JOURNAL OF CHEMICAL EDUCATION
the order of 0.001. While this method uses concentrations the required error analysis insists that total equivalents be considered for the more rigorous derivation.
To the Editor: I have used a technique for the Hittorf transferencenumber experiment which represents a hybridization of the Levy ( J . CHEM.EDUC.,29, 384 (1952)) and Angus (ibid., 30, 155 (1953)) ideas. This method is approximate and its pedagogical value lies in the student's making a rigorous analysis of the approximations involved. The anode and cathode compartments are drained into dry 100-ml. flasks and then filled to the mark with distilled water measured with a buret. Analysis i s carried out in any appropriate manner. The drainage is neglected. Let V be the volume in ml. of Hz0 added to the flask, e the total equivalents found in the flask, N the normality of the original solution, Ae the apparent change in the number of equivalents in the compare ment, and f the number of faradays during electrolysis. Then Ae = ( 1 0 0 1 =
(f
- V)N/1000 - e
* Ae)/f
By assuming that the drainage-loss solution has the same composition as the solution drained off and that there is no volume change during electrolysis the student can show that Ae' = A e ( l 0 0 - V - A V ) / ( 1 0 0 - V) where Ae' is the true change in the number of equivalents in the compartment and AV is the ml. of solution lost due to drainage. Experimentally, AV is found to be less than 0.1 ml. for well-cleaned cells. Since V for the usual apparatus is about 90 ml. the error in t is of
To the Editor: I am writing with reference to the article entitled "The algebra of simultaneous equilibria," by D. Davidson and K. Geller, which was published on page 238 of the May issue of the JOURNAL OF CHEMICAL EDUCATION. Like that by R. N. Boyd [J. CHEM.EDUC., 29, 198 (1952)],which it criticizes, this article is concerned with the question of teaching students how t o "solve" problems involving more than one equilibrium constant. The equilibrium constants involved, and their combinations, here presented as final simple solutions of the "problems," are simple algebraic equations. Given such an algebraic equation, or set of algebraic equations, the student presumably knows, from highschool elementary algebra, how to substitute numerical values and calculate an unknown. This "algehraic operation" is not a chemical problem; certainly it is not a problem of chemical education, but rather of elementary algebra and simple arithmetic. The chemical question is whether or not the final simple algebraic equation applies, in its simple form, to the particular equilibrium problem under consideration. The subject of the "solving" of problems in college courses should be concerned not with algebraic substitution in a simple formula but. with the question of the applicability of the formula. If only the simplest formulas or simplest approximations are taught and used, it is up to the instructor, in testing his students, to make sure that his test problems can legitimately he solved by these simple formulas. The result is that the student is tested on algebraic and arithmetical manipulations, while the instructor is being self-tested in chemical principles and chemical judgment.
The significance of these criticisms may be seen by .specific reference to the 14 "problems" used as illustrations in the article by Davidson and Geller. Six of these problems, Nos. 4, 5, 6, 8, 10, and 12, require merely t h e statement of the definitions and their combinations as algebraic equations. No. 14, the calculation of the concentration of zincate ion when zinc hydroxide "is equilibrated with 0.1 M sodium hydroxide," is also merely a question of the definitions, if "0.1 M sodium hydroxide" means the final concentration of hydroxyl ion. The simple solutions presented for these seven problems, therefore, are explicit and exact, and may be said to apply correctly, by definition, for all possible values of the numerical data. I n each case the solution 'involves no approiimations or assumptions; it requires, in other words, no appraisal of the situation, no exercise of judgment. The remaining seven problems, on the other hand, do involve assumptions and approximations, requiring appraisal of the relations to ascertain that the simple solution used is applicable to the situation. These assumptions and approximatious, however, are not stated in the article. Hardly a student reading the article will be aware of them, or will realize that the final simple solution presented applies only for certain values of the numerical data. In fact, there will be more than one instructor who will not realize these limitations. Problem 1, the calculation of t,he pH of sodium bicarbonate solution, is based on the product of the ionization constants of a dibasic acid, or KIK2 = [H+I2 [D--]/[HID], a combination of definitions. In the previous discussion, the authors write "if [H,D] = ID--] (which is true in an aqueons solution of NaHD)," the result is, of course, [H+] = dK,K,. Although this
this problem is whether or not we may cancel out [NH,] and [CN-1, not the algebraic difficulty in calculating the answer if these concentrations may be cancelled. The student would be, chemically speaking, much more educated if he knew how to think through, even slowly, the general problem in one general case, than he would be if he knew how to "solve" glibly a hundred problems suitably arranged by the instructor for the applicability of the approximate solution, so as to require no real thinking on the part of the student. Problem 7, for the pH "of a saturated solution of barium carbonate in 0.02 M barium chloride," is based on the combination of definitions Then, the authors write, "since HC03- and OH- ions are formed in equivalent amounts," it follows that iH+lZ= (0.02)KxKU/K.,
But it is not pointed out that it is being assumed that [HC03-] [OH-], and that the assumption holds only approximately and only for certain values of the data. If the student assumed the same approximate solution of the corresponding problem for zinc sulfide in 0.02 M zinc chloride, for example, he would calculate = 1.3 X (for the values of the constants as used in the article), an obviously absurd result. In problem 9, for the concentration of ammonia required to dissolve 0.01 mol of AgCl per liter, the assumption is simply that K.,/c may be neglected relative t o c, the number of mols of silver chloride being dissolved per liter. This is quite generally valid for AgCI, although it would not do for AgBrOa or Ag2S04. I n problem 11, x mol of AgCl is shaken with 1 liter of NaBr solution of molarity c, and the student is to cal, culate the proportion of sodium bromide converted to [D--1. He should know uhy this approximation silver bromide. The answer given is based on the simholds well for NaRCO?but not for NaHSOPorNaHS,and ple relation [Cl-]/[Br-] = K,,/K.,,, holding if the sohow and why it holds better for high than for low concen- lution is saturated with both salts. But unless trations. TheassumptionsmadeunderProhlem2,forthe x is greater than a certain value, the solution pH of ammonium ryanide solution, are almost exactly is saturated only with AgBr. For c = 0.1 the analogous to those of Problem 1. The interesting required value of x is 0.09952 (for the constants as used question is that of the range of applicability of these in this article), so that with x = 0.1 the answer given assumptions, not the "algebraic solution" of the final does apply. But given x = 0.09, for example, what principle will the student use to see whether the "soluapproximation. Problem 3 asks for the ratio of the complex ions tion" of the problem is applicable? In Problem 13, the pH a t which zinc sulfide will dis[Ag(CN)2-] and [Ag(NH&+] on addition of 0.01 equivalent of silver ion to a solution "containing one solve to form a 0.1 M Zn++ solution is to be calculated. equivalent each of cyanide ion and of ammonia." With By definition, IAg(CN)n-I INHII'/[A~(NH~)~+I[CN-I~ = .K,/Kn by definition, the answer is given as Ki,/Kt,. But in which K , ( = 0.1) is the concentration of unionized what would the student do for something like the addi- HIS in a solution saturated with H,S. I t happens that tion of 0.01 equivalent of silver ion to a solution con- if ZnS dissolves to form 0.1 M Zn++, the total concentaining 0.01 equivalent of cyanide ion and 0.02 equiv- tration of the dissolved forms of RPS will be 0.1 so alent of ammonia, or to a solution containing 0.01 that the solution will be practically saturated with H a equivalent each of cyanide ion and of ammonia-parif [H+] is not very low, and then the above expression ticularly, in these cases, if the two instability constants will just apply. But if the problem had called for were nearly equal? The real question in "solving" 0.01 M Zn++, the equation would be absurd; and L--"-
m+]
