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GEORGE
California Stale Un~versity Carson. CA 90747
The Nernst Equation in High School Textbooks Daniel M. Perrine, S. J.' St. lgnatius College Prep. 1076 W. Roosevelt Road. Chicago. IL 60606
Almost all high school chemistry texts include a chapter on electrochemistry in which the student learns how to use a table of reduction notentials to nredict the voltage nroduced from a cell formedby joining two standard half c&s.*~hecells must bestandard for this process to work; i.e., they must he of unit activity-or, with the usual simplification, unit molarity. If the same student eoes on to take chemistrv in colleee or advanced placement &emistry in high schoo1,'he or shewill almost certainly be exposed to the Nernst equation (NE), which allows the cell voltage to he calculated when the half-cell concentrations are other than standard. The latest edition of a t least one excellent high school text has included the NE in its chapter on electrochemistry. Since this follows the trend of the last decade in making the high school chemistry curriculum more quantitative, we can perhaps expect to see the NE presented in other high school texts in the near future. With this there need be no quarrel; certainly the availability of inexpensive calculators makes logarithmic expressions such as the NE far less formidable a challenge than they once were. Besides, theoretical as the NE seems, it is one of the most practical things that high school students can learn: it explains why most of the "batteries" of everyday experience start out with a certain maximum voltage which eraduallv ,. , drons until the hatterv is "dead." Howewr, tht. high schml text just mentioned nut only giws exsmnles of hou [he N K can he used t o ralctllatr (.l~n.rrode poteniials a t nonstandard cell concentrations, but also it uses the NE to calculate electrode potentials a t nonstandard temperatures-and in a manner which leaves the student with the erroneous impression that the functional dependence of the electrode potential on temperature is given by the variable T i n the NE. Below is an example problem modeled after the example problem given in the text of this otherwise fine book.
(8.314)(318.15)(2,302) 2.00 lag (2)(96487) 0.500
Why Example Problem One is Incorrect
T h e obvious implication of the format and procedure of example prohlem one is that the variable T in the Nernst equation governs variations in cell voltage resulting from nonstandard temperature just as the concentration variable Q governs those resulting from nonstandard concentrations (activities). "Nonstandard concentrations? Change Q. Nonstandard temperature? Change T'-is how any high school student is likely to react to example problem one. Yet this cannot be so. If it were, any standard cell would be independent of the temperature, since log Q = 0 for Q = 1. But this certainly cannot he true, since Eo is related to the free energy change, AGO, by a constant:
and AG is clearly temperature dependent, being a function of temperature
.
~
Example Problem One: Nonstandard Concentrations at Nonstandard Temperature Problem: What would he the voltage of the cell ZnIZn2+(2.00M)IICu2+(0.5M)M)ICu
-
Zn(cj + Cu2+(aq) Cu(cj + Znz+(aq) and from the table of standard potentials we see thnt
-
+
Eo = Eo(Cu2+ Cuj - Eo(Zn2+ Zn) = 0.34 V 0.76 V
= E"
dG = VdP - S d T This equation is to be integrated for a change in pressure restrict the conditions to those of constant temperature, so that d T vanishes, and we have dG = VdP. Using the ideal gas law to express the volume as a function of pressure gives an equation integrable in terms of P , provided, again, that the temperature is constant. After several other steps (2) we finally have AG
= +l.lOV
Using the Nernst equation, E = Eo
If we were to accept the NE as giving the dependence of the cell voltage on temperature, we would be forced to conclude that all standard electrochemical cells, regardless of their chemical makeup, have a net cell reaction with a molar entropy change of zero, since AG could only be independent of changes in temperature if AS were identically zero. But this conclusion is very unlikely, to say the least. More fundamentally, the implication of example problem one is erroneous because i t nenlects the orininal process whereby the YE was derived. The starting point for ;he drrivation 11) is thr diffcrrntinl form uf the comhined first and second laws:
(= concentration for a perfect gas). T o allow this, we must
a t 45.00°C? Solution: The net cell reaction is
E
= 1.10 V -
- RTInF In Q, we have
- RT (2.302) log([Zn2+]/[Cu2+]) nF
= AGO
+ RT lnQ
and substitution of AG = -nFE gives the NE, Present address: Loyola University, Jesuit Community, 6525 North Sheridnn Road, Chicago, IL 60626. Volume 61 Number 4
April 1984
381
Hence the validity of the NE depends throughout on the assumption of a constant temperature. I t is true that the temperature, T, appears in the final result, hut this T is that unique temperature corresponding to Eo, i.e., that constant T a t which the integration was performed. If we use Eo for a cell of standard concentration at 298.15 K, then T must he set equal to 298.15 in employing the NE; but, if we are using E0 for a standard cell a t 350.15 K, then T must he set equal to 350.15. How To Do It Right How can the cell voltage he calculated if both concentrations and temperature are nonstandard? T o answer a problem like Examole Problem One correctlv, we must first find E0 for astandard rrll at 4S°C, EDnld.M'icannot use the ordinary tableof reduction potentials for this, as the textbook orohlem does, because these are for cells of standard conce&ration (andlor pressure) a t 298.15 K, Eozgs.~s How then do we find E0318?We can use a tahle of thermodynamic properties (3)to calculate AHo and AS" for the net cell reaction. Since these thermodynamic functions are essentially constant over this small temperature range, we can solve for AGO, using AGO = AHo - TASo, hut with T equal to 318 K. Using AGO= -nFEo, we then find E0318. If we follow this procedure for Example Problem One, we find that ASo is quite small, only -3.84 callmol K, and hence Enzlsis not changed significantly by the temperature change. The example in the textbook was a lucky one-hut all the more misleading for that! Below is a similar example prohlem-this time solved correctly by taking the change in Eo into account. Example Problem Two: Nonstandard Concentrations at Nonstandard Temperature Problem: What would he the voltage of the cell CuICu2+(2.00M)I IAg+(0.500M)IAg a t 45"C? Solution: T o use the NE, we must have Eo for 45'C = 318 K. From a tahle of thermodynamic properties (4) we find
Then
Finally,
For comparison, at thermodynamic standard temperature, this AgICu cell has an E0298of 0.46 V (we can determine this value from AG02gs or from a table of standard electrode potentials). At the same temperature, hut at the nonstandard concentration specified by problem two, it would have a voltage of
or 0.43 V. Finally, as we have just seen, with standard concentrations hut a t 45' i t has Eoala ..... = 0.44 V: and with nonstandard concentrations and nonstmldard temperature, it has a voltaer of 0.41 V. These results are presented in tabular form below. ~ g + ( M), l Cu2+(l M)
Ag+(0.500M). CU~+(Z.OOM)
0.46 V 0.44 V
0.43 V 0.41 V
298 K 318 K
Conclusion The procedure of Example Prohlem Two is quite lengthy and involves some material which is not covered in any high school text, even the one under discussion, which is of unusually extensive breadth and depth. Consequently, i t seems that if the NE is taught to students in their first year of high school chemistry, it would he more advisable to use the simple form E=EO-- 0.05917 n
Then for the net cell reaction,
-
+
Cu(c) + 2Ag+(aq) 2Ag(c) Cu2+(aq) AH" = (15.48) - (2)(25.234)= -34.99 kcal/mol
AS"
=
(2)(10.17)+ (-23.8)
- (7.923) - (2)(17.37)
At 318 K,we have
10s Q
emphasizing that this equation holds only for cells a t the standard thermodynamic temperature, 298.15 K. If a more inquisitive student asks for an explanation of the origin of the constant 0.05916, he or she can he told that it is equal to RT(1n 10)lF. .. . hut that the derivation of this must await a later course. However, the process of calculating variations in E for (small) variations in temperature which is described in Examole Problem Two can eive eood oractice in the use of " . thermodynamic tables and-the manipulation of the standard thermodvnamic eauations. Problems of this sort could provide worthwhile matehal for high school advanced placement or first-year college chemistry courses. Llterature Cited (1) Mshan, Bruce H., "Univemiw Chemievy." 3rd ed..Addison Wesley, Reading, MA, 1975. " . I , " *"",.
(2) Ref ( I ) . pp. 337342. (3) Dean, John A,, (E&for)."Lange'sHandbookofChemistw,"12th ed., McCrsw-Hill Bmk Co., New York,1979, pp. 9-20 and 9-62. (4) Ref. (3).pp. 9-49and 9-62.
382
Journal of Chemical Education
