Journal of Solution Chemistry, Vol. 4, No. 4, 1975

Glass-Electrode Measurements over a Wide Range of Temperatures: The Ionization Constants (5-90~ and Thermodynamics of Ionization of Aqueous Benzoic Acid John G. Travers, 1 Keith G. McCurdy, 1 Douglas Dolman, ' and Loren G. Hepler ~ Received August 23, 1974; revised October 15, 1974 Glass-electrode measurements have been applied to the determination of the ionization constant of aqueous benzoic acid from 5 to 90~ with the experimental proeedure and calculations described in detail. The results lead to AH~ = +60 cal-mole -1 for the standard enthalpy of ionization at 298~163 which is eompared with AH~ values from earlier investigations.

KEY W O R D S : Glass-electrode measurements; benzoic acid ionization; thermodynamics of acid ionization.

1. I N T R O D U C T I O N

We have undertaken this investigation of the ionization of aqueous benzoic acid from 5 to 90~ because of a specific interest in benzoic acid and also to test the applicability of glass-electrode measurements at temperatures higher than "normal." Benzoic acid-sodium benzoate buffers have been used occasionally near room temperature and can be more often used for higher-temperature work where the solubility of benzoic acid is larger than at room temperature and where its relatively low volatility (compared to acetic acid, etc.) is an advantage. For this use of benzoic acid-sodium benzoate buffers it is necessary to have reliable pK values for benzoic acid at all temperatures of interest. 1 Department of Chemistry, University of Lethbridge, Lethbridge, Alberta, Canada. 267 9 1975 Plenmn Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. N o part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by ar~y means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission o f the publisher.

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Travers, McCurdy, Dolman, and Hepler

Benzoic acid is the " p a r e n t " compound in many studies of effects of substituents on "reactivity" of organic compounds. For example, the most commonly used Hammett ~ parameters are based on the difference between pK values for benzoic acid and substituted benzoic acids. In recent years there has been increasing interest in the investigation of substituent effects over a range of temperatures, which in this case requires that we have reliable pK values for benzoic acid over the temperature range of interest. Various investigations of the thermodynamics of ionization of benzoic acid, reviewed elsewhere, (~) have led to reported enthalpies of ionization at 298~ ranging from A H ~ - 4 2 0 cal-mole -1 to +169 cal-mole -~ Several recent calorimetric investigations were the principal basis for a choice
Glass-Electrode Measurements over a Wide Range of Temperatures

269

gases, and stored in a nitrogen atmosphere. Tests with KI and starch indicated absence of permanganate contamination. Analar benzoic acid from Fisher was used as received to prepare stock solutions that were about 0.01 m. Buffer solutions were prepared from weighed amounts of analar potassium acid phthalate (KHP), Na2HPO4, and KH2PO4 in distilled water described above. Known amounts of KCl(c) were added to these buffer solutions to give a final concentration of KCl(aq) of 0.005 to 0.015 m. A 50~o (by weight) solution of NaOH was prepared from Fisher analar pellets and allowed to stand until a small quantity of precipitate (presumably Na2CO3) settled out. Aliquots of this solution were removed and diluted to yield solutions that were approximately 0.1 m. These solutions were standardized against oven-dried (120~ analar KHP. Analar KNO3 was added to some solutions to vary the ionic strength. All measurements were carried out with cell solutions under an atmosphere of CQ-free N2. Measurements were made on standard buffer solutions and on benzoic acid solutions, both containing known concentrations of KC1. Benzoic acid-KC1 solutions (50 ml, about 0.01 m in each) were titrated with standardized N a O H solution with voltages from the glass electrode/silver-silver chloride cell being read after each addition of NaOH. Volumetric data were converted to masses by way of densities of solutions.

3. M E T H O D

AND

RESULTS

The cell we have used can be represented by glass ]] solution (a~, ac~) l AgC1, Ag

(1)

in which aH and ac~ represent activities of H +(aq) and C1- (aq) in the solution. The potential E of this cell is related to the standard potential E ~ by E = E~-

(2.303RT/F)log(a~acl)

(2)

Using 7 to represent an activity coefficient and a = ym, we put Eq. (2) in the form -log(aayc~) = F ( E -

E~

+

log(mcl)

(3)

We represent the ionization of benzoic acid by HB(aq) = H+(aq) + B-(aq)

(4)

and express the equilibrium constant for ionization as K = aaaB/aaB

(5)

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Travers, McCurdy, Dolman, and Hepler

We combine pK = - log K with (5) to obtain pK = -log(ai~,cl) - log(a~/aHB) + log(ycl)

(6)

In order to relate the desired pK to measurable quantities, we define an " a p p a r e n t " equilibrium constant (denoted by a prime) by p K = pK' - log(y~/TaB) + log(7cl)

(7)

which we substitute into (6) to obtain pK' = -log(a~Tc~) - log(mB/m~)

(8)

The second term on the right side of (8) is evaluated by application of material- and charge-balance equations to knowledge of quantities of chemicals added to the cell. We evaluate - log(aEsop) for use in (8) by way of separate measurements on cell (1) containing standard buffer solutions described by Bower and Bates (5> and partly neutralized solutions of benzoic acid. Combination of the general equation (3) with equations given by Bower and Bates (5~ leads to -log(a~yc~) = S ( E ' -

E*) + pHs + am*~ - log(7"1)

(9)

in which the symbols have the following meanings. If cell (1) were described exactly by the Nernst equation, we would have S = F/2.303RT. Because there is good reason to be cautious about attributing exact Nernstian behavior to glass electrodes, we have evaluated S by way of measurements on two different standard buffers at each temperature, both before and after each " g r o u p " of measurements. The largest difference between experimental and Nernst values of S was 2 ~ at 90~ which leads to an entirely negligible error when the benzoic acid-benzoate solution and the reference buffer solution have approximately the same pH. We have used E' and E* to represent the potentials of cell (1) containing benzoic acid-benzoate solution and standard buffer solution, respectively. Values of pHs and a for the various standard buffers have been listed by Bates et al. ~,6~ We calculate log(~,*l) from log(7"1) = - A ~ / ; / ( 1 + Bd~/-~)

(10)

in which ~ represents the ionic strength of the buffer at mol = 0, as specified indirectly by Bower and Bates. (5~ Following measurements with buffer solutions leading to evaluation of E* and S, further measurements and calculations were carried out as follows for benzoic acid. Standardized N a O H solution was added to HB solution (containing KC1, but no added KNOs) until m ~ / m ~ ~ 0.7, at which point E' was measured. Then further E' readings were obtained after six further additions of NaOH until rnB/mE~ ~-- 1.4. Each E' value leads by

Glass-Electrode Measurements over a W i d e Range of Temperatures

271

way of (9) to a corresponding log(aH~,cl) and thence with (8) to a value of

pK" at a known ionic strength. Thus each potentiometric titration leads to seven pK' values at seven slightly different ionic strengths (difference between highest and lowest /z less than 0.002 m within each group of seven p K ' values). The whole procedure was then repeated twice with benzoic acid solutions to which KNOa had been added to give two different ionic strengths, leading to 14 more pK' values. Finally, we plotted p K ' values against corresponding ionic strengths to obtain the desired standard-state pK at the intercept tz = 0. This extrapolation is based on the assumption that I/HB = 1.0 in very dilute solution and that deviations from Eq. (10) applied to log(7'cl/7'B) are linear in tL.~7~ Ionic strengths ranged from approximately 0.01 to 0.06 m. Results of all our measurements are summarized in Table I. We emphasize that the _+ values listed in Table I are our estimates of total uncertainties and that our pK values are based on the "molat scale" for solutes. Our pK = 4.204 at 25~ is in good agreement with earlier results, which led to a choice ~ of pK = 4.204 as the " b e s t " value at this temperature. We also note that the standard free energy of ionization at 298~ reported by BoRon, Fleming, and Hall (3~ corresponds to pK = 4.202. Now we turn to calculation of A H ~ of ionization by way of application of the van't Hoff equation to our pK results. Because this calculation is not one that can be done properly in routine fashion, we explain in detail our procedures and results. Clarke and Glew ~8~have developed one useful procedure for calculating A H ~ and AC~ values from equilibrium constants at several temperatures, as recently described by Bolton59~ This method, which is based on a series expansion of AC:~, is well suited to computer calculations. We have sent our results to Dr. Bolton, who has carried out Clarke-Glew calculations with them. His results with our pK values are A H ~ = +53.5 cal-mole -1 at 298~ with "standard error" of 35.5 cal-mole -1, and temperatureTable I. Experimental and Calculated pK Values for Aqueous Benzoic Acid pK t, ~

Experimental

C-G-B

5 25 50 70 90

4.223+ 0.010 4,204+ 0.005 4.224+ 0.005 4,266+_ 0,010 4.336+ 0.015

4,224 4,202 4.223 4.269 4.334

Eq. (11) Eq. (lla) Eq. (llb) Ec/. (12) Eq. (13a) 4,222 4.206 4,223 4,266 4.336

4.216 4.204 4.221 4.260 4.321

4.228 4.207 4,224 4.272 4.351

4.224 4.203 4.223 4.268 4.335

4.207 4.189 4,221 4,278 4.358

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Travers, McCurdy, Dolman, and Hepler

independent AC~ = --34.9 c a l - ~ -1 with " s t a n d a r d e r r o r " o f 1.6 cal_OK-l_mole-1. Because AC~ is f o u n d to be independent of temperature, the C l a r k e - G l e w - B o l t o n analysis is equivalent to a three-parameter fit of the experimental p K values. Calculated p K values are listed in Table I, designated C - G - B . These calculated p K values are in satisfactory agreement with the experimental values, with no significant trend in the differences. A l t h o u g h we have been unable to find any two-parameter equation that is consistent with our experimental p K values, we have found several satisfactory three-parameter equations. One such equation is o f the f o r m InK=

AT 2 + BT + C

(11)

with A = - 7 . 6 7 2 8 x 10 -5, B = 0.04612, and C = - 1 6 . 6 t 4 . Calculated p K values are listed in Table I. This equation leads to A H ~ = R(2AT 3 + B T 2) and AC~ = 2 R ( 3 A T 2 + BT) and thence to A H ~ + 6 5 cal-mole -1 at 298~ We also obtain AC~ = - 2 7 c a l - ~ -~ at 25~ ZXC~ = - 2 0 cal-~ -z at 5~ and AC~ = --54 c a l - ~ -z at 90~ In order to gain information about effects of uncertainties in p K values on derived quantities, we have intentionally introduced systematic temperature-dependent bias into the p K values. First, we have maintained our " b e s t " p K a t 25~ while lowering all other p K v a l u e s to near the limits of our quoted uncertainties. This leads to another equation of the form of (11), which we now designate (lla), with A = - 6 . 5 4 6 7 x 10 -5, B = 0.03914, and C = - 1 5 . 5 3 0 . Calculated p K values are listed in Table I. With these parameters we find A H ~ = + 18 cal-mole -~ at 298~ and AC~ = - 2 3 c a l - ~ -1 at this same temperature. F o r another " e r r o r - l i m i t " calculation we have maintained our " b e s t " p K at 50~ (our mid temperature) and increased all other p K values to near the limits of our quoted uncertainties. This leads to still another equation of the form of (11), which we designate (1 lb), with A =-8.836 x 10 -5, B = 0.05334, and C = - 1 7 . 7 3 6 . Calculated p K values are again listed in Table I. These parameters lead to A H ~ = + 115 cal-mole -~ and AC~ = - 3 0 cal-~ both at 298~ Because the systematic temperature-dependent biases introduced in the paragraph above are only consistent with a rather pessimistic assessment o f our experimental results, we conclude that Eq. (11) leads to A H ~ = + 65 cal-mole- ~ at 298~ with total uncertainty no more than _+50 cal-mole -1 A n o t h e r three-parameter equation that leads to a g o o d fit is in K = A T + (B/T) + C

(12)

for which we find A = - 2 . 7 6 2 6 x 10 -2 , B = - 2 4 8 5 . 7 , and C = 6.895. Calculated p K values are listed in Table I. This equation leads to A H ~ = R ( A T 2 - B) and AC~ = 2ART. We calculated A H ~ + 5 9 cal-mole -1 and AC~ = --33 c a l - ~ -~ at 25~ with AC~ = --31 and - 4 0

Glass-Electrode M e a s u r e m e n t s over a W i d e Range of T e m p e r a t u r e s

273

cal-~ -1 at 5 and 90~ respectively. Calculations similar to those leading to Eqs. (lla) and (llb) again lead to the conclusion that the total uncertainty in A H ~ at 298~ is no more than + 50 cal-mole -1. We have also done calculations with the Harned-Robinson equation In K = ( A / T ) + B In(T) + C

(13)

that lead to AH ~ = R ( B T - A) and AC~ = RB. Because this AC~ is independent of temperature, Eq. (13) is in this case equivalent to the ClarkeGlew-Bolton treatment. Our calculations with this equation are consistent with the results of the calculation by Bolton referred to earlier. In a further attempt to obtain information to support or contradict the previously reported ~a~ A H ~ --67 caI-mole -~ and AC~ = --42 cal-~ -~mole- ~ at 25~ we have tried to fit our results with an arbitrarily selected ZXH~ - 7 0 cal-mole -1 at 298~ and a temperature-independent AC~ = --40 cal-~ -1 in Eq. (13), which we now designate (13a). The calculated pK values are listed in Table I. Not only are differences between calculated and experimental pK values much larger than for any other equations considered [even for (llab) that contain intentional bias], but also there is a markedly temperature-dependent trend in these differences. On the basis of all of these calculations, we now select AH ~ = + 60 _+ 50 cal-mole- ~ at 298~ as the " b e s t " value consistent with our experimental pK values (Table I). Here + 50 cal-mole -1 is our estimate of the total uncertainty. Although it is mildly disturbing that there is not better agreement between this value and the previously selected (~ " b e s t " (largely calorimetric) zXH~ = +110 + 20 cal-mole -1, the present work does seem to provide clear evidence that the recently reported (a~ AH ~ - 6 7 + 19 cal-mole -~ at 298~ contains an appreciable error. We have used our Eqs. (11), (12), (13) [(13) is equivalent in this case to the C - G - B calculation] to calculate AH ~ of ionization of aqueous benzoic acid at other temperatures. For 50~ all of these equations are in agreement with the average calculated AH ~ = --777 cal-mole-1, which is in excellent agreement with the calorimetric (~~ AH ~ = 787 cal-mole -1. Poorer agreement at sti)l higher temperatures can be attributed to larger uncertainties in both calorimetric and calculated AH ~ values. In the absence of adequate empirical or theoretical knowledge of the temperature dependence of AC~, it is necessary to combine some arbitrary expression for AC~ = f ( T ) with d A H ~ = AC~ and d In K / d T = A H ~ 2 for subsequent evaluation of AH ~ from K values at several temperatures. It has recently been fashionable and entirely reasonable to express AC2 as a power series of the form (a + b T + . . . ) , with the series cut off at the first term that permits an adequate fit of the experimental K values. But there is no fundamental reason why (a + b T + . - .) must be better or worse than

274

Travers, McCurdy, Dolman, and Hepler

some other polynomial. For example, we might equally well choose to express AC~ by the polynomial ( a T + b T 2 + . . . ) , which leads to Eq. (11). Because at least three equally arbitrary three-parameter equations are equally successful (only trivial differences in standard deviations) in fitting the experimental K values, we have no way of choosing one and rejecting the others. All of these equations lead to practically the same A H ~ values (based on the first derivative with respect to temperature); we may therefore accept all of these A H ~ values as being properly calculated from the experimental results. Although the A H ~ values obtained by differentiation of the several equations that describe the experimental K values equally well are in excellent agreement with each other, it should be noted that there is less good agreement between the AC~ values calculated by way of a second differentiation of these different equations. It is therefore necessary to be cautious about detailed interpretation of the magnitude and especially the temperature dependence of this important but elusive quantity. In conclusion, we point out that this investigation shows that it is possible to use the glass electrode in determination of pK values (5-90~ that are accurate enough to lead to AH ~ values with uncertainties significantly less than + 100 cal-mole -1. ACKNOWLEDGMENT

We are grateful to the National Research Council of Canada for support of this research. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

T. Matsui, H. C. Ko, and L. G. Hepler, Can. J. Chem. 52, 2906 (1974). F. Rodante, F. Rallo, and P. Fiordiponti, Thermochim. Acta 9, 269 (1974). P. D. Bolton, K. A. Fleming, and F. M. Hall, or. Am. Chem. Soc. 94, 1033 (1972). R. G. Bates, Determination o f p H : Theory and Practice (John Wiley and Sons, New York, 1964). V. E. Bower and R. G. Bates, J. Res. Nat. Bur. Std. 59, 261 (1957). W. J. Hamer and S. F. Acree, J. Res. Nat. Bur. Std. 32, 215 (1944); R. G. Bates and S. F. Acree, J. Res. Nat. Bur. Std. 34, 373 (1945). E. J. King, Acid-Base Equilibria (Pergamon Press, Oxford , 1965). E. C. W. Clarke and D. N. Glew, Trans. Faraday Soc. 36, 539 (1966). P. D. Bolton, J. Chem. Educ. 47, 638 (1970). T. Matsui, H. C. Ko, and L. G. Hepler, Can. J. Chem. 52, 2912 (1974).