THERMODYNAMICS OF THE SECOND IONIZATION OF ORTHOPHOSPHORIC ACID
May, 1958
designed to test the electrode at more alkaline pH's are described in Part I1 of this work. The author wishes to acknowledge his indebtedness to Drs. E. M. Crook and S. P. Datta for their invaluable advice on many of the problems en-
555
countered in this work, and to Mr. J. Trendall for making the glass apparatus. He also wishes to thank the Central Research Fund of the University of London for a grant to cover part of the cost of the equipment.
THE STANDARD POTENTIAL OF THE CALOMEL ELECTRODE AND ITS APPLICATION I N ACCURATE PHYSICOCHEMICAL MEASUREMENTS. 11. THERMODYNAMICS OF THE SECOND IONIZATION OF ORTHOPHOSPHORIC ACID BYA. K. GRZYBOWSKI Department of Biochemistry, university College, London, England Received August 88, 1967
The second ionization constant of ortho hosphoric acid has been determined a t 5" intervals from 5 to 50" using cells of the type Pt, Ht; KHZPOl(ml), K N & H P & ( ~ z )NaCl(ms), , HgzClz; Hg. The function relating p K z to temperature is pK2 = 1775.812/2' 3.9762 0.01750892'. A careful comparison has been made of thevaluesof thepK2and theassociated thermodynamic quantities determined in this work and those of earlier investigators, namely, Nims' and Batea and Acree.2,a
-
+
I. Introduction The purpose of this work was to determine the usefulness and limitations of the calomel electrode in experiments designed to obtain the thermodynamic ionization constants of acids. When such constants are known a t a number of temperatures they can be used to calculate the thermodynamic quantities associated with ionization, Le., the standard free energy change, entropy change, etc. I n some biochemical reactions occurring in a medium of constant pH, e.g., the hydrolysis of organic phosphates, the change in the ionization of the reactants and products may contribute a substantial part to the over-all free energy change of the reaction. The knowledge of the dissociation constants is also useful in the investigation of the kinetics of such reactions. I n order to test the calomel electrode, it was decided to employ it in the accurate measurement of the second dissociation constant of orthophosphoric acid. Phosphoric acid was chosen because its pK2 was known already with considerable accuracy from the determinatioiis by Bates and Acree2s3who used cells without liquid junction containing hydrogen and Ag;AgCl electrades over the range 0-60". Comparison of the values obtained with similar cells but with the calomel electrode in place of the Ag ;AgC1 electrode should yield information about the accuracy obtainable with the former electrode in measurements of this kind. 11. Experimental Procedures The phosphate buffers were prepared by the addition of standard NaOH to A.R. grade KHzPO4 recrystallized twice from water and dried over P z O ~ . The standard NaOH was made from a saturated solution in which NazCOa is insoluble prepared from A.R. grade NaOH. The solution after the solid NaOH and Na2CO3 had settled was transferred by pressure of COz-free nitrogen into an appropriate volume of conductivity water which had been bubbled with CO2-free nitrogen beforehand, care being taken to exclude COZ at (1) L.F. Nims, J. Am. C h e n . Soc., 56, 1946 (1933). (2) R. G. Bates and S. F. Acree, J . Research NatE. Bur. Standards, 3 0 , 129 (1943). (3) R. G . Bates and S. F. Acree, i b i d . , '34, 373 (1945).
every stage of the transfer. The dilute NaOH solution was subsequently standardized by weight titrations with A.R. potassium hydrogen phthalate three times recrystallized from conductivity water. Agreement among four separate analyses was = t O . l % . NaCl free from bromide and iodide was prepared accqrding to the method described by Pinching and Bates,4 z.e., a saturated solution of NaCl was bubbled with chlorine gas and then boiled to remove any bromine, iodine and excess chlorine. The NaCl was subsequently precipitated twice with gaseous HC1, and dried over PzOS. Finally to remove any excess HCI the dry NaCl was fused in an electric furnace. A11 concentrations were expressed on the molal scale, and each solution was prepared by weighing out the appropriate amount of a stock solution of the materials mixed in the correct proportions and diluting it with a known weight of conductivity water, all weights being corrected for the buoyancy of air. Only one stock solution was prepared consisting of KHzP04 partly neutralized with NaOH, plus NaCl, the ratio of KNaHPOd: KHzPOa: NaCl being 1 : 2.0527: 3.3349. Care was taken in the preparation of the solution and in their transfer to the cells to prevent their contamination with COZ. The e.m.f.'s of the cell P t ; Hz,KHzPOd(ml), KNaHP04(mZ)? NaCl(m3), HgzC11; Hg were measured in the way described for HCl solutionR in Part I of this work.
111. Theory The second thermodynamic dissociation constant of orthophosphoric acid may be calculated from eq. 1, Le. -log K
- PZ
(E
- Eo)F+ log mCLmHeP04- +
= In 1o R2'
mHPo,a-
2AZ1/2
I
+ Ba*I1/z
By plotting the R.H.S. against I and extrapolating to I = 0, -log K = intercept is obtained. a* (the so called mean distance of closest approach of the ions) and p are arbitrary constants; A is the Dehye-HuckelS limiting lam constant. (4) G . D. Pinching and R. G. Bates, J . Research NafZ. Bur. Standards, ST, 311 (1946). ( 5 ) P. Debye and W. Hdckel, Physik. Z . , 24, 185 (1923).
A. K. GRZYBOWSKI
556
Vol. 62
The pKz values determined at 5" intervals from 5-50" were fitted to an equation suggested by Harned and Robinson,6 Le.
.-
9 z
-log K = $ - D + C T
c
.4 '3
(2)
The constants A , D and C can be evaluated by means of orthogonal polynomials.' Also it is relatively simple to compute the effect of the experimental errors on the pK's and the thermodynamic function derived from them; the method used was that described by Please.8 The thermodynamic functions AFO, AHo, ASo and ACpo can be obtained from the temperature coefficient of -log K. 0 and Ke, the temperature of maximum ionization and the maximum ionization constant, respectively, are given by eq. 3 and 4
%
z
log Ke = -2(AC)*/s
+D
(4)
IV. Results and Discussion The behavior of the Hg;HgzC12electrode in cells without liquid junction follows superficially at any rate the same pattern as that of the Ag;AgCl electrode, ie., the accuracy of the e.m.f.'s decreases as the temperature increases. This is reflected in the much higher standard error in the pK's d e termined at higher temperatures. Although the calomel electrode appears to be somewhat more sensitive to elevated pH's than the Ag;AgCl electrode, the agreement between the values of the second pK of orthophosphoric acid found in this work and those of Bates and Acree2s is very good (Table 11). Up to 40" there is no significant difference between the two sets; above 40" the values determined by Bates and Acree are somewhat higher. This makes the temperature coefficient of the pK determined in the two investigations somewhat different, i.e., there is a discrepancy in the values of the parameters of the Harned and Robinson equation. (Bates and Acree3-eq. 5 ; this investigation-eq. 6.)
+ 0.0209122' 1775.812 pK2 = ___ - 3.9762 + 0.01750892' T pK2
0
rl
0
m *
3
b
It
2073*0 - 5.9884
(5)
(6)
This disparity at higher temperatures leads to somewhat different temperatures of maximum ionization (Bates and Acree3-41.7"; this work -45.3") as well as causing discrepancies in the values of AHo and ACPo as both these quantities are very sensitive to variations in the temperature coefficient of the p K . However, the agreement between the, values of AFDand ASo found in the two experiments is quite satisfactory especially a t 25". I n Fig. 1 the extrapolation to zero ionic strength of one of the sets of results obtained by Bates and Acree2 and those found in this work are compared. The two intercepts are at 25", 7.197 (6)H. S. Harned and R. A. Robinson, Trona. Faroday Soc., 36, 973 (1940). (7) R. A. Fisher, "Statistical Method8 for Research Workers,'' 10th Ed., Edinburgh, 1948. (8) N. W. Please, Biochsrn. J . (London),66, 196 (1954).
May, 1958
THERMODYNAMICS OF THE SECOND IONIZATION OF ORTHOPHOSPHORIC ACID
557
TABLEI1 p & of orthophosphoric acid; pKz
was obtained from equation 1; P&.oalod..was calculated from equation 2. are the arbitrary constants of equation 1, used in this investigation. obsd.
a* and p
Bates and Acreea
0
5 10 15 20 25 30 35 37 40 45 50 55 60 uav. 3
7.2797 7.2525 7.2305 7.2129 7.2004 7.1902 7.1828
13 13 13 10 11 14 12
7.1783 7.1758 7.1764
13 19 27
i 1 6 X
pKz = ___ 1775312 T pKe = 7.1759
7.2782 7.2530 7.2318 7.2142 7.2002 7.1895 7.1820 7.1798 7.1775 7.1760 7.1771
+15 -5 -13 -13 2 7 8
3.6 3.7 3.8 3.8 3.8 3.8 3.8
-0 ,0817 - .0815 - .0929 .0845 - .0936 ,0942 ,0851
-
3.8 4.0 4.0
-
+
+ + +8 -
2 7
Aav. = i l l X
lo-*
- 3.9762 + 0.0175089T e = 45.3"
-
.0831 ,1062 .lo83
PKlobad.
PKzas1.d.
7.3131 7.2817 7.2537 7.2312 7.2130 7.1976 7.1891 7.1850
7.3129 7.2810 7.2540 7.2315 7.2134 7.1994 7.1893 7.1829 7.1813 7.1800 7.1806 7.1843
7.1809 7.1809 7.1831 7.1870 7.1944 A av. = A12 X
pK2 =
A X 10'
-
2 7 3 3 4 +18 2 -21
+ + + + -
9 3 +12
2 E -o 5.9884 T
+
0.020912T pKe = 7.1798 9 = 41.7" values from 0 t o 50" only used in the derivation of above eq.
7.2178 7.2058 7.1973 7.1918 7.1890 7.1888 7.1912
7.2177 7.2060 7.1974 7.1917 7.1903 7.1889 7.1888 7.1912
-1 $2 $1 -1 -1 0
0
A av. =
+
0.0165569T pKs = 7.1885 9 = 42.7"
1
and 7.200, respectively, the values of the slopes 7.18 being -0.153 and -0.094. The difference between IIthe intercepts is not significantly greater than the sum of the statistical errors for the two sets (d= 0.0018 and 0.001 1). The values of pK2 measured by Nimss from 20 to 50" are consistently higher than those found here and by Bates and Acree. The thermodynamic functions obtained from the temperature coefficient also differ more than would be expected from the experimental errors. Since Nims used largely the same method, the reason for the discrepancy is obscure. Also puzzling is the markedly lower value of AHo a t 25" determined by Pitzerlo using a careful calorimetric technique. A number of determinations of the second ionization constant of orthophosphoric acid have been made using cells with liquid junctions. Most of these values (compiled by Bates and Acree2) are of comparatively low accuracy with the possible exception of that of Bjerrum and Unmackll who found pKz (extrapolated to zero ionic strength) to be 7.227, 7.207 and 7.165 at 18, 25 and 37") respectively. (The interpolated value at 18" obtained in this investigation is 7.221.) 0.02 0.04 0.06 0.08 0.10 It is interesting to estimate the effect of slightly Ionic strength. different values of the fundamental constants on the Fig. 1.-The second dissociation of orthophosphoric acid. value of p l G . Bates and Acree used values of Some extrapolations to zero ionic strenrrth (eq. 1): 0 , this RT/F and the Debye-Huckel constants A and B work; X , Bates and Acree. Top P I X . e , t 50"; middle listed by Hamer, Burton and Acree12and the values curve, 25'; bottom curve, 5". of EaA,;Agcl determined by Harned and Ehlers13 whereas in this experiment the value of E o ~ g ; ~ g z ~ l r (9) L. F. Nims. J . A m . Chem. Soc., 66, 1946 (1933). as well as R T / F , A and B were computed using the ( I O ) K. 9. Pitzer, ibid., 66, 1946 (1933). fundamental constants compiled by DuMond and (11) N. Bjerrum and A. Unmack, Kul. Danske Videnskab. Selskab. Jlat-PUs. Medd., 9, 129 (1929). Cohen.14 To carry out a strict comparison between (12) W. J. Hamer. J. 0. Burton a n d S. F. Acree, J . Research Null. the two determinations, the value of E o ~ g ; ~ ~ g A ~ Bur. Standards, 84,269 (19.40). (13) H. S. Harned and R . W. Ehlers. J. A m . Chem. Soc., 64, 1350 (14) J. W. M. DuMond and E. R. Cohen, Rev. Mod. Phye., 86, [3] (1932).
691 11953).
,
A. K. GRZYBOWSKI
558
Vol. 62
7.28 .
7.26
*
7.24
-
7.22
.
7.20
.
%
7.18 . L
'
10
20
30
40
50
t, "C.
Fig. 2.-Variations of the second pK of orthophosphoric acid with temperature: 0 , this work; X , Bates and Acree; o, Nims.
at 25" was recalculated using the same fundamental constants as those employed by Harned and Ehlers, and was then used together with the constants of Hamer, Burton and Acree in the calculation of the second pK of orthophosphoric acid. The value was identical with that obtained with the new constants to three decimal places (7.200), the difference being in the fourth decimal place only, Le., much less than the experimental error. Thus the calomel electrode appears to be quite satisfactory in measurements involving phosphahechloride buffers. Unfortunately further experiments in this Laboratory have shown that its applicability is not as general as that of the Ag;AgCl electrode. For example, attempts to measure the second ionization constant of glycerol-1-phosphoric acid using the calomel electrode have failed because mercury ions (Le., Hg2++, Hg++, or possibly some chloromercury ions) appear to interact with the glycerol-1-phosphate ion causing a drop in the pH of the buffer solution. The values of E - Eoare much lower than those obtained with the Ag ;AgC1 electrode and the e.m.f.'s themselves are much more erratic. However, the agreement between the values of the second ionization constant of orthophosphoric acid found in this investigation and those of Bates and Acree appears to be sufficiently good to justify the conclusion that the value of the standard potential of the calomel electrode determined in the first part of this work is consistent a t least within f100 pv. with the value of the standard potential of the Ag;AgCl electrode used by Bates and Acree and that the calomel electrode is capable of giving reproducible e.m.f.'s comparable, in the case of
B
I
i;l
Q
I
I4
0
rn
M
E?
0
May, 1958
FREEZING POINTS IN MULTICOMPONENT SYSTEMS
simple acid buffers, to those obtainable with the Ag ;AgC1 electrode. The author wishes to acknowledge his indebtedness to Drs. E. M. Crook and S. P. Datta for their invaluable advice on many of the problems en-
559
countered in this work, and to Mr. J. Trendall for making the glass apparatus. He also wishes to thank the Central Research Fund of the University of London for a grant to cover part of the cost of the equipment.
A PRECISION METHOD WITH AUTOMATIC RECORDING FOR THE STUDY OF FREEZING POINTS IAT MULTICOMPONENT SYSTEMS BY ROBERT H. DETTRE'AND DONALD H. ANDREWS Department of Chemistry, The Johns Hopkins University, Baltimore,Maryland Received October 11, 1967
An improved method has been developed for measuring the temperature of equilibrium between the crystal and solution phases of multicomponent systems using a constant temperature differential to control rate of cooling and automatic recording of temperature as a function of time. This makes possible the determination of true equilibrium temperature without corrections for undercooling. Where crystallization velocities and rate of attainment of equilibrium are small, this enables one to obtain the true equilibrium temperature from a study of the effect of cooling rate on the temperature displacement of the cooling curve. The method has been applied to the system diphenylmethane-diphenyl ether and compared with other methods.
As pointed out by Guggenheim,2 there has long been a need for more extensive and precise measurements of the equilibrium properties of simple liquid mixtures. This is important not only to improve our theories of mixtures, but also to expand our knowledge of the liquid state itself, for which no adequate theory has yet been developed. One of the most promising approaches lies in the study of the equilibrium between a liquid mixture and the crystalline phase of one of its components, since our knowledge of the nature of crystals has far outdistanced our knowledge of liquids, and in this type of system we have an exchange of molecules between an ordered and a disordered state, both being a t approximately the same density. Ideally one would like to have a sufficiently precise knowledge of the composition and the corresponding equilibrium temperature to permit an evaluation of activity coefficients, and to be of help in calculating potential functions and various statistical properties. Generally speaking this demands a measurement of equilibrium temperature to a t least 0.01'. The early methods of measurement were far from this goal, though later improvements have made it possible to attain this precision, especially in cases where there is a relatively small concentration of the component not in equilibrium with a crystalline phase. Particularly noteworthy work in this field has been done by Skau and Saxby Mair, Glasgow and R o ~ s i n i ,by ~ Witschonke6 and by Stull.'j ( 1 ) Thia article is based on a dissertationsubmitted in partialfulfillment of the requirement for the Ph.D. degree a t The Johns Hopkins University. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under contract No. A F 18(600)-765. Grateful acknowledgment is made of fellowships from the Procter and Gamble Company (1954-55) and from the Kennecott Copper Corporation (1955-56). (2) E. A. Guggenheim, "iMixtures," Oxford University Press, London, 1952, Preface. (3) E. L. Skau and B. Saxton, THIS JOURNAL, 37, 183 (1933). (4) B. J. Mair, A. R. Glasgow and B. D. Rossini, J . Research Natl. Bur. Standards, 26, 591 (1941). (5) C. R. Witschonke, Anal. Chem., 24, 350 (1952). (6) D. R. Stull, Ind. Eng. Chem., Anal. Ed., 18, 234 (1946).
There still remains a large number of interesting systems, however, where small rates of crystallization in some concentration ranges make the present methods inadequate. For this reason we have undertaken the study reported here in order to develop a method where automation results both in improved precision and a greater ease of operation, which will be beneficial in making an extensive survey of different types of systems. This method uses a temperature differential between sample and shield automatically held a t a constant value, while the temperature of the sample is observed with a recording potentiometer with an absolute accuracy of f0.002'. The advantages of this method will be discussed further in connection with studies we have made to compare it with some of the previously used methods. Apparatus and Test Materials.-The temperature sensitive element is a six-junction copper-Advance wire thermopile. ("Advance" wire, manufactured by the DriverHarris Company has approximately the same proportion of copper and nickel as constantan wire). High accuracy and sensitivity are obtained by maintaining the reference junction of the thermopile within 0.2" of the temperature that is being measured. The resulting output of the thermopile (0 to 50 microvolts) is amplified and recorded using a stabilized d.c. amplifier in combination with a recording potentiometoer, with full scale deflection of the latter corresponding to 0.2 . The reference junction temperature can be kept constant to 0.001' or better at any temperature from -40 to +40°. The working junction and reference-junction assemblies are enclosed in an insulated box which is cooled by circulation of cold dioxide gas produced by the evaporation of "Dry Ice" located in the left side of the box. The box temperature is controlled to f1 using a heater placed in front of a circulating fan and thermoregulator and it is usually kept 6 to 7 " below the temperature of the reference junction. Figure 1 shows the details of the thermopile and its location in the sample holder; six junctions were chosen as the optimum number for the desired sensitivity and space requirements, for minimal heat conduction along the wires and for small heat capacity. The work of White' and Robertson and LaMerEserved as a guide in the construction of the thermopile. The elements consist of No. 32 copper wire and No. 30 advance wire. T.le whole assembly is stationary and all the alloy wires are inside the constant-tempera(7) W. P. White, Phus. Reu., 31, 135 (1910); W. P. White, J . Am. Chem. Soc., 36, 2292 (1914). (8) C. Robertson and V. I
