Henry Freirer and Quintus Fernando University of Arizona Tucson
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I
Teaching
SILLEN, L. G. in "Treatise on Andytical Chemistry," KOLTI. M.AND ELVING, P. J., Editors. Part I, Vol. 1, chap. 8. Interscience Publishers, Inc., New York, 1959. This excellent chapter present8 an historical background in addition to an exposition of graphical methods for complex formation, solubility, redox, and acid-base equilibria. Q. "Ionic Equilibria in Ansr FREISER,H. AND FERNANDO, lvtical Chemistry," John Wiley & Sons, Inc., New York, 1963. " BUTLER,J. -N., ('Ionic ~quilibrium,A Mathematical Approach," Addison-Wesley Publishing Co., Inc., New York, 1964. Also see the paperback volume by the same author "Solubility and pH Calculatiom," Addison-Wesley Publishing Co., Inc., New York, 1964. 1
Equilibrium
Use o f log-chart transparencies
Heretofore many of those interested in rigorous ionic equilibrium calculations have been preoccupied with elaborate mathematical expressions. To develop fifth or sixth order equations on the basis of simple principles is not difficult in most cases. To obtain reasonably accurate solutions from such equations without tedious calculation, however, requires an appreciation of factors other than the simple arithmetic relationship of the various terms in a high order equation. The use of graphical methods brings the problem of significance of various terms in complicated expressions into proper focus. Furthermore, a pictorial representation permits the student to see a t a glance how the concentrations of various species in a system a t equilibrium change with conditions. Although graphical methods for the presentation of equilibrium calculations have been advocated for many years, particularly among Scandinavian chemists such as Hagg and Sillen,' these have not generally been used by any but research workers in this country. Only in the last few years have educators begun to incorporate this powerful approach in their undergraduate teaching. I n 1963, the first undergraduate text using graphical methods a p ~ e a r e d . ~Shortly thereafter, there appeared another, more advanced text, promoting these method^.^ Over the past several years, we have regularly taught ionic equilibrium calculations using this combined approach to sophomores and (on an experimental basis) to freshmen and advanced placement high school groups. These students were able to solve problems that required a degree of understanding usually expected only from far more advanced students. This method can be illustrated by its application to acid-base equilibria and pH calculations. After an introduction to acid-base reactions from the Br$nstedLowry viewpoint, the concentrations of the components of a solution containing a weak monoprotic acid H A are described as a function of two factors, the total concentration, C., and the pH of the solution. This pH
HOFF,
hit
dependence can be expressed in a manner that is independent of C,, by using or values defined as follows':
+
Since oro orl = 1, suhstitution in terms of the dissociation constant, K., gives:
For a diprotic acid, H 2 A , the three or values are:
Care is taken to point out to the student that the denominators in all of the or expressions for a particular acid are identical. Furthermore, each term in the denominator arises from the concentration of a particular component present. Thus, for a diprotic acid, the three terms in the denominator represent factors proportional to [ H 2 A ] , [ H A - ] and [ A - ] respectively. Therefore, it is possible to generate or functions for any acid. First, the denominator is written as a descending power series in [ H + ] . Second, the numerators in each of the expressions are obtained from the appropriate term of the denominator. Recognition of the contribution of each species to the mathematical expressions for or greatly assists the student in dealing with all phases of the calculations. At this point, the graphical approach is introduced by showing the student charts of the plots of log or versus pH (Fig. 1, 2, 3). Logarithmic expressions of the or values are employed because, since the pH is a logarithmic function, it is convenient to have other concentrations similarly expressed and the concentration of any component can be obtained by simple addition of terms, e.g., log [ A = ] = log orz log C,. Also, these loplog plots are simple linear functions except for limited regions where the pH is close to the pX. This simplicity results from the predominance of a single species in a region of p H , 1.0 or more units away from a pK value. The slopes of the linear portions have integral values, i.e., 0,1,2, etc., reflecting the number of protons
+
J
H. A. "Chemical Analysis," 4 See for example LAITINEN, H., McGraw-HiU Inc., New York, 1960, p. 36. Seealso FREISEE, and FERNANDO Q:, lac. eit., p: 95 Volume 42, Number 7, January 1965
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2 Figure 1.
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PH pH vs log cr for acetic odd.
Figure 2.
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pH pH vr log a for cwbonic odd.
gained or lost in going from the species predominant a t the particular pH to the one under consideration. Auseful teaching aid is obtained by constructing log a versus pH charts, on the same scale, on sheets of transparent plastic. By superimposing the appropriate log a versus pH transparency on a base chart (also transparent), which consists of a ruled grid representing -log C as ordinate and pH as abscissae (Fig. 4), it is possible readily to translate a values to concentrations. A set of such charts called the EQUILIGRAPH (which comprises seven &in. hy 10-in. transparencies suitable for overhead projection for the systems acetic, carbonic, hydrosulforic, phosphoric acids, a generalized monoprotic acid, and ammonia) is available at theFreiser EquiligrapH Co., P.O. 4900, Tucson, Arizona. An instruction manual ia included.
Thus, if a solution of an acid is 0.10 M the a chart is placed so that the upper horizontal line is a t -log C = 1.0 (Fig. 5); for 0.02 M, a t -log C = 1.7, etc. Also shown on the base chart are two diagonal lines representing [H+] and [OH-] respectively. The diagonal from upper left to lower right is simply drawnin accord with the definition of pH = -log [H+]. The [OH-] line is drawn from the logarithmic expression of the ion product of water, -log [OH-] = pH - 14.0. A pocket size student version of the EquiligrapH, Figure 6, can be used to great advantage by the student in his evaluation of the importance of various species
Figure 3.
involved in the problems. In addition, the studens should develop facility in drawing simplified versiont of these diagrams. It is not the purpose of this paper to reiterate the discussion now afforded this approach in the readily available textbook literat~re.'-~ Those familiar with the essential approach can devise a set of "rules" by which a student can be guided into competence with the method as a tool for calculation. Our experience has been that the student soon gets the feel for the problem and sees the approach in terms of "common sense" (albeit translated into somewhat graphical mathematical language). Proton Balance Equation
From the combination of charts such as illustrated in Figure 5, it is possible to show how the concentrations of various species in acids and bases change with pH and to indicate which species are important and which have negligible concentration a t a given pH value. It is also possible to calculate the pH values of such solutions quickly and accurately with the help of the relation called' the "proton balance equation." The proton balance equation (PBE), which serves to define the pH of the solution, matches the concentrations of species which release protons with those which consume protons. Let us first consider the PBE for H,O not only because it is the simplest, but
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pH Figure 4.
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pH vr log C bore chart.
lournol of Chemical Education
pH pH vr log or for phosphoric ocid.
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pH Figwe 5. pH vr log C for 0.10 M acetic acid. (Fig. 1 mperirnpored on Fig. 4 with limiting values of cro, an set ot -log C = 1.0.)
also since it will be involved in all other PBE's as well. [H'] = [OH-] i.e., Protons consumed
=
Protons released
The reason the hydroxide ion concentration measures the concentration of protons released is that when water acts as an acid, i.e., a proton-releasing species, it produces a hydroxide ion for each proton released. When water acts as a base, it forms one H,O + for every proton consumed. The hydronium ion concentration, abhreviated as [H+], is a measure of the proton consumption of water. In a solution of a monoprotic acid H X (strong or weak), as for any aqueous solution, the PBE is derived using the PBE for water as a starting point. In this case, the dissociation of HX releases one proton which is measured by the X- formed in the same process. Thus, the PBE for HX is iH+] = [OH-] [X-] In a solution of a strong base such as KOH, the PBE is [R+]+ [K+] = [OH-] Using the neutral KOH as our starting point, its dissociation releases an amount of K + equivalent to the OH-. The [K+] is a measure of the [OH-] released and therefore equivalent to the protons consumed in the dissociation. Of course, there are many bases which do not directly release hydroxide ions (Bransted bases). In the case of NHa for example, protons are consumed to form an equal number of NH4+ ions and the PBE for aqueous NHs is [H+]+[NH& = [OH-]
+
In solutions of strong electrolytes, (MX where M is not hydrogen), the dissociation into M + and X - neither consumes or releases protons. One or both of these ions, however, may subsequently consume or release protons. Thus, a solution of a salt such as NaCl will have the same proton balance equation as that of water.
Figure 6. Demonstrating the use of the student EquiligrapH in s&ufating the OH of 0.01 M NH, solution.
In NH4C1, the NHa+ ion is an acid whose proton release is measured by [NH3],but the C1- does not act as a base in water. Therefore, the proton balance equation for NH4C1 is: Similarly for NaOAc, in water, it will be seen that its PBE is: [H+I + [HOAc] = [OH-] A case of a strong electrolyte in which both ions are involved in proton balance is that of NH4CN. Here the PBE is [H+] IHCNI = [NHal [OH-] This principle applies equally well to polyprotic acids and bases. In these cases, the concentration terms in the PBE are multiplied by the number of protons consumed or released in the formation of the species in question from the starting material. For example: The PBE for HnCOsis [H+] = [OH-] 2[CO3-I IHCOs-l The PBE for NaHCOs is [Ht] + [HGOJI = [OH-] + [CO,-] The PBE for NanCOaia [H+] [HCOs-I + 2 [ H C O r ] = [OH-I The method here suggested for any pH calculation centers around the proton balance equation. I t has the advantage of emphasizing the chemical aspects of the problem before the introduction of complex mathematical terms. Other valid approaches to such an equation, such as the charge and mass balance equations, can often be as convenient as the PBE. The information established by the PBE is combined with that obtained from the -log C versus pH graph as follows. The intersection of the line corresponding to one of the components on the left-hand side of the PBE with the line corresponding to one of the components on the right-hand side of the PBE at the highest value of the concentration is noted. (The intersection closest to the top of the diagram is called the principal intersection.) This gives the approximate pH of the solution. The approximation can be refined further. The PBE is simplified by discarding all terms that are seen to be less than 5% of the main components. The 5% criterion is related to the experimental reliability of 20.02 for ordinary pH measurements.) For the remaining terms, the appropriate Ca expressions are substituted and the equation is further simplified by discarding terms in the denominators of the various a expressions, corresponding to species of insignificant concentration (15%). The resulting equation now leads to an appropriate algebraic expression for the pH of the solution that is reliable to ~t0.02. Use can be made, of course, of the simple means of correcting pK, values for the ionic strength of the solution.' In most cases, including rather difficult problems, the final algebraic expression is a simple equation. This method has another advantage in dealing with those unusual problems in which the final algebraic expressions are not so simple. Not only are derivations of such expressions straightforward, but the components responsible for the complexity can he clearly identified.
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FREI~ER, H.,
AND
FERNANDO, Q.,bc. cit., chap. 3.
Volume 42, Number I, January 1965
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Example 1.
Colculote the pH of 0.10 M HOAc IpK, = 4.70)
+
The PBE for this system is [HtI = [ O H 1 [OAc-I. Reference to Figure 5 gives the approximate pH d u e at the principal intersection, that of the ]H+] and [OAc-I lines, namely pH = 2.9. The diagram shows that the corresponding [OH-] is negligible, so the simplified PBE is [H+] = [OAc-I. Substituting CGax for [OAc-] :
At pH 2.9 [OAc-I is seen to he much less than [HOAcl (Fig. 5) and therefore from the denominator the corresponding K. term can be removed. This simplification leads to:
- log C,)
Exmmple 3.
Calculate the p H of a roiulion containing the following:
0.01 M NaNHmO,, 0.04 M NaOAc, 0.02 M NH,Cl and 0.02 M NaCOs. (pK. for NH,+ = 9.27; pK.' for HOAc = 4.65; pK, = 1.65; and pKx = 66.7 for HaSOs; and pK,' = 2.04, pK2' = 6.98, and pK8' = 12.1 for HaPO4) The PBE for this system is [A+] [HOAc] [HSOp-] 2[H,SOs] [HnP04-] 2[HsPO~u n l w o,rre.wd f l r ionir s~rengrh p t i v~luesUP used in rhr construct~u~l
