Teaching the Truth about pH Stephen J. Hawkes Oregon State University, Corvallis, OR 97331 What do you tell a student who measures the pH of 0.1 M HCl a n d finds t h a t ( 1 ) i t i s 1.1when s o u have just taught him a method of calculation t h a t saysit is l.OO?kny explanation reduces to a n admission t h a t s o u deceived him about the definition of pH. When the student calculates from t h e published densities t h a t log [HI] varies by only 0.01 between 0 and 90°C, why does t h e pH vary (2)from 1.08 to 1.13? If he should ever find out t h a t the pH of 0.01 M HCI i n 5 m LiCl is 0.8 (3)instead of t h e 2.00 t h a t you taught him to calculate, what will he think of your professional competence? When we teach methods of calculation t h a t give answers t h a t a r e substautially different from experimental reality, we accept a grave responsibility. When we teach a definition of pH that h a s been obsolete for half a century, we are guilty of professional malpractice. Yet no introductory text t h a t I have seen suggests that log [H+lis only a n approximation to pH. Conversely, when they discuss the measurement of pH they do not say t h a t the reading on a pH meter is only a n approximation to t h e log [Ht' t h a t they have given incorrectly as the definition of pH. Students who have been taught t h a t p H i s log [H+lwill obviouslv exoect t h a t their o H meter will read log 1Htl. Even p r ~ f e s ~ owho r s have t c e usual background i n physical chemistry have t h e same expectation. Arecent puhlished experiment (4) contained this expectation i n the second decimal! Students who make lab measurements carefullv and compare their result with the expectation can he ex~ e c t e dto cheat. The consequences are worse when they apply their understanding of DH to realworld Droblems after thevrrmduate. Our teaching has lead them into error. Such teaching i s unnecessary.

How to Tell the Truth Any approach to p H must be simple enough for introductory students to understand, must not be misleading, and must lead correctly into whatever calculations we think i t necessary to teach. But let u s understand what i s important. MacInnes, one of t h e leading workers on p H theory in his time, made this clear (5)when h e said: In possibly all but one case in a thousand, it is not necessary to consider the meaning of pH in terns of solution theory at all, but only to accept the numbers as a practical scale of acidity and alkalinity. So how much should we teach about pH? Are our calcnlations necessary? Whatever our answer to this, i t is clear t h a t we need to emphasize the practicality of t h e pH scale a s far more important than any calculations for which i t may be used. Moreover, our decision to teach calculations should not compromise the essential simplicity of the concept of pH.
The following sequence puts the concepts i n a n order in which they can be taught logically, with t h e most import a n t concepts first, and without resorting to untruths. It can he halted a t whatever level a teacher thinks appropriate to the students. pH is a measure of the acidity of a solution. pH decreases as acidity increases. acidity is the tendency of a solution to supply Hi to a reaction. the tendency of a solution to supply H+is represented in calculations by a quantity called the "activity" of Hi in the solution. the "activity" is directly proportional to the concentration of Hi in the solution and is also a function of any other substances in the solution whose molecules are close enough to H+ ions to have any effect on them and is also a function of the solvent itself, which is usually water. It varies slightly with temperature even when the hydrogen ion concentration does not change. other ions in the solution affect the activity of Ht more than neutral molecular species because of the interaction of their electric field with that of the Hf. DHdecreases bv one unit for each tenfold increase in activitv. pH may, therefore. be measured hy rhc rxrrnr ro which o solution interact* \nth a n electrode th;lt is scnairi\,r t o H'. ur how it affect3 the tquhbnum h e t w e n n rolorrd arid and its conjugate base. the potential at the electrode is the conventional measure of pH. The electrode is calibrated against standard solutions of which the H+ activity has been determined as accurately as possible by methods that are outside the scope ofthe course. Standard solutions have been recommended hy national and international authorities with recommended pH values for calibration of the electrodes. .because the ~otentialat an electrode varies emonentiallv poses as 10P". in solutions so dilute and so pure that Hi ions are not influenced bv anvthinn exceDt . the solvent.. DH . = log 1Htl. in mos&al;tians, pH differs from log IHt1 in the first deeimal, but this may sometimes be near enough for very approximate calculations. in most calculations the activity must be used. the activity often is represented as lgammaiIH+l where (gamma1 represents the effects of the solvent and the other substances in the solution: in aqueous solutions it is between 0.05 and 13 in every case that I have found but is usually between 0.5 and 1.0. It is 0.7 in seawater u~
~
The Definition of pH The International Union of Pure and Applied Chemistry (6)h a s defmed pH essentially a s the reading on a pH met e r t h a t has been standardized against standard buffers. That i s
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This is the same definition as that used in the standards of all national bodies that have ruled on the question. The actual standard buffers varv from one authoritv to anbuffother. Most use NRS ~NationhBureau of ~tandarhs) ers as standards. while IUPAC and some others use 0.05 m potassium hydrdgen phthalate, but all give the same values of pH within 0.005 unit. They are chosen so that their recommended pH equals log aH+as closely as theory permits the activity to be calculated. The symbol pH has its origin in a paper by Sorensen (7) in 1909 about the effect ofH+on the action of enzymes. The concept of activity was not available to Sorensen in 1909, so he believed that the output from his electrodes bore a Nemstian relation to [HI] and when inventing the useful term pH for this paper he naturally defined it as log [H'I. It is ironic that the natural and inevitable misconception of this pioneer haunts our introductory texts nearly a century later, even though their authors have the benefit of a mode m education. Sorensen himself came to realize his misconception within a decade and invented (8)a new term paH, where aH is the activity of Ht, but this became shortened (9) in common usaee to DH. In time. it was realized that even the activity was too'dificult to measure and involved uncertain methods of interoretine exoerimental data. So one national body after anbther,&d'eventually IWAC, agreed on the unambiguous and simple definition that depends only on an experimental datum. Theoretical electrochemists need alternative definitions that are usually given special subscripts and refer to various methods of defining activity. There is a useful discussion of such approaches in the recent text (10) by Galster, but it is not necessary to include such refinements in introductory courses. The simple definition as the output from a pH meter is the most useful for the practical purposes that will be valuable to graduates of an introductory chemistry course. Eauilibrlum Calculations Equilibrium calculations involving H' must necessarily involve the activity rather than the concentration, both because it is the activity that affects the equilibrium and because in any real equilibrium it is the activity, not the concentration, that is experimentally measurable through the pH. The activities of the other substances in the solution usually are not known, but their concentrations are known from the stoichiometry. The useful definition of the equilibrium constant, therefore, involves the activity of H+ and the concentrations of everything else. The acid dissociation constant K, for example would be
aH+[Al
&=
[HA]
This is variously known as the "mixed" or "Brmsted" or "practical" constant. In dilute solutions with negligible ionic strength it is identical with the concentration ratio and also with the activity ratio which is the thermodynamic definition. But few realworld problems involve such solutions. Indeed, most of the problems that appear in textbooks have ionic strengths greater than 0.1, so that the thermodynamic constant is useless, and the mixed constant is essential and may be only 213 of the concentration ratio, as is the case in seawater. The name "practical constant" should convince readers that it is unethical to use any other variation of K, in an introductory course. Introductory students will have no use for the esoterica of the electmchemist, but will need a practical method of getting correct answers to practical problems if they need such calculations at all. Where are such values tabulated? They are the original data from which other forms of K,, such as concentration ratios or the ratio of activities are calculated, so all that is 748
Journal of Chemical Education
needed is to reverse the calculation. The most useful source of critically evaluated constants is the sixvolume work by Smith and Martell (111, but they give concentration ratios. They explain that they do this so as to make the definition fall into line with the definitions of the stability constants of metal complexes that are the main part of their compilation and these are most usefully defined as concentration ratios. They give the method of reversing their calculation to obtain the above "practical" definition, by adding a constant, which is tabulated in their introduction, to the log of the concentration ratio. Of course, both the concentration ratio that is usually given in introductory texts and the more useful practical constant have the complication that they vary with the ionic strength of the solution especially if the acids are ions as in NH4+or HP04". Usually this is ignored in introductory texts, although the concentration ratios used in them to define K. are even more dependent on ionic strength than the practical constants. This deceives students into believing that they can be used in situations where they are useless, and this deception stays with them even after they have become full professors. This problem will be elaborated in later papers. I have calculated pK.(practical) for the acids most commonly quoted by teachers of introductory chemistry, using the data and the correction in (11)and they are compiled in the table. Rationalization
The error is ingrained so deeply from our own training in high school and college that we seek a rationalization that releases us from it. The one I have heard most often is that there are two definitions of pH, one involving concentration and the other involving activity. Atenn can be dep& (Practical) for Acids Commonly Quoted in Texts
Ionic strengthC 0.0 H3P04 HzP04i ~ ~ 0 4 ' 
2.15 7.20 12.38
HzC03 HCO3CsH5(COzH)z CsH5(COzH)COz(COZH)~ HOzCC02NH4' CH3NH3* HF CBHSCOZH CHjCOzH HCN HCOzH HN02 H20a
6.35 10.33 2.95 5.41 1.25 4.27 9.24 10.64 3.17 4.20 4.76 9.21 3.75 3.15 14.00

13.89
13.89
13.93
'for H20. Ib(Mactica1I is the Dractical Kw, that is awJOH1. Where I O H l
occurs in a n y &i'culation;it may, therefore, be replacsd bj. ~,ipractical)iam.~ b ~ ~ decimal n d not available or varies too greatly with the nature of the interfering ions to be generally significant. Tonic strength = 0.5oi[(molaritiesofion$+(chargeof ions)?]
fined in any way we choose, but this dual definition has no foundation in either convention or lopic. I t is logical to define pH in a way that is useful in practice. Because the concentration of Hi cannot be obtained from the experimental pH, a definition in terms of concentration makes subsequent calculations impossible, while definition in terms of activity lends itself to calculation of other quantities. I can find no evidence of a convention allowing both definitions. I n the nine reference works (6,1219) that I have consulted, not one quotes both definitions a s valid. Three of them quote log [H*] as a n approximation, and two give it as the only definition. The remainder give the definition correctly.
authors of textbooks someday will indicate that the expression pH =log [HI1is not a definition and that it may be anly a very rough approximation. His "forlorn hope" has been realized in most handbooks, but is not yet fulfilled in any introductory text that I have seen. Perhaps he was too nice. Let us lay it on the line. Authors that define pH a s log [H+I are deceiving their readers and leading them into serious error. Those of us who have taught this error for decades, a s I have, are guilty of professional malpractice and need to correct our teaching in the future. The error can be avoided easily, a s has been shown above, by a trivial change in pedagogy that will not cause difficulty to teachers or students.
How Close is Activity to Literature Cited The solution being tested and the standard solution usually differ from each other in ionic strength and in other ways. This makes even the determination of activity a n Verlag, Berlin. approximation because of junction potentials in the cell, 3. Peterson.0. Chomklng. nchn. 1368.40.76. 4. Cawley J. J.J. Cham. Edue. 1993.70. 596598. althoueh " it is much closer than DH is to loe IHil.  Is it a 5. MacInnes, D.A Seieneo 1948,108,693. good enough approximation? T L ~IUPAC document (6) 6. Mills, I.: Cvitaa, T:Kallay, N.; Homann, K; Knchitsu, K. Quonliiies, Units, and puts the difference between pH and log a H + a s no more Symbols in Physiml Chemistry: Blackwell Scientific: Lpandon, 1988: pp 5P55. 7. Sorenren, S. P L. Bioehrm 2.1909.21, 131 and 201. Pvblifhed simultaneovsly in than 0.02 for aqueous solutions more dilute than 0.1 M. I n Compt Rend. l h u . Lab. Carisberg 1909,8,1. more concentrated solutions the difference may be greater 8. Sorensen, S. P. L.; LindersVomLsng, K. Compl. Rend. l h o . Lab. Corlsbwg 1924. 15, No.6. but may be offset by compensating errors. In standard 9. Hamed. H. S.:Owen. B. B. Th4 Physical Chrmistry ofElectrolytie Solutions; Reinseawater. for examole. the difference (20) is about 0.05. hold: New York, 1943; p 317. oceanographers either specify standard buffers that are 10. Galater H.pHMeosvremen(;VCH:New York, 1991, pp 4 7 5 6 . 11. Mamll, R. M.; Smith, A. E. Critical Sfobilih Comlonfs.Vols. I 4 ; Plenum: New specifically designed to eliminate this error when measurYork. 19751989. ing the pH of seawater, or they use sophisticated methods ~T J.~compmhensiue ~ , nictionory of physicoi cheminry; bentice 12. m i a x L.; K Hall, 1992; p 306. of correction that are outside the scope of a n introductory R. J. Condensed Chpm&l Dictionary: Van Nostrand Reinhold: New York, 13. Lewis, course, or determine the activity d ~ c ~ t r o ~ h o t o m e t r i r a ~ ~ y . ,003."" SO" The official procedure (211 for determining the pII of blood 14. Bevan. S. C.;Gregg, S. J.; Rosseinsky, A C~onciseElymologiedDictionary ofCheminy;Applied Science Publishem: London, 1976; p 2 8 4 involves secondam standards referred to the NBS stand~~ 15. K q t . D.Dcctionary of Chem~callkchnology; Elsevier. 1980. p 284. ard buffers, with solutions that simulate those properties 16. Pa~khh,S.P.M&rowHill Enwclopedia dChcmisiry. 2nd ad.: MeGrswHill, 1993. of blood that affect pH measurement. 17. Lide. D.R., Ed. CRCHondbook orchemistry ondPAysles, 74thed.; CRCPuhlishing,

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1993. pp G34.35.
Conclusion I t has been nearly half a century since MacInnes (5) wrote
The relation pH = lag [Htl is an approximation anly, and using it as a definition may lead to serious errors. It is the author's rather forlorn hope that compilers of handbwks and
18. Dean, J. A. Langds Hondbwk ofChrm&try. 14th ed.; MeGrawHill, 1992: pp 8103
to 8108. 19. Parhr, S. P McGmwHill Dictionary ofScipnLifi and lkcknicol h s , 4th ed.; MGrawHill: New York, 1989: p 1404. 20. Culbcrson, C. H. In M m i m Electmch~mlsfry:Whiffield. M.; Jagner,D.,Eds; Wiley New York. 1981: p 200. 21. Maaa,A. H.J.:Wei6berg.H. F.:Bumett,R. W.:Mullc~Plathe,0.;Wimber1ey.P. 0.: Zijlstrs,W G.; Durst, R.A,: SigaardAnderson, 0 . J. Clin. Chem. Ciin. Biochem. 1987,281289.
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