THE TEACHING OF THE THEORY OF THE DISSOCIATION OF ELECTROLYTES. 11. THE DEFINITION OF pH
The definition of pH i s discussed i n an attempt to give a clear picture of the problem of the determination of hydrogen-ion concentration and "actinrity." Buffer solutions are defined. The relationshigs existing between the various dissociation constants appearing i n the literature on acids and bases are presented. ,
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Practically every textbook on general inorganic, analytical, or physical chemistry defines pH, and the symbol is so much used by chemists, hiologists, and others that its definition is fundamentally important. Yet its definition varies from worker to worker. And very often the teacher and research worker think and talk one definition and measure a quantity which has another dehition. The definition most often taught to the student is pH = log l/ca+ = -log ca+
(1)
where cH+ represents the concentration of hydrogen ion in moles per liter. Yet the experimenter does not often report his results in terms of the actual hydrogen-ion concentration. A review of how this came about may make the situation a little clearer. Originally the term was introduced by Sorensen (1)because of the form of the equation relating the free energy to the hydrogen-ion concentration. At that time it was thought that the freg energy of a given component of an electrolytic solution could be expressed by the equation
where F is the partial molar free energy of the component, c its concentration, and B a constant dependent on the temperature and pressure and on the component in question. For the concentration cell
where E represents the electromotive force of the cell, corrected for the liquid junction potential, cH+and cfH+are the hydrogen-ion concentrations of solutions S and S', and R, T , N, and F have their customary significance. The standard hydrogen-ion concentration was chosen equal to unity, and the term log l / c H +was called the hydrogen exponent or pH. Sorensen's reference solution was a 0.1 N solution of hydrochloric acid, and the calomel electrode in 0.1 N potassium chloride solution was referred to the reference solution by means of the cell 1010
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DISSOCIATION OF ELECTROLYTES. I1 Pt. HZ
I
HCI 0.1 N
I' 11
KC1 3.5 N
1
KCI, 0.1 N HgCL
I
1011
Hg
where E again represents the electromotive force of the cell corrected for the liquid junction potential. On this basis there has been developed a considerable literature in which pH has been spoken of and defined as -log c ~ + . For the sake of clarity, however, let us call these values pH (Sorensen) values. They comprise the values determined by means of the cell above, using Sorensen's EQ,. or by comparison with Sorensen's standard buffer solutions. It was later realized that in the method outlined there are two other difficulties besides the difficulty of evaluating the liquid junction potential. In the first place, Sorensen based his calculation of the hydfogen-ion concentration of 0.1 N hydrochloric acid solution upon the conductance data. In other words, Sorensen took the X/Xo ratio as the degree of ionization of the acid, and since for 0.1 N hydrochloric acid X/Xo is 0.92 a t IS0, he obtained a hydrogen-ion concentration of 0.092 N and a pH of 1.04 for the reference solution (2). It is now known that the hydrogen-ion concentration of a 0.1 N hydrochloric acid solution is practically 0.1, pH = 1.00. The second difficulty arises in the application of the classical equation relating concentration to free energy for solutions of electrolytes. With the realization that the classical expression did not hold, Lewis introduced the concept of activity and defined it in terms of free energy as follows: P=RTlna+B
*
(4)
Equation (3) then becomes AF = -FE = R T In ae+/a'a+
(5)
so that, with a proper choice of a standard, and provided certain non-thermodynamic assumptions are made, one can obtain a value of the hydrogenion activity.*
* It is now realized that measurements of cells with or without liquid junction do not yield any information about ionic free energies. From cells without liquid junction one obtains the product of the activities of the ions of the electrolyte, anuaxq,where R,X, dissociates t o give @ cations and p anions, or the ratio of the activities of two cations or two anions, an1/"/an'l/~where the ions have valences v and p, respectively. By making some arbitrary assumption, such as ax+ = an- in a solution in which potassium chloride is one of the components, one can obtain numerical values of the activities of the ions present. In regard to cells with liquid junction, it is thermodynamically impossible to evaluate the liquid junction potential without howing the activities of the individual ions, and conversely it is impossihle to evaluate from electromotive force measurements the individual ionic activities without knowing the liquid junction potential. Taylor (3) has shown that the electromotive force of a cell with transference (liquid junction) is a function of molecular free energies only. I t fallows therefore that in order t o obtain
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JOURNAL OF CHEMICAL EDUCATION
Sum, 1932
By means of equation (5) a new pH, designated as p.H, was introduced (5) p.H = - log ex+
(6)
In order to determine p.H from measurement of the electromotive force
a method of correcting for the liquid junction potential must be agreed upon and Eomust be fixed. EOnow represents the electromotive force of the cell above when the h$drogen electrode dips into a solution in which the hydrogen-ion activity is unity. Bjerrum and Unmack (6) measured the electromotive force of cells of the type given, where S represented a solution of potassium chloride and hydrochloric acid, or sodium chloride and hydrochloric acid, the alkali metal halide being present in great excess over the acid. By a method of extrapolation to infinite dilution they obtained EQfor a number of temperatures. Other workers have used different values of Eo. If the solution S be made 0.1 N hydrochloric acid, for which the product aH+ac,-is known, and if it be assumed that an+ = ac,-, a value of Eo can be calculated. Needless to say, for each EOemployed one obtains a different "p,H" for a given solution. In particular, if Sorensen's value is employed, the resulting quantity [which we have called pH(Sorensen)] is given as nearly as can be determined by the equation. pH (%reusen) = -logl.laH+
(7)
a t room temperature. Let us summarize what has been said regarding pH, pH(Sorensen), and p,H. 1. pH = -log c ~ + . This is the definition in terms of concentration. It has been seldom used. 2. pH(Sorensen) = -log l.laH+. From E, the measured electromotive force of the cell
corrected for the liquid junction, and from Sorensen's value of Eo,the pH(Sorensen) of the solution S may be calculated according to the equation pH (Siirensen) = (E - Eo)F/2.303RT ionic activities from the electromotive force of a cell with liquid junction, some arbitrary convention must again be adopted (4). Practically speaking, this means that in determining the "hydrogen-ion activity" by means of the cell commonly used, 3.5 N 0.1 N KC1 Hg, Pt. HZ Soh. S KC1 HgCl Ea must be fixed and the liquid junction potential must he corrected for consistently
1
1
1
1
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DISSOCIATION O P ELECTROLYTES.
I1
1013
As already mentioned, a considerable literature of pH(S6rensen) values has been developed. 3. p.H = -log an+. This is the definition in terms of the hydrogenion activity.* It is not surprising that this state of affairs has led to confusion among the workers in the field. The question is-What shall we teach the elementary student so that the teacher in the more advanced course may have something on which to build? The authors would advocate teaching the student the definition based upon the hydrogen-ion concentration, and continually emphasizing the definition. Too many workers in the field forget that pH represents the logarithm of a quantity, and too often the student thinks that the difference between two determinations such as pH = 5.00 and pH = 5.05 is one per cent in hydrogen-ion concentration. I t is possible to carry out in the laboratory experiments in which the hydrogen-ion concentration is determined by electrometric, colorimetric, and kinetic methods (7). And in many cases a knowledge of the concentration of the hydrogen ion is more important than a knowledge of its "activity." The various definitions of pH have led to various forms of the mass law equation; these will be discussed after a discussion has been given of acidity and buffer systems in general. Two recent papers in this JOURNAL have called attention to the advantages of the modern definition of acids and bases as put forward by Bronsted and Lowry (8). According to this definition, an acid is defined by the formal equation c
A=B+Ht
in which A represents an acid and B the conjugate base with electrical charge less by one than that of A. The hydrogen ion, H30+, is therefore only one of a large number of acids and the hydroxyl ion is only one of a large number of bases. Let us confine ourselves for the time being to aqueous solutions. Since the solvent water has basic properties, there exists in an aqueous solution of an acid a double acid-base equilibrium. For example, if hydrogen chloride is dissolved in water add HCL
+
base
H20
=
acid
Haof
base
f
C1-,
where it is indicated by the size of the type that in this case the reaction proceeds practically completely to the right. In other words the acid, hydrogen chloride, gives up its proton to the base, water. If hydrogen bromide were used instead of hydrogen chloride the reaction would proceed even more completely to the right. But the reaction in the case of hydrogen chloride is already so nearly complete that we are unable to determine the amount of molecular hydrogen chloride present. This leveling
* See the footnote on page 1011.
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JUNE. 1932
effect of the solvent makes all strong acids appear to be of equal strength in aqueous solution. Now let us take an acid which does not give up its proton so readily, acetic acid, for example. Here acid
HAc
base
+
HeO
=
acid H,O+
+
base Ac-.
At equilibrium there is an appreciable quantity of molecular acetic acid, a fact of which we take advantage in making buffer solutions. If we increase the concentration of the base acetate ion
we cause a decrease in the hydrogen-ion concentration and a t the same time make the system less sensitive to changes in hydrogen-ion concentration upon the addition of acids or bases. In other words, we have buffered the solution in respect to the hydrogen ion. Of course we can also buffer the solution in respect to the acetate ion by increasing the hydrogen-ion concentration. This can be done by adding a strong acid.
Now let us consider the problem of acid strength. If we had used formic acid instead of acetic we should have had a diierent equilibrium, for formic acid gives up its proton to the base water more readily than does acetic acid. Consequently we could define acid strength by the equilibrium constant of the formal process A e B +&I' KA' = csca+/c*
(8)
and basic strength by the reciprocal of this constant. Since, however, the concentration of the proton in aqueous solution is of the order of 10-1" moles per liter and cannot be accurately measured, we relate the constant KA1to the equilibrium constant KOwhich defines the acid property of the hydrogen ion, H30+.
+
HsOC=H20 Ht KO= c a m c ~ + / c ~ ~ o +
(9)
Dividing (8) by (9) one obtains Considering the concentration of the water as constant and transferring it to the other side, CRCH.O*/C~
=
C H ~ K I ' I K=OKA
(11)
K g , the constant of basic strength, is the reciprocal of KA. KA is the same as the classical dissociation constant KGso that in aqueous solution the classical dissociation constant can be taken as a suitable measure of the acid strength. As was pointed out in the previous paper K, is
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DISSOCIATION OF ELECTROLYTES. I1
1015
dependent on the ionic concentration. If we know K, for a partkular concentration of any electrolyte we can calculate the hydrogen-ion concentration C ~ U *=
Kd
(12)
c a
where A represents the acid in question and B its conjugate base. Defining pH in terms of the hydrogen-ion concentration and letting pK, represent -log I(,, (12) becomes pH
=
PK,
+ log c d c n
(13)
Equation (13) is based on the definition of pH in terms of concentration. If we use the definition p,H = -log a ~ we + employ a dissociation constant which is best referred to as the incomplete dissociation constant* K' = aE+c~/c*
(14)
e ~ + = K'cA/ca
(15)
From (14) and upon setting p,H = -log ax+ and pK' = -log K' P.H
=
PK'
+ log c d c *
(16)
Using the definition pH(Sorensen) = -log 1 . 1 ~ ~=4 -log aa+(Sorensen) we have corresponding to equations (14), (15), and (16), respectively, the equations K,= a~+(S6rensen) cs/c* (17) an+(S6mnsen) = K,CA/CB pH(Sdrensen) = pKl +'log CB/CA
(18) (19)
The greater part of the literature on buffer sys&ms is based on equations (17), (la), and (19). The prevailing confusion is due largely to the failure of authors of papers to state which definition of "pH" they are using. In addition a confusing point arises upon extrapolation of data to infinite dilution. K,, and K'o, the limiting values aproached by K, and K', respectively, are equal to each other and equal to the thermodynamic dissociation constant. However K,., the limiting value approached by K,, does not coincide with the thermodynamic dissociation constant. For IG, the relationship pKrD= pK'a = pK,,
+ 0.04
(20)
is valid. It is essential that one knows which dissociation constant is under consideration. Confronted with the problem of presenting the ideas outlined to the elementary student, the authors would recommend presentation from the
* This name arose from the fact that K'falf* = Kca where f, and f* are the activity coefficients of B and A, respectively, and K, is the thermodynamic dissociation constant, the limiting value approached by KC as the solution becomes more and more dilute.
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JOURNAL OI; CHEMICAL EDUCATION
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standpoint of concentration. In a third paper it is proposed to discuss the colorimetric determination of pH and to point out that if definitions are kept clearly in mind and medium effects are taken into consideration the pH values determined electrometrically, catalytically, and calorimetrically are the same within the experimental error of the measurements. Literature Cited Compl. rend. Lab. Cnrlsberg, 8, 23 (1909). ( 1 ) SORENSEN, J. CHEM.EDUC;.9,840 (May, 1932). ( 2 ) KILPATRICK. I.Phys. Chem.. 31, 1478 (1927). (3) TAYLOR, (4) GUGGENHEIM, ibid., 34, 1758 (1930). (5) CLARK, "The Determination of Hydrogen Ions," 3rd edition. Williams & Wilkins Co., Baltimore, Md., 1928. Chap. 23 gives a detailed djscussion of the term. ~ UNMACK.Del. Kgl. Danske Videnskeb. Selskob., Malh.-fyr. Medd., ( 6 ) B J E R RAND IX,N o . 1 (1929). (7) KILPATR~CKAND CHASE,5.A m . Chem. SOL.,53, 1732 (1931); CHASEAND KILPATRICK, ibkd., 53, 2589 (1931). J. CHEM. EDUC.,7,782 (ISRO); KILPATRICK, ibid.. 8, 1566 (19.31). (8) HALL,
