Teaching Conductometry Another Perspective Maarten C. A. Donkersioot Depanment of Chemical Technology, Eindhoven University of Technology, Eindhoven, The Netherlands Usually the concepts of conductometry are taught along more or less historic lines. First, the easy conduction of electric current hv some liauid solutions is put forward as important evidence for the-existence of dyr&nically independent ions; second, molar conductivity, A, is defined and the concentration dependence of this quantity is used to demonstrate the existence of, and the difference between, strong and weak electrolytes. In order to do so, use is made of two seemingly separate theoretical models: with weak elece is shown to he the instrumentrolytes i n ~ & ~ l e iionization tal factor, whereas with strong electrolytes it is the influence of various effects of nonideality of ionic solutions that is responsible for the concentration dependence of molar conductivitv. The Gproach sketched above has some drawbacks. Often the theory of conductometry is presented to students who have already acquired a good understanding of the basics of electrolyte solutions. In this situation it seems undesirable to overemphasize the dichotomy of strong and weak electrolytes; it should rather he understood that between "strong" and "weak" all gradations are possible and that an electrolyte that is completely ionized in one solvent may very well he poorly ionized in another. In addition the definition of molar conductivity in terms of conductivity, K , and analytical concentration, c:~A= xlc, is convenient only from the experimentalist's point of view: hoth quantities involved can he readily measured. Logically, however, something is wrong: K should, rather, he divided by some measure of the actual total ionic concentration' if it is to he reduced to a meaningful molar ionic quantity. A further problem is that the definition cannot he applied in a straightforward manner to solutions of more than one electrolyte: x is not additive in terns of molar conductivities and analytical concentrations. In particular, the conductometric behavior of very dilute aqueous solutions cannot be satisfactorily treated this way bicause there the solvent discernibly contrihutes to the conduction process. It is possible to present the concepts of electrolytic conduction in such a way that the difficulties mentioned above are clearly visible and can be dealt with openly. This can be done by teaching the theory from the beginning in terms of the molar ionic conductivity, A, instead of the awkward molar conductivity, A. In this paper we want to sketch such an approach. ~~~
~~
Concentration Dependence ot Conductlvlty Having derived K from the resistance, R, of the conductivitv cell with the studied solution, we recognize that n is e&entially a colligative property, therefore in firstinstance linearly proportional to the number of ions present between the electrodes. As not everytype of ion contrihutes to K to the same extent, n must be written as a linear combination of contributions:
where i labels all distinct ionic species in the solution, [i] denotes the equilibrium concentration of ionic species i, and 136
Journal of Chemical Education
A; is the molar ionic conductivitv of ion i. related to the electric charge of ion i and the mobility of the solvated ion in its environment of solvent molecules and other ions present. The following facts about X should be stressed. In the first place, X is not a real observable because all solutions are electrically neutral, and therefore only linear combinations of X's for cations and anions can he measured. However, as a theoretically defined entity A is well established by the work of Debye, Huckel, Ousager, and others.24 In the second place, although a great deal of the concentration dependence of n has been accounted for hy eq 1,Xis still dependent on the comnosition of the solution. that is to say dependent on hoth the concentration and the specific nature of all ionic svecies present. In varticular. A values that somehow have he& determined for ions in one'specific solution may only he used for another solution if the approximate nature of this transfer is clearly noted. This aspect gains importance as the solutions studied become more conrentrated. However, for sufficiently diluted solutions A, may be considered a constant that belongs ro the fully solvated ionic species i. Equation 1can he seen as a generalization of Kohlrausch's law of the independent migration of ions. In the third place, it should he stressed that for the conductometric properties of a solution the nature of the parent electrolvte is of no imnortance once the eouilihrium concent r a t i ~ n ~the o f ionic species have been established. The contribution of each ionic species may he regarded as a product of two factors. The first, Xi, represents the physics of a chareed varticle and its environment in an electric field. the second, 61, represents the colligative aspects of the solition considered.
Concentration Dependence of Molar ionic Conductivity From experimental results (Kohlrausch) as well as from theoretical considerations (Dehye, Hiickel, and Onsager) it has been deduced that for the molar ionic conductivity of ions in very dilute solutions we may write:
In this expression I stands for the ionic strength of the solution; the dependence on flillustrates how X; dkpends on the equilibrium concentrations of all ionic species present in the solution:
where Zi is the charge number of ion i. A! and bi are parame-
' Adamson. A. W. A Textbook of Physical Chemism: 2nd ed.; Academic: New York. 1979. Harned, H. S.; Owen. B. B. The Physical Chemistry of Electrolyiic Solufions, 3rd ed.; Reinhold: New York, 1958. Crow, D. R. Principles and Applications of Electrochemistry3rd ed.; Chapman and Hail: New York, 1988. 'Korvta. J.; Dvorak, J.; Bohadkova V. Electrochemistry,Methuen:
ters that are independent of concentration but are dependent on the nature of the ions (charge and size of the solvated species), the nature of the solvent jviscosity, dielectric constant), and the temperature. A! is the molar ionic conductivity of ion i a t "infinite dilution", that is, a t such dilutions that I approaches the value zero. A! is a temperature-dependent property of the fully solvated ionic species i. I t should be emphasized that eq 2 is only valid for very dilute solutions. At higher concentrations extra terms have to be added to the expression. In practice, however, these extra terms are added to the observable linear combination of X's. The nature of these terms is mostly empirical. The theoretical background of eq 2 is well knownz4 but not relevant here. Notice that in an actual teaching context eqs 1and 2 provide ample opportunity to discuss the details of the underlying theoretical models. Notice also that, with eas 1and 2 in hand, the practical applications of conductometry (determination ~ f ~ u i l i h r i u m c o n s t a n tdeterminas, tion of the end point of titrations, etc.) are mere exerrises in the calculation of equilibrium concentrations in terms of equilibrium constants and analytical concentration^.^ All relevant knowledge about ionic conduction is contained in eqs 1and 2. We will substantiate this claim with the following example and demonstrate how molar conductivities can be introduced in order to yield well-known results. We wish to determine the concentration dependence of x of a weak acid HA. We consider the following equilibria: HAeHt+AH,O
Ht
+ OH-
with K = [Ht] . [A-]/[HA] withKw= [Ht] .[OH-]
We have a mass balance equation: [HA] = [OH-] charge balance equation: [H+] solution, according t o eq 1: K
+ [A-] = c and a + [A-1. For this
+ [OH-]. A., + [ A ] . A., = ([OH-] + [A-I) .AH, + [OH-] .Ao,. + [A-1. A,= [OH-]. (AHl + AoH-)+ [A-] .(AH+ + A*-) + a . c .(AH+ + XA-) = [OH-]. (AH+ + = [H']. A,+
(3)
It can be seen that the A's combine in pairs corresponding t o electrically neutral combinations of ions. In addition i t is clear that eq 3 is perfectly valid for a strong acid HA, provided the condition a = 1 (K >> [H+]) for all values of c is applied. It can also be seen that eq 3 automatically accounts for the contribution from the autoprotolysis of the solvent, which will be important a t very low values of a c. When a .c >> [OH-], this contribution can be neglected (note that [OH-] Assuming a.c >> [OH-], applying has an upper limit of eq 2 and using I = a. c we can write:
z).
AHA = a . (A:+
+ hi-) - (bHt + bA.). 1-
This expression is still valid for solutions of both strong and weak acids. Specializing for solutions of strong acids, cr = 1:
This equation demonstrates the usefulness of defining quan.A: and BHA= b ~ +b ~and - plotting AHA tities ARA = A;+ as a function of fi from such a plot ALA can be read as the extrapolated value and BHAas the slope of AHAVS. & a t c = 0
+
+
*.-. M
In the same way for solutions of weak acids: This corresponds to the Arrhenius equation: AHA= a. AiA, provided the product a c is small enough for the term BHA to be negligible. The concentration dependence of a is easier dealt with in the context of a general treatment of electrolytic equilibria5 than in a course on conductometry. A value for ARAcan he derived from measurements on selected A&& strong electrolytes, for example: ALA = A& We stress that, although this alternat~veanalys~sy~elds the same results as the traditional approach, now all approximations are explicitly mentioned and introduced where they are logically required.
.
.
+
Summary
We have shown that the main features'of electrolytic conduction may be presented with the help of two equations. Equation 1clearly expresses how in first approximation K denends on ~-~the eauilibrium concentrations of all ionic species present. The kquilibrium concentrations can be c&ulated for solutions of almost any composition by methods already well known to most chemistry student^.^ In fact, all problems concerning weak vs. strong electrolytes and mixtures of electrolytes, including the contribution of the solvent. are sorted out once the equilibrium concentrations have' been calculated. Eauation 2 describes t o what extent the conduction process itself is complicated by the fact that every ion in solution exists in an environment of solvent molecules and other ions. We have restricted the discussion to the very simplest model, but in principle any further refinement of the model may be dealt with h i adding more terms to eq 2. In short, the presentation of conductivity phenomena can become more accurate, more transparent, and more explicit if use is made of molar ionic rather than molar conductivities. ~
where a is the degree of ionization of HA, defined by:
.
With the definition AHA= X / C we now get:
Skoog. D. A,: West, D. M.: Holler. F. J. Fundamentals of Analyiicai Chemistry, 5th ed.; Saunders: New York, 1988.
Volume 68 Number 2
February 1991
137
