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THE DEBYE-HUCKEL THEORY ND ITSAPPLICATION IN THE TEACHING OFQUANTITATIVE ANALYSIS

BARNET NAIMANCollege of the City of New York, New York

N A recent paper1 the advantages of using the Br@n-sted theory in the calculation of the pH values of acids,bases and salts were pointed out. It is the purposeof the present communication to show how anothermodern concept, the Debye-Hiickel theory, may betaught to advantage in quantitative analysis coursesand how the idea of activity, activity coefficient, andionic strength may be introduced into calculations in-volving ionic equilibria.Shortly after Arrhenius announced his theory ofionization in 1887, the anomalous behavior of strongelectrolytes became apparent through the work of suchinvestigators as A. A. Noyes, Milner, Bjerrum andmany others.Arrhenius had assumed that upon solution in waterall electrolytes dissociated to varying extents intooppositely charged parbides (ions). A condition ofequilibrium was established between the un-ionizedmolecules of the solute and it s ions. The extent towhich substances ionized varied all the way from a verylow value for substances such as hydrocyanic acid tothe highest values (almost complete ionization) for thestrong electrolytes such as hydrochloric acid and thetrue salts. The degree of ionization of weak electro-lytes, as calculated from conductivity data, agreedreasonably me11 with the values calcnlated from vaporpressure and osmotic pressure data and from the relatedmeasurements of freezing point and boiling pointchanges. The weak electrolytes also followcl Ost-mald's dilution law. However, no such agreement heldfor t,he strong electrolytes.Further, it was shown th at strong electrolytes do not.follow the law of mass action in the slightest degree andthat the weak acids and bases follow the law only to alimited extent. Although the addition of a commonion causes a decrease in the concentration of theoppositely charged ion, the decrease is not proportionalto the increase in concentration of the common ionexcept for very slight increases. The so-called massaction constant increased in value as the total ion con-centration increased.Following shortly after Arrhenius, Ne&st2 advancedthe idea th at at constant temperature the solubility ofslightly soluble electrolytes in aqueous solution, withor without other electrolytes, is dependent o n a con-stant, the solubility product constant, which is pro-

NAIMAN. THISOURNAL5 454 1948).YERNSTW. Z p b y ~ i k . hem. 4, 372 1889).

portional to the concentrition of the ions of the slightlysoluble electrolyte, each concentration raised to a powerequal to t,he number of these ions arising from onemolecule.In accordance with the ~rrheniusheory it wasassumed that in the case of a saturated solution of anelectrolyte the postulated molecular species remainedconstant under all conditions. This notion soon cameunder the attack of many investigators. I t was shownthat the solubility product does not remain constanta t various concentrations of salts and therefore themolecular species does not remain constant.As early as 1904, A. A. Noyes, and later Milner aswell as Bjerrum, came to the conclusion that strongelectrolytes are completely ionized and that the devia-tions from the various ideal laws were due to the at-traction between the ions in solution. In 1914 Rragg

and Bragg presented proof, based on X-ray analysis ofsodium chloride crystals, that salts in the crystallinesta te contain no molecules but are made up of alternatepositive and negative ions held together by electro-static abtraction. It seemed obvious that upon solu-tion of a salt in a solvent of high dielectric const,ant,such as water, there shonld be no association of theions into definite molecules.f t,here are no molecules, why then do the st,rongelect,rolytes not behave as if they were 100 per centionized? Further, why cannot equilibrium constantsbe calculated for strong electrolytes and why do thesolubility products of slightly soluble salts vary withvarying salt concentrations?

The answer to these questions was found in theattraction between the ions in solution. The ions donot act with maximum efficiency in conducting theelectric current nor in affecting the vapor pressure ofsolutions and related properties for the following rea-sons. As a positively charged particle approaches anegatively charged particle, they mutually slow oneanot,her down or draw together according to Coulomb'slaw which states tha t the force of attraction betweenoppositely charged particles is directly proportionalto the size of the charges, and inversely proportionalto the square of the distance between the particles andthe dielectric constant of the medium separating them.The higher t,he concentration, the closer are the chargedparticles to one another and therefore the greater theeffect.G .N Lewis even as early as 1901, and in much sub-

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MAY 949

sequent work,3 pointed out the inconstancy of theequilibrium constants of many salts. He introducedthe term "ionic strength," or p , to define the thermo-dynamic characteristics of a solution of electrolytes,"activity," the effective concentration of the ions, and"activity coefficient," a factor by which the stoichio-metric concentration or molality must he multipliedto give the effective concentration. He established therelationship

m x = a (1)where

m = molality (or molarity in dilute solutions)= activity coefficienta activity

I t is customary to designate.the molar concent,rationby brackets and the activity by parentheses, thus

It is the activity which accurately defines the thermo-dynamic concentrations and which must be used toobtain true equilibrium constants.For example, the thermodynamic solubility product,S, for AgCl is

[Agilf*.+ X [CI-lfcl- (Ag+)(CI-) = S (3)I t is the activity of an ion which is determined inpotentiometric measurements.The activity coefficient varies in sufficiently dilutesolutions with the total ionic concentration, or moreaccurately 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 multipliedby the square of the valence, Z, of t ha t ion, divided bytwo (in order to account for the effect of both the posi-tive and negative ions). As the solution approachesa state of infinite dilution, approaches unity, and theactivity becomes equal to the concentration.Debye and Hiickel' were able to correlate the dis-crepancies concerning the anomalous behavior of strongelectrolytes and the effect of diverse ions in their com-prehensive Interionic Attraction Theory. They devel-oped a mathematical expression as a first approxima-tion applicable to dilute solutions for the calculation ofthe Levis activity coefficient,flogf = - 0.51zp.\/;1 0.33 a

where p = ionic strength as defined above, Z = valenceof the ion being considered, a the effective mean ra-dius, in A units, of all the ions in solution. By the useof equation (5), the activity coefficientof an ion may becalculated. It is obvious that the second term in the

denominator is small and may usually be ignored. Ifthe calculated values for f are applied to equation 3),a true solubility product constant may be obtained.It is implied in equation (5) tha t f values vary withthe kind and size of ion, as well as with concentration.This would require many tables off values for each saltand mixture of salts a t different concentrations. How-ever, Lewis showed th at "in dilute solutions, theactivity coefficient of a given strong electrolyte is thesame in all solutions of the same ionic strength." Inshort, equilibrium constants, such as solubility prod-ucts vary with the ionic strength in dilute solutions(up to ionic strengths of about 0.25).I t is found that if p S (-log S) is plotted against ;an almost linear relationship is obtained. In Figurevalues of psareplotted as ordinatesagainst G a n d howhow far from constant t he so-called so ubility product8 constants" are. In the case of BaSOn, for example,S varies from 1.0 X 10-lo at zero ionic strength to 2.2

;LEwrs, G . N., AND M. RANDALlr, Thermodynami's, BY permission imm ~uani i tatlve n e ~ y r i s . .nd ad by W. ~ i e m a n . 0.~ e u s aMcGraw-Hill Book Ca., New York 1923. and B. Naiman. copyrighteb 1942 by McGrav-Hill Bmk Co.. nc:DEBYE,., AND E. HBCKBL, h l~s ik . . 4,185,305 (1923). riwm 1 ps values t 25

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JOURNAL OF CHEMICAL EDUCATIONX a t an ionic strength of 0.25. The value 1.0X 10-lo, true at infinite dilution, is the one com-monly used in textbooks of analytical chemistry forsolubility product calculations involving BaSOI. Thecalculated results are therefore frequently far from theactual experimental results.

By use of the graphs relating solubility products toionic strengths, calculated values more nearly approachexperimental values. An example will ~ll us trate hecalculation of ionic strength and the use of the graphsin solubility product prohlems.5Example In the determination of chlorine the usualprocedure was followed and the following data obtained:2.5mmolsof calcium chloride were dissolved in 139'ml. ofwater, 1ml. of 6 M nitric acid and 60 ml. of 0.1M silvernitrate were added. The h a 1 volume was 200 ml.;the temperature was 25C.

CaCl, 2AgNO. = 2A gC l l C s ( N O &Two and five-tenths mmols of calcium chloride yield 5.0mmols of silver chloride which is removed from solut ion;therefore 5.0 mmols of silver ion and 5.0 mmols ofchloride ion contribute a negligible effect to the ionicstrength. The ionic strengt.h calculation may betabulated as follows:

on Sowce Mmol .?I Z ZP MZ1C a + + C aC l. 2 . 5 0 .0125 2 0 . 0 5 0NOs- A g N O s 6 . 0 0 . 0 3 0 1 0 . 0 3 0& + A ~ N O J 1 .0 0 . 0 0 5 1 0 . 0 0 5HNOs 6 . 0 0 . 0 3 0 0 . 0 30K O a- 11X 03 6 . 0 0 .030 1 1 0 . 0 3 0

MZ' = 0 . 1 4 5

From Figure 1 pS ,,, = 0.52.[AgCl[Cl-I = 3.0 X lo- '[Agt l = 1.0 mmol per 200 ml. 5.0 X 10-a mnrol/ml.

thereforeRIEMAN,W., J D. N~us s , ND B. NAIMAN,QuantitativeAnalysis, 2nd ed., McGraw-Hill Book Co., N w ork, 1942.

[CI-I = 3'0 10- 0 = 6.0 X 10-ammol/ml. X 200ml . 1 .2 X5.0 X 10-8 10-' mmol.1.2 X 10-5mmol X 143mg./mmol 0.0017 mg. lost asAgC1. If the classical method were used:

= [Ag+][CI-1 = 1.0 X lO- Oand[CI-I = X lo- 2.0 x mmol/ml.5.0 X lo-

2.0 X 10 mmol/ml. X 200 ml. X 143 mg./mol0.00057 mg. loss. A 300 per cent difference is found be-tween the two methods.

The advantages of using the theory developed byLewis and by Dehye and Hiickel in treating solubilityproduct problems and other equilibrium. constantsare:

1 I t makes the various calculations conform moreclosely to conditions actually existing in solution.

2. I t therefore leads to more accurate results in thecalculations.3. It makes the student conscious of the presence ofan ionic environment in th e solution and the effect ofit on t he properties of all ions.4. I t teaches the student t o calculate the ionicstrengths of the solutions.5. I t emphasizes very definitely the inconstancy of

the so-called classical constants.6 It demonstrates clearly the significance ofactv itie s and activity coefficients.7 It satisfies the student with a complete answerto the often asked question: Why use these 'con-stants' and bother to make the calculations if they are

80 far from accurate?8. It acquaints the student. more intimately withthe use of graphs on which he can see a t a glance howt,he constants vary with the ion c strength.The author acknowledges his debt to William Rie-man 111, of Rutgers University, who suggested themethods discussed in this paper.