- Email: [email protected]
On isohydric solutions and buffer pH
Journal of
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 582 (2005) 21–27 www.elsevier.com/locate/jelechem
On isohydric solutions and buffer pH Robert de Levie
*
Chemistry Department, Bowdoin College, Brunswick, ME 04011, USA Received 19 July 2004; received in revised form 20 December 2004; accepted 28 December 2004 Available online 5 February 2005 Dedicated to Ron Fawcett on the occasion of his 65th birthday, for his many contributions to electrochemistry
Abstract A general derivation is given for the isohydric theory of Arrhenius, and its consequences for some aspects of buffer pH are considered. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Isohydric solution; Arrhenius theory; Buffer pH
1. Introduction Electrochemistry started as a science at the end of the 18th century through two pivotal contributions: the discovery of the electrolysis of water to hydrogen and oxygen by Peats van Troostwijk and Deiman [1], and the extensive studies on animal electricity of Galvani [2]. Shortly thereafter, the construction of the first practical battery by Volta [3], the electrolytic preparation of the alkali and earth alkali metals by Davy [4], and the subsequent studies by Faraday [5], put electrochemistry on a solid footing. At the end of the 19th century the development of the theory of electrolyte solutions, by Arrhenius [6], Ostwald [7], van Õt Hoff [8], and Nernst [9], provided a renewed interest in electrochemistry, (and led to the development of physical chemistry as a separate branch of chemistry), culminating in the recognition of the central role played by pH [10] in aqueous solution chemistry through the invention of the glass electrode [11,12], the extensive development of pH-measuring electronics
*
Tel.: +1 207 725 3028; fax: +1 207 725 3017. E-mail address: [email protected]
0022-0728/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2004.12.028
[13–24], and their subsequent commercialization since 1935 by, e.g., Beckman and Radiometer. Now in its third century, electrochemistry is again reinventing itself, turning from its emphasis on the structureless liquid/liquid interface and analytical applications to the nature of the solid/liquid interface which, perhaps, holds the answers to some of its great yet unsolved problems, such as metal corrosion. In connection with his work on electrolytic dissociation [6], Arrhenius [25] formulated a theory of isohydric solutions. Most likely this was the first quantitative paper on pH, long before pH was even defined [10]. The experimental data concerned measurements of solution conductance and freezing point depression, and Arrhenius couched his theory of isohydric solutions mostly in terms of those contemporary experimental parameters, although the word ÔisohydricÕ clearly indicated his focus on protons. Butler [26] stated the Arrhenius theorem in modern language as follows: If two solutions of the same pH are mixed, the pH is unchanged regardless of the compositions of the solutions, provided that no polynuclear proton complexes are formed. As I am unaware of a general proof of this property, even more than a century after its original formulation, one will be provided here. Moreover, the resulting buffer strength will
22
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
be described, again a concept not yet developed in 1888, and several related aspects of buffer pH will be discussed. The pH will be defined here in terms of the free proton concentration, i.e., as pH = log [H+]. Incidentally, in physiology the term isohydric principle is sometimes used to mean that the pH of a buffer mixture can be determined by considering a single buffer system, such as bicarbonate/carbonate, because all buffers in mutual equilibrium correspond to the same pH. This much more limited interpretation (which, at any rate, is self-evident for an equilibrium treatment) does not require any justification or comment, and therefore need not be considered here. 2. Preliminaries: the proton function It is convenient to start from the general statement of the pH of an aqueous solution in terms of its proton function [27], which is simply what one obtains by writing the proton condition [28,29] as a function, by bringing all concentration terms over to the side containing the term [H+], and then deleting its other side. For example, the proton condition for a single monoprotic strong acid HA of C M concentration ½Hþ ¼ ½A þ ½OH ¼ C þ ½OH
ð1Þ
listing proton gainers (left) and losers (right) can be rewritten as þ
½H C ½OH ¼ 0
where the integer coefficients n = 1, 2, and 3 indicate the number of protons lost by H3A in generating the species H3-nAn. For a solute mixture containing C1 M strong acid, C2 M weak monoprotic acid HA, and C3 M triprotic acid H3A 0 we find H
where the proton excess D (a quantity first introduced by Ricci [30] as D) is given by D ½Hþ ½OH
ð4Þ
and where, obviously, the pH can be found by requiring that H
C ¼ 0:
ð5Þ
For an aqueous C M solution of a single weak monoprotic acid HA we have instead the proton condition ½Hþ ¼ ½A þ ½OH ¼
K aC þ ½OH ½Hþ þ K a
ð6Þ
or K aC : ð7Þ ½Hþ þ K a For a C M triprotic acid H3A the proton function is
H
C D ½A ¼ D
H
C D ½H2 A 2½HA2 3½A3
3
2
ð9Þ and so on. The above approach is not limited to acids, but applies equally well to bases and salts. For example, a C2 M aqueous solution of sodium carbonate has the proton function H
C D þ 2½H2 CO3 þ ½HCO 3 ¼Dþ
ð2½Hþ 2 þ ½Hþ K a1 ÞC 2 2
½Hþ þ ½Hþ K a1 þ K a1 K a2
;
ð10Þ
where Ka1 and Ka2 now represent the acid dissociation constants of carbonic acid, H2CO3. Likewise one finds for a C1 M aqueous solution of sodium bicarbonate H
C D þ ½H2 CO3 ½CO2 3 ð½Hþ 2 K a1 K a2 ÞC 1 2
½Hþ þ ½Hþ K a1 þ K a1 K a2
;
ð11Þ
while a mixture containing C1 M NaHCO3 plus C2 M Na2CO3 has the proton function 2
H
C ¼Dþ
½Hþ ðC 1 þ 2C 2 Þ þ ½Hþ K a1 C 2 K a1 K a2 C 1 2
½Hþ þ ½Hþ K a1 þ K a1 K a2
:
ð12Þ H
In all the above examples, the proton function C is the algebraic sum of the proton function of the solvent water (equal to D, the proton excess) plus the contributions FiCi of the individual solutes i, where Fi, symbolizes a specific function of [H+] and the appropriate equilibrium constants. The quantity Fi is positive when, upon dissolution of compound i in water, protons are gained by that compound i, and negative when i loses protons (e.g., either releases them into the solution or loses them to other solution species). We therefore write the proton function in its general form as X H C ¼Dþ F iCi: ð13Þ i
2
¼D
3
½Hþ þ ½Hþ K a1 þ ½Hþ K a1 K a2 þ K a1 K a2 K a3
¼Dþ ð3Þ
3½A0
ð½Hþ K a1 þ 2½Hþ K a1 K a2 þ 3K a1 K a2 K a3 ÞC 3
ð2Þ
C ½Hþ C ½OH ¼ D C;
2
2
and therefore corresponds to the proton function H
C D C 1 ½A ½H2 A0 2½HA0 K aC2 ¼ D C1 þ ½H þ K a
ð½Hþ K a1 þ 2½Hþ K a1 K a2 þ 3K a1 K a2 K a3 ÞC ½Hþ 3 þ ½Hþ 2 K a1 þ ½Hþ K a1 K a2 þ K a1 K a2 K a3
;
ð8Þ
For a single strong monoprotic acid we have Fi = 1, see Eq. (3), for a single weak monoprotic acid Fi = Ka/ ([H+] + Ka), see Eq. (7), for a single strong monoprotic base Fi = +1, for a single weak monoprotic base
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
Fi = + [H+]/([H+] + Ka), for an acid salt such as bicarbonate Fi = ([H+]2 Ka1Ka2)/([H+]2 + [H+]Ka1 + Ka1Ka2), see Eq. (11), etc. As already emphasized, for mixtures the proton function HC is the algebraic sum of the proton functions of the solvent and its solutes. This is so because the proton condition is additive in the concentrations of the chemical species gaining or losing protons, since it can always be derived from the conditions of conservation of mass and charge. It is this additivity of the individual solute contributions FiCi to H C that forms the basis of our treatment of the Arrhenius theory of isohydric solutions.
23
ing up that mixture. The same reasoning can be extended to mixtures of arbitrary complexity.
4. The buffer strength of a mixture of isohydric solutions The buffer strength B of a solution is given by [27] B¼
H d HC þ d C þ ¼ ½H d ln½H d½Hþ
ð20Þ
so that B1 ¼ ½Hþ þ ½OH þ ½Hþ
X
Ci
dF i ; d½Hþ
ð21Þ
Cj
dF j d½Hþ
ð22Þ
i
3. The pH of a mixture of isohydric solutions Now consider two aqueous solutions, labeled 1 and 2, with solutes i and j and proton functions HC1 and HC2, respectively, i.e., X H C 1 ¼ D1 þ F i C i ¼ 0; ð14Þ
B2 ¼ ½Hþ þ ½OH þ ½Hþ
j
and Bmix ¼ ½Hþ þ ½OH þ
i H
C 2 ¼ D2 þ
X
F j C j ¼ 0:
þ
ð15Þ
j
We now mix a volume Va of solution 1 with volume Vb of solution 2. In that case, assuming that there is no volume expansion or contraction upon mixing, and no new types of chemical reactions between the components i and j (such as resulting in polynuclear complexation, gas evolution, or precipitation), we likewise have for the mixture Va X Vb X H C mix ¼ Dmix þ F iCi þ F j C j ¼ 0: VaþVb i VaþVb j ð16Þ Multiplying Eq. (14) by Va/(Va + Vb), Eq. (15) by Vb/(Va + Vb), and adding them, yields V a D1 þ V b D2 Va X Vb X þ F iCi þ F j C j ¼ 0: VaþVb VaþVb i VaþVb j ð17Þ Comparison of Eqs. (16) and (17) yields Dmix ¼
V a D1 þ V b D2 : VaþVb
ð18Þ
When the original solutions 1 and 2 are isohydric, i.e., have the same pH, we write D1 = D2 = D, and therefore find Dmix ¼
V a D1 þ V b D2 V a D þ V b D V a þ V b ¼ ¼ D¼D VaþVb VaþVb VaþVb ð19Þ
so that the pH of the mixture of two isohydric solutions will indeed be the same as that of the two solutions mak-
X
¼
V a ½Hþ X dF i Ci VaþVb i d½Hþ
V b ½Hþ X dF j Cj VaþVb j d½Hþ
V a B1 þ V b B2 VaþVb
ð23Þ
or, in words, the buffer strength of a mixture of two isohydric solutions is the volume-weighted average of the buffer strengths of those two isohydric solutions. The same applies to the buffer index [31] as defined originally by van Slyke, b = B ln(10) 2.3B, the only difference being that we have taken seriously the suggestion of Henderson [32] to delete the factor ln(10) from the definition. The above approach can readily be extended to any arbitrary mixture of isohydric solutions. As before, the validity of the result is constrained by the absence of volume expansion or contraction upon mixing, and the absence of chemical reactions between the components i and j insofar as those reactions were not already present in the solutions before mixing. Mixtures of isohydric solutions therefore make it possible to vary the buffer strength of a solution at constant pH, e.g., by mixing the buffer (depending on its pH) with a dilute strong acid or base of that same pH. The clear implication is that simply diluting that buffer with water does not maintain a constant pH. This is indeed the case, and is most readily seen from Fig. 1, which plots the buffer pH for equimolar conjugate acid–base mixtures as a function of C = Ca + Cb at various pKa-values. It also shows what can be done by diluting a buffer with an isohydric, dilute, strong monoprotic acid or base. When constant proton activity (rather than proton concentration) is desired, the isohydric solutions can be prepared with excess inert strong electrolyte, so that
24
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
5. Numerical solutions In terms of theory, there is little to add to what has been stated above. However, calculating buffer pH is typically a numerical problem, and a few comments may be relevant in this respect, concerning both exact and approximate results. To emphasize the principles involved, we will only consider the case of a simple buffer mixture containing Ca M of a weak acid and Cb M of its conjugate base. For such a buffer mixture one can compute the pH based on the proton function H
C ¼Dþ
½Hþ C b K a C a ¼0 ½Hþ þ K a
ð24Þ
or from the so-called Charlot equation [33,34] Fig. 1. The pH of an equimolar aqueous buffer containing C/2 M acid and C/2 M of its conjugate base, as a function of C, for values of pKa ranging from pKa = 1 (bottom curve) to pKa = 13 (top curve). Also shown (as gray lines) is the pH that can be obtained by diluting the buffer with a strong monoprotic acid or base of pH = pKa and concentration C, thereby extending the range of constant pH by at least an order of magnitude in C. The lowest concentrations that can be reached in this way are emphasized by solid gray points.
the proton activity will also be kept constant, at least to the extent that activity coefficients are only a function of ionic strength.
½Hþ
K a ðC a DÞ ¼ 0: ðC b þ DÞ
ð25Þ
These two expressions are mathematically equivalent, in the sense that one can readily be derived from the other and vice versa, but computationally they are not necessarily equivalent. When one uses, e.g., a nonlinear least squares method to minimize the logarithm of the absolute value of the middle part of Eq. (24), or the left-hand side of (25), use of the proton function with its broad minimum provides the more robust approach, as illustrated in Fig. 2(a),(b).
Fig. 2. The logarithm of the absolute value of (a) the proton function equation (24) and (b) the Charlot equation (25), as a function of pH, for a 0.75 lM ammonium chloride/0.25 lM ammonia buffer. (c) Detail of Figure 2b for 7.4 6 pH 6 7.5. While log |HC| exhibits a wide minimum, the corresponding minimum in the absolute value of the logarithm of Eq. (25) is very narrow at pH > 7.434, and therefore can be missed easily.
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
25
Fig. 3. (a) The proton function HC and (b) the Charlot equation (25) in the pH range close to the zero crossing at pH 7.434 identified by a filled circle, for the same system as in Fig. 2. The Charlot equation also exhibits a discontinuity (dashed line) at pH 7.455 which may complicate the numerical analysis.
The reason why Eq. (25) is less suitable becomes evident in Figs. 2(c) and 3(b), which shows these functions around their zero-crossing at pH 7.434. The proton function can always be written as a monotonically decreasing function of [H+], and consequently has only one zero crossing. Eq. (25) has also only one zero crossing, but in addition it has a discontinuity, where it goes through infinity at pH 7.455. In this case (and, it appears, in general) Eq. (24) is the better computational choice for either bisection or nonlinear least squares fitting [35,36].
6. Approximations Before computers became ready available, approximations used to dominate pH calculations. Even though they are no longer necessary, it may be useful to consider them here. In this context we can distinguish between two types of approximations: those based on chemical plausibility, and those based on mathematical convenience. Both will be illustrated below in connection with Eq. (25). In the first category one finds the approximation D = [H+] [OH] [H+] for a buffer with an acidic pH, and D = [H+] [OH] [OH] = Kw/[H+] for a buffer in the basic pH range. Consequently, for buffer action at pH < 7, Eq. (25) leads to 2
½Hþ þ ðC b þ K a Þ½Hþ K a C a 0 or ½Hþ
ðC b þ K a Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC b þ K a Þ2 þ 4C a K a 2
ð26Þ
ð27Þ
and, for buffer action at pH > 7, 2
C b ½Hþ ðC a K a þ K w Þ½Hþ K a K w 0 or
ð28Þ
½Hþ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðC a K a þ K w Þ þ ðC a K a þ K w Þ þ 4C b K a K w 2C b
: ð29Þ
These relations only fail when Ca and/or Cb are smaller than about 106 M, at which point buffer action has lost most of its usefulness anyway. The ranges of validity of Eqs. (27) and (29) are well-defined and explicitly stated, so that there is no ambiguity in their use. In fact, a reasonable system of approximations applicable to a single conjugate acid–base pair under all possible experimental conditions can be formulated as follows: for pKa < 7, [H+] is given by Eq. (27) as long as that relation yields a value in excess of 107 M, otherwise use [H+] = 107 M. Likewise, for pKa > 7, [H+] is given by Eq. (29) as long as that equation yields a value smaller than 107 M, otherwise use [H+] = 107 M. If one wants to compute buffer pH based on approximations, these are the relations to use. In the second category we find the so-called Henderson approximation [37], which simply deletes the terms D from Eq. (25), and then leads to ½Hþ
K aCa ; Cb
ð30Þ
i.e., to a buffer pH that is independent of the concentrations of the buffer components. It is this approximation that, when used inappropriately (i.e., when extended to dilute buffer mixtures, where the terms in D may no longer be negligible with respect to Ca and/or Cb) may lead to the mistaken conclusion that a buffer can simply be diluted with solvent in order to lower its strength at constant pH. As Arrhenius already knew, such a conclusion would be incorrect, because only mixing with an isohydric solution can keep the pH constant. (It is also this approximation that is sometimes confused [38] with the mass action law of Guldberg and Waage.)
26
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
In Fig. 4 the exact expressions (24) or (25) are compared with the approximations Eqs. (27) or (29), and with the Henderson approximation equation (30). It is
found that Eq. (27) is perfectly satisfactory for acidic buffers down to C 1 lM, and the same applies to Eq. (29) for basic buffers, whereas the Henderson approximation fails at much higher C-values, especially at a more extreme buffer pH.
7. Discussion The concept of isohydric solutions was a very early application of ArrheniusÕ realization that the ions introduced by Faraday do not exist merely at electrodes during electrolysis, but are also present in bulk solutions. In retrospect it is amazing that Arrhenius could base his concept of isohydric solutions on measurements of conductance and freezing point depression, which are not nearly as proton-specific as modern pH measurements. The proton function (like its predecessor, the proton condition) is an additive function of the various solute concentrations in a solution. Consequently, it is a convenient measure to describe proton-related solution properties, such as its pH and buffer strength, as well as its acid–base titration behavior. This is plainly illustrated here, where the Arrhenius theory of isohydric solutions is verified in a few lines, from Eq. (14) through Eq. (19). (Strictly speaking we could have used the proton condition in the above examples. The advantage of using the proton function becomes most obvious when one deals with titrations.) For complexation and redox equilibria, corresponding ligand and electron functions can be used. As can be seen in Fig. 1, the practical usefulness of isohydric solutions is somewhat limited, because one clearly cannot maintain an aqueous solution at, e.g., pH 2 with less than 0.01 M of a strong monoprotic acid. In strongly acidic and basic solutions, the available range of buffer strengths is therefore rather small. Still, it allows one to extend the range of buffer action at constant pH by at least an order of magnitude in C beyond what can be achieved by simply diluting an acid–base buffer mixture with water.
References
Fig. 4. (a) The pH of a 3:1 mixture (Ca = 0.75C; Cb = 0.25C) of ammonium chloride and ammonia (pKa = 9.24), and of a 3:1 mixture of the acid and its sodium salt of (b) acetic acid (pKa = 4.76), (c) iodoacetic acid (pKa = 3.18), and (d) dichloroacetic acid (pKa = 1.3). Solid points and heavy line: exact results according to Eqs. (24) or (25). Open circles connected by a thin line: approximate results according to Eq. (29) for (a), and according to Eq. (27) for (b)–(d). Open triangles connected by a thin line: results according to the Henderson approximation equation (30).
[1] A. Peats van Troostwijk, J.R. Deiman, J. Phys. 2 (1790) 130. [2] L. Galvani, De Viribus Electricitatus in Motu Musculari Commentarius, Bologna, 1791. [3] A. Volta, Phil. Trans. 90 (1800) 403. [4] H. Davy, Phil. Trans. 98 (1808) 1, 333. [5] M. Faraday, Phil. Trans. 123 (1833) 507, 675. [6] S. Arrhenius, Z. Physik. Chem. 1 (1887) 631. [7] J.H. van Õt Hoff, Z. Physik. Chem. 1 (1887) 481. [8] W. Ostwald, Z. Physik. Chem. 2 (1888) 36. [9] W. Nernst, Z. Physik. Chem. 4 (1889) 129. [10] S.P.L. Sørensen, Biochem. Z. 21 (1909) 131, 201. [11] M. Cremer, Z. Biol. 47 (1906) 562. [12] F. Haber, Z. Klemensiewicz, Z. Physik. Chem. 67 (1909) 385.
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27 [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
K.H. Goode, J. Am. Chem. Soc. 44 (1922) 26. D.F. Calhane, R.E. Gushing, J. Ind. Eng. Chem. 15 (1923) 1118. W.D. Treadwell, Helv. Chim. Acta 8 (1925) 89. K.H. Goode, J. Am. Chem. Soc. 47 (1925) 2483. H. Bienfait, Rec. Trav. Chim. Pays-Bas 45 (1926) 166. E. Linde, Svensk Kem. Tids. 39 (1927) 285. J.W. Williams, T.A. Whitenack, J. Phys. Chem. 31 (1927) 519. C. Morton, Trans. Faraday Soc. 24 (1928) 14. L.W. Elder Jr., W.H. Wright, Proc. Natl. Acad. Sci. USA 14 (1928) 936. W.C. Stadie, H. OÕBrien, E.P. Laug, J. Biol. Chem. (1931) 243. S.B. Ellis, S.J. Kiehl, Rev. Sci. Instrum. 4 (1933) 131. L.D. Goodhue, Iowa State Colloid J. Sci. 10 (1935) 7. S. Arrhenius, Z. Physik. Chem. 2 (1888) 284. J.N. Butler, Ionic Equilibrium, a Mathematical Approach, Addison-Wesley, Reading, MA, 1964, p. 141. R. de Levie, Chem. Educator 6 (2001) 272.
27
[28] G. Ha¨gg, Kemisk reaktionsla¨ra, Geber, Uppsala, 1940; Die theoretischen Grundlagen der analytischen Chemie, Birkha¨ser, Basel, 1950, p. 58. [29] L.G. Sille´n, P.W. Lange, C.O. Gabrielson, Problems in Physical Chemistry, Prentice-Hall, New Jersey, 1952, p. 201. [30] J.E. Ricci, Hydrogen Ion Concentration, Princeton University Press, 1952, p. 2,4. [31] D.D. van Slyke, J. Biol. Chem. 52 (1922) 525. [32] L.J. Henderson, J. Biol. Chem. 52 (1922) 565. [33] G. Ha¨gg, Kemisk reaktionsla¨ra, Geber, Uppsala, 1940; Die theoretischen Grundlagen der analytischen Chemie, Birkha¨ser, Basel, 1950, p. 84 Eq. (3). [34] G. Charlot, Anal. Chim. Acta 1 (1947) 59. [35] K. Levenberg, Quart. Appl. Math. 2 (1944) 164. [36] D.W. Marquardt, J. Soc. Ind. Appl. Math. 11 (1963) 431. [37] L.J. Henderson, Am. J. Physiol. 21 (1908) 173, 427. [38] R. de Levie, Chem. Educator 7 (2002) 132.
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 582 (2005) 21–27 www.elsevier.com/locate/jelechem
On isohydric solutions and buffer pH Robert de Levie
*
Chemistry Department, Bowdoin College, Brunswick, ME 04011, USA Received 19 July 2004; received in revised form 20 December 2004; accepted 28 December 2004 Available online 5 February 2005 Dedicated to Ron Fawcett on the occasion of his 65th birthday, for his many contributions to electrochemistry
Abstract A general derivation is given for the isohydric theory of Arrhenius, and its consequences for some aspects of buffer pH are considered. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Isohydric solution; Arrhenius theory; Buffer pH
1. Introduction Electrochemistry started as a science at the end of the 18th century through two pivotal contributions: the discovery of the electrolysis of water to hydrogen and oxygen by Peats van Troostwijk and Deiman [1], and the extensive studies on animal electricity of Galvani [2]. Shortly thereafter, the construction of the first practical battery by Volta [3], the electrolytic preparation of the alkali and earth alkali metals by Davy [4], and the subsequent studies by Faraday [5], put electrochemistry on a solid footing. At the end of the 19th century the development of the theory of electrolyte solutions, by Arrhenius [6], Ostwald [7], van Õt Hoff [8], and Nernst [9], provided a renewed interest in electrochemistry, (and led to the development of physical chemistry as a separate branch of chemistry), culminating in the recognition of the central role played by pH [10] in aqueous solution chemistry through the invention of the glass electrode [11,12], the extensive development of pH-measuring electronics
*
Tel.: +1 207 725 3028; fax: +1 207 725 3017. E-mail address: [email protected]
0022-0728/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2004.12.028
[13–24], and their subsequent commercialization since 1935 by, e.g., Beckman and Radiometer. Now in its third century, electrochemistry is again reinventing itself, turning from its emphasis on the structureless liquid/liquid interface and analytical applications to the nature of the solid/liquid interface which, perhaps, holds the answers to some of its great yet unsolved problems, such as metal corrosion. In connection with his work on electrolytic dissociation [6], Arrhenius [25] formulated a theory of isohydric solutions. Most likely this was the first quantitative paper on pH, long before pH was even defined [10]. The experimental data concerned measurements of solution conductance and freezing point depression, and Arrhenius couched his theory of isohydric solutions mostly in terms of those contemporary experimental parameters, although the word ÔisohydricÕ clearly indicated his focus on protons. Butler [26] stated the Arrhenius theorem in modern language as follows: If two solutions of the same pH are mixed, the pH is unchanged regardless of the compositions of the solutions, provided that no polynuclear proton complexes are formed. As I am unaware of a general proof of this property, even more than a century after its original formulation, one will be provided here. Moreover, the resulting buffer strength will
22
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
be described, again a concept not yet developed in 1888, and several related aspects of buffer pH will be discussed. The pH will be defined here in terms of the free proton concentration, i.e., as pH = log [H+]. Incidentally, in physiology the term isohydric principle is sometimes used to mean that the pH of a buffer mixture can be determined by considering a single buffer system, such as bicarbonate/carbonate, because all buffers in mutual equilibrium correspond to the same pH. This much more limited interpretation (which, at any rate, is self-evident for an equilibrium treatment) does not require any justification or comment, and therefore need not be considered here. 2. Preliminaries: the proton function It is convenient to start from the general statement of the pH of an aqueous solution in terms of its proton function [27], which is simply what one obtains by writing the proton condition [28,29] as a function, by bringing all concentration terms over to the side containing the term [H+], and then deleting its other side. For example, the proton condition for a single monoprotic strong acid HA of C M concentration ½Hþ ¼ ½A þ ½OH ¼ C þ ½OH
ð1Þ
listing proton gainers (left) and losers (right) can be rewritten as þ
½H C ½OH ¼ 0
where the integer coefficients n = 1, 2, and 3 indicate the number of protons lost by H3A in generating the species H3-nAn. For a solute mixture containing C1 M strong acid, C2 M weak monoprotic acid HA, and C3 M triprotic acid H3A 0 we find H
where the proton excess D (a quantity first introduced by Ricci [30] as D) is given by D ½Hþ ½OH
ð4Þ
and where, obviously, the pH can be found by requiring that H
C ¼ 0:
ð5Þ
For an aqueous C M solution of a single weak monoprotic acid HA we have instead the proton condition ½Hþ ¼ ½A þ ½OH ¼
K aC þ ½OH ½Hþ þ K a
ð6Þ
or K aC : ð7Þ ½Hþ þ K a For a C M triprotic acid H3A the proton function is
H
C D ½A ¼ D
H
C D ½H2 A 2½HA2 3½A3
3
2
ð9Þ and so on. The above approach is not limited to acids, but applies equally well to bases and salts. For example, a C2 M aqueous solution of sodium carbonate has the proton function H
C D þ 2½H2 CO3 þ ½HCO 3 ¼Dþ
ð2½Hþ 2 þ ½Hþ K a1 ÞC 2 2
½Hþ þ ½Hþ K a1 þ K a1 K a2
;
ð10Þ
where Ka1 and Ka2 now represent the acid dissociation constants of carbonic acid, H2CO3. Likewise one finds for a C1 M aqueous solution of sodium bicarbonate H
C D þ ½H2 CO3 ½CO2 3 ð½Hþ 2 K a1 K a2 ÞC 1 2
½Hþ þ ½Hþ K a1 þ K a1 K a2
;
ð11Þ
while a mixture containing C1 M NaHCO3 plus C2 M Na2CO3 has the proton function 2
H
C ¼Dþ
½Hþ ðC 1 þ 2C 2 Þ þ ½Hþ K a1 C 2 K a1 K a2 C 1 2
½Hþ þ ½Hþ K a1 þ K a1 K a2
:
ð12Þ H
In all the above examples, the proton function C is the algebraic sum of the proton function of the solvent water (equal to D, the proton excess) plus the contributions FiCi of the individual solutes i, where Fi, symbolizes a specific function of [H+] and the appropriate equilibrium constants. The quantity Fi is positive when, upon dissolution of compound i in water, protons are gained by that compound i, and negative when i loses protons (e.g., either releases them into the solution or loses them to other solution species). We therefore write the proton function in its general form as X H C ¼Dþ F iCi: ð13Þ i
2
¼D
3
½Hþ þ ½Hþ K a1 þ ½Hþ K a1 K a2 þ K a1 K a2 K a3
¼Dþ ð3Þ
3½A0
ð½Hþ K a1 þ 2½Hþ K a1 K a2 þ 3K a1 K a2 K a3 ÞC 3
ð2Þ
C ½Hþ C ½OH ¼ D C;
2
2
and therefore corresponds to the proton function H
C D C 1 ½A ½H2 A0 2½HA0 K aC2 ¼ D C1 þ ½H þ K a
ð½Hþ K a1 þ 2½Hþ K a1 K a2 þ 3K a1 K a2 K a3 ÞC ½Hþ 3 þ ½Hþ 2 K a1 þ ½Hþ K a1 K a2 þ K a1 K a2 K a3
;
ð8Þ
For a single strong monoprotic acid we have Fi = 1, see Eq. (3), for a single weak monoprotic acid Fi = Ka/ ([H+] + Ka), see Eq. (7), for a single strong monoprotic base Fi = +1, for a single weak monoprotic base
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
Fi = + [H+]/([H+] + Ka), for an acid salt such as bicarbonate Fi = ([H+]2 Ka1Ka2)/([H+]2 + [H+]Ka1 + Ka1Ka2), see Eq. (11), etc. As already emphasized, for mixtures the proton function HC is the algebraic sum of the proton functions of the solvent and its solutes. This is so because the proton condition is additive in the concentrations of the chemical species gaining or losing protons, since it can always be derived from the conditions of conservation of mass and charge. It is this additivity of the individual solute contributions FiCi to H C that forms the basis of our treatment of the Arrhenius theory of isohydric solutions.
23
ing up that mixture. The same reasoning can be extended to mixtures of arbitrary complexity.
4. The buffer strength of a mixture of isohydric solutions The buffer strength B of a solution is given by [27] B¼
H d HC þ d C þ ¼ ½H d ln½H d½Hþ
ð20Þ
so that B1 ¼ ½Hþ þ ½OH þ ½Hþ
X
Ci
dF i ; d½Hþ
ð21Þ
Cj
dF j d½Hþ
ð22Þ
i
3. The pH of a mixture of isohydric solutions Now consider two aqueous solutions, labeled 1 and 2, with solutes i and j and proton functions HC1 and HC2, respectively, i.e., X H C 1 ¼ D1 þ F i C i ¼ 0; ð14Þ
B2 ¼ ½Hþ þ ½OH þ ½Hþ
j
and Bmix ¼ ½Hþ þ ½OH þ
i H
C 2 ¼ D2 þ
X
F j C j ¼ 0:
þ
ð15Þ
j
We now mix a volume Va of solution 1 with volume Vb of solution 2. In that case, assuming that there is no volume expansion or contraction upon mixing, and no new types of chemical reactions between the components i and j (such as resulting in polynuclear complexation, gas evolution, or precipitation), we likewise have for the mixture Va X Vb X H C mix ¼ Dmix þ F iCi þ F j C j ¼ 0: VaþVb i VaþVb j ð16Þ Multiplying Eq. (14) by Va/(Va + Vb), Eq. (15) by Vb/(Va + Vb), and adding them, yields V a D1 þ V b D2 Va X Vb X þ F iCi þ F j C j ¼ 0: VaþVb VaþVb i VaþVb j ð17Þ Comparison of Eqs. (16) and (17) yields Dmix ¼
V a D1 þ V b D2 : VaþVb
ð18Þ
When the original solutions 1 and 2 are isohydric, i.e., have the same pH, we write D1 = D2 = D, and therefore find Dmix ¼
V a D1 þ V b D2 V a D þ V b D V a þ V b ¼ ¼ D¼D VaþVb VaþVb VaþVb ð19Þ
so that the pH of the mixture of two isohydric solutions will indeed be the same as that of the two solutions mak-
X
¼
V a ½Hþ X dF i Ci VaþVb i d½Hþ
V b ½Hþ X dF j Cj VaþVb j d½Hþ
V a B1 þ V b B2 VaþVb
ð23Þ
or, in words, the buffer strength of a mixture of two isohydric solutions is the volume-weighted average of the buffer strengths of those two isohydric solutions. The same applies to the buffer index [31] as defined originally by van Slyke, b = B ln(10) 2.3B, the only difference being that we have taken seriously the suggestion of Henderson [32] to delete the factor ln(10) from the definition. The above approach can readily be extended to any arbitrary mixture of isohydric solutions. As before, the validity of the result is constrained by the absence of volume expansion or contraction upon mixing, and the absence of chemical reactions between the components i and j insofar as those reactions were not already present in the solutions before mixing. Mixtures of isohydric solutions therefore make it possible to vary the buffer strength of a solution at constant pH, e.g., by mixing the buffer (depending on its pH) with a dilute strong acid or base of that same pH. The clear implication is that simply diluting that buffer with water does not maintain a constant pH. This is indeed the case, and is most readily seen from Fig. 1, which plots the buffer pH for equimolar conjugate acid–base mixtures as a function of C = Ca + Cb at various pKa-values. It also shows what can be done by diluting a buffer with an isohydric, dilute, strong monoprotic acid or base. When constant proton activity (rather than proton concentration) is desired, the isohydric solutions can be prepared with excess inert strong electrolyte, so that
24
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
5. Numerical solutions In terms of theory, there is little to add to what has been stated above. However, calculating buffer pH is typically a numerical problem, and a few comments may be relevant in this respect, concerning both exact and approximate results. To emphasize the principles involved, we will only consider the case of a simple buffer mixture containing Ca M of a weak acid and Cb M of its conjugate base. For such a buffer mixture one can compute the pH based on the proton function H
C ¼Dþ
½Hþ C b K a C a ¼0 ½Hþ þ K a
ð24Þ
or from the so-called Charlot equation [33,34] Fig. 1. The pH of an equimolar aqueous buffer containing C/2 M acid and C/2 M of its conjugate base, as a function of C, for values of pKa ranging from pKa = 1 (bottom curve) to pKa = 13 (top curve). Also shown (as gray lines) is the pH that can be obtained by diluting the buffer with a strong monoprotic acid or base of pH = pKa and concentration C, thereby extending the range of constant pH by at least an order of magnitude in C. The lowest concentrations that can be reached in this way are emphasized by solid gray points.
the proton activity will also be kept constant, at least to the extent that activity coefficients are only a function of ionic strength.
½Hþ
K a ðC a DÞ ¼ 0: ðC b þ DÞ
ð25Þ
These two expressions are mathematically equivalent, in the sense that one can readily be derived from the other and vice versa, but computationally they are not necessarily equivalent. When one uses, e.g., a nonlinear least squares method to minimize the logarithm of the absolute value of the middle part of Eq. (24), or the left-hand side of (25), use of the proton function with its broad minimum provides the more robust approach, as illustrated in Fig. 2(a),(b).
Fig. 2. The logarithm of the absolute value of (a) the proton function equation (24) and (b) the Charlot equation (25), as a function of pH, for a 0.75 lM ammonium chloride/0.25 lM ammonia buffer. (c) Detail of Figure 2b for 7.4 6 pH 6 7.5. While log |HC| exhibits a wide minimum, the corresponding minimum in the absolute value of the logarithm of Eq. (25) is very narrow at pH > 7.434, and therefore can be missed easily.
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
25
Fig. 3. (a) The proton function HC and (b) the Charlot equation (25) in the pH range close to the zero crossing at pH 7.434 identified by a filled circle, for the same system as in Fig. 2. The Charlot equation also exhibits a discontinuity (dashed line) at pH 7.455 which may complicate the numerical analysis.
The reason why Eq. (25) is less suitable becomes evident in Figs. 2(c) and 3(b), which shows these functions around their zero-crossing at pH 7.434. The proton function can always be written as a monotonically decreasing function of [H+], and consequently has only one zero crossing. Eq. (25) has also only one zero crossing, but in addition it has a discontinuity, where it goes through infinity at pH 7.455. In this case (and, it appears, in general) Eq. (24) is the better computational choice for either bisection or nonlinear least squares fitting [35,36].
6. Approximations Before computers became ready available, approximations used to dominate pH calculations. Even though they are no longer necessary, it may be useful to consider them here. In this context we can distinguish between two types of approximations: those based on chemical plausibility, and those based on mathematical convenience. Both will be illustrated below in connection with Eq. (25). In the first category one finds the approximation D = [H+] [OH] [H+] for a buffer with an acidic pH, and D = [H+] [OH] [OH] = Kw/[H+] for a buffer in the basic pH range. Consequently, for buffer action at pH < 7, Eq. (25) leads to 2
½Hþ þ ðC b þ K a Þ½Hþ K a C a 0 or ½Hþ
ðC b þ K a Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC b þ K a Þ2 þ 4C a K a 2
ð26Þ
ð27Þ
and, for buffer action at pH > 7, 2
C b ½Hþ ðC a K a þ K w Þ½Hþ K a K w 0 or
ð28Þ
½Hþ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðC a K a þ K w Þ þ ðC a K a þ K w Þ þ 4C b K a K w 2C b
: ð29Þ
These relations only fail when Ca and/or Cb are smaller than about 106 M, at which point buffer action has lost most of its usefulness anyway. The ranges of validity of Eqs. (27) and (29) are well-defined and explicitly stated, so that there is no ambiguity in their use. In fact, a reasonable system of approximations applicable to a single conjugate acid–base pair under all possible experimental conditions can be formulated as follows: for pKa < 7, [H+] is given by Eq. (27) as long as that relation yields a value in excess of 107 M, otherwise use [H+] = 107 M. Likewise, for pKa > 7, [H+] is given by Eq. (29) as long as that equation yields a value smaller than 107 M, otherwise use [H+] = 107 M. If one wants to compute buffer pH based on approximations, these are the relations to use. In the second category we find the so-called Henderson approximation [37], which simply deletes the terms D from Eq. (25), and then leads to ½Hþ
K aCa ; Cb
ð30Þ
i.e., to a buffer pH that is independent of the concentrations of the buffer components. It is this approximation that, when used inappropriately (i.e., when extended to dilute buffer mixtures, where the terms in D may no longer be negligible with respect to Ca and/or Cb) may lead to the mistaken conclusion that a buffer can simply be diluted with solvent in order to lower its strength at constant pH. As Arrhenius already knew, such a conclusion would be incorrect, because only mixing with an isohydric solution can keep the pH constant. (It is also this approximation that is sometimes confused [38] with the mass action law of Guldberg and Waage.)
26
R. de Levie / Journal of Electroanalytical Chemistry 582 (2005) 21–27
In Fig. 4 the exact expressions (24) or (25) are compared with the approximations Eqs. (27) or (29), and with the Henderson approximation equation (30). It is
found that Eq. (27) is perfectly satisfactory for acidic buffers down to C 1 lM, and the same applies to Eq. (29) for basic buffers, whereas the Henderson approximation fails at much higher C-values, especially at a more extreme buffer pH.
7. Discussion The concept of isohydric solutions was a very early application of ArrheniusÕ realization that the ions introduced by Faraday do not exist merely at electrodes during electrolysis, but are also present in bulk solutions. In retrospect it is amazing that Arrhenius could base his concept of isohydric solutions on measurements of conductance and freezing point depression, which are not nearly as proton-specific as modern pH measurements. The proton function (like its predecessor, the proton condition) is an additive function of the various solute concentrations in a solution. Consequently, it is a convenient measure to describe proton-related solution properties, such as its pH and buffer strength, as well as its acid–base titration behavior. This is plainly illustrated here, where the Arrhenius theory of isohydric solutions is verified in a few lines, from Eq. (14) through Eq. (19). (Strictly speaking we could have used the proton condition in the above examples. The advantage of using the proton function becomes most obvious when one deals with titrations.) For complexation and redox equilibria, corresponding ligand and electron functions can be used. As can be seen in Fig. 1, the practical usefulness of isohydric solutions is somewhat limited, because one clearly cannot maintain an aqueous solution at, e.g., pH 2 with less than 0.01 M of a strong monoprotic acid. In strongly acidic and basic solutions, the available range of buffer strengths is therefore rather small. Still, it allows one to extend the range of buffer action at constant pH by at least an order of magnitude in C beyond what can be achieved by simply diluting an acid–base buffer mixture with water.
References
Fig. 4. (a) The pH of a 3:1 mixture (Ca = 0.75C; Cb = 0.25C) of ammonium chloride and ammonia (pKa = 9.24), and of a 3:1 mixture of the acid and its sodium salt of (b) acetic acid (pKa = 4.76), (c) iodoacetic acid (pKa = 3.18), and (d) dichloroacetic acid (pKa = 1.3). Solid points and heavy line: exact results according to Eqs. (24) or (25). Open circles connected by a thin line: approximate results according to Eq. (29) for (a), and according to Eq. (27) for (b)–(d). Open triangles connected by a thin line: results according to the Henderson approximation equation (30).
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