Textbook Errors, 706 Omer Robbins, Jr. Eastern Michigan University Ypsilanti, Michigan 48197
The Proper Definition of Standard Electromotive Force
In practically every textbook of general chemistry, be it written for a terminal course or for use by science students, the electromotive force of a galvanic cell is discussed. The importance of this quantity is understandable since it is used as a measure of the amount of electrical work obtainable from the cell and an aid in obtaining the equilibrium const,ant and other thermodynamic quantities, e.g., reaction free energy changes, entropy changes, and activity coefficients. The discussion invariably leads to thc concept of the standard electromotive force, Eo, and the Nernst Equation. Becausc of the importance of the Envalue and its appearance in tabulated data it is imperative that this concept be presented free from misconceptions. Most advanccd texts in physical chemistry treat this problem correctly and a t length but many authors of beginning texts do not. It is the purpose of this paper to prcsent the correct definition of the standard state for electrolytes; to define E0 both rigorously and from a teaching point of view and finally to show the magnitude of the difference between the EO's obtained from a correct and an incorrect standard state definition. Only aqueous electrochemical cells without liquid junction potentials will be considered and a specificd temperature of 25°C is also assumed. The Standard State
A review of twenty-four textbooks written for oneyear, non-terminal courses disclosed that the majority of authors associated EO values widh improperly described standard states. Table 1 summarizes the results; only three of the authors of these texts use the proper definition of the standard state. Such s t a t e ments as these occur frequently: '' . . . Numerical values of the oxidation potential apply to aqueous solutions at 25°C in which the concentration, or more exactly, the activity of dissolved species is 1 molal"; "The standard conditions that have been chosen arc 1 molar concentration for all dissolved materials . . . (footnote: Actually not 1 molar hut 1 molal)"; Presented before the ACS Southwest Section Meeting., December 4, 1968. Suggestions of material suitable for this column and guest columns suitable for publication directly should be sent with as many details as possible, and particularly with reference to modern textbooks, to W. H. Eberhardt, School of Chemistry, Georgia Institute of Technology, Atlanta, Ga. 30332. Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the sources of errors discussed will not be cited. I n order to he presented, s n error must occur in s t least two independent recent standard books.
and "Standard voltages are determined at 25'C for cells mith all reactants and products at unit activity." The standard state is an important concept hccausc it provides a reference point upon which to establish all numerical values of thermodyuamic quantities; e.g., I' . . . the standard state must be clearly and unambiguously defined before the activity can be given a defiuit,~numerical value" (1). It, however, is not the reference state. This lattcr statc is one that can be approached experiment,ally although not necessarily achieved, such as that state of a gas where it,s fugacity is equal t,o it,s partial pressure, namely zero pressurc. On the other hand, the standard state, again for a gas, is the ideal gas at 1 atm pressure. Thus, standard states may be selected quite arbitrarily and do not always correspond to states that can bc obtained under experimental conditions. Specifications of t,hc standard state of a chemical species vary from system to system, i.e., it may be different for a pure substance as a gas, liquid, solid, or in solution. For the purposes of this discussion it will suffice to give the definition of the standard state for electrolytic solutes using the molal concentration scalc. This concentration scalc is chosen because it is temperature independent and practically all Eo values are expressed with reference to it. A general statement t.hen is as follows: the standard state of an clectrolytic solute is that of a hypothetical solution at a mean ionic molality of unity where the reference state is so chosen that the mean ionic activity coefficient approaches unity as the concentration of the solute approaches zero (24). Before a more dct,ailcd discussion of the definition is undertalcen, it is necessary to define the mathematical relationships between the quantities involved in the above statement. For the general solute, A,B,, they are a, = a*.+"+.aB.-"-=
Table 1.
(a*)" = (rn*"*)"
Standard State Definitions Found in General Chemistry Textbooksa
Definitions
Number of texts
1. Unit molarity 2. Unit molality 3. Unit moldity (qualified via footnotes that refer to activily) 4. Unit activity of electrolyte in the solution 5 . Hypotheticalor ideal solution of unit activity 6. Noncommittal Total ~
a
(1)
4 5 4
7 3
~
1 24
For science students. Volume 48, Number
I I , November 1971
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737
Table 2. Solute
tme
1-1. 2-2
Formulas for E l e c t r o l y t e Activities
Example
KCI, MgSO. BaCL
3-1
Lacla
11.2' 22.11 11.3'
3-2
AI.ISO.I3
2'-3"
2-1 1-2
NslS01
mt from
viY+-u-"11.11
= = = =
1
4'1% 4'1% 27 2i1/'m 108 108'/% 4 4
Eqn. (3)
m = 1.5874 m = 1.5874 m = 2.2195 m = 2.5508 m
rz* from
Eqn. (4)
rn'?iP 4ma-/,'
4m+*' 2irn'yi' 108m67l
and where a2 is the activity of the solute in the actual solution; a*, mi, and r* are the mean ionic activity, molality and activity coefficient, respectively, m is the molality of the solute and v+, u-, and v are the numbers of positive, negative, and total ions, respectively, per formula of the solute. Combining eqns. (1) and (3) yields the usual formula for the activity of an electrolyte in terms of its activity coefficient and molality as = (v+"*.v-"-)(muJ
(4)
Hereafter m+ and r+will be referred to as the mean molality and the activity coefficient, respectively. Table 2 contains the results of an application of these equations to some representative solutes. The form of the relationship involving molality, column 4, Table 2, is dictated by the criterion for the reference state for electrolytic solutes; namely, that the limit of a+/m, = r+ = 1 as m + 0. Lewis and Randall (7) illustrates this quite nicely for a 2-1 electrolyte. Returning now to the definition of the standard state given earlier, it is desirable to indicate why unit mean molality was chosen rather than infinite dilution. Some properties such as the partial molal enthalpy and partial molal heat capacity have nearly the same values both a t infinite dilution and at unit mean molality (8), thus either state would be a satisfactory standard for these properties. On the other hand, properties such as the partial molal free energy (the chemical potential) and the partial molal entropy approach negative infinity and positive infinity, respectively, at infinite dilution. Thus, no specific values can be assigned in this limit and a finite concentration must be utilized; unit mean molality is a natural choice among finite values. Figure 1 provides a graphical illustration of the
hypothetical solution
of
definition of the standard states for the case of a 1-1 electrolyte. The dashed line corresponds to an activity coefficient of unity a t infinite dilution, or the reference state. The hypothetical solution representing the standard state corresponds to an extension of the limiting slope to a mean molality of unity. Thus, in the standard state m, = 1 and r* = 1, or point A in Figure 1. Because of these two requirements, az = (m,r+)= will also he unity. Now that the standard state has been defined, some indication as to what the standard state is not may be helpful It does not correspond to solute of unit activity in the actual solution, point B, Figure 1. It does not correspond to solute of unit molality in the actual solution, point C, Figure 1. 3. It does not correspond to a solution at infinite dilution, point 0,Figure 1, or the reference state.
As indicated, Figure 1 is a schematic representation of the way activity, az, varies with molality. Figures 2 and 3 show these relationships for actual solutions; namely HC1, a 1-1 electrolyte, and CaC12, a 2-1 electrolyte. The concentration range covered in these latter two figures is too large to show that az/m*' = 1 asmj0. To summarize, two conditions must be met in establishing the standard state, namely that the activity coefficient and the mean molality both equal unity and and the only way for this to occur is to postulate a hypothetical solution. Thus, those definitions used by the authors of the texts categorized in Tahle 1, that did not include some reference to an ideal or hypothetical solution, were avoiding the true concept of the standard state. In all fairness, it should he said that the use of unit
Figure 2.
Activity versus mean mololity for HCI rolution (10).
Figure 3.
Activity versus mean m o l ~ l i t yfor CaCln solutions ( I I ) .
; 2 + .,
-+
2 I I
I
I
i
0
1.0
Figure 1. Schematic diagram of the ostivity as a function of mololity for a 1-1 electrolyte (9).
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Journal of C h e m i c a l Education
activity is more correct than unit molality or molarity. And, in fact, the standard state does involve unit activity. However, one of the criteria of this state is that the activity coefficient also he unity. Figures 2 and 3 demonstrate clearly that for unit solute activity in the actual solutions, points B, t.he activity coefficient corresponding to the square and cube roots, respectively, of the slope at these points, is definitely not unity. Standard Electromotive Force, E0
For the general reaction taking place in a cell in which a solution of an electrolytic solute A,B,, is produced from its elements, where any gases involved are at unit fugacity and liquids and solids a t unit activity, the Nernst equation takes the form
Here the activity of the electrolyte, az, is that for the actual solution. Combining eqns. (1) and (5) and using the notation x = u+ and y = v-, one obtains
or from eqns. (4) and (5)
For solutions of unknown activity coefficient, there are two variables that must be specified, namely Eu and 7,. To determine both quantities from measurements of E, extrapolation of a function of E versus a function of molality to infinite dilution is necessary. A variety of functions have been used depending on the concentration range (13-14). However, since this paper is concerned more with the concept of Eo than the details of its experimental determinat,ion, the reader is referred to the references cited for these details. Combining the definition of the standard state discussed in the previous section with eqn. (6), Eo is seen to be that electromotive force for a cell containing a solution in which m , = y, = 1. Necdless to say, this is not a definition that has much meaning for freshman students. However, it does include the characteristics possessed by En,namely, association with solutions of zero ionic strength, as reflected in y+ = 1, and a finite value of concentration, not zero, as reflected by m, = 1. This latter condition is dictated by the logarithmic form of the Nernst equation and is directly related to the partial molal free energy. To reiterate, the only type of solution that can simultaneously satisfy the requirements that m, = y, = 1, is a hypothetical one. As mentioned earlier, in connection with Tahle 1, Enis frequently described in terms of unit activity of the electrolyte in the actual solution. When this is done, the third term in eqn. (5) disappears and E = EU. However, it should be clearly understood that this does not represent the standard state, and strictly speaking the emf so obtained should not be called the
standard electromotive force. Admittedly it is a subtle point because the numerical values of E 9 b tained from eqn. (5) when a? (actual) = 1 are the same as those obtained from eqn. (6) when m , = y, = 1. In light of these comments, what then is a definition of Eo that can be taught to freshman students and still make these necessary distinctions? A possible definition might he: the standard electromotive force is that potential associated with a reaction conducted in a cell in which the reactants and products are in their standard states, which for solutions of n-n electrolytes is represented by an ideal solution of unit molality. This statement must be supplemented by an explanation of n-n; namely z+ = z- = n or 1-1, 2-2, . . . , type electrolytes. To obviate this restriction on the definition, it would then be necessary to say "an ideal solution of unit mean ionic molality," thus requiring elucidation of an even more difficult concept, i.e., m,. The use of ideal rather than hypothetical ensures the conditions for y, = 1. The definition used by Latimer (16), is quoted frequently, i.e., "a potential is referred to as an Eu value if all gases involved in the reaction are at a fugacity (thermodynamic pressure) of 1 atmosphere and all dissolved substances at an activity (thermodynamic concentration) of 1 molal." This definition states that one can measure Eo directly using a solution of unit activity. Such is the case if the activity coefficient as a function of concentration is known, see eqn. (7). However, as pointed out earlier, such a solution does not represent the standard state, thus the Eu so obtained should not properly be called the standard electromotive force. These comments about Eu were made not so much to take issue with the use of unit activity, but rather to pave thc way for an analysis of the effect on electromotive force calculations when the standard state is described in terms of unit molality or unit molarity. Unit Molality and EO
Without specifying a hypothetical or ideal solution and an n-n type solute, it is incorrect to define Eo utilizing a standard state in terms of unit molality or molarity, e.g., Tablc 1. It will be thc purpose of this section to compare known E0 values with those E values calculated using such an incorrect definition for some selected electrolytes. To point up the distinction between solutions of unit molality and unit activity, Tahle 3 lists the activity corresponding to unit molality, i.e., m z = 1, for several electrolytes. An examination of this table shows that the activity deviates markedly from unit molality for Table 3.
Electrolvte
Activity of One Molol Solutions of Selected Electrolvtes
Tv~e
r~for ml = 1*
an
( e m . (4))
HC1 HBr HI NaCl NaI MgCL MnCh NiCL NkCrO, ZnSO$ a
From ref. ( 1 7 ) . Volume 48, Number 11, November 1971
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739
Table 4.
A Values for Some Selected Electrolytes [A =E' - P )
Electrolyte
'able 5.
Difference, A = E' - P, as a Function of Activity Coefficient
Electrolyte type
---
r* range for
Lower .rl
1 m solution*
A for----
Higher 7 ,
From ref. (17)
Summary
!-2 electrolytes, less so for the 1-2 type and varies be,ween 0.4 and 0.9 for the 1-1 and 2-1 electrolytes :hosen. Thus, it is clearly seen that unit molality does lot correspond to unit activity. What then will be ,he E corresponding to an incorrectly specified stanlard state of unit molality? At 25°C the Nernst equation becomes
here n is the number of "moles of electrons" involved n a reaction. A comparison can be made between E' (that electromotive force corresponding to unit nolality) and E0 (the standard electromotive force :alculated from tables of standard half reaction poteniials) by defining a diierence A=E,-Eo=
- -0.0591 log an(m = 1) n
(9)
Fable 4 lists A values for the electrolytes of Table 3 tgain for 1 molal solutions, along with E0for comparison ourposes. At least one observation can be made from data in Table 4. As the activity coefficient, y*, decreases, 30 does the activity corresponding to a one molal ~olutionresulting in a larger and larger difference, E' - EO. Hydrogen iodide, a* = 0.93, has a A value of ).001 V, whereas zinc sulfate with a, = 0.0019 yields h = 0.081 V. Table 5 contains data for a larger number of electrolytes than Table 4, e.g., 30 for 1-1 type, 26 for 2-1 type and 8 each for 1-2 and 2-2 types. The activity coefficient range represents the smallest and largest y, for each of the above groups. The third and fourth columns correspond to the difference E' - Eo, using az (m = 1) for the smallest and largest y,, respectively. It can be seen that E' calculated from a definition of
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the standard state as one molal, differs from En, corresponding to a proper definition of the standard state, by as much as 0.08 V. As expected, the diierence decreases for solutions that more nearly approximate ideal behavior, e.g., 0.0054 V for a 1-1 type solute of y* = 0.9. Admittedly A values are small for high y,, but regardless of this fact, we teachers of freshman chemistry are not justified in using a faulty definition of the standard state. The use of unit molarity to describe the standard state is open to even greater criticism, principally because most tabulated values of standard half reaction potentials were obtained utilizing solutions expressed in molality. For solutes ranging in molecular weight from 100 to 400, the molality of a 1 M solution corresponds to 1.02 m and to 1.05 m, respectively. Thus a larger A is obtained when using this concentration scale to define the standard state. Of course, the obvious remedy is not to use molarity when discussing EO'sobtained from the molality scale.
Journal of Chimical Education
It has been pointed out that many authors of general chemistry textbooks do not properly specify the standard state for electrolytes when discussing the standard electromotive force. If these definitions are taken literally the calculated E values can differ markedly from known E0 values depending on electrolyte type and activity coefficient. The challenge that faces textbook writers is how then to present this topic a t a beginning level without introducing misconceptions. A suggested definition of EOwas proposed; namely, that potential associated with a reaction conducted in a cell in which reactants and products are in their standard states, which for solutions of n-n electrolytes is represented by an ideal solution of unit mean ionic molality. Acknowledgment
My thanks go to E. F. Westrum Jr. and A. J. de Bethune for reviewing the manuscript in its preliminary form and making valuable suggestions. Literature Cited (1) HARNBD,H., A N D OWENS.B., "The Physical Chemistry of Electrolytic Solutions" (3rd ed.). Reinhold Publishing Corp., Nsw York, 1958, pp. 7-8. (2) Ibid p. 425. (3) ROBINBON. R., AND STORE& S., " E l e c t d y t e Solutions" (2nd ed.), Butterworth, London, 1959, p. 28. (4) MACINNEB. D., "The Prinelplea of Eieatrachemistry," Dover Puhiioations.Ina., NewYork. 1 9 6 1 , ~ 183. . (5) Komdm, G.. "Treatise on Eleotroehemistry" (2nd ed.). Elsevier Publishing Co., New York, 1965, p. 59. (6) Lswrs, G.. A N D RANDALL. M. (revised by Pnwm. K.. AND BREWER, L.), "Thermodynamics" (2nd ed.), MoGraw-Hill Book Co., New Ynrk. -~~ 1961. n.247 ( 7 ) Ibid.. p . 311. (8) G L A ~ ~ T O N9.. . . "Themadynamics for Chemists." D. van Noatrand, New York, 1947. p. 366. (9) KLOTZ, J.. "Chsniioal Thermodynsmios." Pmntice-Hall Inc., Englewood Cliffs, N. J., I950,p.318. M., op.cit., p.317. (10) Levla, G., AND RANDALL, (11) Ibid., p. 653. (12) Ibid., pp. 314-319. S..o ~ ) . c i t .~n.363-368. . (13) GLABBTONE. j14j H ~ B N ~ D H.,, *& o~kbs,B., o p . it., pp. 430-433. (15) Lnwra, G., AND RANDALL, M.. 09. cil., Chapter 22. W. M.. "Oxidation Potentials" (2nd ed.), PrantiobHall Ino., (16) LATIMER, EnglewoodCliffs. N. J.. 1952, p. 2. (17) Rosmson, R.. A N D STOKES, S..op.cil., pp. 491-502.
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