COMBINING HALF-REACTIONS AND THEIR STANDARD ELECTRODE POTENTIALS SIDNEY I. MILLER,' University of Michigan, Ann Arbor, Michigan
F R O M his first year of chemistry through his analytical courses the student uses an activity series and standard electrode potentials before he understands the free-energy relationships involved. At all levels the usefulness of such a series in the teaching of chemistry has long been recognized. To increase the value of such a table of standard electrode potentials a new method of combination of half-cell reactions is proposed. The proposed method of combining half-reactions and their corresponding electrode potentials is based on the method of volt equivalents ( I ) . With the early introduction of this method as soon as half-reaction potentials are presented, the later transition to the freeenergy concept should be natural and logical. No claim is made for the originality of this approach, but if the difficulties inherent in the alternative method are generally rerognized, then an important aim of this paper will have been satisfied. The data in the table on page 141 and the conventions followed are taken from W. M. Latimer's "Oxidation Potentials" (1). The relationship
in which AF is the free-energy change, E the oxidation potential, 5 the faraday, and n the number of equivalents changed, mill be referred to in the discussion. The number of volt equivalents, nE, is directly proportional to - A F and is commonly used instead of it for the calculation of E from the combination of any two half-reaction potentials (1). It is not suggested, however, that "free energy" need necessarily be brought into class discussion at.the elementary level. I n the combination of half-reactions and half-reaction potentials two working "rules" or conventions are often followed (I). The first of these states that a complete chemical reaction is obtained by subtracting one halfreaction from the other; the standard e. m. f . for the complete reaction is obtained by subtracting the corresponding standard electrode potentials:
multiplying the first e. m. f . by 3 and the second by 2 t o get the e. m. f . of the combined cell of example I. In qualitative and quantitative analytical courses the more general Nernst relation (6),
which shows how the half-cell potential changes with concentration, may be given (4). The ri are coefficients and the a,areactivities of the species involved in the half-reaction. The rule of combination is t.hen given as:
There are many cases, however, in which the convention expressed by the first rule or by equation(3) leads to erroneous results. Suppose the student is required to deduce E o for the reaction 2Cu'
= Cu + CU++
(4)
from a table of data containing the half-reactions of copper (1). These are the possibilities:
(IV)
Cu+=Cu+++e Cu e C u + + e 2Cut Cu Cut+
= +
-0.168 -0.522 0.354"
-0.168
-0.522 0.354
The first rule applies in examples I and IV, not because e. m. f.'s are additive as such, but because the e. m. f.'s are directly proportional to the free energies, which are additive. In examples I1 and 111, typical of those first recognized by R. Luther (67,both half-reactions Volt involve a single element in different oxidation states. E" equivalals In such cases the number of electrons transferred in the (I) 3 (hIg e Mg++ + 2e) 2.34 14.04 half-reactions may differ from the number transferred 2 (Au = Auf++ + 36) -1.42 -8.52 in the complete reaction. The e. m. f.'s, no longer pro3 . 7 6 22.56 3Mg + 2Auf++ e 3Mgf+ + 2Au portional to the free energies, may not be manipulated This rule is found in a variety of textbooks (2, 3, 4, 5). like free energies and the first rule cannot be applied. The second mle states that when two half-reactions For the early courses the presentation is, of necessity, arbitrary. Thus, the student must be cautioned against are combined t,o give another half-reaction the standard Incorrect. Department of Chemistry, Illinois Insti1 Present address: tute of Technology, Chicago 16, Illinois.
a
140
Correct.
141
MARCH, 1952
free-energy changes, rather than the standard electrode, potentials, are combined algebraically; from the result the new standard e. m. f. is derived. In these cases the use of volt equivalents shortens the calculations considerably.
intensive properties. He may also know how to convert the extensive into theintensive property by specifying a unit quantity of substance concerned. If the student has understood the differencebetween weight and density he should not find it unnatural to characterize a cell by its standard e. m. f. in volts and Volt work with another unit, the volt equivalent, when he is Eo equivalents combining one half-cell with another. He is now prepared for a general rule applicable to all combinations of half-cells. Manipulate both the chemical reaction and The second rule may be worded so as to include all pos- the volt equivalents in analogous algebraic operations until sible combinations. It is thermodynamically exact the resultant volt equivalents are associated with a change of and, applied to the preceding examples, it invariably one chemical equivalent in the combined reaction. This gives the correct e. m. f. number of volt equivalents can now be regarded as the It seems desirable, therefore, to devise a presentation cell or half-cell potential. Several of the preceding examples are worked by this of standard electrode potentials which will avoid the errors inherent in the first rule, which will be consistent method to obtain standard e. m. f. values. with theory, and yet which can be taught in general Vo't chemistry without making the treatment more difficult or arbitrary than it is a t present. It will be assumed ' / d M g Z Mg++ + 2e) 2.34 -1.42 that Faraday's laws of electrolysis have been presented (VI) '/s(Au Au+++ + 3e) 3.76 L/aAu+t+ '/&Ig++ + ~ / % A U '/2Mg before the standard electrode potential is introduced. E" = 3 . 7 6 volts The following treatment is similar to that of Eastman and Rollefson (7) but a fundamental modification is introduced in the rule of combination. A standard electrode potential is an intensive property of a given half-cell, and as such is nonadditive it is independent of the number of chemical equivalents or faradays involved. On the other hand, when Notice that the first answer obtained in VII from addistandard electrode potentials are to be combined they tion must be doubled to give a reaction in which only can be regarded as quantities of free energy in units of one electron is transferred. 23,100 cal., since Eo is proportional to AFo if n is taken Volt eouiualents as unity. Therefore, a table of standard electrode X '/dCr Cr+++ 3e) X0.71 potentials is also a series of numbers in volt equiva- (VIII) '/*(Cr++ Crf++ + e ) X 0 .'41 lents, characteristic for a half-reaction in which one '/.(Cr Cri+ + 2e) 0.86 chemical equivalent is involved. It is convenient to Eo = 0 . 8 6 volt write the chemical half-reactions as in the following table so as to associate a one-faraday change with the Finally the combination of a half-cell with a complete standard electrode potential; whenever these half- cell reaction is also accessible by this method. Vnl, reactions are combined the standard electrode poten. """ equivalents tials should be termed "volt equivalents" and treated '/&MnO,-2H20 e MnOl (IX) additively. 7Mnn.n nA * IAOH-l
-
+
=
== =
+
+
'/1(40H2H,O
+
++'2e)MZ; Mn0,--
+ = Mn04- + e MnOl--
E"
=
0 . 5 4 volt
"."A
-0.58 -0.54
If the half-reactions are written as they appear in the table, the rule of combination can be reformulated for more advanced students as
To a firstyear student the distinction between the volt and the volt equivalent need not be confusing. He has already learned to differentiate between some of the following related properties: weight and density, mols and concentration, force and pressure, heat capacity and specific heat, the speed of a reaction and its specific rate constant, in short, between extensive and
where n represents the number of electrons transferred in the complete reaction. In many cases n = 1, but when n f 1 the calculation requires an extra step, as in example VII. With this. rule of combination even the first-year student can use e. m. f. values to predict whether or not any reaction will go, and to makeup new half-reactions.
142
JOURNAL OF CHEMICAL EDUCATION
Isolated facts in the chemistry of an element may take Professor Robert W. Parry for several helpful dison new significance; for example, the disproportiona- cussions. tion of the cuprous ion in aqueous solution and the r e LITERATURE CITED quirements for stable cuprous compounds can be discussed in the light of the equilibria given in example (1) LATIMER,W. M., "Oxidation Potentials," Prentice-Hall, Inc., New York, 1938. VII. It is emphasized that, if electrode potentials are (2) PIERCE, W. C., AND E. L. HAENISCE,"Quantitative Analto be presented, the operations involved in the voltysis," 3rd ed., John Wiley & Sons, Inc., New York, 1948, equivalent method of combining half-reactions are no p. 269. more arbitrary or difficult than those associated with (3) TIMM,J. A,, "Generel Chemistry," 2nd ed., McGraw-Hill Book Co., Inc., New York, 1950, ~ ~ 4 7 5 . the first rule. Once learned it can easily he applied in (4) CURTMAN, L. J., "Semimicro Qualitat~veChemical Anslysia," topics in general and analytical chemistry. Finally, The Macmillan Co., New York, 1947, p. 103. the volbequivalent method is thermodynamically (5) Mooan, W. J., "Physical Chemistry," Prentice-Hall, Inc., New York. 1950. sound so that the student will have nothing to unlearn (6) LUTHER,R., AND D. R. WILSON,2. Physik. Chem., 34, 488 later. (1900); LUTAER, R., ibid., 36, 385 (1901).
ACKNOWLEDGMENT
The author would like to express his appreciation to
(7) EASTMAN. E. D.,
AND G. K. ROLLEFSON, "Physical Chemistry." McGraw-Hill Book Ca., Inc., Xew York, 1947, p.
452.
