D. N. Bailey, Owen A. Moe, Jr., and J. N. Spencer Lebanon Valley College Annville. Pennsylvania 17003
On the Relationship between Cell Potential and Half-Cell Reactions
It is commonly assumed that a unique En may be associated with each cell reaction. In fact, different electrochemical processes with the same net chemical reaction are possible. Because AGO must be the same for identical reactions, the difference between the electrochemical processes can only he in the number of electrons, n, transferred during the reaction. Many examples of the same net cell reaction with different n values are known so that i t is insufficient for subsequent calculations to give only a net cell reaction and an En value; the half-cell reactions must be specified. Jenkins and Marks' and Eberhardt2 have previously touched on this point, but the problem is sufficiently bothersome to students that it seemed relevant to address it in some detail. I t may seem contradictory that cells for which the same net reaction occurs can have different potentials, hut when the detailed electrode processes are considered the contradiction is removed. For example, in the discussion of redox reactions, it is not uncommon to see such statements as: "the standard electrode potential, E n , for the reaction
if such cells could be constructed, voltages of 0.337, 0.521, and 0.153, respectively, would be obtained. For these cells and many others, the tabulated potentials are not obtained from direct measurements on the cells but rather are calculated from thermodynamic data. For each cell the copper half-cell is more positive than the hydrogen half-cell. If any of these cells were short circuited, the electron flow in the external circuit would be from the hydrogen electrode to the copper electrode. The half-reactions for cell (8) would be
giving the spontaneous net reaction Hl
+ Cd+
=
+
2 ~ + Cu"
Now consider the following battery in which the Cu electrode of half-cell b is connected to the P t electrode of halfcell c.
+
2cu+ = CU'+ Cu" (1) is 0.368V." From this information it is usually assumed that the standard free energy change, AGO, or the equilibrium constant, K, can be readily calculated by making use of the standard relationships
.
AG"
-nFEU = -RT In K. 12) If no more information is provided than that given above, a value for n in eqn. (2) must he assumed. The choice of n must be consistent with the particular combination of halfcell potentials used to obtain the given Eo. Two different half-cell combinations may be used to obtain reaction (1). The first is CU+
=
+ e = Cu" Cu+ = Cu'+
2Cu' = Cu"
+ + Cuo P
E"
=
0.521 V
(3)
E" E"
=
-0.153 V 0.368 V
(4)
=
from which it is obvious that n = 1 and that A@' = -(I) (23.060) (0.368) = -8.49 kcal mole-'. The second comhination of half-cells is given by ?[Cut
+ e = Cu"]
C' =
0.521 V
(3)
Cu" = 2e + Cu2+ En = -0.307 V (5) 2Cu+ = Cuo +Cult En = 0.184 V The net cell reaction is the same in both cases but for halfcell combinations (3) and (51, n = 2 and Eo = 0.184. AGO is calculated to be -8.49 kcal mole-' as before, but n and EO are different. Thus Eo for the reaction of eqn. (1) could have been given as 0.184 or 0.368 V. The standard potentials of the half-cells given by eqns. (3-5) could be obtained from the following cells
b
c
d
+O.52l V
fO.521V
+ 0.368 V
a
arbitrary 0
Because voltages are relative, the P t electrode (a) may he chosen as the arbitrary zero. The copper electrode in half-cell b is then 0.521 V more positive than the platinum electrode in half-cell a. Because the platinum electrode in half-cell c is physically connected to the copper electrode in half-cell b, it must also be 0.521 V more positive than the arbitrarily chosen reference point. In cell c the platinum electrode in the copper half-cell is 0.153 V more positive than the platinum electrode in the hydrogen half-cell and since, under standard conditions, this physical relationship always holds regardless of the direction of current, the platinum electrode in half-cell d of the battery must be 0.153 V more negative than the connection between half-cells b and c making it 0.368 V more positive than the reference zero. The same conclusion could he reached by using Kirchoff's laws and by recognizing that the two cells are connected in opposing directions. If the P t electrodes of half-cells a and d are connected, the electron flow in the external circuit will be from a to d. The corresponding half-reactions which would occur on the passage of one Faraday are
+
Half-cell a 1/2HI = H+ e b Cu+ + e = Cu" e Cu+ = CulC e d H+ e = 1/2H,
+
+
and the net reaction is 2cu+
=
CU'+
+ CuU
The free energy change is
Although the actual construction of cells (6) and (7) is not possible because of the instability of Cu(1) ion in solution,
'Jenkins, D. A., and Marks, D. J., Educ. Chem., 2,213 (1965). 2Eberhardt, W. H., J. CHEM. EDUC., 48,829 (1971).
Volume 53, Number 2,February 1976 / 77
= =
AGO = -(I) (23.060) (0.W) = -4.24 kcal mole-'
(-1) (23060) (0.368) -849 kcal mole-'
This is the same net reaction and free energy change which would be observed in the cell ~tfcu+,cu'+~~cu+~cu"
This cell also corresponds to the half-cell comhination (3) and (4). A similar argument can be applied to determine the potential of a battery constructed from cells (6)and (8).
0
0:521 V
0.521V
0.184 V
The simplified cell derived from this battery
corresponds to the half-cell combinations (3) and (5).The net potential is 0.184V and the Cu(1) half-cell is the more positive. The half-cell reactions which occur on the passage of one Faraday are H+ + e = 1/2H2 1/2cuo = 1/2Cu2+ CU+
+e
=
l/ZH, =
+e
Cuo
H+ + e
Acknowledgment
The net reaction is cuC
which is one-half that found for the previous battery and, because the net reaction is halved, A@ is precisely what would he expected on thermodynar+c grounds. For the same amount of chemical reaction, ACnwould obviously he the same for both batteries. Thus the "same" chemical reactions corresponding to half-cell combinations (3) and (4), and (3) and (5) have different electrochemical processes. Half-cell combination (3) and (4) requires twice as much reaction to occur on the passage of one Faraday as half-cell comhination (3) and (5). Neither n nor En are pathway independent properties as is the thermodynamic function AG. The value of n depends on the cell construction as just demonstrated. Thus E0 must also depend on cell construction because AGO = -nFEn. It will often occur that a given net reaction will have a unique value for n and En.However, when the same chemical element appears on the same side of the chemical equation in two different oxidation states, a unique n and En will not he found. This requires special care in the discussion of the cell potentials for disproportionation and dismutation reactions. I t must be made clear which halfcell reactions were used in the calculation of En. Such statements as that made in the second paragraph of this article should be avoided because the value of n is not obtainable from the information given.
=
l/2Cu'+
+ 112 Cu"
Because one Faraday has been passe2 through the cell with a potential difference of +0.184V, ACnis
78 / Journal of Chemical Education
The authors are very much indebted to W. H. Eherhardt, Editor, Textbook Errors Column for his most helpful discussion of electrochemical processes and for his many suggestions which hrought this article to its present form.
