THE DEBYE-HUCKEL THEORY AND ITS APPLICATION IN THE TEACHING OF QUANTITATIVE ANALYSIS BARNET NAIMAN College of the City of New York, New York
IN
A recent paper1 the advantages of using the Br@nsted theory in the calculation of the pH values of acids, bases and salts were pointed out. It is the purpose of the present communication to show how another modern concept, the Debye-Hiickel theory, may be taught t o advantage in quantitative analysis courses and how the idea of activity, activity coefficient, and ionic strength may be introduced into calculations involving ionic equilibria. Shortly after Arrhenius announced his theory of ionization in 1887, the anomalous behavior of strong electrolytes became apparent through the work of such investigators as A. A. Noyes, Milner, Bjerrum and many others. Arrhenius had assumed that upon solution in water all electrolytes dissociated to varying extents into oppositely charged parbides (ions). A condition of equilibrium was established between the un-ionized molecules of the solute and its ions. The extent to which substances ionized varied all the way from a very low value for substances such as hydrocyanic acid to the highest values (almost complete ionization) for the strong electrolytes such as hydrochloric acid and the true salts. The degree of ionization of weak electrolytes, as calculated from conductivity data, agreed reasonably me11 with the values calcnlated from vapor pressure and osmotic pressure data and from the related measurements of freezing point and boiling point changes. The weak electrolytes also followcl Ostmald's dilution law. However, no such agreement held for t,he strong electrolytes. Further, it was shown that strong electrolytes do not. follow the law of mass action in the slightest degree and that the weak acids and bases follow the law only to a limited extent. Although the addition of a common ion causes a decrease in the concentration of the oppositely charged ion, the decrease is not proportional to the increase in concentration of the common ion except for very slight increases. The so-called mass action constant increased in value as the total ion concentration increased. Following shortly after Arrhenius, Ne&st2 advanced the idea that at constant temperature the solubility of slightly soluble electrolytes in aqueous solution, with or without other electrolytes, is dependent o n a constant, the solubility product constant, which is pro-
NAIMAN, B., THISJOURNAL, 25,454 (1948). YERNST,W., Z. p b y ~ i k .Chem., 4, 372 (1889).
portional to the concentrition of the ions of the slightly soluble electrolyte, each concentration raised to a power equal to t,he number of these ions arising from one molecule. In accordance with the ~ r r h e n i u stheory it was assumed that in the case of a saturated solution of an electrolyte the postulated molecular species remained constant under all conditions. This notion soon came under the attack of many investigators. I t was shown that the solubility product does not remain constant a t various concentrations of salts and therefore the molecular species does not remain constant. As early as 1904, A. A. Noyes, and later Milner as well as Bjerrum, came to the conclusion that strong electrolytes are completely ionized and that the deviations from the various ideal laws were due t o the attraction between the ions in solution. I n 1914 Rragg and Bragg presented proof, based on X-ray analysis of sodium chloride crystals, that salts in the crystalline state contain no molecules but are made up of alternate positive and negative ions held together by electrostatic abtraction. It seemed obvious that upon solution of a salt in a solvent of high dielectric const,ant, such as water, there shonld be no association of the ions into definite molecules. If t,here are no molecules, why then do the st,rong elect,rolytes not behave as if they were 100 per cent ionized? Further, why cannot equilibrium constants be calculated for strong electrolytes and why do the solubility products of slightly soluble salts vary with varying salt concentrations? The answer t o these questions was found in the attraction between the ions in solution. The ions do not act with maximum efficiency in conducting the electric current nor in affecting the vapor pressure of solutions and related properties for the following reasons. As a positively charged particle approaches a negatively charged particle, they mutually "slow" one anot,her down or draw together according to Coulomb's law which states that the force of attraction between oppositely charged particles is directly proportional to the size of the charges, and inversely proportional to the square of the distance between the particles and the dielectric constant of the medium separating them. The higher t,he concentration, the closer are the charged particles to one another and therefore the greater the effect. G . N. Lewis even as early as 1901, and in much sub-
MAY, 1949
denominator is small and may usually be ignored. If the calculated values for f are applied to equation (3), a true solubility product constant may be obtained. It is implied in equation (5) that f values vary with the kind and size of ion, as well as with concentration. This would require many tables off values for each salt and mixture of salts a t different concentrations. However, Lewis showed that "in dilute solutions, the activity coefficient of a given strong electrolyte is the same in all solutions of the same ionic strength." In short, equilibrium constants, such as solubility prodwhere ucts vary with the ionic strength in dilute solutions m = molality (or molarity in dilute solutions) (up to ionic strengths of about 0.25). f = activity coefficient I t is found that if pS (-log S) is plotted against 4; a = activity an almost linear relationship is obtained. In Figure 1 I t is customary to designate.the molar concent,ration values of psareplotted as ordinatesagainst G a n d show by brackets and the activity by parentheses, thus how far from constant the so-called so!ubility product 8' constants" are. In the case of BaSOn, for example, S varies from 1.0 X 10-lo a t zero ionic strength to 2.2 It is the activity which accurately defines the thermodynamic concentrations and which must be used to obtain true equilibrium constants. For example, the thermodynamic solubility product, S, for AgCl is sequent work,3 pointed out the inconstancy of the equilibrium constants of many salts. He introduced the term "ionic strength," or p, to define the thermodynamic characteristics of a solution of electrolytes, "activity," the effective concentration of the ions, and "activity coefficient," a factor by which the stoichiometric concentration or molality must he multiplied to give the effective concentration. He established the relationship mxf =a (1)
[Agilf*.+ X [CI-lfcl- = (Ag+)(CI-) = S
(3)
I t is the activity of an ion which is determined in potentiometric measurements. The activity coefficient varies in sufficiently dilute solutions with the total ionic concentration, or more accurately with the ionic strength of the solution. It, mas defined by Lewis thus,
i. e . , the ionic strength equals the sum of the molarity (more accurately the molality) of each ion multiplied by the square of the valence, Z, of that ion, divided by two (in order to account for the effect of both the positive and negative ions). As the solution approaches a state of infinite dilution, j approaches unity, and the activity becomes equal to the concentration. Debye and Hiickel' were able to correlate the discrepancies concerning the anomalous behavior of strong electrolytes and the effect of diverse ions in their comprehensive Interionic Attraction Theory. They developed a mathematical expression as a first approximation applicable to dilute solutions for the calculation of the Levis activity coefficient,f: logf =
- 1 0.51zp.\/; + 0.33 a
4
where p = ionic strength as defined above, Z = valence of the ion being considered, a = the effective mean radius, in A. units, of all the ions in solution. By the use of equation (5), the activity coefficientof an ion may be calculated. It is obvious that the second term in the LEwrs, G . N., AND M. RANDALlr,"Thermodynami's," McGraw-Hill Book Ca., New York, 1923. ' DEBYE, P., AND E. HBCKBL,Phl~sik.Z.,24,185,305 (1923).
4;
BY permission imm "~uaniitatlve~ n e ~ y r i s2nd . . ad.. by W. ~ i e m a n1. . 0. ~ e u s a and B. Naiman. copyrighteb 1942 by McGrav-Hill Bmk Co..lnc: riwm 1. ps values at 25%
JOURNAL OF CHEMICAL EDUCATION
X a t an ionic strength of 0.25. The value 1.0 X 10-lo, true at infinite dilution, is the one commonly used in textbooks of analytical chemistry for solubility product calculations involving BaSOI. The calculated results are therefore frequently far from the actual experimental results. By use of the graphs relating solubility products to ionic strengths, calculated values more nearly approach experimental values. An example will ~llustratethe calculation of ionic strength and the use of the graphs in solubility product prohlems.5 Example. I n the determination of chlorine the usual procedure was followed and the following data obtained: 2.5mmolsof calcium chloride were dissolved in 139'ml. of water, 1ml. of 6 M nitric acid and 60 ml. of 0.1 M silver nitrate were added. The h a 1 volume was 200 ml.; the temperature was 25°C. CaCl, 2AgNO. = 2AgCll t Cs(NO&
+
Two and five-tenths mmols of calcium chloride yield 5.0 mmols of silver chloride which is removed from solution; therefore 5.0 mmols of silver ion and 5.0 mmols of chloride ion contribute a negligible effect to the ionic strength. The ionic strengt.h calculation may be tabulated as follows: -
Ion Ca++ NOs-
&%+ KOa-
Sowce
CaCl. AgNOs A~NOJ HNOs 11X03
Mmol 2.5 6.0 1.0 6.0 6.0
.?I
0.0125 0.030 0.005 0.030 0.030
Z 2
ZP
1 1
1
1
1
4
1 1 1
ZMZ'
From Figure 1, pS
,,,
=
MZ1 0.050 0.030 0.005 0.030 0.030 = 0.145
0.52.
S = [AgCl[Cl-I = 3.0 X lo-"' [Agtl = 1.0 mmol per 200 ml. = 5.0 X 10-a mnrol/ml.
therefore
' RIEMAN,W., J. D. N ~ u s s AND , B. NAIMAN, "Quantitative Analysis," 2nd ed., McGraw-Hill Book Co., N w York, 1942.
[CI-I =
3'0 10-'0 = 6.0 X 10-ammol/ml. X 200ml. = 1 . 2 X 5.0 X 10-8 10-' mmol.
1.2 X 10-5mmol X 143mg./mmol = 0.0017 mg. lost as AgC1. If the classical method were used: S = [Ag+][CI-1 = 1.0 X lO-'O
and [CI-I =
X lo-'' = 2.0 5.0 X lo-"
x
mmol/ml.
2.0 X 10" mmol/ml. X 200 ml. X 143 mg./mol = 0.00057 mg. loss. A 300 per cent difference is found between the two methods. The advantages of using the theory developed by Lewis and by Dehye and Hiickel in treating solubility product problems and other equilibrium. "constants" are: 1. I t makes the various calculations conform more closely to conditions actually existing in solution. 2. I t therefore leads to more accurate results in the calculations. 3. It makes the student conscious of the presence of an "ionic environment" in the solution and the effect of it on the properties of all ions. 4. I t teaches the student to calculate the ionic strengths of the solutions. 5. I t emphasizes very definitely the inconstancy of the so-called "classical constants." 6. It demonstrates clearly the significance of actvities and activity coefficients. 7. It satisfies the student with a complete answer to the often asked question: "Why use these 'constants' and bother to make the calculations if they are 80 far from accurate?" 8. It acquaints the student. more intimately with the use of graphs on which he can see at a glance how t,he constants vary with the ion c strength. The author acknowledges his debt to William Rieman 111, of Rutgers University, who suggested the methods discussed in this paper.
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