JOURNAL OF CHEMICAL EDUCATION
A "DERIVATION" OF THE NERNST EQUATION FOR ELEMENTARY QUANTITATIVE ANALYSIS LOUIS MEITES Yale University, New Haven, Connecticut
SOPHOMORES and juniors taking elementary quantiConsider the half-reaction connectina ferrous and tative analysis without previous instruction in physical ferric ions: Fe++ = FeC+++ e chemistry frequently h d their attempts to understand (1) and apply the Nernst equation hampered by its pres- The corresponding equilibrium constant would be entation to them as a fait accompli. Knowing nothing of the considerations entering into its derivation, they are forced to use it in equilibrium constant and similar Having already learned that acid-base reactions concalculations without comprehending the fundamental relationship between the standard potential of a half- sist essentially of the transfer of protons, while redox reaction and the equilibrium constant of a completed reactions involve the transfer of electrons, the student equation. Consequently one all too often finds that is inclined to accept a parallel between the significance even the students who can perform the algebraic manip- of the proton concentration in acid-base reactions and ulations successfully do so as though performing a that of the "electron concentration" in redox reactions. well memorized series of arbitrary mathematical steps. Since in acid-base reaction theory one is concerned with This state of affairs is the logical result of the fact the logarithm of the hydrogen ion concentration,. that a rigorous derivation of the Nernst equation is the analogy can be pursued further by solving equation beyond the scope of such a, course. The writer has (2) for the logarithm of the "electron concentration." for several years been experimenting with a "derivation" This gives IFe++l which, although clearly beneath criticism on thermodylog [el = log K + log [Fe+++l (3) namic grounds, produces the Nernst equation in a manner which the student can easily appreciate. Having The next step is purely a matter of convention: found that their understanding of the applications of [Fe+++I the equation is thereby greatly enhanced, he is impelled log [el = log K - log [Fe++l to present it here for the information of others troubled by the considerations outlined above. At about this point it is necessary to discuss the mean-
MARCH, 1952
ing of the "electron concentration." Of course an aqueous solution containing ferrous and ferric ions contains no free electrons. (The diierence between metallic and electrolytic conductors may be reviewed here.) So the "electron concentration" is not a physically significant quantity. It is merely a measure of the tendency of the solution to take up or give off electrons. If a solution contains a high concentration of ferric ion and a low concentration of ferrous ion, equation (1) would suggest the qualitative conclusion that reduction be much easier, and oxidation much more difficult, than if the relative concentrations were reversed. This is the same conclusion that would be reached on the basis of assuming a low "electron concentration" in the first solution and a much higher one in the second. Thus, although the "electron concentration" is fictitious, its use as an aid in reasoning leads to conclusions which are qualitatively correct. Now, if this "electron concentration" is a measure of the tendency of the solution to react with or give up electrons, it should be possible to measure the difference between this tendency of one solution and the corresponding tendency for another solution. Let us arbitrarily assume that a solution containing 1M hydrogen ion in equilibrium with hydrogen gas a t one atmosphere pressure behaves as though it contained one mol of electrons per liter. (This is equivalent to defining the standard potential of the hydrogen: hydrogen-ion couple as zero. The instructor may now wish to digress to explain the difficulties involved in defining a single electrode potential, and to stress the fact that the h a 1 comparison between two solutions cannot depend on the standard to which the properties of both are referred.) If an inert metal mire is immersed in this solution, it might be expected to assume the same charge density (i. e., the same potential) as the solution itself. If another similar wire is immersed in a second solution, it would he expected t.o assume the potential of that solutiou. Now if the two wires are connected to the terminals of a high-resistance voltmeter, so that no current can flowtoalterthe compositions of the solutions during the measurement, the difference in potential can easily he measured. I t may be pointed out that a diierence in potential means that a redox reaction would occur if the two solutions were mixed, and that a large potential difference indicates that the reaction would he more nearly complete than if the potential difference were smaller. Suppose that such a measurement is made of the potential difference between our staudard hydrogen: hydrogen-ion solution and another hydrogen: hydrogenion solution containing, say, 10 M hydrogen ion. By comparison with equation (2) it will he found that the second solution will behave as though it contained ten mols of electrons per liter. The values of log [el for the two solutions thus differ by one unit. The voltmeter reading will be found to be 0.05915 volt a t 25'. Then we can write
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E
=
0.05915 log [el
(5)
which when combined with equation (4), gives E
=
[Fe+++l 0.05915 log K - 0.05915 log -[Fe++l
(6)
In a solution 1M in both Fe++ and Fe+++ the second term on the right of equation (6) vanishes. The potential is then the potential under a sort of "staudard condition" like the staudard conditions involved in gas law calculations, and is called the "standard potential," Eo. Since by equation (6) this is equal to 0.05915 log K, it follows that E
=
E"
[Fe+++] - 0.05915 log [Fe++l
(7)
By an entirely analogous procedure with, say, the manganous ion: permanganate couple, it is possible to show the significance of the value of n, and thus complete the derivation. Even the common sign convention can be justified to the student's satisfaction. Consider a solution containing permanganate, manganous, and hydrogen ions, all at 1 M concentration: It is a strongly oxidizing solution, which means that it has a great affinity for electrons, or that it appears to have a very low "electron concentration." By equation (5), therefore, E (which in this case is equal to Eo) is negative. Similarly Eo would be expected to be positive for, say, the chromous :chromic couple. In calculating the equilibrium constant of a redox reaction one can of course set the potentials of the two couples equal at equilibrium and proceed in the conventional fashion. The following method, however, seems to be more readily appreciated by the student. Taking the ferr0us:permanganate reaction as an example: EDa. = 0.05915 log
[Fef++l [el [Fe++l
(8)
Rewriting (8) as Eopo=
0.05915 [Fe+++16[el6 5 log [Fef+]8
(10)
and subtracting (9) from (10) gives, after a little simple algebra, 0.05915 log E n p e - Eoxn= --5
(11)
The theoretical shortcomings of this presentation are so numerous and readily apparent that it seems unnecessary to list or discuss them. Nevertheless, the writer feels that its use is more than justified by the students' learning immediately that the Nernst equation is fundamentally nothing more than a means of describing equilibrium relationships in redox systems, and by their greater confidence in its use.
