An Electrochemical Introduction T. L. Webel, T. E. Milk, and S. A. Sahon' Florida State University, Tallahassee, FL 32306
N. J. Sellev has nronosed a straiehtforward laboratorv experiment t i a t illukrhes the concept of the entropy in chemical reaction ( I ) . Since students eenerallv have a h e t ter-developed senseof the role of energy (really heat) in reactions than of entropy, this experiment is of particular value. The essence of i t is this: When two species X and Y are mixed together, the Second Law of Thermodynamics requires that the entropy of the system increase (2). If the interactions between X's and X's, Y's and Y's, and X's and Y's are nearly the same, then the mixing is said to be ideal-no heat is given off or absorbed by the system-and the increase in the entropy is the sole "driving force" for the process. When X and Y are related ionic species, such as Fe(CN)f3- and Fe(CN)r4-. then the mixine can be accomplished ~ l e c t r o c h e m i c aby l~~ the transfer ofan electron from Fc(C.\I,&'- in one half-rell to Fe(CN),:" in the other halfcell. Although solutions ot'thcse ions a;c not ideal, the principal driving force is still the entropv of mixinr. The student isthereforeahle to observe that forthe "rea&on7'
a
Fe(C~)h-[l]+ Fe(CN)2-[2]
-
+
F~(cN)$~-[z]F~(cN):-[I]
(1)
which has no net gain or loss of any species, there is nonetheless a cell potential or voltage produced. (We use the notation [I] or [2] to label species in each half-cell.) This means, of course, that one could use this "mixing reaction" as a battery and obtain useful work from it. In his discussion Selley chooses to emphasize the connection of entropy to statistical concepts. This is an important aspect that students need to grasp if they are truly to understand the sienificance of e n t r o.~ vHowever. . we feel that such a clearly comprehensible experiment can profitably he linked to a t least two other quantities that are of fundamental importance in chemical thermodynamics: chemical potential and activitv. Most students find the concept of chemical potential to he rather abstract. That i t can heso naturally coupled to a concrete measurement like cell voltages ought to he stressed to the students. Similarly, most students regard "activity coefficient'' as a dressed-up expression for "fudge factor" because they have not had much experience dealing with solution thermodynamics nor have they thought a great deal about how molecular interactions affect macroscopic properties of solutions. Again, observing that the cell potential changes when one adds "inert" salts ought t o create a lasting and perhaps even a thougbt-provokinp" im~ression. Now, "concentration cells" have been used in undergraduate laboratow for a lone time (3).However. the -experiments . reaction in eq 1seems to he particurarly well suited for'this purpose, for i t provides a ready chemical connection t o an L a s h interpretable, everyday oicurrence-mixing. There are many variations on the original Selley experiment that one can think of to illustrate the thermodynamic concepts mentioned above. We first derive the expressions
' Author to whom correspondence should be addressed.
Activity is a dimensionless quantity. Thus, c,should really be me concentration in molarity divided by 1 M. However, we will continue to label it with molarity units because it is often convenient to think of c, as if it did have units. 492
Journal of Chemical Education
for the cell potential, free energy, and entropy of mixing appropriate for reaction 1 by thermodynamic arguments. Then, as one kind of example, we present some results for the cell potentials carried out as suggested hy Selley and after adding 0.25 M KN03 and 0.25 M KzSOl to one of the half-cells. These dataare compared with the "ideal" solution cell potentials and with the "ideal" values corrected for activity coefficients calculated from the Dehye-Huckel theory (4). Cell Thermodynamics A particularly clear exposition of this subject is given by Levine (5). Our development here is adapted from his approach. The chemical potential of species i in a solution is defined as
where G is the Gihbs free energy of the whole system, nj is the number of moles of component i, nj#j refers to the numbers of moles of the components other than i, T is the absolute temperature, and P i s the pressure. The activity of i is defined by the expression where fit" is the chemical potential of i in its standard state. We can choose that aj be given by cjyj, where cj is the molaritv of i and -G is the activitv coefficient of i in the . (molIdm3) . s ~ l u t i o n In . ~ this case becomes unity as the solution hecomes more dilute and n;" is the chemical notential of i as the solution becomes infinitkly dilute extrapdlated hack to ci = 1 M. Or, in other words. UP is the chemical potential i would have at c, = 1 M if the&)lecular interactinm were the same at this concentration that thev are at infinite dilution. For the case that i is an ionwith charge z (e.g., for K+, z = +l;for F ~ ( C N ) G ~z -=, -3), the free energy of the ion also depends on the electrical potential, 9 , in the medium. This means that the total free energy for a system which includes ions, G", must include an additional term zFpni, where F is the Faraday or the charge of one mole of unit positive charges (-96500 Clmol), for each ionic species. The "electrochemical" potential of i can thus he written as
4
In a conducting medium that is not carrying current there is no electric field and therefore the electrical notential is constant (6). In the usual aqueous solution with electrolytes s role since 0 is constant and the extra term in ea.3 ~ . l a.v no the requirement of electrical neutrality ensures that these "extra" terms will cancel out for the total free energy; i.e.,
An electrochemical cell usually consists of two half-cells connected by a salt bridge; each half-cell includes a metal electrode. Thus, if the two electrodes are not in direct contact, there are three separate phases: a single liquid phase
and two separate electrode phases. The half-cell reactions corresponding to reaction 1can be written as, for half-cell 1, Fe(CN):-
+ e-
and for half cell 2
-
Fe(CN)f
Fe(CN),' Fe(CN)?or the total reaction can he written as
(4)
+ e-
(5)
When the volume of each half-cell is V, then the change in number of moles can he related to the change in concentration of each species bv vVdc, where c is in molldm3. V is in dm3, and v is the stoichiome&ic coefficient of each species in reaction 1with the convention that v is positive for ~ r o d u c t s and negative for reactants. Using eq 2 fbr the activities, eq 8 becomes
The notation az[N] is shorthand here for the activity of F e ( C N p in half-cell N. The cell potential is then when the electrodes in each half-cell are not connected together. Under the conditions of extremely high resistance between electrodes, the current flow is essentially zero and this closed system must he in equilibrium (not chemical equilibrium in the usual sense!). At constant T and P the change in the Gibbs free energy for all the components of reaction 6, including the electrons in the separate electrode phases,
We can rewrite this summation, separating out the terms for the electrons, as
where the prime on the left-hand side means the sum only over solution components. This quantity is the same as dG" for the species in reaction 1, which is what we would like to focus on. When an infinitesimal amount of current is allowed to flow, there is an infinitesimal change in the numher of moles of reactants and products of reaction 6 such that for each chemical species dn[l] = -dn[2]. Now, since the electrical potential rpe of the liquid phase is essentially constant under these conditions, i t is easy to see that the terms containing rpP will cancel out so that
where G is now the usual free energy of the chemical species in reaction 1.At the same time on the right-hand side, when the electrode materials are both the same, say platinum electrodes, then the electrochemical potentials of the electrons can differ only because the electrical potentials in the two unconnected electrodes differ. That is, the right-hand side becomes -F(q' - rp2)dne[2]. We now can define the electromotive force of the cell, or cell potential, as the difference in the electrical potentials of the electrodes: E = q1 -rp2. If one now recalls that electrical work consists in moving charges across a potential difference, one sees that the right-hand side of eq 7 represents the electrical work that the cell is capable of under reversible conditions, namely its maximum possible work. This result can be summarized as dG = l ' p , d n j
= -FEdn,[Z] = dw,,
(8)
i
One should be aware that this is not the same as that forme mixing of solutions of equal volume and concentration of K3Fe(CN)B and K,Fe(CN),. This is because the electrochemical mixing does not involve the K+ Ions. For the mixing of the actual solutions one would need to include in eq 13 terms for the initial and final K+ concentrations. This result isafactor of 2smallerthan the AGand Asobtained by Selley ( 1).
which is an example of the Nernst Equation (7). Finally, using the relation between activity and concentration, we write eq 10 as two terms
The notation here of @ [ N ] and - f [ N j is analogous to that above for activity. One should note that the second term in eq 11 becomes zero for an ideal solution. In a completely mixed system where each half-cell has the same composition, the concentration and activity coefficient ratios are unity. From eq 11one can see that E must therefore be zero even if the solution is not ideal. This is not true, however, when "inert" salts are added to one half-cell; for the activity coefficient ratios are then no longer the same. The AG of mixing can be ohtained by integrating eq 8
where c represents the change in concentration from the "pure" reactant concentrations or the concentration of the products as reaction 1progresses. Initially c = 0;after the mixine is c o m ~ l e t ec = c3-11112 = c4-12112 if one starts with equal;olumeiand concen&kions of tjlereadants. For ideal solutions the expression in ea 11can be suhstituted for E and integrated an&ically. o n i can also obtain the result by so that recognizing that A&, = Gmired- Gunmixed
where the index j refers to each species in each solution. When one mixes ideal solutions of Fe(CN)& a n d Fe(CN)&, starting with equal volumes V and equal initial concentrations c;, the final volume is 2Vand final concentrations are ciI2, and one obtains from eq 12 or eq 13 that AG,i, = -2Vc;RT 1112. T o get the AG per mole of ions, one just divides t h ~ e s u l by t the total number of moles 2Vc;, with the result AG,i, = -RT In2 = -1.69 kJImol a t 293K.3,4 For an ideal solution AG,:, = -TAS,k (8).Thus, per mole = +R In2 = 5.76 JK-moL4 of ions for the case above, aSmimi Experimental
A standard electrochemical cell was employed for these experiments, very similar to that described hy Selley (I). The half-cell compartments were connected t o the salt bridge compartment by a type of sintered-glass connection. The electrodes were shiny platinum foil strips approximateIv 1cm2.sealed into dass. The cell ~otentialswere measured 6 y a R K Precision ~odel2.806digital multimeter which has a lo9-ohminout impedance and a resolution of 0.1 mV on its 200 mV input range. Commercial analytical grade K8Fe(CNh, K4Fe(CN)~, KCI, KN03, and KzSOa were used t o make stock solutions for use in the experiments. In the experiments with salt addedto balf-cell 1, the concentration of theKN03 or K2SO4 Volume 63 Number 6 June 1986
493
was always 0.25 M. In all the experiments the sum of the Fe(CN)& and FE(CN)& concentrations was kept a t 0.1 M. Measurements were taken with the concentrations of each species in each half-cell chosen t o correspond to a different stage of mixing, starting with c = 0.005 M u p to the completely mixed value of c = 0.05 M. The volume in each half-
cell was always 20ml. Saturated KC1 was used exclusively in the salt bridge. The experiments were not carried out in a constant temperature bath so that the temperature varied slightly from day t o day. The.range was approximately 17-22°C, which probably leads to errors of about 1-270. Results and Discussion
The results of the measurements are shown in Figures 1-5.
CONCENTRATION CHANGE c (mol/dm3) Figure 1. Cell potential as a function of the concentration change c. The experimentalresults(., -)are compared with ideal potentials (0. --)and with the ideal potentials wnected for activity coefficients as in eq 14 of the text ( 0 . .). To calculate the A&.. the experimental cell potemial curve has been analytically extended lo c = 0 M(cwrerp0ndingto 0.1 MFe(CN)e3and Fe(CNh4- solutions), as described in the text. The errws in the measure ments of Eere estimated to be about +3 mV.
..
Figure 2 Ceii pmsmlai as a lunction at wncentratlon change c. when the 0 1 MFe(CN)$- hail-cell 8s 0 25 Min KNO, The flgure lo atherw sa composed as in Figure 1
494
Journal of Chemical Education
CONCENTRATION CHANGE c
Imolhm3)
Fngure 3 Ceii potenttai as a tunction at wncentratlon change c. when the 0. 1 MFe(CNIs4- haitcell is 0.25 Min KNOo The logure 8s otherwise composed as in Figure 1
Figure 4. Cell potential as a hrnction of concentration change G when the 0.1 MFe(CN)e3-Kalt-cell Is 0.25 Min K2S04. The figure is otherwise composedas in Figure 1.
In these figures the data are compared with the ideal values of E (open circles), and the values of E calculated by assuming that the activity coefficients in eq 11 can be obtained from the Debye-Hueckel theory ( 4 ) (open squares),
5.)
Iny =-1.174z2 -
where I is the ionic strength of the solution in each half-cell.5 In eeneral, the ameement is not bad.. narticularlv in Fieures 1,3, and 5: One should note that E becomes negative before c = 0.05 M in Figures 3 and 5. This means that in these cases the cell would reach equilibrium (i.e., when E = 0) before there is complete mixing. Similarly, for Figures 2 and 4, equilibrium would not be reached until well after c = 0.05 M. This is a conseouence of the chanee in activities and chemical notentials or the species in haG-cell1 d x t o the added salts. Also given in each figure is the AGmi.. This quantity corresponds to the work that could he obtained from the cell running reversibly to achieve complete mixing. We obtained the results shown by graphical integration of E, as in eq 12, from c = 0.005 to c = 0.05 M. T o obtain the part of the integral for the unmeasured region from c = 0 to 0.005 M, we assumed a form (large dashed curve) for E
which is suggested by eq 11 and which can be integrated analvticallv. We fit the constant term Cn bv matchine.. eo. 15 to the measured curve at c = 0.005 M. ~ h ais.i Co represents the difference between the measured valueof E and the ideal
value at c = 0.005 M. Equation 15 in effect assumes that the activitv coefficient ratios do not chanee annreciablv in this range of c, which is probably a good a&ro&natio~.~ In Figure 1,the value obtained for AG- differs only by a few percent from the ideal value (-1.69 kJ1mol). For the other cases, however, there are substantial deviations. One should note that our integrations include the negative contributions in Figures 3 and 5. If one actually ran these cells, one would have to put energy back in to get t o the c = 0.05 M point because, as noted above, the cell would stop operating when the concentrations are such that E = 0. By using eq 13, one can compare the experimental with that expected when the activities are given by the Debye-Huckel theory. For Figures 1-5, respectively, we obtain -1.47 kJ/mol, -1.54 kJImol, -1.46 kJ/mol, -1.67 kJ/mol, and -1.39 kJImol. Although these results tend in the right direction.. thev . are verv far off the mark. This suewsts that the Debye-Huckel the& is better at giving ratiosol activity coefficients than absolute values of activity coefficients.
zmi,
Conclusions We haveshown here thesuhstantialeffecrsofadding inert salts to electrolytic cells, which, it is hoped, convinces students that concentration and activity are-not necessarily the same, that the chemical potentials, which are the proper quantities to consider in examining the driving force of a chemical reaction, can give rise to electrical potential differences in oxidation-reduction reactions and that the extent of a chemical reaction may depend on the strength of the interactions of nonreactine-species with the reactants and rod. ucts. Variations on these experiments can easilv be worked out. One could, for example. have students invesiigate more carefully the dependence of the deviation of E from ideality on ion& strength by using a number of different inert salts a t various concentrations. In labs with comuuter facilities available, the students could practice numer&al integration techniques in calculating the AG,i, for the different cases. They could also explore the effect of varying the parameters in the models for activity coefficients (4) in matching the theoretical values of E and AGmi, to their experimental results. A final suggestion is to measure the cell potentials in constant-temperature baths over a range of temperatures from, say, 20 to 40 "C. The temperature dependence of AGmi, is related to the AHmi,. For an ideal mixing process (8) AHmi, = 0 so that the deviations from ideality can be observed in this way as well.
(3) See, f o r e m p l e , the experiment in Liuingeton, R. "Phyaim Chemical Experimentz": M a c m i l h New York, 1939: p 222. (41 Ref. 2, p 268. ( 5 ) Ref. 2, chap 14. ( 6 ) For example, see Sehwartz, M. "P"nciples of Eieetrodynamies"; MGran-Hill: New York. 1972: p34. (71 See ref 2, p 389. (8) See rot 2, P 239.
CONCENTRATION CHANGE c ( ml/dm3) Figure 5. Cell pdentlal as a function of eoncenlration change c, when the 0.1 MFe(CN)$-half-cell is 0.25 Min K2S04.The figureis otherwisecomposed as in Figure 1
The error introduced by using Icalculated from molarities rather than molalities here is not significant at these concentrations in comparison with the approximations in the Debye-Huckel model itself.
Volume 63
Number 6
June 1986
495
