product of the reaction between a hemin digest andp-dimethylamino-benzaldehyde in acetic acid, in relation to the spectrum of a reagent blank. A sample of hemin of iron content 8.45% was taken as a basic standard for the present work. The molar absorptivity of the final color with respect to the hemin was 192 X l o 3 liter cm-' mole-', and with respect to the iron was 194.5 X lo3 liter cm-l mole-'. The iron contents of the various blood samples were assumed to be entirely due to the presence of hemoglobin. This meant that the color yield could be related to the iron content irrespective of the presence of serum or of the additional material owing to formalin fixation. Freeze-dried human whole blood, heparinized red cells, or red cells citrated and then washed free of citrate, all gave molar absorptivities of about 165 X l o 3 liter cm-1 mole-', based on their iron contents-ie. about 85 % of the hemin figure. When these freeze-dried materials were heated overnight at 105 "C, the color yield dropped by about 2 0 z and on further heating continued to drop by about 6 % of the previous value per day thereafter. The yield also dropped when the materials were stored some months at room temperature. The formalin-fixed lungs for which blood content values were to be determined, had been dried at 105 "C for one to two days. Their color yield might well have been affected by this treatment. A number of the lungs were heated at 105 "C for several more days. Some of these lungs were moistened with four times their weight of water or formolsaline during heating. The color yields were not significantly affected. This suggested that formalin fixation might prevent the drop in color yield which resulted when freeze-dried blood was heated. This theory was reinforced when it was found that the color yield of an unfixed freeze-dried human lung was affected by heating at 105 "C in ; same way as the freeze-dried blood samples.
.
However, when various blood samples were fixed in formolsaline, their color yields were just as sensitive to heat as they were before fixation. Formalin fixation alone was evidently not enough to protect the crucial blood component from inactivation by heat. It appeared that there might be a material present in the lung tissue which protected the crucial blood factor, but only after formalin fixation. A human lung was minced and fixed in AnalaR formolsaline, and the entire suspension was freeze-dried. The color yield of this material fell less than 10% during one day of heating at 105 "C and less than 3 % of the previous value per day thereafter. Another portion of the minced, fixed lung was dried at 105 "C for two days with the entire formolsaline suspension. This time the color yield fell by only 3 % when heated for a further day, and about 0.3 of the previous value per day thereafter. The lungs whose blood contents were to be determined had been dried in a vacuum oven for one to two days at 105 "C and so were unlikely to have lost even as much as 15% of their hemin color yield. Such an effect is not serious, especially in the context of biological variations, when the hemin iron values are used in conjunction with total iron values to calculate the nonhemin iron in pneumoconiotic lungs; the hemin iron figures for a series of eighty coal-workers' lungs were on average less than a tenth of the total iron figures. The other main use of hemin analyses in the present work was as a correction to allow analytical values for dust, for instance, to be referred to blood-free lung tissue. The average computed whole-blood content of a series of coalworkers' lungs was about 20%, so the possible 15% coloryield loss owing to oven-drying was not likely to be serious in this context either. RECEIVED for review December 30, 1970. Accepted April 27, 1971.
Direct Potentiometric Measurement of Hydrogen Ion Concentrations in Sodium Chloride Solutions of Fixed Ionic Strength G. R. Hedwig and H. K. J. Powell Department of Chemistry, Uniuersity of Canterbury, Christcliurch, New Zealand The calibration of the cell glass electrode ' , H + (aq)~calomel electrode as a [H+} probe in the p H range 2.0 to 10.3 and at I = 0.04, 0.10, 0.15, and 0.20M (NaCI) is described. Calibration is effected against dilute HCI solutions and the buffer solutions ethylenediamineethylenediammoniumchloride and sodium acetateacetic acid, all of known [H+]. Plots of pH' (measured pH) against p[H+] are colinear for the three systems and coincident, within experimental error, for each ionic strength. The relationship pH' = (0.9951 =t 0.0003) p[H!] (0.088 =t0.002) was observed. From this calibration, [H+] can be accurately determined from pH' for NaCl solutions at these ionic strengths. I n contrasi, calibration against standard buffers and conversion of pH' to p[H+] by use of the Davies equation for mean ionic activity coefficients involves significant assumptions concerning residual liquid junction potentials and activity coefficients. The two approaches are shown to give quite different results for [H+] measurements. The described method of calibration i s applied to the determination of the protonation constants for 1,5,8,12-tetraazadodecane.
+
1206
THEREIS WIDESPREAD use of glass electrodes for the measurement of solution pH. The electrode is commonly used in a cell with liquid junction and its response to solution pH is determined relative to response to one or more standard buffer solutions of defined (conventional) hydrogen ion activity ( I ) . For the standard buffer, the emf of the cell glass electrodel 1 solution~KC1(saturated), Hg2C12(S),Hg(1) is given by E,
=
E"
=
E"
+ E,, + ELJ - ( R T . I ~ ~ A + ) / F + E,, + ELJ+ (2.303 RT.pH,)/F
(1)
where E,, and EL^ are, respectively, the asymmetry potential of the glass membrane and the liquid junction potential for the cell, and E" is the emf of the reference electrode with KC1 (satd) chosen as the standard state. (1) R. G. Bates, "Determination of pH Theory and Practice," John Wiley and Sons, New York, N. Y . , 1964, p 31.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
For a solution of unknown acidity,
E’
=
E”
+ E,, + E‘LJ+ (2.303 RT.pH’)/F
=
E,
+ (E’LJ- E M )- 2.303RT (pH, - pH’)/F
(2)
Thus pH’
=
pH, -
+
(Ez - E’) (E’LJ- ELJ) 2.303 RT/F
(3)
T o obtain a correct measure of hydrogen ion activity from E‘ for a cell standardized in this way, it is necessary that E‘LJ = ELj. This will seldom be the case. For example, the majority of equilibrium constant measurements which involve determination of solution pH have been made on solutions with I = 1.0, 0.1, or 0 with respect to some background electrolyte (2). On the other hand, the standard buffers most commonly used for cell calibration, potassium hydrogen-phthalate and sodium tetraborate have I = 0.053 and 0.02M, respectively. For equilibrium constant measurements the pH of the solution is often determined to give a measure of the hydrogen ion concentration, [H+], (pH’ = p[H+] - log YE+, where Y ~ is~ the + hypothetical single ion activity coefficient for H+). [Hf] is then used for substitution in mass balance equations and directly in the expression for the concentration quotient k . To make direct use of pH meter readings in these calculations, it is necessary to assume that the liquid junction potential for the cell containing the standard buffer solution is the same as for the cell containing the test solution (i.e., ElLJ ELj = 0) and that some simple equation [e.g. Debye-Huckel or Davies (3) equation] accurately defines the (hypothetical) single ion activity coefficient Y H - for (probably) a mixed electrolyte solution, and allows precise conversion of measured pH’ to [H+]. Neither of these assumptions is fully tenable and together they introduce uncertainties in p[H-] and errors in the derived concentration quotients. These assumptions can be avoided and errors can be minimized by calibration of the cell against solutions of known p[H-] and with the same ionic strength and ionic background as the test solutions. Alternatively, the error due to the different liquid junction potentials for cells containing standard and test solutions can be avoided by use of a cell without liquid junction and having an Ag, AgCl reference electrode in solutions which all contain a “spike” of C1- ( 4 ) . McBryde ( 4 ) and Irving et ai. ( 5 ) have recently reviewed the use of pH measurements in the estimation of equilibrium constants and have described the calibration of cells glass electrodel 1 solution’calomel electrode with dilute acid solutions of known [H+]. However, these calibrations are limited to solutions with pH < 4, they involve solutions of low buffer capacity, and it is not clear whether the calibration curve pH’ = Mp[H+] C can correctly be extrapolated into higher pH regions [Since preparation of this article McBryde has reported an extension of his work to dilute NaOH solutions (W. A. E. McBryde, Analyst, in press)]. In this paper we describe the calibration of the cell glass electrodel solution’calomel electrode against solutions of
+
~
(2) A. Martell and L. G. Sillen, “Stability Constants of Metal-ion Complexes,” Chemical Society London, Special Publication No. 17, 1964. (3) C. W. Davies, “Ion Association,” Butterworths,London, 1962, p 41. (4) W. A. E. McBryde, A m l y s t , 94, 337 (1959). ( 5 ) H. M. Irving, M. G. Miles, and L. D. Pettit, Anal. Chin7. Acta, 38, 475 (1967).
known hydrogen ion concentration in the p H range 2.0 to 10.3 and at ionic strengths 0.04, 0.10, 0.15, and 0.20M at 25 “C. The standard solutions used were ethylenediamineethylenediammonium chloride and sodium acetate-acetic acid buffers, for which concentration quotients are accurately known, and dilute hydrochloric acid, all in NaCl medium. To make comparison between results obtained using the Davies’ equation to calculate YH+ and results obtained using the pH’/p[H+] calibration curve, the protonation of the base 1,5,8,12-tetraazadodecanehas been studied at 25 “C and at the above ionic strengths. EXPERIMENTAL
Materials and Solutions. ETHYLENEDIAMMONIUM CHLORIDE was recrystallized from water-propan-2-01. (Found : C, 18.17; H, 7.53. Calcd for C2H10N2C12: C, 18.25; H, 7.52x.) 1,5,8,12-TETRAAZADODECANEWas prepared by the addition of 1,3-diaminopropane (67 ml, 0.81 mole) to a solution of 1,2-dibromoethane (13 ml, 0.15 mole) in ethanol (50 ml). After refluxing (1.5 hr), KOH (40 grams) was added and the mixture further refluxed for 1.5 hr. The unreacted KOH and KBr were precipitated by ether (50 ml) and removed by filtration. Unreacted reagent was distilled from the filtrate and the product was twice fractionally distilled under vacuum using a short Vigreux column. The fraction was collected at 136-1 38 “C (1.5-2 mm mercury). The primary and secondary standard buffers, with the exception of the carbonate buffer (6),were prepared from AnalaR reagents by the method of Bates (7). Procedure. All pH measurements were recorded using a Beckman Research pH meter with a Beckman E2 glass electrode type 39004 and Beckman frit junction calomel reference electrode, type 39071 (saturated KC1). The flow of’ KCI into the test solution increased the ionic strength by ca. O.O04M/ hour. Attempts were made to reduce this flow by placing a glass sleeve with capillary outlet around the calomel electrode. A variety of different media were tried in the sleeve: agarKC1 (saturated, I M , O.lM), KC1 (saturated, l M , 0.1M), test solution; however, in each case unstable emf readings resulted. The assembly was standardized using the standard 1 :1 phosphate buffer (pH, 6.865 at 25 “C) before and after each set of measurements. The linearity and slope of the NBS conventional activity scale was checked by measuring the response of standard buffers with respect to the 1 :1 phosphate buffer. These standards are internally consistent (8) for cells with and without liquid junction (9). The carbonate, borax. tetroxalate buffers, and the HCljKCl solution gave pH’ values in agreement with the pH, values tabulated by Bates (10) to within +0.003 pH. However, for the 0.05M potassium hydrogen phthalate and saturated potassium hydrogen tartrate, there was a small discrepancy (phthalate; pH’, 4.026; pH, 4.008; tartrate; pH’, 3.570; pH,, 3.557). On the basis of these observations, the pH’ values in this region were corrected ( e . g . , pH” values in Table I). A titration technique was used to obtain the calibration curve pH’ data. The titration cell was similar to Perrin’s (11). Standard sodium hydroxide was added to the test (6) D. J. Alner, J. J. Creczek, and A. G. Smeeth, J . Chem. SOC.( A ) , 1967, 1205. (7) R. G. Bates, “Determination of pH Theory and Practice,” John Wiley and Sons, New York, N. Y., 1964, p 124. (8) Ibid., p 87. (9) R. G. Bates, G. D. Pinching, and E. R. Smith, J. Res. Nut. Bur. Stand., 45, 418 (1950). (10) R. G. Bates, “Determination of pH Theory and Practice,” John Wiley and Sons, New York, N. Y . , 1964, p 76. (1 1) D. D. Perrin and I. G. Sayce, Ckern. Znd., 1966,661.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
1207
Table I. pH’, p[H+] and pH” Data from Acetic Acida-NaOH Titrations in NaCl Media at I I = 0.20M I = 0.10M ___ pHIh pH’IC p[H+I P” pH“ PW+I 3.877 3.855 3.826 3.789 3.814 3.913 3.887 3.954 3.915 4.039 4.020 4.000 3.979 3.963 4.066 4.026 4.110 4.090 4.149 4.129 4.090 4.166 4.209 4.189 4.126 4.249 4.229 4.201 4.301 4.283 4.219 4.428 4.413 4.346 4.301 4.306 4.388 4.371 4.594 4.577 4.512 4.480 4.472 4.454 4.389 4.758 4.743 4.656 4.564 4.636 4.553 4.841 4.826 4.761 4.619 4.646 4.702 4.635 4.929 4.915 4.719 4.850 4.810 4.945 4.880 4.808 4.894 5,025 5.010 4.986 a Initial concentration of acetic acid -9.3 x 10-3M. * pH’ = pH (measured). c pH” = pH’ + correction for nonlinear response to NBS buffers. Table 11. Interpolated ki Values for Ethylenediamine and Interpolated k.4 Values for Acetic Acid, Each at 25 “C and I = 0.20, 0.15, 0.10, 0.04M NaCl Media Ethylenediamine“ I
Log kl
0.20 0.15
Log kr
9.996i0.003 9.970 =k 0.003 0.10 9.960i0.003 0.04 9.941 i 0.005 a From reference 23. * From reference 12.
7.189i0.003 7.152 i 0.003 7.105i0.003 7.027 i 0.005
0.20,0.10, and 0.04M I = 0.04M
pH” 3.792 3.942 4.070 4.181 4.282 4.462 4.548 4.629 4.794 4.970
P [H+I 3.722 3.876 4.005 4.119 4.220 4.402 4.487 4.569 4.733 4.909
in which C, is the initial concentration of acetic acid, cs is the concentration of sodium acetate (added OH-), and K,, is the ionic product for water in NaCl solution at a given ionic strength (13). The term K,/[H+], a correction for hydrolysis of acetate ion, was approximated to K,/antilog( - pH’) and the resultant quadratic equation in [H+] from Equation 4 was solved to give p[H+] a t each point in the titration curve buffer region. Data from titrations against hydrochloric acid-sodium chloride solutions ( I = 0.20, 0.10, and 0.04M) are given in Table 111. Values of p[H’] were calculated from the analytical concentrations of acid and alkali. Near the end point, readings became progressively less stable and data a t pH’ > 4 were not considered. A plot of pH’ against p[H+] was linear in the range pH’ = 2 to 3. However, as the pH’ values increased further, the plot became curved (tending to larger pH’/p[H+] values as pH’ approached 3.5, as observed by McBryde ( 4 ) . Data from titrations against ethylenediammonium chloridesodium chloride solutions ( I = 0.20, 0.15, 0.10, and 0.04M) are given in Table IV. Experimental data were used from the most buffered regions, ca. pk, 1 0 . 6 . k , (protonation constant) values are given in Table 11. The [H+] a t each titration point was calculated by solving the derived cubic Equation 5 in which TBand TH
Acetic acid” k A X lo” (moles. liter-1) 3.090i0.005 2.975 i 0.005 2.809i0.005 2.489 i 0.005
-
=
solution (49.90 i 0.05 ml) using an “Agla” micrometer glass syringe (total delivery 0.5 ml; accuracy =kO.lOZ). Data were reproducible t o i0.003 to 0.005 pH. RESULTS
Data from titrations of standard sodium hydroxide solution against acetic acid-sodium chloride solutions, at total ionic strengths ( I ) of 0.20, 0.10, and 0.04 are given in Table I. Only data from the most buffered region of the titration curve (pkl 1 0 . 7 ) were considered. The p k values ~ for acetic acid (Table 11) were interpolated from the data of Harnea and Hickey (12). The [H+] for each data point was calculated from Equation 4
(4) ~~
are the total concentrations of ethylenediamine and ionizable hydrogen, respectively, and [OH-] ’ is the concentration of
~
(12) H. S . Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” 2nd ed., Reinhold Publishing Co., New York, N. Y . ,1950, p 523.
(13) Zbid.,p 578.
Table 111. pH’ and p[H+] Values from HCI/NaOH Titrations in NaCl Media at I I P“
2.083 2.150 2.256 2.321 2.395 2.485 2.597 2.750 a
2.016 2.081 2.185 2.248 2.322 2.410 2.521 2.671
0.067 0.069 0.071 0.073 0.073 0.075 0.076 0.079
2.133 2.187 2.250 2.320
=
PW+I 2,061 2.115 2.177 2.247
pH‘ = pH (measured). A = pH’ - p[H+].
~~~~
1208
~
~~~~~
~~
=
0.10M
~
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
0.20, 0.10, and 0.04M I = (1.04M
A
0.072 0.072 0.073 0,073
2.084 2.151 2.254 2.317 2.393 2.480 2.589 2.741 2.970
2.016 2.081 2.185 2.248 2.322 2.410 2.521 2.671 2.899
0.068 0,070 0.069 0.069 0.071 0.070 0.068 0.070 0.071
Volb 0.050 0.060 0.070 0.080
0.090 0.100 0.160 0 . I70 0.180 0.190 0.200 0.210 0.220 0.230
Table IV. pH’ and p[H+] Values from Ethylenediammonium Chloride-NaOH Titrations in NaCl Media. at Z = 0.20, 0.15, and 0.04M Z = 0.15M Z = 0.20M P” PW+l Ac Vol P“ PW+l 7.016 6.956 0.060 0.040 6.836 6.774 7.147 7,089 0.058 0.060 7.108 7.052 0.059 0.080 7.366 7.309 7.276 7.217 7.406 7.346 0.060 0.090 7.501 7.445 0.060 0.100 7.651 7.595 7.541 7.481 7.692 7.632 0.060 0.120 8.055 8.000 9.368 9.327 0.041 0.160 9.351 8.304 9.535 9.500 0.035 0.170 9.524 9.478 9.675 9.642 0.033 0.180 9.667 9.621 9.800 9.767 0.033 0.190 9.790 9.746 9.912 9.880 0.032 0.200 9.903 9.861 10.017 9.986 0.031 0.210 10.008 9.969 10.118 10.087 0.031 0.230 10.209 10.172 10.215 10.185 0.030 I
Vol
=
0.10M
Z
=
-
A
0.062 0.056
0.057 0.056 0.056 0.055
0.047 0,046 0.046 0,044 0.042 0.039 0.037
0.04M
il PW+l 6.634 6.571 0.063 0.030 6.547 6.483 0.064 6.798 6.740 0.058 0.040 6.710 6.651 0.059 0,050 6.942 6.886 0.056 0.050 6.852 6.796 0 056 7.075 7.021 0,054 0.060 6.983 6.929 0.060 0.054 0.070 7,204 7.151 0.053 0.080 7.237 7.187 0.050 7.337 7.284 0.053 0.100 7.525 7.475 0.080 0.050 0.090 7.475 7.424 0.051 0.120 7.938 7.890 0.048 0.100 7.633 7.582 0.051 0.140 8.748 8.704 0.044 0.150 0.041 9.176 9.142 0.034 0.160 9.326 9.285 0.160 9.394 9.355 0.039 0.180 9.645 9.605 0.040 0.170 9.561 9.521 0.040 0.190 9.772 9.733 0.039 0,180 9.701 9.660 0,041 0.200 9.887 9.850 0.037 0.190 9.825 9.785 0.040 0.220 10.102 10.068 0.034 0.200 9.938 9.899 0.039 0.240 10.310 10.278 0.032 10.047 10.008 0.039 0.210 10.151 10.113 0.038 0.220 10.253 10.216 0.037 0.230 10.354 10.317 0.037 0.240 Initial volume of ethylenediammonium chloride-NaC1 solution 49.9 ml. Initial concentration of ethylenediammonium chloride 2.977 x lO-3M(I = 0.20,0.15M), 2.961 x 10-3M(I = 0.10M) and 2.972 X 10-3M(Z = 0.04M). Volume (ml) of NaOH titer, concentration 1.098M(Z = 0.20,0.15, and 0.04M) and 1.114M(Z = 0.10M). A = pH‘ - p[H+]. pH
PW+l
J
Vol
A
P“
0.030 0.040
I
hydroxide formed by hydrolysis of ethylenediamine. This equation was solved by the Newton-Rapson method (14) using the experimental value, antilog (- pH’) as an approximate solution for [H+]. An alternative treatment was to form a quadratic Equation 6 C(1
+ k,[H+l + kik?[H+12)
=
TB(~I[H+] 4-2kik2[H+I2) ( 6 )
+
where C = TH [OH-]’ - [H+]. For the term C, [H*] was initially estimated from the experimental pH’ value. The quadratic equation was solved for [H+] and an iterative procedure used to obtain improved values of [H+]. Comparison of the Three Buffer Systems. A single plot of pH’ against p[H+] for each ionic strength gave a straight line for the hydrochloric acid data and the two sets of ethylenediamine-hydrochloric acid data. The acetic acid data did not lie on this line. However, using the pH” values for acetic acid (pH ’ corrected for the nonlinear response to standard buffers in this region) a good fit was obtained within experimental error. A linear least squares analysis of the pH data C gave the using an equation of the form pH’ = Mp[H+] following results. For 100 data points containing pH’ acetic acid values
+
M
=
0.9932
* 0.0003, C = 0.106 * 0.002, u = 0.011
where u is the standard deviation of pH’ values from the computed curve. For 100 data points containing pH” acetic acid values
M = 0,9951 zt 0.0003, C
=
0.088 + 002, r~
=
0.005.
For 70 data points from the hydrochloric acid and ethylenediamine sets only
M
=
0.9953
f 0.0002,
C
=
0.086 i 0.002,
r~
=
0.005.
1,5,8,12-Tetraazadodecane Protonation Constants. Data from the titrations of standard NaOH against 1,5,8,12tetraazadodecane and excess hydrochloric acid in sodium chloride solution ( I = 0.20, 0.15, 0.10, and 0.04M) are reported in Table v. Two or three titrations were needed to obtain data reproducible to within *0.002-0.005 pH. As an additional check on systematic errors, fresh stock solutions were prepared and the titrations repeated. At each point on the titration curve, there are equilibria among the species H+, B, BH+, BHZ2+,BH33+,and BHd4+in solution:
(14) N. I. Vilenkin, “Successive Approximation,” Pergamon Press, Oxford, 1964. ANALYTICAL CHEMISTRY, VOL. 43, NO. 10,AUGUST 1971
1209
Table V. Representative Data. from Titrations of 1,5,8,12-Tetraazadodecane-HCl Solutionsb against NaOH at Ionic Strengths of 0.20, 0.15, 0.10, and O.O4M, NaCl Media I = 0.10M I = 0.15M I = 0.04M I = 0.20M NaOH, NaOH, NaOH, NaOH, ml ml fi ml ml a fic ii P[H+l P[H+I P[H+I P[H+ld 4.938 3.826 3,827 3.828 5.000 3.832 0.160 0.160 0.160 5.057 0.215 4.689 5.329 3.632 3.633 3,634 0.170 5,461 4.920 5.399 3.723 0.170 0.170 0.220 5.503 3.534 3,436 3.436 5.750 3.656 0.175 0.180 0.180 5.816 0.223 5.048 5.679 6.173 3.237 3.238 5.171 3.435 0.190 0.190 6.568 0.226 3.588 0.180 5.876 6.935 3.336 3.038 3.038 3,427 0.200 0.200 7.033 5.461 0.185 0.233 6.116 7.681 3.237 2.840 3.265 0.210 7.745 2.840 0.210 0.240 5.808 0.190 6.925 2.641 2.642 3.033 8.138 6.743 8.082 3.038 0.220 0,220 0.250 0.200 7.637 2.444 8.445 0.260 8.392 2.839 2,444 2.800 0.230 0.230 7.552 0.210 2.249 8,734 0.270 7.851 8,685 2.248 2.568 0.240 0,240 7.979 2.740 0.215 8.026 2.641 2.336 0.250 9.020 0.280 8.339 8.978 2.056 2.058 0.220 0.250 1,873 0.260 9.291 8.338 9.257 2.443 1.870 2.220 0.230 0.260 0.285 8.524 1.691 9.521 8.485 9,493 2.345 1.696 0.270 0.270 8.714 2.106 0.235 0.290 8.636 1,527 9.711 1.519 9.688 2.247 0.280 0.280 9.089 1.882 0.240 0.300 1.356 1,366 0,290 9.873 9.396 8.789 9.854 2.150 0.245 0.290 1,665 0.310 1.213 10.014 1.200 8.941 8.994 2.054 0.300 0.300 9.525 0.250 1,560 0.315 1.959 1.072 1.457 10.141 1,059 9.640 9.088 10.129 0.255 0.310 0.310 0.320 9.227 1.866 0,945 10,261 0.926 0.330 9.844 10.252 0.260 1.259 0.320 0.320 10.364 1.686 0.830 1.072 10.369 0.810 10.020 0.340 9.468 0.270 0.330 0.330 9.667 0.729 10,468 0.707 10.183 10.465 1.512 0.902 0.280 0.340 0.340 0.350 0.645 10.560 0.622 9.836 10.562 1.345 0.290 0.350 10.333 0.360 0.753 0.350 10.644 10.649 9,984 1.187 0.573 10.473 0.550 0.629 0.300 0.360 0.360 0.370 10.120 1.041 10.721 0.496 10.730 0.514 0.310 0.370 0.370 10.597 0.527 0.380 0,446 10.245 0,907 0.466 0.320 10.791 10.709 10.801 0.451 0.380 0.380 0.390 10.361 0.330 10.806 0.788 0.392 0,400 10.911 0.388 0.400 10.468 0.684 0.340 10.569 0.600 0.350 10.659 0.528 0.360 10.817 0,428 0.380 a The log k values reported in Table VI are the average values obtained from a number of titrations. Composition of solutions: I = 0.04; [NaOH] = 1.268M, TB = 9.653 x lO-aM, TB = 1.088 x lO-3M, initial volume = 49.9 ml. I = 0.10,0.15,0.20; [NaOHl = 1.269M, TH = 7.752 X 10-3M, TB = 1.103 X lO-3M, initial volume = 49.9 ml. ii is the degree of formation; F. J. C. Rossotti and H. Rossotti “The Determination of Stability Constants,” McGraw-Hill, New York, N. Y . , 1961, p 40. p[H+]values were calculated from the calibration curve.
The charges have been omitted for clarity. The mass balance equations for this system are:
+ [BHI + [BHzI + [BHd + DH41 [B] (1 + ki[Hl + kikz[H12 + kikzkdH1’ +
Te = [Bl =
kikzk&4[H14) (7) and TE
+ [OH-]’ - [HI = [BH] + 2[BHzl + 3iBH.d + 4[BH41 [B]ki([H] + 2kz[H12 + 3kzk3[W3 + 4kzkak4[H14) (8) =
From Equations 7 and 8 on elimination of [B], Equation 9 was derived TH
+ [OH]’- [HI
=
TBfXH1, k d
(9)
and was solved by a nonlinear least squares procedure using a Fortran program ORGLS, (adapted from program ORGLS of W. R. Busing and H . A. Levy, Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tenn., 1962) on an IBM 360/44 computer. The LHS of this equation (termed F,) is readily calculated from the analytical concentrations and the corrected p H r values and has an uncertainty termed u. If the RHS is termed F,(x,y) then the least squares procedure varies the kl values to minimize an error square sum M(Z,Y, =
54 (Fob - [FdX,Y)li
i= 1
1 2
In Table VI, the log k data obtained from the pH’ measurements by the application of the pHr/p[Hf] calibration curve (extrapolated to p H r = 10.8) are compared with those obtained from the application of the Davies equation. The thermodynamic constants reported in Table VI were obtained by extrapolation of the log k data against some function of Z (see Table footnote) to I = 0 such that the plot was linear and of small slope. DISCUSSION
Liquid Junction Potentials. At each ionic strength, the standard solutions used for calibration contained a low concentration of buffer (0.003 to 0.009M) compared with the background electrolyte (0.04 to 0.20M). If this is also the case for any test solutions, then it can be assumed that at the same ionic strength and pH, the standard and test solutions each generate the same liquid junction potential in the cell. It also follows for these solutions that (standard) = Y~~ (test) for a given ionic strength, so that standard and test solutions with the same pH’ have the same value of p[H+]. These two conditions are necessary for an application of the pH’/p[H+] calibration curve. The most important features of this method of electrode calibration are that p[H+] for an unknown NaCl solution with I between 0.04 and 0.20M is independent of ELJfor the standard buffer (phosphate) and independent of E’LJfor the test solutions, and derivation of p[H-] does not require use of any empirical equations for single ion activity coefficients.
where wt
= 1/u2.
1210
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
Log kiValues for Protonation of 1,5,8,12-Tetraazadodecanein NaCl Media log ki log ks pH’/p[H+]caliba Davies eqb pH‘/p[H+]calib 10.62 f 0.03c 10.25 i 0.02 9.82 =t 0.02 10.57 f 0.03 10.29 f 0.02 9.79 i. 0.02 10.53 f 0.03 10.33 + 0.02 9.77 f 0.02 10.50 f 0.02 10.37 i 0.02 9.69 i 0.02 10.46d=!C 0.03 10.42df 0.03 9.51e i 0.02
Table VI. I(mll) 0.20 0.15 0.10 0.04 0.00
log ka log ka I(m/l) pH’/p[H+]calib Davies eq pH ’/p[H+]calib 0.20 8.41 i. 0.02 8.31 i: 0.02 5.76 f 0.03 0.15 8.36 =!C 0.02 8.26 f 0.02 5.65 i 0.02 0.10 8.30 + 0.02 8.23 f 0.02 5.59 f 0.02 0.04 8.12 f 0.02 8.08 =t 0.02 5.34 i 0.03 0.00 7.81, =!C 0.02 7.79, i: 0.02 4.868 i 0.03 a From the application of the pH’/p[H+]calibration curve. From the application of the Davies equation for single ion activity coefficients. All errors estimated as the standard deviation from a number of independent determinations. From a plot of log kl cs. I . e From a plot of log k2 - P Z / ( l + P 2 ) US. I. f From a plot of log k B - 2P2/(1 P Z cs. ) I. From a plot of log ka - 3I1l2/(1 P 2 )cs. I.
Davies eq 9 84 L 0.02 9.80 i 0.02 9.76 =t 0.02 9.68 i 0.02 9 . 9 i 0.02 Davies eq 5.67 I-t 0.03 5.60 i 0.02 5 . 5 5 i 0.02 5 . 3 3 i 0.03 4.900 0.03
*
+ +
Q
If Q ” H + = antilog(-pH”) is the true thermodynamic hydrogen ion activity for the test solution, then
If log YE+ and ( E ’ L ~ ELj) are functions of I only under the conditions chosen, then plots of pH’ against p[H*] at different ionic strengths should be parallel and with a separation
where A refers to the change in the quantity in brackets, from one ionic strength to another. Our observation is that, within experimental error, the plots are parallel and coincident. This convenient result, which implies that A(E’LJ)/AZ A(1og -y=+)/AI, indicates that the cell emf varies linearly with log [H+] and is independent of ionic strength in the range 0.04 to 0.20M, for NaCl solutions. This was observed, in an earlier study by Powell and Curtis (15) for a smaller number of data points in the ethylenediamine-ethylenediammonium system up to I = 0.35M and has been confirmed for this system by an independent study (16). The observed slope (0.995) and intercept (0.088) for pH’ = (slope)p[H+] intercept, agree well with the data published by McBryde ( 4 ) for dilute HC1 solutions (slope = 0.990, 0.994, 0.995, and intercept = 0.088, 0.105, 0.097, a t I = 0.05, 0.10, and 0.20M, respectively, NaCl medium). The non-unit slope implies that the operational activity coefficient y ’E+ given by Equation 11 varies with solution pH.
-
+
This implies that either E I L ~- ELJ or log
YE+
(11) or both these
(15) H. K. J. Powell and N. F. Curtis, J . Chem. SOC.( B ) , 1966, 1205. (16) M. W. Morgan and H. K. J. Powell, University of’canterbury, Christchurch,New Zealand, unpublished results.
quantities are a function of solution pH. It is noted that the activity coefficient of HC1 in an HC1-NaC1 mixture at constant total ionic strength 0.1M varies little with HC1 concentration (17). However, it has been predicted that liquid junction potentials for the junction solution-KCl(satd) will vary with solution p H (18). For solutions of high acidity or alkalinity, the ions H 3 0 +and OH-, which have exceptionally high mobilities, will make significant contributions to the junction potential. The predicted effect (9) is to make pH’ high in the acid region and low in the alkaline region with respect to calibration against neutral buffer solutions of the same ionic strength. Studies on cells with and without liquid junction indicated a pH dependence of ELJat pH < 2 and p H > 10 but the sign and magnitude of the potentials determined were not uniformly consistent with the simple theory outlined above (9). However, the results reported here and by Childs (19) suggest that there is the expected dependence of ELJon solution pH. Use of Single Ion Activity Coefficients. Alternative to the methods described in this paper, the estimation of [H+] in solution from the use of the aHL-sensitive glass electrodes commonly involves use of (hypothetical) single ion activity coefficients yH+ = aH+/[Ht]. For a mixed electrolyte system, Y H ~ ,which is approximated to yHcI, will be dependent on all specific ionic interactions in solution, as related in the Guggenheim equation (20). This equation is seldom used because it is cumbersome and it is necessary to know all the interaction coefficients Pm ,z . from measurements on single electrolyte solutions at the same ionic strength. Commonly, the empirical Davies equation (3) is used, -log YE+ = AZ+2(Z1’2/(1 PIz) - PI), /3 = 0.30. Although the equation was derived for solutions of pure electrolytes only, it has in certain systems of mixed electrolytes led to thermodynamic (acidity) equilibrium constants k O in good agreement with those obtained by extrapolation of concentration quotients
+
(17) H. S . Harned,J. Amer. Chem. Soc., 48,329 (1926). (18) R. G. Bates, “Determination of pH Theory and Practice,” John Wiley and Sons, New York, N. Y., 1964, p 58. (19) C. W. Childs, Znorg. Chem., 9,2465 (1970). (20) E. A. Guggenheim and J. C. Turgeon, Trans. Faraday SOC., 51, 747 (1955).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10,AUGUST 1971
1211
Table VII. Comparison of Operational Activity Coefficient y 'H + and Empirical (Davies Equation) Activity Coefficient 1 Solution for the Cell Glass ilNaCl medium KCl(satd.), Hg2C12,Hg ~
- log
I
P" 9 800 7.276 4.090 9.938 7.337 4.134 9.887 7.237 3,942 I
-1% ?'Hi (Davies) 0.124 0.124 0.124 0.105
Y'HCI
0.20 0.20 0.20 0.10 0.10
0.064 0.039 0.053
0.10
0.068
0.04 0.04
0.037 0,050
0.04
0.066
0,033 0.059
0.105 0.105
0.077 0.077 0.077
k to zero ionic strength (21). Use of this equation for a cell with liquid junction incorrectly implies an equality, Equation 12 -log
Y H ~(Davies)
=
-log
t
Y H ~
E'LJ- ELJ = pH' - p[H+] (12) 2.303 RTJF and can give very erroneous estimates of p[H+] in solution. This point is illustrated in Table VI1 where -(pH' - p[H+]) = log yH+ is compared with log YE+ (Davies) for solutions of different pH and ionic strength. Somewhat better agreement results when the Debye-Huckel equation is used with an ion size parameter a = 9 p\ for H+ as suggested by Kielland (22) (uiz: -log Y E + = 0.081, 0.065, and 0.052 at I = 0.20, 0.10, and 0.04M). Protonation Constants for 1,5,8,12-Tetraazadodecane. Table VI indicates that significantly different log k, values are obtained from experimental pH' data by application of the Davies equation or the pH'/p[H+] calibration curve. In the latter case, the relationship log k, " = Lt(1og k,) will hold I+O
because in Equation 13.
the term in parenthesis is unity at I = 0. However, from application of the Davies equation to data from a cell with liquid junction the derived concentration quotient is k,'
=
[BHnn+l [BH,_i("-')+ [H+]'
-
where [H+]' # [H+] because of residual liquid junction potentials. The question arises, does [H+]' + [H+] as I O? That is, does E'LJ = ELJat infinite dilution? The convergence of graphical extrapolations of log k, to I = 0 (results in Table VI) suggests that this is the case, but this must be considered fortuitous, as E'LJand EL^ refer to very different liquid junctions. The agreement between k,
(-
)
[H+ln-n-~i~
and k,'
is better for constants determined from data in the acid region (k4,k3) than in the alkaline region (kl;kz an exception) and improves with decrease in ionic strength. This observation correlates with the agreement between log YE+ (Davies) and log ~ ' H C Ivalues in Table VII. Choice of Concentration Quotients. The method of electrode calibration described in this paper requires a knowledge of concentration quotients, ki, for an acid-base system which (21) J. R. Brannan and G. H. Nancollas., Trans. Faraday SOC.,58, 354 (1962). (22) J. Kielland,J. Amer. Chem. SOC.,59, 1675(1937). 1212
buffers in a suitable pH range. The quotients must be known with precision. They must be determined either from cells without liquid junction or from matched cells with matched liquid junctions. For the system ethylenediamine-hydrochloric acid in NaCl medium, Everett and Pinsent (23) determined concentration quotients kl and k2for ethylenediamine protonation at a series of ionic strengths (0.07 to 0.30M) with the cell
1
I
I
Pt,H2 HCl (l), en,NaClI KCl ( 3 . 5 M ) HC1 (2), NaCl Hz,Pt The ionic strengths in the two half cells were kept equal, making the hypothetical single ion activity coefficient YH+ equal for the reactions at the two electrodes and giving similar liquid junction potentials at the two NaCl (as) KCl(aq) interaH+(1) RT - ELJ(P) - RT - In= faces. For this cell, E = ELJ(I) F a~+(2) F [H+ (1>1 lnat zero buffer concentration. Errors introduced by [H+ (a1 small differences in the activity coefficients and in the liquid junction potentials were eliminated by extrapolation to zero buffer concentration. The constants obtained had an uncertainty of *0.5 (*0.002 in log k). ki values were reported at 0" to 60 "C at 10" intervals and ionic strengths 0.07 to 0.30M. For this study, values were interpolated for 25" and I = 0.04,0.10,0.15,and 0.20M (Table 11). The concentration quotients used for acetic acid were determined by Harned and Hickey (24) using an unbuffered cell without liquid junction
1
I
I
I
1
H z HAc(M)NaCl(m) AgC1-Ag-AgC1 HCl(O.01) H2. For each ionic strength, data were extrapolated to [acetic acid] = 0 so that experimentally determined values of ~ " H C I (YHCI at zero HCl molality in NaCl medium) could be used in the equation relating emf and [H+]. For the very dilute (3.0 to 9.0 X 10-3M) acid solutions used in this study, it can be assumed that k values are independent of solution composition-Le., under reaction conditions the product(s) of the activity coefficients of the species in equilibrium is constant, e.g. Y ~ ~ H + / Y ~ ~ . Y=H kl"Jk1 + is constant. Previous Work. Earlier Powell and Curtis (15) used the ethylenediammine-ethylenediammonium buffer system for glass electrode calibration. Their work involved the assumption that kf values derived for NaCl solutions would be applicable to NaC104-Ba(C104)2 solution of the same ionic strength. They used a cell involving a liquid junction between saturated KC1 solution and perchlorate solution. Because of the insolubility of KClO4, this could have led to variable liquid junction potentials. Also their choice of interpolated ki values (25) is at slight variance with ours; this and the fact that they have given weight to data in acidbase composition regions of low buffer capacity probably accounts for the slight curvature in their calibration plot. Rajan and Martell (26) have used the acetic acid-sodium acetate buffer system and standard HC1 and NaOH solutions to calibrate a cell containing glass and calomel electrodes. Experimental details were not given. RECEIVED for review November 16, 1970. Accepted April 5 , 1971. (23) D. H. Everett and B. R. W. Pinsent, Proc. Roy. SOC.(London), ,4215, 416 (1952). (24) H. S.Harned and F. C. Hickey, J . Amer. Chem. SOC.,59, 1284 (1937), and references therein. (25) H. K. J. Powell, Ph.D. Thesis, Victoria University of Wellington, New Zealand, 1965. (26) K. S. Rajan and A. E. Martell, J . Inorg. Nucl. Chem., 26, 789 (1964).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
