Potentiometric pH Measurements of Acidity Are Approximations, Some

In the Classroom

Potentiometric pH Measurements of Acidity Are Approximations, Some More Useful than Others Robert de Levie Chemistry Department, Bowdoin College, Brunswick, Maine 04011 [email protected]

A few years ago, McCarty and Vitz (1) described a number of pH paradoxes, “demonstrating that it is not true that pH = -log[Hþ]”. They are certainly right in that it is no longer true, provided that we follow the IUPAC recommendation on pH. This is because IUPAC has moved away from the original definition of pH as equal to -log[Hþ], proposed more than a century ago by Sørensen (2) and subsequently reiterated by Sørensen and Linderstrøm-Lang (3). IUPAC now defines pH as an approximation to -log(aHþ) (4), for which Sørensen and Linderstrøm-Lang (3) had proposed the separate notation paH. In other words, it is more a matter of paraphrase, which my Webster defines as “a rewording of the meaning of something”, than of paradox. Incidentally, the quantity -log(aHþ) was still called paH by Bates, the major architect of the current IUPAC recommendations, as “recently” as 1968 (5). But the matter involves more than a game of words and is sufficiently important for the teaching and practice of chemistry that an alternative viewpoint should be presented. McCarty and Vitz correctly state that their paradoxes arise only when one confuses the original (Sørensen and Sørensen and Linderstrøm-Lang) definition of pH with its subsequent (IUPAC) definition. In fact, their demos show that the pH, as measured with a pH meter and interpreted according to a muchexpanded reading of the IUPAC definition, is not equal to -log[Hþ], where we use [Hþ] to denote the hydrogen ion concentration. This is because their examples consistently fall outside the limits of validity specified by IUPAC, that is, a pH between 2 and 12 and an ionic strength, I, not exceeding 0.1 mol kg-1 (see their Note 1). Even the calibration of their pH meter is questionable, because IUPAC favors bracketing the measured pH by buffer solutions at higher and lower pH. But there are no IUPAC buffer solutions below pH 1.5. Just consider their opening paragraph: “When equal volumes of 5 M lithium chloride solution and hydrochloric acid solution with pH = 1 are mixed, the hydrogen ion concentration decreases and we might presume that the solution becomes `less acidic'. But the pH actually decreases! The acidity, as measured correctly by indicators or a pH meter, increases!” The following comments are in order.

• The mixture of equal volumes of 5 M LiCl and approximately 1 M HCl will have an ionic strength I of about 3; the pH measured with a pH meter is only defined by IUPAC (4) for I e 0.1, an ionic strength that is a factor of 30 smaller, whereas the hydrogen ion concentration [Hþ] is a factor of 50 larger than the IUPAC limit of pH 2. • Figure 1 in their article shows the pH decreasing with increasing volume of added LiCl. But the anomalous behavior of HCl upon addition of LiCl has been known at least since Huckel briefly mentioned it in 1925 as a small discrepancy (geringe Diskrepanz) (6) and was confirmed by Hawkins, who commented that “no

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satisfactory explanation for this behavior is available” (7). But why treat a known anomaly, occurring only at ionic strengths well beyond 0.1, as an example of a general rule? • “The acidity, as measured correctly by indicators” yields pH = -log[Hþ], because Beer's law is based on particle densities, that is, concentrations. The implication that the acidity, as measured with indicators or a pH meter, would give the same result, when the former measures concentrations whereas the latter is defined as -log aHþ, is not only incorrect, but also contradicts the main point McCarty and Vitz try to make. Guggenheim showed long ago that their claim is generally incorrect (8, 9).

Some Inherent Limitations It is important to guard against such cavalier use of the IUPAC definition and to emphasize (and stay within) its inherent limitations. McCarty and Vitz described as their goal “to produce a dramatic effect that is easily demonstrated in a classroom and demonstrates a principle that is true at all concentrations; that is, pH is not -log[Hþ].” Yet they admitted that “the ionic strength of solutions and concentrations of acid used in these demonstrations are often outside of the range where IUPAC defines pH”, in which case it is not clear what precisely would be the significance of their instrumental readings. When an approximation (such as the Bates-Guggenheim expression that forms the basis of the IUPAC recommendation) is extended beyond its specified range of applicability, the results obtained with it are not necessarily applicable either. McCarty and Vitz have clearly demonstrated that, under their extreme conditions where the IUPAC definition does not apply, their pH meter fails to yield chemically meaningful numbers. That is hardly surprising. In their zeal for a dramatic effect, their demonstration has lost any purpose beyond illustrating the inherent limits of their chosen instrumental method. Where do those limits originate? A pH meter with an indicator electrode, plus an external reference electrode (i.e., separated from the sample solution by a liquid junction), and calibrated with the usual reference or standard buffer solutions, can only yield an approximation for -log(aHþ) as currently defined by IUPAC, for the following reasons. • The ionic activity coefficient is, admittedly, an immeasurable quantity and therefore needs an approximation. Well beyond the estimated range of validity of the chosen Bates-Guggenheim (10) approximation, which IUPAC set at I e 0.1 mol kg-1, the method becomes powerless. At I = 0.1 mol kg-1, it already has an estimated uncertainty of at least (10%. • The measurement specifically includes a liquid junction to make the measured response maximally independent of the anions in the sample. Unfortunately, the resulting liquid junction potential is also

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In the Classroom

immeasurable and can only be estimated by approximate models that neglect the difference between ionic concentration and activity. Enough is understood about the origins of the liquid junction potential to know the conditions that make it small, and this led to the use of liquid junctions containing maximally concentrated nearequitransferent salts and to restricting the sample pH to values between 2 and 12, a restriction McCarty and Vitz consistently violate. At the much lower pH values used by McCarty and Vitz, proton transport can be expected to affect the liquid junction potential strongly.

How Large a Difference? The paradoxes of McCarty and Vitz aim to emphasize the distinction between the definitions of pH by Sørensen (2) and Sørensen and Linderstrøm-Lang (3) on the one hand and by IUPAC (4) on the other. We will therefore put the distinction between -log[Hþ] and -log(aHþ) as defined by IUPAC in a numerical context. IUPAC (4) specifically recognized that its definition “involves a single ion quantity, the activity of the hydrogen ion, which is immeasurable by any thermodynamically valid method”, and therefore based its definition on an assumed formula (10) for the equally immeasurable chloride activity coefficient γCl. For this it used a Debye-Huckel (11) fit to the directly measurable mean NaCl activity coefficient γ( (12) in the interval 0 e I e 0.1 mol kg-1. To facilitate comparison with the modern literature on pH measurements, we will here use molality, the concentration unit preferred by physical chemists because of its temperature independence and now used exclusively in the IUPAC definition of pH (4). (The only exception will be in the section on kinetics, where the molarity formalism is more commonly used.) For our argument, which concerns dilute aqueous solutions at room temperature, the small difference between concentration scales in terms of molality (moles per kg of solvent) or molarity (moles per liter of solution) is rather insignificant. The Debye-Huckel equation (11) is a monotonic function of I and the Bates-Guggenheim expression (4, 10) chosen by IUPAC assumes that, in solutions containing chloride ions, √ the chloride √ activity coefficient γCl is given by log(γCl) = -(A I)/(1 þ 1.5 I) -1/2 1/2 kg at 25 C. It follows that log γCl f 1 for where A ≈ 0.5 mol I f 0 in a 1,1-electrolyte, and log γCl f -A/1.5 ≈ -0.33 for I f ¥. Note that γCl is defined only for ionic strengths not exceeding 0.1 mol kg-1, because species-specific effects not represented in the Debye-Huckel formula start to become important at higher concentrations. That same limit, therefore, applies to the IUPAC definition of -log(aHþ). The more severe restriction of pH to the range between 2 to 12 has to do with the additional assumption that the liquid junction potentials are the same with either sample or standard in the measurement cell, in view of the high aqueous mobilities of Hþ and OH-. Because the liquid junction potential is not directly measurable, we can only use models to estimate its magnitude. The most often used model is that of Henderson (13, 14). It uses a number of assumptions that make it unsuitable for use in highly concentrated solutions, but Harper (15) has more recently extended it to concentrated solutions. Although we have no means to verify such computed numbers, here are some relevant estimates Harper listed: about -76 mV (or 1.28 pH units) for a liquid junction between 0.0585 and 2.82 m HCl, about -99 mV or 1.67 pH between 0.0585 and 5.00 m HCl. Interposing a

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concentrated KCl bridge will greatly reduce these numbers but is unlikely to bring the liquid junction potential close to zero in view of the combined high concentration and high mobility of protons in such concentrated hydrochloric acid solutions. The limitation to I e 0.1 mol kg-1 means that pγCl, the negative logarithm (p) of that IUPAC-assumed activity coefficient γCl at room temperature, ranges from 0 (at I = 0) to at most 0.11 (at I = 0.1 mol kg-1) for aqueous solutions of the alkali chlorides. Now the product γMγCl (where M is any monovalent cation, including Hþ) is a directly measurable quantity, γ(2. Moreover, for I e 0.1 mol kg-1, the ionic activity coefficients for monovalent anions X and cations M are typically assumed to be equal; that is, γM ≈ γX ≈ γ( . According to the IUPAC recommendation (4), “the only consistent, logical way of doing it is to assume γH = γCl and set the latter to the appropriate Bates-Guggenheim conventional value.” Likewise, the IUPAC Compendium on Analytical Nomenclature (16) states that “any other convention would be unjustified”, whereas Bates and Guggenheim (10) write that “any other convention would be far-fetched.” The known values for pγ( of HCl and the alkali metal chlorides at m = 0.1 mol kg-1 lie between -log(0.796) = 0.099 for HCl and -log(0.756) = 0.121 for CsCl (17). The IUPAC convention uses NaCl as a middle ground (12), for which -log(γ() = -log(0.778) = 0.109 (17). The likely magnitude of the maximal difference paH - pmH = -log(γH) ≈ -log(γCl) is therefore estimated as 0.11 ( 0.01 for the usual alkali cations and as less than 0.20 ( 0.02 for the earth alkali cations Mg2þ and Ca2þ. The “dramatic” effects demonstrated by McCarty and Vitz clearly fall outside those limits, because they extrapolate the Bates-Guggenheim expression far beyond its intended range. Can the pH Meter Be Used To Estimate Hydrogen Concentrations? Chemical equilibrium expressions must be written in terms of activities, and this applies to the Nernst equation. That point was emphasized by McCarty and Vitz when they wrote that “pH cannot be defined as -log[Hþ], because these operational definitions are based on cell potentials, which in turn are dependent on activities, not concentrations.” This is certainly true, but by the same token one cannot define pH as -log(aHþ) because measurable cell potentials never depend on the activity of a single ionic species either. They can yield values for salt activities, that is, mean activities of independently variable, thermodynamic solution components (which ions are not), but these always depend on the chemical nature of at least two ionic species and therefore would not yield a convenient, hydrogen-specific measure of acidity. An external reference electrode linked to the sample by a liquid junction tends to reduce the effect of the second ionic species on the measured cell potential, at the cost of introducing a liquid junction potential and unverifiable assumptions regarding ionic activity coefficients. This is precisely why such pH measurements cannot yield exact values for either hydrogen concentration or hydrogen activity. But even if we are willing to settle for plausible estimates, the question remains: can a properly used pH meter yield estimates of pH = -log[Hþ] as originally defined by Sørensen (2) and Sørensen and Linderstrøm-Lang (3)? To answer this question, we consider the method of Hamer and Acree (18), which is used to specify the numerical values of paH = -log(aHþ) for the IUPAC reference and standard buffer solutions. Internal

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(i.e., junction-free) hydrogen and chloride electrodes are placed in a cell containing a chloride-free buffer plus known amounts of an added chloride-containing strong 1,1-electrolyte such as HCl, NaCl, or KCl, so that the measured potential can yield the corresponding values of p(aHaCl) = p(mHmClγHγCl). The known values of mCl are then used to compute the quantities p(mHγHγCl) = p(mHmClγHγCl) - p(mCl), because p(mCl), the negative logarithm of the chloride ion concentration, is known. Extrapolation of these quantities to zero mCl yields p(mHγHγCl)o, where the superscript identifies the absence (or presence in negligible, trace amounts) of chloride. So far, everything is strictly thermodynamic and unambiguous, but leads to the usual product of a concentration mH and the square γ(2 = γHγCl of a mean salt activity coefficient, not what we are looking for. The Bates-Guggenheim equation (10) is therefore introduced to define γCl; this equation disregards the chemical nature of the counterions present (otherwise it would not be helpful for use in a variety of buffer solutions) and assumes that the chloride activity coefficient is merely a function of the ionic strength. Subtracting the assumed pγCl at the ionic strength of the buffer then yields the desired estimate, paH = p(mHγH) = p(mHγHγCl)o - pγCl . IUPAC claims validity of the approximate expression for pγCl as long as the ionic strength is limited to I e 0.1 mol kg-1. Within that same range, we can therefore carry the just-described process one step further by once more subtracting pγCl, so that we obtain pmH = paH - pγH ≈ paH - pγCl because pγH ≈ pγCl . In other words, the very same assumptions that make it possible to estimate paH = -log(aHþ) also let us estimate pmH = -log[Hþ] as paH - pγCl. Moreover, the difference between such estimates of pmH and paH is small and is unlikely to exceed 0.11 ( 0.01 with the usual monovalent inorganic cations. The “demonstration” by McCarty and Vitz is therefore an exaggeration of a relatively minor effect. And, to answer the question posed in the heading of this section: yes, the pH meter can indeed be used also to estimate hydrogen ion concentrations, based on the same assumptions used by IUPAC to estimate hydrogen ion activities. Chemical Equilibrium Although ionic activities are convenient in writing the Nernst equation for a single electrode, the potential of such a single electrode is not measurable, just as it is impossible to clap with one hand. No measurable potential differences of electrochemical cells depend on the immeasurable potential of a single electrode or on the equally immeasurable activity of just one ionic species. Equilibrium constants can often be written conveniently in terms of ionic activities, but these always occur in combinations that can be measured (8, 9). For acetic acid, we can write 2 HAc / Hþ þ Ac- and KQ a = aHþaAc-/aHAc = (mHþmAc-γ( )/ (mHAcγHAc), where the latter form does not contain any ionic activity coefficients. The mean electrolyte activity coefficient γ( and the activity coefficient γHAc of the undissociated acid are thermodynamic quantities that are in principle directly measurable, and KQ a is the thermodynamic equilibrium constant. Likewise, for the ammonium/ammonia equilibrium NH4þ / Hþ þ NH3, we find KQ a = aHþaNH3/aNH4þ = (mNH3mHþ/ mNH4þ)γNH3(γHþ/γNH4þ). Both the product γHþγAc- = γ(2 and the quotient γHþ/γNH4þ are directly measurable, thermodynamic quantities, and again no immeasurable activity coefficients of individual ionic species are needed. 1190


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Whereas the proton activity is an immeasurable quantity, the proton concentration is well-defined. In their first paradox, McCarty and Vitz used a strong (i.e., fully dissociated) acid such as HCl and then added a concentrated strong, also fully dissociated electrolyte such as LiCl or CaCl2. In that case, [Hþ] can simply be calculated as equal to the original concentration of HCl after correction for its dilution by the added salt solutions, and we can also measure [Hþ] directly, by titration with a strong base. These two approaches, within experimental accuracy, must yield the same result. In other words, such ionic concentrations are well-defined quantities, which we can compute as well as measure. More than a century of analytical experience testifies to the fact that, for such strong electrolytes, the computation and direct measurement of [Hþ] agree. After all, determining concentrations is the central focus of quantitative chemical analysis, and volumetric titrations are among its oldest, most extensively tested methods. Spectrometry We may be tempted to take a purely metrological approach and argue that the measurement, whatever its theoretical significance might or might not be, still can have physical relevance. What could be easier than getting an estimate of the acidity of a sample solution by sticking a combination pH electrode into that sample solution and using a pH meter to read off the resulting potential difference, calibrated in convenient units? As long as we are willing to elevate the pH meter and the associated standard buffer solutions to the position of sole, absolute arbiters of acidity, this might be fine. But there is a very practical reason to be hesitant to follow IUPAC in this: the pH as defined by IUPAC not only lacks any predictive power and in that respect fails the test of a scientifically useful concept, it also ignores a large body of spectroscopic (UV-vis absorption, fluorescent emission and nuclear magnetic resonance) measurements that directly relate to [Hþ] and hence -log[Hþ], by measuring the fraction or ratio of protonated or deprotonated species in UV-vis absorption or fluorescence or by determining that fraction from a shift in the magnetic resonance frequency of atomic nuclei neighboring the acid moiety. Beer's law is derived (and must be written) in terms of concentrations, not activities. There are of course activity effects on the pKa values of colorimetric pH indicator dyes because equilibrium constants are correctly written in terms of activities. However, as we have already seen, these never require the activities of single ionic species, only combinations thereof that are, at least in principle, measurable, such as their mean electrolyte activities a(. Already some 75 years ago, von Halban and Kortum (19, 20) used precision spectrometry to determine the hydrogen ion concentrations [Hþ] and pKa values of 2,4dinitrophenol, showed that their results were in quantitative agreement with the Debye-Huckel theory (11) and could readily be extrapolated to the thermodynamic, infinite-dilution value pKQ a of pKa. (It would be inadvisible to use 2,4-dinitrophenol, a metabolic poison that uncouples oxidative phosphorylation in mitochondria, in an undergraduate lab experiment, but safe equivalent dyes such as bromophenol blue are available.) This clearly illustrates that spectrometry can indeed yield the two thermodynamic values, viz. the hydrogen ion concentration [H þ] and the activity-based pKQ a value; unfortunately it


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cannot furnish aHþ. Guggenheim, who defined ionic activities thermodynamically, also made this argument and showed that the hydrogen activity does not play a role as such in colorimetric pH determinations but only as an integral part of in principle measurable combinations such as the mean activity a( (8, 9). Again, there are activity effects, and the addition of otherwise “inert” salt can indeed shift chemical equilibria, but these are all describable in terms of mean activities and do not require the activities of single ionic species. Of course, optical absorption is not the only spectroscopic method used to determine acidity: fluorescence and NMR are used increasingly to determine pH, and they likewise determine pH in terms of hydrogen ion concentrations, that is, according to the original Sørensen definition. Moreover, spectrometry should be used correctly. McCarty and Vitz reported that methyl green in about 0.1 M HCl turned from green to yellow upon the addition of concentrated CaCl2 or, to a lesser extent, with concentrated LiCl. Therefore, we will briefly look at this type of problem and consider how to explain such an observation. McCarty and Vitz did not specify whether they used the methyl or ethyl form, purified the dye to remove the usual crystal violet contamination, or gave any other specifics, and apparently treated methyl green as a monoprotic acid HIn, that is, describable with a single equilibrium constant Ka = [In-] [Hþ]/[HIn]. Because it does not change the nature of the qualitative argument, we will do so likewise, even though methyl green is usually considered a diprotic acid, which makes it more likely that a strong salt effect is observed. Spectrometric measurements of the titration of 0.1 M HCl with concentrated CaCl2 or LiCl can yield the ratios [HIn]/[In-] = [Hþ]/Ka, from which [Hþ] can be obtained provided that the change of Ka during that titration is also known. In fact, a standard procedure to determine the thermodynamic equilibrium constant KQ a takes exactly the opposite approach, in that each spectroscopic ratio [HIn]/[In-] is combined with the known value of [Hþ] to obtain the formal Ka value as Ka = [In-][Hþ]/[HIn], whereupon the various Ka values so obtained are extrapolated to zero ionic strength to obtain KQ a. It is well-known that formal equilibrium constants in general depend on ionic strength, and more so when the equilibrium involves diprotic ions, as they would be in the equilibrium H2In / HIn- þ Hþ. In either case, the way formal equilibrium constants depend on the presence of added salt can be described quantitatively in terms of the mean activities in such mixtures and does not require single ion activities (8, 9). It is obviously incorrect to imply that a shift in color signifies a change in [Hþ] without considering what will happen with Ka, that is, by assuming Ka to remain constant. On the other hand, once the dependence of Ka on solution composition is established, spectrometry can be used to determine [Hþ] directly. With their simplistic interpretation of indicator color change, McCarty and Vitz merely confused the issue. And by focusing exclusively on the pH meter, at the exclusion of spectroscopy, the IUPAC definition is unnecessarily self-limiting. Spectroscopy allows the pH range to be expanded more than 4-fold from its present range from 2 to 12 to starting at less than -30, by what is called the Hammett function H0 (21), using a scale specifically adjusted to fit the original, Sørensen definition of pH (22).


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Kinetics In their third paradox, McCarty and Vitz argue that Birk and Walters (23) used an incorrect rate law for the acid-catalyzed rate of bromination of acetone, and they “demonstrated” this by adding 9 g of solid MgCl2 3 6H2O to 10 mL of 1.0 M HCl, making a final solution of about 3 M MgCl2 with an ionic strength of about 9.6 mol L-1. Such an experiment illustrates nothing more than that one cannot derive scientifically meaningful data when operating an instrument so far beyond its stated limits of interpretability. McCarty and Vitz, however, offered another explanation: according to them, the expression for the reaction rate R, which is usually formulated as R = k[C3H6O][H3Oþ], instead should be written as R = kaC3H6OaH3Oþ. In the rate law for an irreversible chemical reaction such as A þ B f, the terms -dcA/dt and -dcB/dt in the rate expression derive from the mass balance equations, that is, from counting particles and therefore have nothing to do with energetics, that is, with activities. But how about the other side of the rate equation, often written as = kcAcB? Because k characterizes the driving force of the reaction, energetics are clearly involved. Then should we not formulate the rate R as R = -dcA/dt = -dcB/dt = kaAaB? Brønsted (24, 25) extensively studied this question and concluded that in this case the rate law must be written as Rc = -dcA/dt = -dcB/dt = kKQcAB = kcAcBfAfB/fAB, but not as Ra = kKQaAB = kaAaB = kcAcBfAfB, where the subscripts c and a on R distinguish the two formalisms, the thermodynamic equilibrium constant KQ = aAaB/aAB refers to an assumed thermodynamic equilibrium between the reagents A and B and a transition state AB, and k is the rate constant for the dissociation of that transition state AB into products. The only special case in which Ra is applicable is when fAB = 1, which (within the limits of the Debye-Huckel theory) occurs when the reagents have opposite charges of the same magnitude. The bromination of acetone is not such a special case. The above rate expressions can of course be written also in terms of the corresponding molal concentrations m and activity coefficients γ. Shortly after the Brønsted papers, Christiansen (26) showed that the same result can be obtained without assuming a transition state, simply on the basis of the Debye-Huckel equation (11) and the effect of ionic strength on the effective reaction distance of the reagents, assuming it to be equal to their distance of closest approach. The difference between the two formalisms, Rc = k cAcB fA fB/fAB and Ra = k cAcB fA fB might seem small, but it is quite informative, because it allows us to differentiate unambiguously between these two rate expressions. We will here consider the logarithms of the measured rates R and restrict the discussion to sufficiently low ionic strengths where the √ Debye-Huckel limiting law log f = -z2A I (where z is the charge of the ion considered) applies. Most such experiments are performed in this range of ionic strengths to avoid so-called secondary salt effects, which we will encounter below. Secondary salt √ effects exhibit a linear dependency of log R on I rather than on I and would therefore complicate the analysis at higher ionic strengths. √ Substituting log f = -z2A I into the rate laws we obtain √ log {-z A2 - z B2 þ (z A þ zB)2 }A I = R c = log(k0 cA cB) þ √ log(k0 cAcB) þ 2zAzBA I because√zAB = zA þ zB, whereas log Ra = log(k0 cAcB) - (zA2 þ zB2)A I. The expression for log Rc predicts that the sign of the primary salt effect will depend on whether zA and zB have the same or opposite signs and yields no


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primary salt effect when at least one of the reagents is uncharged, as acetone is. By contrast, log Ra always predicts a negative primary salt effect and requires only one charged reagent (such as Hþ) to show it. The Brønsted-Christiansen model has been confirmed extensively and quantitatively at very low ionic strengths, see Figure 2 in La Mer (27). Perhaps the most convincing evidence of its direct applicability and predictive power came in the identification of solvated electrons as the reactive species formed by electron bombardment of ionic solutions (28). An equation of the form advocated by McCarty and Vitz would fail to explain the various signs of the primary salt effect in those data, let alone their magnitudes. The sample reaction chosen by McCarty and Vitz is actually quite illuminating because the Brønsted-Christiansen model deals with the so-called primary salt effect, which is the effect of an otherwise (to the reaction components) chemically inert salt on the reaction rate of ionic reactions. We have just seen that the primary salt effect is zero in the rate-limiting step C3H6O þ H3Oþ f because acetone is uncharged, a result that would not follow if we were to write the reaction rate as R = aAaB, as suggested by McCarty and Vitz. But at high ionic strengths, such as used by McCarty and Vitz, the situation is more subtle. There the secondary salt effect applies because the inert salt also (but much more weakly) affects neutral compounds such as acetone, as well as all electrolytes, through so-called salting-in or salting-out. The usual description of secondary salt effects is based on the relations of Brønsted (29) and Huckel (6), who added a term proportional to the ionic strength I to the Debye-Huckel expression for log f or log γ. This extension of the Debye-Huckel model was subsequently adopted by, for example, Guggenheim (8, 9) and Davies (30), and first given a solid chemical basis by Stokes and Robinson (31). Such species-specific secondary salt effects are rather insignificant at I < 0.1 mol kg-1 but often become dominant at higher ionic strengths. They are specifically excluded from the Bates-Guggenheim convention that forms the backbone of the IUPAC recommendation, which is the very reason why that recommendation is explicitly limited to low ionic strengths. Secondary salt effects illustrate that effects dominant at high ion concentration are not necessarily prevalent at much lower I values, and why, in this specific case, it is not permissible to extrapolate results obtained at high ionic strengths to lower ionic strengths. The demonstrations of McCarty and Vitz involve concentrated solutions of HCl, LiCl, MgCl2, or CaCl2, all of which exhibit mean salt activity coefficients that are nonmonotonic functions of ionic strength. Starting from I = 0 their activity coefficients decrease with increasing ionic strength, then reverse that trend and increase at higher ionic strengths, eventually exceeding their starting values of 1 at around 2 mol kg-1 (17). It is, therefore, invalid to use highly concentrated solutions of HCl, LiCl, MgCl2, CaCl2, and their mixtures to demonstrate, by implication, some properties at low ionic strengths. The Use of Hypothetical Quantities for Practical Purposes The activity of a single ionic species, such as that of hydrogen ions, is a purely hypothetical quantity. According to the most recent pH-defining IUPAC document (4), the hydrogen ion activity “is immeasurable by any thermodynamically valid method and requires a convention for its evaluation.” It is 1192


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still a useful hypothetical construct in calculations, just as imaginary numbers are useful hypothetical constructs in solving polynomials, or electronic wave functions are in quantum mechanics. IUPAC merely defined a metrological procedure for approximating paH under rather restricted conditions (I e 0.1 mol kg-1, 2 e pH e 12), using what it assumed to be a plausible estimate of pγCl. But that means that there is also no testable interpretation or deeper meaning of that measurement other than whether it satisfies its own prescribed procedure. Some scientific concept! It is no accident that McCarty and Vitz argued that pH 6¼ -log[Hþ] and even exhorted their readers to replace concentrations by activities but did not check whether the readings on their pH meter agreed with -log(aHþ). The reason is obvious: although [Hþ] is readily computed, there is no known way in which aHþ can be determined, other than by deriving it from the output of a pH meter, a transparently circular argument. Let me be clear: electronic wave functions are helpful because we can use them to compute energies and electron densities that are testable by comparison with experiment, even though the wave functions themselves are not. Imaginary numbers are useful because they allow a convenient, formal handling of higher-order polynomials, or of vectorial quantities such as an electrical impedance or a complex dielectric permittivity. In the same sense, ionic activities are useful in describing the response of an individual electrode. But there are rules that specify how to go from, say, imaginary or complex (and therefore nonmeasurable) √ numbers to real ones. The magnitude M = (a2 þ b2) = √ √(a þ bj)(a - bj) of an imaginary number a þ bj (where j = -1) is real, that is, as the geometric mean of the product of that number and its complex conjugate, as is the phase angle φ = arctan(b/a) of a complex number, based on the ratio of its coefficients a and b. In the same way, specific products and ratios of ionic activities are real and measurable, namely, those that constitute the activities of weighable, neutral compounds or some of their combinations, or that derive from the equilibrium responses of measurable cell potentials. But here is the catch: because neither aHþ, nor its negative logarithm is directly measurable by itself, it is fundamentally impossible to demonstrate whether it agrees with a practical quantity such as the readout of a pH meter. McCarty and Vitz have certainly illustrated that pH meters can produce readings beyond the limits of the IUPAC-specified pH but have not shown what such readings might mean. They clearly do not yield -log[Hþ] and, according to IUPAC, not -log aH either. A Question of Usefulness Now that we have a choice of using the pH meter to estimate either pmH or paH, an option that in fact was acknowledged by Buck et al. (4), we consider the question of usefulness. Here is how Bates, the architect of the final IUPAC compromise between the United Kingdom and Japan on one side and the United States on the other, and his co-worker Gary (32) summarized the situation, with [] denoting omitted literature references: “Although the proton activity cannot be evaluated thermodynamically, the pH value of aqueous solutions derives some fundamental meaning from the arbitrary convention on which the numerical values of pH are based []. If one wishes to estimate thermodynamic equilibrium data through measurements of acidity, the individual activity of the proton or hydrogen ion would be much less useful than the hydrogen ion concentration


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In the Classroom

or certain other related functions []. The simplest “acidity function” is therefore probably mH, the hydrogen ion concentration, or its negative logarithm, which has been termed pcH [].” It appears that the activities of single ionic species are not only nonthermodynamic, immeasurable quantities, but are also unnecessary. If it were otherwise, and just one measurement would be known where a single ionic activity would be crucial, then a method could be based on that one measurement to extract that particular activity coefficient, and this could then be used to determine all of them. Lewis and Randall already stated in their 1923 book (33) that “It is evident that if we ascertain, or if we arbitrarily assume, the individual activity coefficient of some one ion, at a given value of the ionic strength, we can then proceed to determine the values for other ions.” Yet not a single such measurement has surfaced since Lewis and Randall, with some trepidation, introduced the concept of an ionic activity (34) more than a century ago, at that time already warning that “from the nature of the ions we are never able to determine the numerical values of their activities”. In a later article with Randall (35), we find this statement for the activity coefficients γ- of anions and γþ of cations: “It remains for us to consider whether these separate values can be experimentally determined. This is a problem of much difficulty, and indeed we are far from any complete solution at the present time.” That statement is still true today. Nonetheless, IUPAC did “arbitrarily assume”, for a given range of ionic strengths, an expression for the individual activity coefficient γCl of chloride ions (10), an assumption that Huckel (6) had already argued against as incompatible (unvereinbar) with the Debye-Huckel theory. A Philosophical Note As indicated above, it is possible with the usual potentiometric measurements (a pH meter, an indicator electrode, and an external reference electrode connected via a salt bridge) to obtain an estimate of the negative logarithm of the hydrogen ion concentration (in the original meaning of the term pH), based on the very same assumptions that are used to obtain the negative logarithm of the assumed hydrogen activity, subject to the same constraints, and with similar accuracy. Both are therefore approximate. However, beyond the greater chemical usefulness of pmH (32), there is another significant difference with paH: the hydrogen ion concentrations so obtained are independently testable (e.g., by UV-vis, fluorescence, and NMR spectroscopies, and sometimes even by titration), whereas the hydrogen ion activity is not so testable, because there is no known way to measure it directly. Using Popper's criterion (36), the hydrogen ion activity as defined by IUPAC is not falsifiable and therefore falls outside the demarcation that separates science from nonscience. That is why the problem lies with IUPAC rather than with the original Sørensen definition. Summary It would be grossly unfair to hold anyone other than the authors of the IUPAC rules responsible for any perceived deficiencies in those rules. A discussion of whether the IUPAC recommendation is good, bad, or indifferent is a quite different


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topic, one that will require more space and will indeed be forthcoming in a separate communication (37). IUPAC certainly did not explain how a practical measurement could ever be based on an immeasurable quantity. However, McCarty and Vitz used examples that went far beyond the explicit, self-imposed IUPAC limits. Did they really believe that the Guggenheim-Bates definition of pγH and a 4.2 M KCl salt bridge were designed to withstand a change from 0.1 M HCl (I = 0.1 mol kg-1) to 0.05 M HCl and 2.5 M CaCl2 (I = 7.55 mol kg-1) without a substantial challenge of the validity of the Guggenheim-Bates expression and/or a change in the liquid junction potential? They also advocated uses in spectroscopy and kinetics that not only were well beyond those IUPAC limits but were also basically unsound. Suffice it to conclude that the interplay between activities and concentrations is much more complicated, subtle, and interesting than the approach advocated by McCarty and Vitz. Literature Cited 1. McCarty, C. G.; Vitz, E. J. Chem. Educ. 2006, 83, 752–757. 2. Sørensen, S. P. L. Biochem. Z. 1909, 21, 131–304. 3. Sørensen, S. P. L.; Linderstrøm-Lang, K. C. R. Trav. Lab. Carlsberg 1924, 15, 1–40. 4. Buck, R. P.; Rondinini, S.; Covington, A. K.; Baucke, F. G. K.; Brett, C. M. A.; Cam~oes, M. F.; Milton, M. J. T.; Mussini, T.; Naumann, R.; Pratt, K. W.; Spitzer, P.; Wilson, G. S. Pure Appl. Chem. 2002, 74, 2169–2200. 5. Hetzer, H. B.; Robinson, R. A.; Bates, R. G. Anal. Chem. 1968, 40, 634–636. 6. Huckel, E. Phys. Z. 1925, 26, 93–147. 7. Hawkins, J. E. J. Am. Chem. Soc. 1932, 54, 4480–4487. 8. Guggenheim, E. A. J. Phys. Chem. 1929, 33, 842–849. 9. Guggenheim, E. A. J. Phys. Chem. 1930, 34, 1758–1766. 10. Bates, R. G.; Guggenheim, E. A. Pure Appl. Chem. 1960, 1, 163– 168. 11. Debye, P.; Huckel, E. Phys. Z. 1923, 24, 185–206. 12. Bates, R. G. Determination of pH, Theory and Practice; Wiley: New York, 1954; p 57. 13. Henderson, P. Z. Phys. Chem. 1907, 59, 118–127. 14. Henderson, P. Z. Phys. Chem. 1908, 63, 325–345. 15. Harper, H. W. J. Phys. Chem. 1985, 89, 1659–1664. 16. Inczedy, J.; Lengyel, T.; Ure, A. M.; Gelenczer, A.; Eulanicki, A. IUPAC Compendium of Analytical Nomenclature, Definitive Rules 1997, 3rd ed.; Blackwell: London, 1997; http://old.iupac.org/ publications/analytical_compendium/ 17. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions: the Measurement and Interpretation of Conductance, Chemical Potential and Diffusion in Solutions of Simple Electrolytes, 2nd ed.; Butterworth: London, 1959; pp 491-509. 18. Hamer, W. J.; Acree, S. F. J. Res. Natl. Bur. Stand. ( U.S.) 1936, 17, 605–613. 19. von Halban, H.; Kortum, G. Z. Phys. Chem. 1934, 170, 351–379. 20. von Halban, H.; Kortum, G. Z. Elektrochem. 1934, 40, 502–507. 21. Hammett, L. P.; Deyrup, A. J. J. Am. Chem. Soc. 1932, 54, 2721– 2739. 22. Hammett, L. P.; Paul, M. A. J. Am. Chem. Soc. 1934, 56, 827–829. 23. Birk, J. P.; Walters, D. L. J. Chem. Educ. 1992, 69, 585–587. 24. Brønsted, J. N. Z. Phys. Chem. 1922, 102, 169–207. 25. Brønsted, J. N. Z. Phys. Chem. 1925, 115, 337–364. 26. Christiansen, J. A. Z. Phys. Chem. 1924, 113, 35–52.


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In the Classroom La Mer, V. K. Chem. Rev. 1932, 10, 179–212. Hart, E. J. Science 1964, 146, 19–25. Brønsted, J. N. J. Am. Chem. Soc. 1922, 44, 938–947. Davies, C. W. Ion Association; Butterworths: London, 1962; p 41. Stokes, R. H.; Robinson, R. A. J. Am. Chem. Soc. 1948, 70, 1870–1878. 32. Bates, R. G.; Gary, R. J. Res. Natl. Bur. Stand. ( U.S.) 1961, 65A, 495–505.

27. 28. 29. 30. 31.

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33. Lewis, G. N.; Randall, M. Thermodynamics and the Free Energy of Chemical Substances; MacGraw-Hill: New York, 1923; p 381. 34. Lewis, G. N. Proc. Am. Acad. Arts Sci. 1907, 43, 259–298. 35. Lewis, G. N.; Randall, M. J. Am. Chem. Soc. 1921, 43, 1112– 1154. 36. Popper, K. R. The Logic of Scientific Discovery; Basic Books: New York, 1959. 37. de Levie, R., to be submitted.


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Potentiometric pH Measurements of Acidity Are Approximations, Some

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