STUDIES OF CELLS WITH LIQUID-LIQUID
JUNCTIONS. V
STANDARDS FOR HYDROGEN IONMEASUREMENTS AND THE SECOND DISSOCIATION CONSTANT OF PHOSPHORIC ACID E. A. GUGGENHEIM AND T. D. SCHINDLER Department of Chemistry, Stanjord University, California Received July 5, 1033
I.
INTRODUCTION
A common type of cell used in the electrometric determination of hydrogen ions is the following:
1
Hg I HgCl, 0.1 N KC1 13.5 N KC1 I solution S Hz
(1)
The exact thermodynamic treatment of such cells has been discussed in a previous paper (6). The electromotive force is a definite but rather complicated function of the concentrations and transport numbers of all the ionic species and of the mean activity coefficients of all the electrolytes present, not only in the electrode solutions and the bridge solution but also in the transition layers between. The formulas, though thermodynamically exact, are of little practical use. To use them it would be necessary first to make measurements with a large (theoretically infinite) number of cells without liquid-liquid junctions in order to obtain the required values of mean activity coefficients, not to mention the required knowledge of transport numbers. It is therefore expedient to use the formula
where E is the electromotive force of the cell, Eo is a constant, E, is the ideal value for the diffusion potentials for the whole cell, and CH+is the concentration of hydrogen ion in the solution S, and f H + may be called the activity coefficient of the hydrogen ion in the solution S. This formula is actually a conventional definition of Eo and of fH+, the value of either depending on the value conventionally assigned to the other. However as long as the solution S is so dilute that the activity coefficient has approximately the same value for various electrolytes of the same electric type, and as long as the solution S contains only small concentrations of either hydrogen ion or hydroxyl ion, it is possible to assign to Eo a value such 533
534
E. A. GDGGENHEIM AXD T. D. SCHINDLER
that the value of f H + defined and measured according to equation 2 will to within a few units per cent be equal to that of f k , the mean activity coefficient of a uni-univalent electrolyte in solution S. It is only to this approximation that there is any sense in regarding the activity coefficient fH+ measured by such a cell as a thermodynamic quantity. Even at dilutions so great that the activity coefficients, f*, of all uni-univalent electrolytes have the same value to within one per cent or less, it is possible for fH+ to differ by as much as three or four per cent from fi of uni-univalent electrolytes. For this reason there does not seem to be much sense in determining values of - log f5"+fHT, or paH1 as it is usually denoted, to a greater accuracy than 0.01, or at the most 0.005. If the liquid-liquid junctions are made in a suitable manner (7,9, 16, 19) the electromotive forces of these cells are about as reproducible and stable as most cells without such junctions and the value of E, may be calculated by using Henderson's formula (12). Bjerrum and Unmack (3) have made extensive measurements of cells of type 1 and have used formula 2 in their computations. The treatment of our own data is in principle the same as that of Bjerrum and Unmack. We wish however to draw attention to a few differences of detail. First in computing E, by the use of Henderson's formula, Bjerrum and Unmack use the value 0.497 for the transport number tK+ of the potassium ion in potassium chloride. They discuss in some detail how their computations would be affected by the choice of a different value. Meanwhile accurate determinations of this quantity a t 25°C. have been made by Longsworth (13) and by MacInnes and Dole (15). According to their measurements this transport number has the constant value 0.490 at all concentrations up to half molar. As most of the contribution to E, comes from the more dilute end of the transition layer, we have used this value for tK+. Our values of E , therefore differ somewhat from those of Bjerrum and Unmack. For the ions in the more dilute solutions S, the ionic conductivities require to be known only roughly. Second, we do not believe there is anything to be gained in attempting to compute either fH* or related quantities with an accuracy greater than corresponds to the difference between the mean activity coefficients of two uni-univalent electrolytes. Third, whilst in entire agreement with Bjerrum and Unmack as regards the principle to be used in choosing the best value for E,, we make a different choice of solutions to which we attach most weight. 11. MEASUREMEKTS
All the cells measured were of the type 1, the composition of the solution
S being one of those given in table 1 and the temperature being 25°C. I
paH is a conventionalized form of
~ U H .
STUDIES O F CELLS W I T H LIQUID-LIQUID
JUNCTIONS. V
535
The solutions were prepared and standardized as described in the following paper (8). The experimental set-up was in all essential respects that used by Unmack and Guggenheim (19). In particular the junction was made by dipping the side-tube of the hydrogen electrode vessel into a U-tube containing 3.5 molar potassium chloride. The levels of the solutions in the U-tube and the electrode vessel were so adjusted that on opening the stopcock between the electrode and the side tube the liquid from the U-tube always slowly flowed about halfway up the side tube. The electromotive force reached a constant value usually within half an hour. This was reproducible within at least a tenth of a millivolt and stable for at least a day. Four calomel electrodes were prepared and these checked within 0.1 millivolt. TABLE 1 Hg 1 HgCl, 0.1 M KCl 1 3.5 M KCl 1 solution S H, at 25'C. COMPOSITION O B SOLUTION
s
E. Y.F. XEABUREMENTS*
TOTAL
IONIC
Acid constituent
Basic constituent
!TRENGTI
Neutral salt
Our
values
Values determined by
Bjerrum and
Unmack moles per lzter
(a) 0 01 HC1 (b) 0.10 HC1 (c) 0 01 HAC (dl 0 01 HAC ( e ) 0 10 13.40 (fJ 0.0025 NaHzPOa ( g ) 0 025 NaHzP04 (h) 0 050 NaHzPOI
moles per liter
0.01 NaAc 0 . 0 1 NaAc 0.10 NaAc 0.0025 NazHPO4 0,025 NapHPOa 0.050 Na?HPOI
moles per Izter 0.09 KCl
0.09 KC1 0 00242 NaCl 0.0242 NaCl 0.0485 NaCl
0.10 0.10 0.01 0.10 0.10 0,0125 0.125 0,250
mu.
mv.
-458.30 -400.80 -613 90 -609.75 -610.10 -753,40 -738.80 -732.45
-458.40 400.90
-
-753.10 -738.20
2.10 1.07 4.80 4.67 4.66 7.07 6.835 6.735
*Positive value of E means that electrode on right is positive (American convention).
The results of our measurements are recorded in table 1. The first four columns give the composition of the hydrogen electrode solution S. The fifth column gives the measured electromotive lorce corrected to the mean value of the calomel electrodes and to a hydrogen pressure of one atmosphere. The sixth column gives some electromotive force values of Bjerrum and Unmack (3), which agree fairly satisfactorily with our values. The last column will be discussed in the next section. 111. CHOICE OF
Eo
The value of Eo originally proposed by Sorensen was based on a misinterpretation of conductivity data at a time when it was believed that solutions of electrolytes obey the ideal laws. It is now generally agreed
536
E. A. GUGGENHEIM AND T. D. SCHINDLER
and admitted by Sorensen (18) that this value of -337.6 millivolts at 25°C. is not the most desirable. Various authors have suggested other values and some of these are discussed at length by Clark (4). But there has been a natural reluctance to adopt any other value until one could be chosen likely to be strongly established and permanent. Giintelberg and Schiodt (10) discuss the unfortunate confusion due to the continued use of the obsolete Sorensen value and estimate that values of paH or -loglo CE+f ~ calculated + from the Sorensen value are too low by 0.05 =k 0.015, but they agree that an alternative standard value should not be adopted until it could be fixed to within 0.01 in paH. We wish to put forward what we believe to be a strong case for the adoption of the value
Eo
=
-333.7
millivolts at 25°C.
according to which the Sorensen values of pH, assuming the computation of E , has been correctly carried out, will be too low by 0.065. We shall 0.09 M show that for the five solutions (a) 0.01 M hydrochloric acid potassium chloride, (b) 0.10 M hydrochloric acid, (c) 0.01 M acetic acid 0.01 M sodium acetate, (d) 0.01 M acetic acid 0.01 M sodium acetate 0.09 M potassium chloride, and (e) 0.10 M acetic acid 0.10 M sodium acetate, this value of EOleads to most reasonable values of fHC and fAc-. Solutions a and b have often before been suggested as standards. One of the objections raised against the use of solution b is the difficulty of obtaining definite values for E owing to the large diffusion potential. Actually there is no difficulty in obtaining reproducible and stable values if the junction is made in the correct manner. Solutions c, d, and e have become suitable for standards only quite recently, thanks to the accurate determination of the dissociation constant of acetic acid by Harned and Ehlers ( l l ) , who used cells without any liquid-liquid junctions. Let us first consider cells made in the solutions a and b. In both these the value of CHt is known, being equal to the stoichiometric concentration of hydrochloric acid. According to the best values of the universal con-
+
+
tants compiled by Birge (1) the value of national millivolts.
~
+
+ +
RT log,lO at 25°C. is 59.151 interF
We therefore have, according to equation 2,
59.15 l o g , o f ~ += E
- Eo - E, -
59.15 loglo CH+
(3)
Inserting the numerical values of E, Eo, and E, we obtain for cell a 59.15lOgiof~f= -458.3
+ 333.7 + 0.4 + 118.3 = -5.9
and for cell b 59.15 l0glofH' = -400.8 f 333.7
+ 3.64 .f 59.15 = -4.3
The corresponding values of fH- are 0.795 in solution a and 0.85 in solution b.
STUDIES O F CELLS WITH LIQUID-LIQUID
JUNCTIONS. V
537
An entirely different convention sometimes used (14, 17) for assigning values to ionic activity coefficients is to set the activity coefficient fcl- of the chloride ion in both solutions a and b equal to the value 0.76 of the mean activity fKclof potassium chloride in 0.10 M potassium chloride. This convention leads to values of fH+ equal to 0.80 in solution a and 0.84 in solution b. The agreement between these values and those obtained by the use of our value of Eo is extremely satisfactory. Let us now consider the three acetate buffers c, d, and e. Since each buffer contains equal concentrations of acetic acid and sodium acetate, we have, according to equation 2,
where KHAcis the thermodynamic dissociation constant of acetic acid. Its value determined by Harned and Ehlers is 1.75 X at 25°C. From the experimental value of E and the known values of EO,E,, and K E A C , we are able to calculate fAc-/fHAC. The computations are shown in table 2. If we ignore the deviation of fHAc from unity, then we obtain the values of fAc- given in the last row of the table. It is to be noted that the value 0.90 for fAc- a t the ionic strength 0.01 is exactly equal to f k for any typical uni-univalent electrolyte at the same ionic strength. Moreover the values 0.80 and 0.81 offAo-a t the ionic strength 0.10 are within the range of values of the mean activity coefficients of typical uni-univalent electrolytes at the same ionic strength. We see then that the value we suggest for Eo leads to eminently reasonable values f0i-f" in the solutions a and b, and equally reasonable values for fAo- in the solutions c, d, and e. The corresponding values for -loglo CH-fH+ or paH for all five solutions are given in the last column of table 1. The rather large discrepancy between the value -333.7 which we have chosen for Eo and the value -336.0 selected by Bjerrum and Unmack is due partly to their use of the value tKT = 0.497 in potassium chloride and partly to their attaching greater weight to much more dilute solutions of hydrogen chloride, where the experimental uncertainty is likely to be great. Even if the experimental values at these high dilutions are reliable, it seems to us preferable to choose for EOa value which leads to reasonable values of fH+ and fAc- in solutions of concentrations in the more generally useful range of 0.01 M to 0.1 M . It should not be necessary to mention that in order to obtain the best results it is essential not to ignore E,. Whereas its value is usually not more than about one millivolt, there seems to be a surprisingly widespread belief that it is vanishingly small. In the case of measurements on protein solutions and the like, where it is not possible to calculate E,, one can ob-
538
E. A. GUGGENHEIY AND T. D. SCHINDLER
tain a rough estimate of its magnitude by the procedure proposed by Bjerrum (2), of comparing the values of E obtained by using as bridge solution first 3.5 M potassium chloride and second 1.75 M potassium chloride. Bjerrum and Unmack (reference 3, p. 50) give an emphatic warning against exaggerated ideas of the reliability of this procedure. It is to be recommended only when it is not possible to calculate E,. TABLE 2 Hg [ HgCI, 0 . 1 M KCI I 3.5 M KCI solution S [ Hz at 25°C. ~
i
Composition of solutions S,. . . . . . . . . . . . . . .
E .........................................
E - E D . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bo.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
lOg,ofAc- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SAC-.
......................................
paH = - l o g , & ~ + f ~.+. ... . . . . . . . . . . . . . . . .
C
d
e
1.01 NaAc 0.01 HAC
3.10 NaAc 3.10 HAC
-613.9 $1, 85 -3.33 -1.48 -612.42 -333.7
1.01 NaAc 1.01 HAC 1.09 KC1 -609.75 $1.85 -1.85 0.00 -609.75 -333.7
-278.7
-276.05
-275.8
-281.4
-281,4
-281.4
-610.1
1-1.85 -2.46 -0.60
-609.5 -333.7
+2.7
f5.36
f5.6
-0.045 0.90 4.80
-0.090 0.81 4.67
-0.095 0.80
4.66
* The value +l.S5 given first represents the contribution of the junction 0.1 M KCl I3.5 M KC1. The value given next is the contribution of t h e junction 3.5 114 KC1 1 solution S. The third value, representing EDfor the whole cell, is the algebraic sum of the first two. IV. SECOND DISSOCIATION CONSTANT O F PHOSPHORIC ACID
Having chosen a value for Eo we use our electrometric measurements in the phosphate buffers f, g, and h to compute values of CHAfH+ by exactly the procedure used by Bjerrum and Unmack. As the ratio CHPO~--/ CHIPOIis unity in each case, the thermodynamic dissociation constant KH2poa-is given by
(5)
STUDIES O F CELLS WITH LIQUID-LIQUID
JUNCTIONS. V
539
As already mentioned it is possible to compute such a quantity as K H , p 0 4 from measurements of cells with liquid-liquid junctions only with an accuracy that ignores specific differences between the activity coefficients of two electrolytes of the same type. In a paper to be published shortly it will be pointed out that the activity coefficients of numerous electrolytes of all valence types are correctly given within a few units per cent up to an ionic strength of 0.1, if the activity coefficient of each ion of valency 2 is calculated according to
- log,,f
= 0.50 22
dF -
1+dF
where r is the total ionic strength of the solution. This formula is of the Debye-Huckel ( 5 ) type in which the parameter a, the “mean ionic diameter” is set equal to 3.0 A.U. According to this formula we have
Combining the values of fHpOa--,lfH2po4- thus calculated with the values of CH+fH- obtained from the electrometric measurements we are able to obtain values of KH2pO4-. The complete computations are given in table 3. The three independent values obtained for K H 2 p 0 4 - agree remarkably well. As the formula (7) for activity coefficients is not expected to be accurate at an ionic strength as high as 0.25, the agreement of the value of K H2p04- given by the third solution with those given by the other two is probably fortuitous. Taking the mean of the values given by the two most dilute solutions we obtain as the value of KH2p04-,5.97 X lo-* with an estimated accuracy of 2 or 3 per cent. The value estimated by Bjerrum and Unmack (reference 3, p. 132) was 6.20 X
Eo AT OTHER TEMPERATURES As already pointed out by Bjerrum and Unmack the selection of the most likely value for Eo depends on the value assumed for tK+in potassium chloride. This uncertainty is however pronounced only in very dilute solutions. If then we choose standard solutions of total concentration 0.10 M we may without serious error assume that tK- in potassium chloride has the same value, 0.490, at other temperatures as at 25OC. On this basis and using Bjerrum and Unmack’s measurements we come to the conclusion that the most reasonable values of Eo at the various temperatures of their measurements are those given in table 4. We believe that these values are definite to within about one quarter of a millivolt, that is to within 0.005 of a pH unit or within 1 per cent infH+. In the same table are given the values of fH- in three standard solutions corresponding to this choice of Eo V. VALUES FOR
540
E. A. GUGGENHEIM AND T. D. SCHINDLER
and also the values of fH+ calculated on the convention of setting fcl- equal tofKCl in 0.1 M potassium chloride. The agreement between the two independent conventions is most satisfactory. TABLE 3 Hg 1 HgC1, 0.1 M KC1 13.5 M KC1 1 solution S I H, at 86°C. f
[ 0.01 Na+ 0.0025 H ,PO4Composition of solutions S. . . .
0.0025 HPOa-0.0025 C1ionic strength r . . . . . . . . . . . . . . . 0.0125 ~ r . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.112 -753.4 E . . . . ... .. ......... . . ... . ...., f1.85 -3.24 ED *, , . . . . . . . . . . . . . . . . . . . . . . . . . -1.40 -752.0 E - ED . . . . . . . . . . . . . . . . . . . . . . -333.7 Eo ........................ . . . . . -418.3 E Eo - Ea... , , . . . . .. .. . . .. . 7.072 paH = -lOglocH+fH+ . . . . . , . . . 0.847 x 10CH+fH..........................
[
.
-
-1og10-
i -
.
fH2POd-
fHPOr-'
KH,PO,-
,
g
h
0.10 Na+ 0.025 H2PO40.025 HP04-0.025 C10.125 0.353 -738.6 $1.85 -2.35 -0.50 -738.1 -333.7 -404.4 6.837 1.45 x 10-
0.20 Na+ 0.05 HzP040.05 HPOa-0.05 C10.25 0.50 -732.45 $1.85 -2.21 -0.35 -732.1 -333.7 -398.4 6.735 1.84 10-
0.151
x
0,500
0.391
' '
. . ....., .. ..... ... . ,
..I
7,222 6.00 X 10-8
5.94
7.226 10-
x
5.82
7.235 10-8
x
* The value f1.85 given first represents the contribution of the junction 0.1 M KC1 13.5 M KC1. The value given next is the contribution of the junction 3.5 M KC1 I solution S. The third value, representing ED for the whole cell, is the algebraic sum of the first two. TABLE 4 VENTION
O'C. -
18°C.
25°C.
37". fcl-=f~cl -
Eo. . . . . , . . . . . . . , . . . . . . . . . . , . , . . . . . . . -334.2 -334.2 -333.7 -332.7 0.80 0.80 0.80 0.80 fH+ in 0.01 M HC1 f 0.09 M KC1.. . . . 0.83 0.82 0.82 0.82 fH+ in 0.01 M HC1 0.09 M NaC1.. . , 0.84 0.85 0.85 fHt in 0.10 k f HC1. , , . . . . . . . . . .. . . . . . .
+
0.80 0.82 0.84
We are pleased to learn from Professor Bjerrum that he is in complete agreement with the contents of the present paper and that he accepts our Eovalue at 25°C. as the most reasonable one t o use on the available experimental material.
STUDIES O F CELLS WITH LIQUID-LIQUID
JUNCTIONS. V
541
REFERENCES (1) ,BIRGE:Rev. Modern Phys. 1, 1 (1929). (2) BJERRUM:Z. physik. Chem. 63, 428 (1905); Z. Elektrochem. 17, 389 (1911). (3) BJERRUM AND UNMACK: Kgl. Danske Videnskab. Selskab Math. fys. Medd. 9, No. 1 (1929). (4) CLARK:The Determination of Hydrogen Ions, 3rd edition, p. 474. The Williams & Wilkins Co., Baltimore (1928). Physik. Z. 21,186 (1923); 26,97 (1924). (5) DEBYEANDHUCKEL: (6) GUGGENHEIM: J. Phys. Chem. 34, 1758 (1930). (7) GUGGENHEIM: J. Am. Chem. SOC.62, 1315 (1930). (8) GUGQENHEIM AND SCHINDLER: J. Phys. Chem. 38,543 (1934). (9) GUGGENHEIM AND UNMACK: Kgl. Danske Videnskab. Selskab Math. fys. Medd 10, No. 14 (1931). (IO) G ~ N T E L B EAND R G SCHIODT:Z. physik. Chem. 136,436 (1928). (11) HARNED AND EHLERS:J. Am. Chem. SOC.66, 652 (1933). (12) HENDERSON: Z. physik. Chem. 69, 118 (1907). (13) LONQSWORTH: J. Am. Chem. SOC.62,1897 (1930). (14) MACINNES: J. Am. Chem. SOC.41,1086 (1919). (15) MACINNES AND DOLE: J. Am. Chem. SOC.63, 1357 (1931). (16) MACLAGAN: Biochem. J. 23,309 (1929). (17) SCATCHARD: J. Am. Chem. SOC.47,696 (1925). (18) S ~ R E N S EAND N LINDERSTROM LANG:Medd. Carlsberg Lab. 16, No. 6 (1924). (19) UNMACK A N D GUGGENHEIM: Kgl. Danske Videnskab. Selskab Math. fys. Medd. 10,No.8 (1930).
