The Uncertainty of pH Guy Schmitz Universite Libre De Bruxelles, CP165, 50 Av. F. Roosevelt, 10%I Bmxelles, Belgium During a kinetic study in acetate buffers, we obsewed that the measured pH values were always a little lower than the ones expected, based on calculations. The explanation was the complex relation between the pH meter reading and the activity of hydrogen ions. where [Hi] is concentration, and yH-is the activity wefficient on the molar scale. A brief recall of the theory will clarify the problem. Chemical eauilibria and electromotive forces of cells depend on activities, not concentrations. So the original definition of pH by Sorensen was subsequently modified to This is only a notional definition because a single ion activity is immeasurable. Experiments only give mean activities wefficieuts. For example, in HC1 solution we can measure
but not the individual values y ~and * ycr. The left-hand side of the following equation can be known exactly, but the two terms on the right-hand side cannot be separated without a nouthermodynamic assumption. This is why the practical pH scale is based on standard reference solutions selected by the National Bureau of Standards (1,2).The procedure used to obtain the pH of these solutions (pH(S))illustrates the problem. The NBS pH Scale Consider the following cell without a liquid junction. Pt I Hz (1atm) I buffer, Cl- I AgC1 I Ag (i) Its potential is given by
Rearranging, we get
The pH scale is then defined by the following equation for some standard buffers. PHW)= (-log ~ H Y C I ~+ O(logycl-)~ It is clear that pH(S) is close to -log ar only if we can use eq 2, and the meaning of the pH on this scale is only operational. pH Measurements Usually, pH values are measured with the following cell (6). calomel glass electrode I KC1 (sat. or 3.5 M) 1 I solution I ele
(ii)
This cell is calibrated with a buffer of known pH on the NBS scale, and the pH is calculated by
where S denotes the standard andX the test solution. The value so obtained is correct only if the liquid junction potential is the same with both solutions. It is generally admitted that the differencebetween the liquid junction potentials is small and reproducible because it is determined bv the hieh .. KC1 wncentration and is quasi-independent of the composition of the measured solution (11,'l'his is onlv true w ~ t hcarefullv desiened linuid junctions (3, 4) when the ionic strengthsof t h i standard and the test solution are not too different. Use of a cell without liquid junchon does not prevent the problem due to the effect of ionic strength. For example, if we use cell (i) and eq 1, we must know the activity coefficient of the chloride ions. However, we must remember that activity coefficients are affected by all the ionic constituents of the solution. Tabulated values for pure salts are of little help for real mixtures, and eq 2 is only valid for low ionic strengths. Even the pH on the operational NBS scale cannot be obtained exactly using eq 3 for cell (i) because the activities of chloride ions should be the same in the test and the standard solutions. An Example
From the measured E and the standard potential E , log (azycl-)is obtained. This quantity is measured for several values of [Cl-I, plotted against [C13 and extrapolated to [Cl-I = 0, giving (log ( a m c l)),,. Because y c l depends on all the ions in the solution, (ycl-)ois not equal to 1. Here anonthermodynamic assumption must be introduced: the Bates-Guggenheim convention (4).
where A is the Debye-Huckel limiting slope, and I is the ionic strength.
We have measured the pH of a series of acetate buffers at 25 'C with a glass electrode (Tacussel TG100) and a calomel referenceelectrode (Tacussel TR100). All reagents were from Merck (PA). The system was calibrated with Merck Titrisol buffers of pH 4 and 7 using a bracketing procedure. The slope of the electrode was near the Nernst slope. For each series of measurements we took the same concentrations of NaAc and HAc but different amounts of NaN03 to vary the ionic strength. Some results are given in the table, where a , s, and z are the numbers of mol1L of added HAc, NaAc, and NaN03. The measured values (pH,,) decrease when the NaN03 concentration increases. Let's compare this observed effect with the expected effect of the ionic strength. The Hf concentration can be calculated exactly resolving the following three equations by Volume 71 Number 2 February 1994
117
Effect of the Ionic Strength on pH ,
.
1.02
2.00
0.54
4.81
4.94
0.13
pHmar:experimental; pHml:calculated wllh D = 0.18;ApH = ~H,I-~H,..
an iterative method if the apparent dissociation constant K. is known. a = [Ht] + [HA=] (4) a + s = [AcI+ [HAcl (5) [H'IIAe-I = KJHAcl (6) The thermodynamic value of the dissociation constant is well-known (5),but we need the apparent value K. for the ionic strength I = i + [Ac-I.
Kilpatrick and Eanes (6) have measured K, as a function of the ionic strength in KC1 and NaCl solutions. The two sets of values are very similar and probably close to the values in NaN03. Alternatively we can use the Davies (7) equation below with a suitable value of the semiempirical constant D and take K. =K/yD2.
For D = 0.18 and I S 0.5 M the differences between the calculated values of K, and those of Kilpatrick and Eanes (6) are smaller than 5%. The calculated values (pK.1) in the table are obtained by resolving eqs 4-6 and taking pK.1 =-log yo [Htl Values of pKal- pH, are reproducible and independent of the pH. They are always positive and lncrease with I. There could bc two reasons for this variation: DH?.I is onlv an approximation of -log a* whose validity i's ligited b; the validity of eq 7; pH,. depends on the residual liquid junction potential. We have noted that K, values calculated with eq 7 are in accordance with the experimental values of Kilpatrick and
118
Journal of Chemical Education
Eanes, and pH,* for i = 0 and a = s = 0.1 or 0.01 differ by less than 0.01 unit from the NBS recommended values (2). A 50% decrease of D gives only a decrease of 0.02 unit of pH,* if i = 0.27. Thus, we wnclude that the differences between pK.1 and pH,. are mainly (but not only) due to the residual liquid junction ptential. The usual assumption that the liquid junction potential is constant is thus a rough approximation even for moderate ionic strength. This potential depends on the kind and shape of the junction, and free-diffusionjunctions give better results than ceramic frit junctions (4, 8).However, in the common practice we use commercial junctions of this last kind, and the error may be large. Stapanian and Metcalf (9) have discussed this problem for low ionic strength wabers. For moderate ionic strength, we see that the error may be larger than 0.1 unit even when the measurements are reproducible. Conclusions Much effort is made developing methods for exact pH calculations (1&12). These are mathematically exact and useful because they can give exact values of -log [HI]. But what is the relation with measured values in the light of the wnsiderations summarized in this paper? I t is important to realize that -log a y cannot be calculated exactly bec a u s e ~ ~ - c a n nbe o tknown exactly. Also, the liquidjunction potential is not always negligible. In the common practice we cannot hope to get differences between calculated and measured pH values less than 0.02, and we should not be surprised if they are much larger. A second wnclusion arises from the start of this paper. Consider a reaction in solution of rate order n with respect to [H+1.For the kinetic constant k, a simple error calculation gives
dk k
-= 2.3n d log
[m
If we use pH values instead of log [Ht1, then the kinetic constant includes a conventional activity wefficient. It is conventionalbecause the pH scale is conventional.Also, an error of 0.05 unit of pH gives an error on k of 11.5% ifn = 1 and 23% if n = 2. It is much more difficult than usually realized to measure kinetic constants for acidity-dependent reactions! Literature Cited 1. Batea. R. G.fiWminafion ofnH. .. .2nd ed.:. Wile?: . New Yak.1973 2. Covington, A. K, Batea, R. G.: Dumt, R.A. Pure and Appl. Chem. 1985,57,533. 3. Covington,A. K h a l Chim Acfo 1881,127, 1. 4. Harbinson.T. . R.:. Davison. W A n d Chem. 1987.59.2450. . . 5. SlnbiliN ConaCznts, The Chemiml S o e Y @ ~ e i a l P u b l i i f i i i i , no. 17 and no. 25: Landon. 1964 and 1971. 6. ?Zlpatrick, M.; Eanes, R. 0. J A m r Chem Soe. ImS, 75,586 7. Davies, C. W. J. Chem. Soe. 1928,2033. 8. Daviaon. W D.: Wmf C.And. Chem. 1985.57.2567. , A.:MefcsU, R.C. J Chom. Edu:. 1990,67,623. 9. ~ t a p a n i kM. Chem.Edue. 1990,67, 10. Herman,O.P:Booth,KK:Par*er,O.J.:Breneman,G.L.J. 501. 11. MalinawsLi, E. R . J Cham. Edue. 1990,67,502. 12. Campmado, J. M.; Ballestems R.J. Chom. Educ. 1990,67,1036.
