Climbing a Potential Ladder to Understanding Concepts in Electrochemistry James R. Runo and Dennis G. peters' Indiana University, Bloomington, IN 47405
Developing teaching tools that assist students to overcome and the nroblems encountered in leamim! - ~ - clarifv - ~ ~ &ctrochemistry has &en of long-standing interest (1-7L Students often ex~eriencedifficulty with certain topics, such as relating &e potential of an klectrode, known &th respect to the standard hydrogen electrode, to some other reference electrode. In considering galvanic (voltaic) cells, students can be uncertain as to which electrode is the anode and which is the cathode. They wonder how changes in the concentration or pressure of a component in one or both half-cells affect the voltage of the cell. Upon encountering electrolytic cells, students are not at ease in analyzing the various factors that influence the potentials of the anode and cathode or the selection of the applied voltage for an actual electrolvsis. In this DaDer. we deal with these and other topics by introducinga .mlfied approach that makes use of ladder. We illustrate how a wtential ladder is used in discussions about oxidation-rehuction reactions and electrochemical cells. This pedagogical aid has benefited hundreds of undergraduates at our university and has been viewed enthusiastically by high school and college teachers who have learned about potential ladders at local meetings. ~
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Construction and Significance of a Potential Ladder Among its many uses, a potential ladder helps organize the numerous half-reactions found in a table of standard potentials. To construct a potential ladder, as shown in Figi r e 1, a selected group if half-reactions are arranged as properly positioned rungs on a ladder according to their relative standard potentials. In agreement with convention, all half-reactions are written as reductions. Near the middle of the potential ladder is the standard hydrogen ion-hydrogen gas half-reaction, or standard hydrogen electrode (SHE), with its assigned potential of exactly 0 V.This half-reaction is the standard against which all other half-reactions are measured to determine their potentials. Above the SHE are half-reactions with positive potentials, whereas hdf-reactions with negative potentials are placed on rung8 below the SHE.
Conventions Concerning Free Energy Changes and Potentials At various places in this paper, we refer to the potential of an overall reaction the voltage of an electrochemical cell the potential of an electrode the potential of a half-reaction Strictly speaking, an overall reaction cannot have a potential, only a free energy change. However, due to the fundamental relationship between the free energy change for an overall reaction and the voltage (emf)for a cell in which 'Author to whom correspondence should be addressed. 708
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Figure 1. A potential ladder consisting of plot of values taken from Bard, A. J.; Parsons, R.; Jordan, J. Standard Potentials in Aqueous Solution;Dekker: New York, 1985.The potential forthe saturated calomel electrode (SCE) is a formal potential. The concentration of potassium chloride at saturation is approximately 4.7 Mat room temperature. that reaction takes dace (AG = -nFE). it is common Dractice to speak of the potential (emf)for the overall readion. Similarly, an electrode in contact with a solution has a potential (versus some reference electrode), but the halfreaction that takes place at that electrode cannot truly be said to possess a potential; instead, the half-reaction has a free energy change (AG) associated with it. Provided we recognize these distinctions, it is convenient to discuss the potential of either an electrode or a half-reaction. Relating Potentials of Half-Reactions(Electrodes) to Other Reference Nectrodes
As mentioned above, the position of each half-reaction on a wtential ladder is arrived at bv com~arisonwith the sE%~, the internationally accepted &nda>d reference electrode. Unfortunately, the SHE presents problems in the laboratory because is both hazardous &d cumbersome
Because the rung for the iron(IIIkiron(1I) half-reaction is above the rung for the SCE, a potential ladder confirms that the potential of the imn(IIIkiron(I1) half-reaction is positive with respect to the SCE.
EO = +0.529V vs. SCE
Half-Reactions below Both the SHE and the SCE Note the position of the tin(II!-tin metal half-reaction versus the SHE in Figure 2.
EO = -0.138 V vs. SHE
To obtain the potential of the tin(I1btin metal half-reaction versus the SCE, find the relative positions of the two rungs corresponding to each of these systems. Then subtract the potential (4.138V) of the tin(1Iktin metal halfreaction (lower rung) from the potential (t0.242 V) of the SCE (higher rung). 0.242 -(-0.138) = 0.380 V Because the tin(1Iktin metal half-reaction is below the SCE on a potential ladder, the potential of the tin(1Iktin metal half-reaction is negative with respect to the SCE.
?,
Figure 2. Portion of a potential ladder showing how to relate the potentials of the iron(ll1)-iron(ll),tin(ll)-tin metal, and CuCI;-copper metal couples to the saturated calomel electrode (SCE).
to use. An easy way to offset these difficulties is to use a more convenient reference electrode, such as the saturated calomel electrode (SCE). However, switching to a different reference electrode causes the potent~alsof all haU'.reactions to shiR. In this situation, a potential ladder proves to be a valuable tool because it makes the conversion of potentials from the SHE scale to the SCE scale easy and physically meaningful, as it does for any other reference-electrodescale. Ng (5) has described an approach to convert potentials of half-reactions from one reference-electrode system to another. His procedure is based on the arrangement of potentials along a horizontal scale, whereas Moran and Gileadi (7) have used a tabular procedure.
= -0.380 V vs. SCE
Half-Reactions Placed between the SHE and the SCE On the potential ladder in Figure 2, the rung for the CuCL--copper metal half-reaction is above the rung for the SHE but below the rung for the SCE.
I!? = +0.177V vs. SHE
To calculate the potential of the CuClz--copper metal halfreaction versus the SCE, the potential (+0.177 V) of the CuClz--copper metal half-reaction (lower rung) is subtracted from the potential (+0.242 V) of the SCE (higher rung). 0.242 - 0.177 = 0.065 V Also, because the CuClz--copper metal half-reaction is below the SCE on a potential ladder, the potential of the CuClz--copper metal half-reaction is negative with respect to the SCE.
Half-Reactionsabove Both the SHE and the SCE Knowing the potential of the iron(IIIkiron(I1) half-reaction versus the SHE, Predicting the Spontaneit of Oxidahon-Reduction &actions
EO = +0.771V vs. SHE
we can determine the potential of this half-reaction with respect to the SCE. Figure 2 is a portion of a potential ladder showing the positions of both the SCE and the iron(III!-iron(I1) half-reactions versus the SHE: the iron(II1)-iron(I1) half-reaction ladder higher than that for occupies a rung on a the SCE. To find the potential of the iron(II1)-iron(I1) halfreaction with respect to the SCE, the potential (+0.242 V) of the SCE (lower rung) is subtracted from the potential couple (higher rung). (+0.771V) of the iron(III~iron(II)
Determine the Ouemll Cell Potential Another application of a potential ladder is to predict whether or not an oxidation-reduction (redox) reaction should o c c d For example, let us determine if coppedII) can oxidize manganese metal, according to the following reaction: whether a ,DroDosed redox reaction should . .can - -onlv , ,orecict ~ , occur spontaneous y. (For tne present a sc~ssion, it is assmed that the concentrat ons ana press~resof a I spec es are Jn ty.) It 1s a ways passole that a tnermoaynam ca y sp0ntaneot.s reanlon w I be loo slow kinetically to be of practical significance. ~~~
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in a southwest position. In other words, Cu should not reduce Mn2+. One way to foretell the possible outcome of this experiment is to combine the two half-reactionsso that their sum is the desired redox reaction.
Galvanic (Voltaic) Cells This paper would not be wmplete without our including a discussion of galvanic cells. Suppose there is a galvanic cell with a cadmium wire immersed in a solution containing 1 M Cd(NH3I4%and 1 M ammonia. It is connected through a salt bridge to a solution containing 1M PuOzZ+, 1M Pu-, and 1M Ht into which a platinum wire is placed. Conventions in Describing GalvanicCells
cu2++Mnscu+Mn2+ Eo=+1.52 V
According to convention, the shorthand representation of this cell can be written as follows:
Because the overall potential is positive, the reaction should proceed spontaneously. Check for a NorthwestSoutheast Relationship An alternative appmach is to look at a potential ladder and to use what we call the northwest-southeast relationship. In the present example, Cu2+and Mn are the reactants. Finding the half-reactions on a potential ladder and connecting the two reactants with a line, as shown in Figure 3A, one sees that cu2+is northwest of Mn (or that Mn is southeast of Cu2+);therefore, the reaction should occur spontaneously. In general, any reaction involving an oxidant that is northwest of the proposed redudant will be thermodynamically spontaneous. On the other hand, Figure 3B shows that a proposed redox reaction will he nonspontaneous if it involves a redudant in a northeast position and an oxidant
The anode is always on the left and the cathode is always on the right. Asingle vertical line ( 1) in the shorthand cell representation designates a phase boundary between an electrode and a solution. At this phase boundary, a potential differenceexists that contributes to the total voltage of the cell. A double vertical line (11) in the shorthand cell representation desirmates a salt bridge that connects two solutions of different-wmpositim. ow ever, the (usually very small) mtentiill diflerences arisinn at the boundaries between the salt bridge and the sol&ions it joins are neglected when the total voltage of the cell is wnsidered. Distinguishing the Anode from the Cathode
One can use a potential ladder to determine by inspection which half-cell corresporids to the anode and which half-cell corresponds to the cathode. Locating the two halfreactions on a potential ladder (Fig. 4.41, one sees the following: For a galvanic cell the lower rung (half-reaction)is
FlgLre 3 Lse of a potentla ladoer to pred~ctthe spontane ty or non spontaneity of a proposw redox reanlon (A) Anotlhwest-sodtneast react on sno~ldbe spontaneous (6)A nonheastsouthwest reacllon should be nonspontaneous.
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Figure 4. Potential ladder showing the relative positions of the anode and cathode of a galvanic cell. Cell voltage is (A) 1.654 V when all species are present at concentrationsof 1 M, (B) larger if [ P U O ~is~ ] ~I (D) smaller if increased, (C) smaller if [ c ~ ( N H , ) ,is~ increased, [ P U O ~is~decreased, ] and (E) larger if [C~(NH,),~+] is decreased. The different line segments are not drawn to quantitative scale, but show the key qualitative trends.
always the anode, and the higher rung (half-reaction) is always the ~ a t h o d e . ~ Computing the Voltage
In addition to identifying the electrodes, one can compute the cell voltage by taking the difference between the potentials of the half-reactions (rungs) on the ladder.
This figure is also the value of the potential for the conventional cell reaction because all species are a t 1M concentration. PUO?
+ 4H++ Cd + 4NH3
c ~ ( N H , )+~fi&
+ 2HZ0
En = +1.654 V
A positive (+) sign is placed before the F to indicate that this cell reaction is spontaneous. The voltage of a cell has a magnitude but no sign, whereas the potential of a half-reaction or the overall cell reaction has both sign and magnitude. Moreover, any one electrode of a cell always has a potential (with magnitude as well as sign) with respect to the other electrode of that cell and with respect to some reference electrode (ex., SHE How Changes in Concentrations (Pressures) of Species Affect the Voltage of a Cell
Increasing the Concentration of a Reactant One can use a potential ladder to predict how changes in the concentration or pressure of one or more species in a galvanic cell will affect its voltage. Any time theconcentration (pressure) of a reactant in a half-reaction is increased, the &iving force for the forward reaction increases. This makes the potential more positive, and therefore raises the position ofthe half-reaction (rung) on the ladder. If the half-reaction involved is the top rung (cathode),the distance between the two half-reactions (rungs)on the ladder is made greater, thereby increasing the voltage of the cell. An example of this would be if the concentration of PuOzZ+is increased from 1 M to 1.5 M (Fig. 4B). If the bottom half-reaction (anode) is the one involved, the distance between the two half-reactions (rungs) is shortened, which decreases the cell voltage. As shown in Figure 4C, increasing the concentration of Cd(NHsjr2+ from 1 M to 2 M decreases the voltage of the cell. Decreasing the Concentration of a Reactant Just the oo~ositeholds true if the concentration (pressure) of a reactant in a half-reaction is decreased. he hrivina force for the reaction is less, the ~otentialbecomes more negative, and the position of the hdf-reaction is lowered on a potential ladder. For example, decreasing the ~ + the position of the cathode concentration of P U O ~lowers half-reaction (higher rung) and decreases the cell voltage (Fig. 4D). However, the cell voltage increases ifthe position of the anode half-reaction is lowered by a decrease in the concentration of Cd(NH,j42+(Fig. 4E). If two runos , far aoarl ~. - - -of interest on a ootential ladder are sufhcientlv on tne basas of the r P va Les 1e.g..at .east 0.3 VJ, t 1s ~n.inelytnat the relafivepos t~onsof these rmgs wdl change it me concentrarlons (pressures)of various species are different from unity (but not some unusually and suspiciousiy small or large values).In such asituation, there is no reason to coiect the posiiions of the rungs from those calwlationed with the Nernst equation, because all predicYins derivedfrom apotential ladder mnstructed on the basis of Pvalues will still be valid. ~~
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Increasing the Concentration of a Product Let us consider briefly what happens if we increase the concentration of Pu4', a produn of the half-reaction associated with the upper Ang. In terms ofthe precedingdiscussion, the driving force for the forward reaction becomes smaller, the position of the rung is lowered, and the ceU voltage is decreased. (Note that the cell voltage will increase if Pu4+lisdecreased, increase if [NH31is increased, and decrease if [N&1 is decreased.) This approach is an extension of Le Chatelier's Principle. However, this qualitative procedure allows one to bypass the task of using the Nernst equation to discover how the cell voltage changes for different concentrations or pressures of reactants and products. Short-circuiting(Discharging) a Galvanic Cell
What happens to a galvanic cell when one short-circuits it by connecting the two electrodes with a metal wire? Because electrons are now allowed to flow from the anode to the cathode, the two half-cells exhibit changes in the concentrations (pressures) of their chemical species. For the previous example, the following conventional cell reaction proceeds:
Looking f r s t at the cathode half-reaction, one sees that PuOzZ+ and H+ions are consumed as Pu4+is formed. Cumulatively, the effect of these changes is to decrease the potential of the half-reaction and to lower it on the ladder, as discussed in the preceding section. At the anode, on the other hand, NH3 is consumed,-andCd(NH2)p is produced, causing an increase in the potential of the half-reaction and pushing it upward on the ladder. Over time, the positions of the two half-reactions (rungs) approach each other, and the cell voltage decreases. Does the Cell Voltage Drop to Zero? It is tempting to consider the possibility that the two rungs will approach each other until they meet somewhere between the& original positions, wherebpon the cell voltage will be zero. However, if even one of the reactants involved in the spontaneous cell reaction is exhausted before the two rungs meet (or before equilibrium is establiihed), current flow throueh the cell will stoo. the two runes will remain separated on a potential lad&, and the ceii voltaee will have some value smaller than the orieinal voltaee. Even if no reactant is totally consumed. the concentration of at least one reactant may become so small that the current flow will bemme very tiny. Then the two runas can only approach each other a t an infinitesimally slowrate. In general, these problems become more important as the original distance of separation of the two rungs on a potential ladder is greater.
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Electrolytic Cells We now turn our attention to electrolytic cells+ells in which electrochemical processes are forced to occur in a nonspontaneous direction by the application of an external voltage across the two electrodes. Consider a cell consisting of a piece of zinc metal that dips into a solution containinn 1 M Zn2+connected via an a; iwpriate salt bridge to anorher solution containing 1 M Cu mto which a piece of copper metal is placed. Locating each of these electrode systems on a potential ladder, one sees that the difference between the two runes is 1.100 Y The mpper(II)-copper metal couple is positiv&th respect to the zinc(II!-zinc metal couple (Fig. 5A).
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If the cell behaves as a galvanic cell, the reaction that occurs as the cell discharges is
However, if this cell is to behave as an electrolytic cell, the reaction must be driven in the o posite direction so that wpper metal gets oxidized to CuE, and Zn2+gets reduced to zinc metal.
To achieve this goal, an external voltage source is placed across the two electrodes; the positive (+I terminal of the source is connected to the copper electrode, and the negative (-) terminal to the zinc electrode. Because copper metal is to be oxidized to Cu2+,the up er rung on the ladder is now the anode, and because Zn& .is to be reduced to zincmetal, the lower rung on the ladder is now the cathode -iust the reverse of what was seen earlier for a galvanic Determining an Appropriate Voltage Now that one knows how to wnnect the external voltage source to the zinc and w m e r electrodes and how to identify the anode and cathodh, how large must the voltage he to cause electrolvsis to proceed at a reawnable rate? What will happen ifthe applied voltage is exactly 1.100 V?Nothing will happen, that is, no current will flow, because the intrinsic voltage of the cell is perfectly balanced with an equal but oppositely applied voltage. Therefore, the applied voltage must be made greater than 1.100 V. What factors must one take into acwunt in making the applied voltage larger? There are three things to consider.
the existence of concentrationgradients near the surface of each electrode t h e effectsof electron-transfer kinetics for the reaction occurring at each electrode the resistance of the solution to current flow (migration of ions) Let us examine each of these factors individually in the context of Figure 5. Concentration Gradients At the anode, copper metal will be oxidized to CuZ+.
Thus, at the sullsceof the anode, the local concentration of Cu2' will be Lvcacer than it is in the bulk of the solution. Because the concentration of a s~eciesat the electrode surface governs the actual potenthi of the electrode, the position of the rung is raised. This means that the potential of the copper electrode must be more positive than before, ed is and this coal can be achieved onlv if the a ~.~ l i voltape greater t L n 1.100 V. A n analogous situation exists at the cathode where Zn2+ will be reduced to zinc metal.
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Thus, a t the surface of the cathode, the concentration of Znh must be lower than 1M. If the concentration of Znz+ is less than 1M, the position of the rung for the zindIIk zinc metal couple on a potential ladder will drop, and the potential of the zindIIkzinc metal electrode will bewme more negative. The occurrence of the electrolytic process requires tbat the rung for the copper(I1kopper metal couple be raised and that the rung for the zinc(I1)-zinc metal couple be lowered. Thus. the separation between these two runes is larger, and the voltage tbat must be applied to causethis effectmust be meater than 1.100 V(Fig. 5B).In fact. to get a faster electr~lysis,you must increase the concentratFon differences at the electrode surfaces. Thus,. . vou must deliberately increase the applied voltage. Electron-liansfer Kinetics When electrons are transferred between two oxidation states of the same element, it is necessary to climb and overcome a n energy barrier, as with any kinetically wntrolled process. Energy barriers must be surmounted to convert copper metal to CuZ+a t the anode and to convert Zn2+to zinc metal at the cathode. Where does the energy come from to accomplish these changes? Simply put, extra energy must be supplied, in the form of additional applied voltage, to get the anode and cathode reactions to proceed at a reasonable rate. Accordingly, the potential for the copper(II)-copper metal couple must be made even more positive: The rung on the ladder must be raised further. The potential for the zinc(1Ikzinc metal couple must be made even more negative: The rung on the ladder must be lowered further. Thus, the distance between the two rungs is increased more. The applied voltage must be made even larger by an amount sufficient to overmme the energy barriers for electron transfer ai the anode and cathode tFlg. ~ C I .
F ~ g ~5r ePotent~aliaader snowmg the re.atlve post~onsof the calnode and anode odrlng operar~onof an electroytc ce I ( A ) at vlnua ly zero cdrrent flow,(0)wltn effensof mncenlratlon grad ents aadw. (C) with effects of energy barrieffiforelectron transfer added, and (D) with iR drop added. The different line segments are not drawn to quantitative scale, but show the key qualititive trends 712
Journal of Chemical Education
Solution Resistance Even relativelv concentrated solutions of electrolvtes " have some resis&ce (perhaps a few ohms). When a current is passed through an electrolytic cell, there will be an ohmic potential drop, the sc-called iR drop. Because this ohmic potential drop is unavoidable, the applied voltage
must be further increased to take this effect into account (Fig. 5D). Summary
When the three factorsjust discussed are taken together, one can arrive at a single equation that shows how the voltage applied to an electrolytic cell is apportioned among these factors.
E,,=E,-E, +iR where E ., is the total applied voltage, E. is the actual potential of the anode, E, is the actual potential of the cathode, and iR is the ohmic potential drop. (E,and E,include the effects of the concentration gradient and the energy barrier for electron transfer.) By looking at a potential ladder, we see that the first two terms on the right side of this equation represent the vertical distance (difference)between the rungs on the ladder
corresponding to the anode and cathode (Fig. 5C). The third term on the right side of the equation reveals how much additional voltage must be applied to the electrolytic cell to overcome the iR drop so that the anode and cathode can have the proper potentials for the electrolysis to proceed at the desired rate. Acknowledgment We thank our colleagues--Stephen E. Creager, George I. Hanania, Steven 0.Russo, and Victor E. Viola-for their valuable comments and suggestions concerning this paper. Literature Cited 1. P e r k , R.I. J. Chem Edm. lW, 62,101&1019. 2. Mdox J. T. J. Chern.Edm. 1 N , 62,1018-1019. 3. Weat,A C. J. C l u m Educ 1 W , 63,609-610. 4. b e & , R W.J. ChrmEduc 1987.64.885, 5 . Ng W.-Y. J. Chem. Ed=. lsB8,65,727. 6.Al-Soudi, H . J. C h . Edue. 1980.66.630, 7. Moran,P J.;Cileadi,E. J. C k r n . E d u . l%B,66,912-916.
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