chemical principle/ ted revi~i
Edited by Charles D. Mickey Texas AbM at Galveston GaI"est0n. TX 77553
Clarifying the Concept of Equilibrium in Chemically W. F. Harris Department of Chemical Engineering, University of the Witwatersrand Johannesburg 2001, South Africa
Introduction Mickey (I) gives a clear In a recent issue of THIS JOURNAL account of the conventional approach to the teaching of equilibrium in chemically reacting systems. While this approach may be adequate for the mechanical solution of problems usually posed in introductory courses it does not, t o my mind, provide as good a foundation as i t might for cnurses a t a hieher .- ~~"~ level. Furthermore. it results in misconceptions that are often difficult t o corrkct. My particular objections are as follows: ~
~
~~
~
Square brackets represent concentration. He then argues that, a t equilibrium, Rater = Rate,
(4)
from which i t follows that
~~
~
Hence
1) The derivation of expressions for the equilibrium constant (K) is
false. 2) Students are misled into assigning units to K. 3) A bewildering array of "equilibrium constants" (K., Kc, Kf, K,, K,, etc.) is introduced whose interrelationshipsare obscure. 4) It is not always clear which K is appropriate to any given
- .-...
n.mh1.m r
. .. .
which is then called the "equilibrium constant." Finally Mickey argues that the concentrations in the "Law of Mass Action," eqns. (2) and (3), should strictly be replaced by activities ai for each chemical species i. I t would follow that
5) The reasons for dropping terms representing pure solids from
expressions for K are seldom understood. The student is not usually alerted to cases in which those terms cannot be dropped and isgiven no clue to a solution in those cases. With reference to Mickey's paper, I first offer a critique of the conventional approach and detail the pitfalls. I then outline what I consider would be a more satisfactory approach. A Critique of the Conventional Approach
The conventional approach to reaction equilibrium matches the historical development of the subject. Equilibrium is commonly not explained in terms of the minimum of some potential, the Gibbs potential ( G )or "free energy," of the reacting system. It is justified, rather, in terms of the kinetic "Law of Mass Action." With the reaction A+B*D+E
(1)
Mickey (I), for example, writes the following kinetic expressions for the forward and reverse portions: Rater = kf [A] [B]
(2)
Rate, = k, [Dl [El
(3)
This feature is almed as a review of basic chemical principles and as a reappraisal of the state of the art. Cwnments, suggestions fw topics. and contributions should be sent to the featureeditw.
K = -ad
Now, although eqns. (6)and (7) are both quite valid the method of deriving them is not. The derivation is faulty for two reasons. First, the "Law of Mass Action," as exemplified by eqns. (2) and (3), is not correct in general, even when the mixture is ideal in behavior. Relations such as ean. (1)renresent stoirhihmet&&d may have nothing to do G t h khetk-s. The same is true of eans. (fil and (7). Several discussions of this ooint at the level of'secdnhary school are to be found in the literature (2,3). Second, even in those reactions where the stoichiometry does happen to reflect the kinetics, the substitution of artivities foi concentrations does not render kinetic expressions like eqns. (2) and (3) strictly correct. Denbigh (4) discusses this point in detail. "A sound relation is being 'deduced' from unsound princinles. and a false conceotion mav in conseouence be imolanted in the minds of thosd taught."his quite from wright (2) aoulies eauallv to each of the two errurs althoueh he had in r&d on^; thLfirst. K. eiven hv ean. (6). is dimensionless. but onlv because the reactron ref&& 6 , ' e q n . (I), happens to have the same number of reactant and product molecules. In all other cases K's represented by equations like eqn. (6) would have units. For example, K for the reaction
-
A=D+E would be written
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Journal of Chemical Education
(7)
a. a b
and would have units of moles per liter. Strictly correct equations, like eqn. (7) and eqn. (19) below, show that K, in fact, is always dimensionless. Boggs (5)made the same point more than 20 years ago. One of mv most serious obiections to the usual annroach is the writingeof different kinds of equilibrium cons'&&. My annroach is to areue .. - .(see below) that there is one K. and onlv one, for any given reaction: it is given hy eqn. (19). ~ i c k e y writes two kinds of equilibrium constants,
and
for the one reaction Nz+3Hz=2NH3
(12)
a t 1000 K. (Again these two K's have units but they are not given. One suspects tbat, in eqn. (lo), K, has units of atm-2 in which case the p's would be given the units atm. But, especiallv because of the growing use of SI units, one might he tempLed, not unreasonably, ti; use kN.m-? in which &e one's numerical conclusions are likely to he far wrong.) Other authors use stU further equilibrium &stants: K,, K; K., ex. Just when each should he used is often not clear. One is even tempted to think that there are several kinds of equilihrium-one for each kind of K. There is great potential for confusion. (To add to the confusion one reference ( 6 )tabulates values for log K with units in one place and without them in another.) Pure solids and liquids taking part in reactions with mixtures are usuallv .dronued .. from exnressions for the eauilihrium constant. In my experience most students never appreciate the reasons. Nor are they aware of the circumstances under which doing so leads to significant error. Even less are they able to remove that error. Mickey's example is
For that reaction Mickey puts
a t 298 K. (Again, it is not entirelv clear what units to use for concentration although, of coursi, one supposes they should be mol.1.-I.) His statement is that "it is customary toexclude the concentration terms for solids from the equilibrium expression." But that does not explain why it might be the correct thing to do. Mickey's elaboration tbat one can write eqn. (14) hecause the concentrations of solid silver and copper are constant is simply invalid. Another common example is represented by for which one often sees
Toward A Modern Approach The proper justification for expressions for equilibrium lies not in kinetics but in the basic principles of thermodynamics. Most programs a t secondary school, however, do not lend themselves to this type of justification. Indeed, like several
other educators (7,8), 1 would he chary of teaching any thermodynamics at that level. Nevertheless, there are advanced prog&ns (9) in which equilihrium is analyzed thermodynamically. An empirical approach (2,3),not based on kinetics nor a detailed thermodynamic analysis, suits the normal course. Let us now sketch a thermodynamic approach for the more advanced student. With it in mind we shall then be in aposition to examine an empirical approach for the beginning student which would lay a suitable foundation for his later studies. More Advanced Treatment A reacting system at equilihrium is an analogue of a rolling hall stopped a t the bottom of a valley. The equilihrium state of the hall is defined by aminimum in the ball's gravitational potential. Similarly the equilihrium of the reacting system is defined by the minimum of another kind of potential. For systems a t defined pressure, P, and temperature, T, the appropriate potential is the Gibbs potential, G. As a direct consequence of the requirement that G be a minimum, one can show (see, for example, eqn. (15) on p. 78 of (9)) that, for reactions like eqn. (8), -= Odae a.
exp (-AG:/RT)
(17)
AG;
exactly a t equilihrium. is the "standard free energy of reaction" a t temperature T. The ai are defined (for example, p. 74 of (9)) in terms of fugacities fi,
and are relative to the same standard states used in Ac:. If i is in its standard state then ai = 1. Provided T is constant, the right-hand side of eqn. (17) is strictly constant regardless of changes in P and concentration. I t is appropriate to abbreviate it to K and call it the equilibrium constant. In other words we define K by
The left-hand side of eqn. (17) is generally not a constant in this sense: it may vary with T, P, and concentration. As the system approaches equilibrium, the left-hand side changes until eqn. (17) is obeyed. I t is not appropriate to regard the left-hand side as an equilibrium constant. K, then, defined by eqn. (19), is the one and only equilibrium constant of the reaction. It is a dimensionless number that has no necessary relationship to partial pressures, mole fractions, or concentrations (10). I find eqn. (19) more instructive than the more common equivalent Now, because of definition (19), the important identity (17) can he written adQe -=
(21)
as Beine merelv a definition. ean. (19) tells us nothine. E m . (21). on the othe; hand, is ver; much more important:-it is'the us: able conseauence of the basics of thermodvnamics. To emphasize th& eqn. (21) is more than mere definition of K we reverse the usual order and consistently put K on the right. For ideal gases oi = p5lP"
(22)
where Pois the standard pressure (1 atm or 101.325 kN.m-2). Combining eqn. (22) and the equivalent of eqn. (21) one fmds, for reaction (12), that Volume 59 Number 12 December 1982
1035
The K is not a new K, called K,, but still the same K defined by eqn. (19). The pressure can be expressed in any suitable units. Of course, if one uses atm then Po = 1atm andPo appears to drop from the equation. But to omit Po,as is usually done, is to invite trouble especially with the more frequent use of SI units. One can replace the pi and rewrite eqn. (23) in terms of mole fractions y i ,
or concentrations,
It is apparent from eqn. (25) that the common expression for Kc,asexemplified byeqn. (ll), is not generally equivalent to the true equilibrium constant and can lead to simificant error if treated as equivalent. With this approach i t becomes clear why one can oftenget away with simply ignoring solids in reactionslike eqns. (13) and (15). A pure solid (or liquid) at a pressure of 101.325 kN.m-Z (that is, P a ) is in its standard state and, hence, has a = 1.Thus, for reaction (15) at P =Powith the assumption of ideal gases and pure solid, we obtain
For the student who wants to know how to correct the problem when P i s not Poone can readily show that the activity of a pure solid or liquid is
exactly, or a = exp
M(P - Po) PRT
if p is effectively constant. (M is the molecular mass and p the density.) Eqn. (28) implies that the error in puttinga = 1for a solid in typical cases is roughly 1%if P = 10 atm and 10%if P = 100 atm. I encourage (10) the more advanced student to try to start with the equivalent of eqn. (21) every time. He then becomes aware of the approximations and assumptions he is making and is, therefore, more likely to recognize circumstancesunder which pa~ticularequations like eqns. (23)-(26) are questionable. I like to believe that, in this manner, his understanding of the subiect is imoroved and that he eains confidence in his ability to Holve problems. Simpinled Treatment Ashmore (3)argues that detailed thermodbamic justification of expressions for equilibrium, like that outlined above, is unnecessary in elementary courses. He believes that equations like (23), (24), and (25) can simply be presented as relations with a sound theoretical base. They should be justified emoiricallv with oublished exnerimental data or. better still, witG meas&eme& the student himself has made. The thermodynamic justification can await a subsequent course of study. The soundness of this approach contrasts sharply with the erroneous conventional a~oroachand its kinetic expressions whose "justification" see& largely historical (the
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Journal of Chemical Education
"Law of Mass Action"). Again we do not define K by equations like (23). We regard K as a dimensionless number.. deoendent on T and nothine" else, which characterizes equilibrium of a particular reaction and whose value can. at least in orinciole. be determined hv experiment. The left-hand side bf eqi. (23) is not generalG equal to K: the equality becomes true only when there is equilibrium.As above, we keep K on the right-hand side of the equation to emphasize that K is not being defined. Any of eqns. (231, (24), and (25) can easily he derived from any other. Thus, we can show that equilibrium can he expressed in terns of p i , y i , or concentrations, but the K remains the same. If one wanted to develop the subject further at this level, one could then generalize from eqn. (23) to eqn. (21) by defming activity by eqn. (22) in a mixture of ideal gases. Without actually calculating a's one could alert the student to nonidealitv and to the fact that em. (23) is not correct for all eases. The concept of the standari state can easily be intio&ced with the standard oressure Po.If reactions of solids and eases were to be considered one would need to justify a= 1for a solid either empirically or by arguing that the solid is at least close to its standard state. It is important to get across the idea that each reaction has a K which generally has nothing to do with p i , y i , and concentration. Under different circumstances the left-hand side of relations representing equilihrium will take on different fmns. That form may be in terms of p i for reacting ideal gases, for example, but forother systems it may he quse different. It is not the K that is changing to Kc, K,, or K y ; it remains plain K while the left-hand side of the equation changes according to the circumstances.
.
Conciudlng Remarks As I have alreadv stated. the more advanced treatment of equilibrium of chemically reacting systems is nut appropriate for the beeinnine student. It best litscoursesat the universitv level and;perhaps, advanced programs at secondary school levels. But at secondarv school. I see the aonroach more as a background for the tekher against whichK can construct a suitable introduction to equilibrium alona- the lines of the simplified treatment above. Much of our teaching of basic science is hampered by a desire to follow the historical development of the subject (11, 12). We have seen it here very clearly where the historicaland often quite wrong "Law of Mass Action" is used to analyze equilihrium. That it gives the right answer is due as much to luck as to science. Acknowledgment I thank D. F. Williams, D. Glasser, A. W. Bryson, and three anonymous reviewers for helpful comments. Literature Cited
Press. London. 1911.o. 417 (5) Boggs, J. E.. J. CHEM.EOUC.,36.30 (19581.
(6) "Handbmkof Chpmlstnisnd Physics~57thed..CRCRrsaCleveland,OH. 19761911. pp. D-82 and D-87. (7) Wrighf,P. G.,Edur. in Chem., 11.9 (1974). (8) Hondebrink,J. G., J. C m M . EDW., 58,963 (1981). (9) Bsuman, R. P.,"An lntmduction to Equilibrium Themcdwmmien." Prentice-Hall, Englewd.Cliiis, NJ.1966. (10) Hsrria, W.F., ChemSA,4,170~1978). (11) Harris, W. F..Inf. J. Mech EnmgEdue.9.317 (1981). (12) Warren, J. W., Eur J. Phrs,I, 1% (19SO).
