THE TEACHING OF THE THEORY OF THE DISSOCIATION OF ELECTROLYTES. 111. THE COLORIMETRIC METHOD UNIVERSITY OF
JR. MARTIN KILPATRICK, PENNSYLVANIA,PHILADELPHIA, PENNSYLVANIA
A great many colorimetric determinations of hydrogen-ion concentration are based on the assumption that equal colorfor equal indicator concentration means equal hydrogen-ion concentration. The error i n this assumption i s pointed out and the correct relationships are derived. The "salt error" i s formulated. The afiplication of indicators to the general problem of acid strength i n non-aqueous solution is discussed.
. . . . . . The purpose of the present paper is to present the principles upon which the colorimetric method for the determination of pH is based, and to point out that if these principles are correctly taught the colorimetric method may have some advantages over other methods in the general study of acids and bases. As the colorimetric method is taught at present, it is usually based upon the assumption that equal color for equal indicator concentration means equal pH. For the moment we shall use the definition pH = -log c ~ + . In practice, the unknown is compared with a set of standards of known pH, and the nearest match is found. A great deal has been written about the choice of the standards, their stability, and the accuracy of the determinations. It is generally conceded that matches can be obtained within 0.02 of a pH unit (5% in hydrogen-ion concentration) and very often the experimental worke; takes great care with the actual manipulation. However, if his work is basedBn the assumption given above he may be making errors that in some cases amount to 0.5 of a pH unit (several hundred per cent. in hydrogen-ion concentration). Let us examine the theoretical basis of the colorimetric method and consider the validity of the assumption upon which most colorimetric pH measurements are based. Indicators (substances used for the colorimetric determination of hydrogen-ion concentration) are weak acids or bases whose acid form or basic form (or both) is colored. Let us for simplicity consider an indicator HI of the type of a monobasic acid. Now HI is a weak acid and in aqueous solution enters into a double acid-base equilibrium acid HI
+
acid
base
H20 e H,Of
+
base
I-
In fact we can make a buffer solution, just as with any weak acid, by increasing the concentration of the base I-. The buffer reserve, that is to say the resistance to change in the concentration of the hydrogen ion upon addition of acid or base, may be smaller but that is due simply to the low concentration of the indicator. The only essential difference is one of color and upon this the method is based. The idea that the indicator forms 1226
VOL.9, NO.7
DISSOCIATION OF ELECTROLYTES. 111
1227
a buffer system becomes of great importance when the pH of unbuffered solutions is to be determined (1). As pointed out in the second paper of this series (2),in aqueous solution the classical dissociation constant is a good measure of the strength of an acid. In a solution of the indicator HI d"+ =
and when
CHI = 61-
Kr ~ ~ ~ / ~ c -
(1)
(the half-change point) CH*
=
Kc
(2)
that is to say, the hydrogen-ion concentration of the solution containing equal amounts of the two forms is equal to the dissociation constant of the indicator. As pointed out in the first paper of this series (3) Kc changes as the concentration of electrolyte changes, so that the sensitivity of the indicator to the hydrogen-ion concentration also changes. If K, increases the sensitivity decreases, and vice versa. To take a specific case, the halfchange point for bromocresol green* in a solution 0.1 M in potassium chloride lies a t a hydrogen-ion concentration of 3.07 X lo-=, and in a solution 0.5 M in potassium chloride it lies a t a hydrogen-ion concentration of 4.03 X 10-"(4). From this it follows that if the standard solution is 0.1 M and the unknown 0.5 M in potassium chloride, and if the change in the dissociation constant of the indicator is neglected and it is assumed that for equal indicator concentration equal color means equal hydrogen-ion concentration, the error is 25%. The change in the dissociation constant may be expressed in relation to I(,,, the dksociation constant a t infinite dilution, by the fraction KJK,. We shall callrthis fraction the "sensitivity" of the indicator; thus if in a given salt solution K, = 2K,, the sensitivity of the indicator is 1/2, or the half-change point lies a t twice as great a hydrogen-ion concentration as in the infinitely dilute solution. As stated, when the indicator is half transformed KO =
L ~ + ( C ~ - / C H I )= C H +
ca*/K, = 1
For solutions 1 and 2 containing the indicator a t its half-change point (or a t any other selected degree of transformation, i. e., cr-/caI ratio) ( c H + / K J=I ( C H + / K J ~
(3)
and on multiplying both sides by I(,, ( c ~ + & d K J t= (cH-KcJ/K&
(4)
which states that for a given cI-/cXr ratio the product of the hydrogenion concentration and the sensitivity of the indicator is constant. Now two solutions of equal indicator concentration have the same color when 'Whether we are dealing with t h e first or second dissociation constant of the indicator does not affect the principle under discussion.
1228
JOURNAL OF CHEMICAL EDUCATION
JULY,
1932
the ratio of basic form to acid form of indicator is the same in the two. Equation (4), or (3), therefore gives the condition which must be fulfilled in order that two solutions of equal indicator concentration have the same color. In other words, the hydrogen-ion concentrations of two solutions of equal indicator concentration, matched in respect to color, are equal only provided the classical dissociation constant of the indicator has the same value in the two solutions. This is of course true for equal concentrations of the same electrolyte.* Letting 1 represent the known, 2 the unknown solution, it follows from (3) or (4) that (cn'h = ( c H + ) I ( K ~ ~ ( K ~ ) I
(5)
and in order to determine ( c ~ +both ) ~ (I(,), and must be known. The dissociation constants of a number of indicators are to be found in the literature, and further determinations are continually being reported. What we have been considering is nothing more than the so-called "salt error" of indicators. For a discussion of the magnitude of the error for indicators of various types the reader is referred to a paper by Giintelberg and Schiodt (5) which presents very clearly the whole problem of salt error in indicator work. Let us formulate the salt error in pH units, it being kept in mind that pH = -log c ~ . . Upon taking logarithms, and multiplying through by - 1, equation ( 5 ) becomes -log
+
-log ( c x * ) , - log ( K J 2 log ( K O ) , pH, = PHI - log ( K h log ( K c ) , ApH = pH2 - PHI = log ( K M K J r ;
(ctr*)~ =
+
((3) (7) (8)
i. e., if the experimental worker assumes &at equal color for equal indicator concentration means equal pH, he assigns to the unknown solution a pH which is in error by log (KC),/(KJ2units. If the definition p,H = -log ax* is employed, an equation of similar as mentioned in the preceding paper form results. The ratio aKbcI-/cIrr, of the series ( 2 ) , is known as the incomplete dissociation constant and is represented by the symbol K'. For solutions 1 and 2 of equal indicator concentration and equal color (atr*/K'), = cHI/CI- = ( a d l K ' ) ~ (a"*)* = ( ~ B * ) ~ ( K ' ) ~ / ( K ' ) I
(9) (10)
Upon taking logarithms and multiplying through by -1 as before, (10) becomes AP.H = P.H. - P.H, = log (K'h/(K')n
(11)
For the definitionpH(Sorensen) = -log aH+(Sorensen)one obtains an analo-
* Due to the fact that Kc for indicators of the acid type first increases with increasing electrolyte eoncentration and then decreases, it would also be true at two different concentrations.
VOL. 9, NO. 7
DISSOCIATION OF ELECTROLYTES. I11
1229
gous equation in which K' is replaced by Kl and p,H by pH(Sorensen) (2). The statement is often made that the electrometricmethod isthestandard method, and the salt error of the indicator is often expressed as the difference between the electrometric and colorimetric determination of pH. This is misleading. To he sure, the electrometric method has been more extensively used than the colorimetric. As already pointed out in the second paper of the series, the electrometric determination of the hydrogenion activity involves certain assumptions. The electrometric determination of hydrogen-ion concentration also involves assumptions ( 5 ) , (6). The standard in the electrometric determination of hydrogen-ion activity or concentration is a solution of hydrochloric acid. And the ultimate standard in the colorimetric determination is the same. In order to use the colorimetric method one must have reference solutions of known hydrogenion concentration or activity to determine the appropriite dissociation constants of the indicators. The reference solutions do not have to be calibrated electrometrically, but can, if concentration rather than activity is employed, be calibrated kinetically. In fact it is possible to start with a solution of a strong acid, assume it completely ionized, and determine the dissociation constant of one indicator, then by successive determinations of the dissociation constant of an acid and of an indicator proceed up the pH scale. In all method~lectrometric,kinetic, and ~~~~~~~~~~~the ultimate standard is a solution of a strong acid. In making colorimetric determinations the principle devised by Friedenthal and used by Salm, Gillespie, and others is of great convenience (7). It makes it unnecessary to match the unknown ylution with a known, and enables one to determine a t once with a colorimeter of the Gillespie type the ratio of basic form to acid form of indicator in the unknown solution. From the colorimeter reading and the appropriate dissociation constant of the indicator the pH, p.H or pH(S6rensen) of the unknown solution can be calculated according to equations (13), (16), or (19) of the preceding paper ( 2 ) . And conversely the principle can be used to determine the dissociation constant of an indicator in a solution of known hydrogen-ion concentration or activity. I t has been mentioned that the hydrogen-ion concentration (but not the activity) can be determined kinetically. The method consists of calibrating the velocity of a reaction in terms of solutions of known hydrogen-ion and electrolyte concentration, and then using the velocity of the reaction to determine the hydrogen-ion concentration of the unknown. A recent comparison of the kinetic and electrometric methods for benzoic acidbenzoate buffer solutions ( 6 ) showed that the two gave results agreeing within 0.01 of a pH unit (395, in hydrogen-ion concentration). In fact with proper attention to definitions and to medium effects all three methods should give the same values.
1230
JOURNAL OF CHEMICAL EDUCATION
JULY,1932
Let us now turn to the more general problem of acid strength in nonaqueous solutions. Here the electrometric method, whether one uses the hydrogen electrode, the glass electrode, the quinhydrone, or the chlorauil electrode, seems to be more limited in usefulness due to difficulties of various sorts with the electrodes. The colorimetric method, however, shows great promise. Let us consider tbeindicator H Ind. in a solvent such as heptane. This solvent has no appreciable acid or basic properties so that there is no double acid-base equilibrium as there is in water. But if the base B is added to the solution of the indicator H Ind. in heptane acid 1 HInd.
+
base 2 B
acid 2
= HBf +
base 1 Ind.-
From the change in color the shift in equilibrium to the right can be estimated, and a measure of the basic strength of B and of the acid strength of the conjugate acid HB can be obtained. By carrying out such experiments for a series of bases their relative strengths, and the relative strengths of their conjugate acids, can be found (8). There are, to be sure, difficulties. Medium effects are much greater than in water. Due to the low dielectric constant of the solvent, ions associate to form salt molecules, and it is known that in some cases molecules associate to form double molecules. A practical difficulty arises from the low solubility of the salt molecules of many acids and bases. Nevertheless the colorimetric method is a powerful one in the study of non-aqueous solutions of acids and bases which has just begun. With a knowledge of the relative acid strengths of the indicators, acid-base titrations can be carried aut in non-aqueous solution just as readily as in water. In conclusion, it may simply be stated that it is important that the teacher of chemistry have the principles clearly in mind, and lay the foundation for the work which is to follow, rather than allow the teaching to become the presentation of a set of vague definitions and empirical rules.
Literature Cited ACREEA N D FAWCETT, Ind. Eng. Chem., Anal. Ed., 2, 78 (1930); KOLTHOPP, Biochem. Z . , 168,110 (1926); J. Am. Chem. Soc., 53,825 (1931). J. CHEM.EDUC., 9, 1010 (June, 1932). KILPATRICK AND KILPATRICK, KILPATRICK, ibid., 9,840 (May, 1932). CHASE,Dissertation. the University of Pennsylvania, 1931. GiiNrsLseRo AND SCHIBDT,Z. phys'ik. Chew., 135, 393 (1928). KILPATRICK AND CHASE,J. Am. Chem. Sot., 53, 1732 (1931). FnreoeNrHaL, 2. Elecfrochem., 10, 113 (1904); SALM,2. physik. Chem., 57, 471 (1907): GILLESPIE, 1.Am. Chem. Soc.. 42, 742 (1920); CLARK,"The Determination of Hydrogen Ions," 3rd edition, Williams & Wilkins Co., Baltimore, Md.. 1928, Chap. 6. BRBNSTED, Ber., 61, 2049 (1928).
