but for the next mosi charged species, [H3P04] l(P04)3-1 [HP04(2-)], 2'[HP04(2-)I is greater to or of similar [HP04(2-)] magnitude compared to other possible choices (i.e., [P04(3-)I, [H30(+)], or [OH(-)]). Thus, the charge [H30+] balance equation was used to solve for [HP04(2-)]. 6Note that equations on the first line will reference Equations to Use in Spreadsheet: either concentrations contained in cells to the left on 6) [H2P04(-)] = 0.05 [H3P04] [HP04(2-)] [P04(3-)] the first line or. if necessarv. initial aooroximate con[H3P04] = [H30(+)] ' [H2P04(-)] I Kal 1) centrattons Tne second I ne wfl refe&ce etther con[H2P04(-)] 3[P04(%)])/2 5) [HP04(2-)] = (.05 + [H30(+)] [OH(-)] centraltons conlaned in ce Is to the left on the second [H30(+)] = Ka2 ' [H2P04(-)] I[HP04(2-)] 2) I ne or, f necessary, concentrat ons contamed in ce Is [OH(-)] = Kw I[H30(+)] on the first line . 4) [P04(3-)] = Ka3 [HP04(2-)] I[H30(+)] ' ~ e f e rto the equilibrium constants with fixed rather 3) than relative references (e.g., in Quatro Pro a 5 before a cell erler or n ~ m b ema r cites a 1 xed reference1 For F gure 1. Statement ot !he prob em ano nital cona lions for tne calculation of the equlexamp e. n FlgJre 3 copylng SC$6 E l 3 from Cell F13 l~brLm concentrations at tne f rst e q valence ~ point for the t'tratton of 25 mL 0.1 M pnosphoric aca witn 0.1 M so0 ~m hydrox de. lo cel F14 r e s is ~ n $C$6 E l 4 in cel Ft4. -u
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Journal of Chemical Education
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Equilibrium Constants
Ka3
7.11~-03 6.34E-08 4.2OE-13
Initial Approximations WPW-)I [HP04(2-)]
[H30(+)1
5.63E-05
Figure 2. Spreadsheet used to calculate the equilibrium concentrationsof all species, referenced instead of cell F3, since the B12 value represents the most current estimate of the H a O a ConcentraDl2 through tion. l'& GI2 cells and the cells on line 13 are similarly calculated, always referring to the cells containing the most curr e n t estimates of concentrations. This process is continued until the values on successive lines agree, that is, line 13is copied down an appropriate number of times. The formulas used in the cells on lines 12, 13, 14, 30, and 31 are provided in Figure 3. The numbers for this problem converged in about 20 lines or 20 iterations, as illustrated in the Figure 4 plot of pH vs # of iterations. After 20 iterations successive numbers were reproducible to the 4th significant digit or better (see rows 30 and 31 in Figure 2). After 80 iterations successive numbers were reproducible to all 14 significant figures available in the spreadsheet. The number of iterations needed for a given precision will vary with
1
Figure 3. Partial spreadsheet with the formulasdisplayed The actual spreadsheet, shown in Figure 2, includes values of equilibrium constants, initial approximations generated in Figure 1, and columns for the equilibrium concentrations of the six species. The B12 cell is first calculated using the first equation (i.e., #6)a t the bottom of Figure 1 and referencing the cells containing the values for the initial approximations. The C12 cell is similarly calculated using the next equation (i.e., # 1)except that cell B12 is
t h e complexity of the problem, the proximity of the initial approximations ta the actual values, and Volume 70 Number 1 January 1993
69
Volume(ml)0.1 M NaOH Figure 4. A plot of pH vs # iterations used to illustrate the process of convergence. the magnitude of values such as initial concentrations and the equilibrium constants; but hundreds of such steps are easily generated in a matter of seconds on a spreadsheet.
Titration Curves and Related Plots Simulated titration curves and related plots for the titration of 25 mL of 0.1 M phosphoric, carbonic, and tartaric acids, respectively, with 0.1 M sodium hydroxide were obtained. Examoles for the titration of tartaric acid are shown in ~ i ~ u G5e a, s b, and c. A detailed description of the macros and spreadsheet modifications used is of limited value and too-lengthy and complex to justify its presence here. However, the authors are willing to correspond with anyone needing such additional information.
Volume(m1)0.1M NaOH
Advantages and Llmitatlons of the Method The method is easilv understood and amlied. reouirine a n elementary understanding of equilib& and'the'use 2 a soreadsheet. but not traditional comouter oroeramminn or Eomplex mathematical models. ~ l m &all sim;ltaneoui equation, multiequilibria type problems attempted from Chapter 6 in Skoog, West, and Holler (12) were successfully solved without difficultv and without the need for accurate initial estimates for- equilibrium concentrations. However, the method does not appear to work for a11 orob.lems. Problems surface when: 1) one cannot avoid the subtraction of two species that are about the same value, or 2 ) equilibria are not successive (e.g., divergence might be a problem if the equilibria involves B - + A and B -t C instead afA -+Band B + C). Suggestions for Problems with the Method 1) Obtain better initial approximations for the equilibrium concentrations. Perhaps calculating equilibrium concentrations for slightly different but more readily calculated conditions could provide reasonable initial approximations. For titration curves this mieht involve interoolation using valuer hefore and aRer the problem area. 2 1 Try changing whwh apcries are solved for using the charge balance and mass balance equations. 3) It is possible far one or more numbers calculated using a mass balance or charge balance equation to become negative during the initial iterations. An error message might then appear in later cells (e.g., trying to find a root of a negative number). If so, redefine the mass balance or charge balance equation such that the absolute value is used. Or, preferably, use "IF statements in those columns that might baome negative (i.e., charge balance expressions and mass balance expressions) ta conditionally use
70
Journal of Chemical Education
Figure 5. Spreadsheet simulated titration of 25 mL of 0.1 M tartaric acid ( k t = 9.2E4,Ka 4.3E-5 (12))with 0.1 M sodium hydroxide: a (top):titration curve, b (middle):equilibrium concentrations vs volume titrant, and c (bottom):distribut~onof species diagram. 5
the previous nonnegative number for that species if the current calculatian would otherwise yield a negative number (e.g., change the cells in column D of Figure 3 such that cell D l 4 becomes
4) TIYapproximations to simplify the simultaneous equa-
tions.
Literature Cited Cobranchi D. P;Eyring, E. M . J C k m . Educ 1991,68,40-41. Oevore, J. A. J. C k m Edue 1988,65,86%870. Vmg, R. J.;Charlson, R. J. J. C k m . Educ. 1985,62, 141-143. Willie, C. W.J. Chem. Educ. 1 9 8 1 , 5 8 , 6 5 ~ 3 . 5 . Stone, E. E. J. Chem. Educ 1966.43.241-244. 6. Stone, E. E. J. C k m Educ 1966,43,241-244. 7 . Pmker, 0.J.; Breneman, G. I.J. Chsm. Educ. 1990,67,A5-A10. U I..harne. S J. C b m . E d ~ r1963 M . N l S - N 4 l Y M c < , . C . D..nn?o. H JI J. (aem Educ 1989. W N d I - n 2 1 2 10 P w k w 0 J Bmneman. C L Spred*hed(.'bh.mwrr). h n r m E n ~ l c v o o d C l m SJ. 1991 11. Atkinson, D. E.; Bmarer, D. C.: MeClard, R.W.;Barkley. D. S. %noModpis in C h i s t w : Simonson: Marina del Rey, CA, 1990. 12. Skoog, D. A,; West, D. M.; Holler, J. M. findomonlola of Anolyfimi Ckmlsfry; Saunders: New Ymk, 1988. 1. 2. 3. 4.
Acknowledgment
The authors gratefully acknowledge the contributions of the NSFIJMU Research Experiences for Undergraduates P r o g r a m Directed bv D a n Downev ( N S F e r a n t ~Hf9000748); the othe;faculty and staffat.1nmes Madison Univeltiitv with s ~ e c i athanks l to Duma Arnenta. Tom Gallaher, ~ a k h ~a o l m Jim , Leary, Rosemarie palmer, and Frank Palocsay; and Audrey Michael, Rebecca Ghent, Melanie Nilsson, and Deanna Yaczko.
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