ACTIVE and PASSIVE MOLECULES and the NATURE of the MASS ACTION EQUILIBRIUM CONSTANT FRANK E. E. GERMANN University of Colorado, Boulder, Colorado
T
HE LAW of mass action was first accurately formulated by Guldberg and Waage, two Scandinavian investigators, in 1867. The work appeared in that year as a university publication under the title "etudes sur les affinites chimiques." In 1879 the same authors made their work more generally available by a second more detailed publication' entitled "Ueber die chemische f i n i t a t . " In this Daaer it is stated that in the reaction the assumption of the existence of attractive forces between the substances or their constituents is not sufficientto explain the two reactions, but that account must also be taken of the motion of the atoms and molecules. They then proceed to develop the idea of a vibration of the atoms in a molecule and suggest that, under certain conditions, this vibration may become so vigorous that a t times the atoms are far enough apart that they actually dissociate. In any case, they would be in a very unstable state. If, a t that instant, two such molecules chanced, due to their kinetic motion, to collide with each other, a reaction might take place. In other words, if a molecule of A collides with a molecule of B a t the instant when the atoms of A and B are far apart, we may anticipate the formation of A' and B'. The authors then point out that, if pand q represent the number of molecules of A and B, respectively, in unit volume, then the frequency of collision of A with B will be proportional to the product pq, and the velocity of the chemical process will be 4pq, if 4 be taken to be a function of temperature. This assumes that all the molecules are in a condition favorable to a reaction. Hence, in accordance with the above argument on atomic vibrations, they postulate that only the fraction a of the b molecules and the fraction b of the omolecules are in a condition favorable to a reaction, so that the velocity with which the reaction pro. ceeds should be represented by @pbq, or mPbq = kPp
(')
idea of active and passive molecules clearly in mind when they developed the mass action law. The idea of active and passive molecules is not to be confused with the term "active mass" introduced by Guldberg and Waage. This latter term is defined in their own words as follows: "We have defined active mass of the material as the quantity of the given substance in unit volume of the body (Kdrper) in which the reaction is taking place." Bv Kdraer, which for accuracv's sake has bein translaied "body," they obviousl; mean phase. Our modern theory of activation is merely a modification in which the activated molecules are regarded as having higher energy contents as the results of displaced electrons. In this sense the ionic state is regarded as an activated state. In reactions of neutralization and precipitation, the fractions a and b of Guldberg and Waage approach unity and the velocity of the reaction becomes immeasurably fast. In accordance with the above conventions, Guldberg and Waage would set up the following equation: +wp=tOqO = +'crrrd6s6
(3)
for the equilibrium condition in the reaction: d f BBQyCf
60
(4)
4 and 4' are temperature functions of the two opposing reactims; p, q, r, and s are the numbers of molecules of A , B, C, and D, respectively, per unit of volume under equilibrium conditions; a, b, c, and d are the fractions of each of the various molecular types which are in a state such that a reaction could take place when a collision occurred and m, 8, 7 , and 6 are the coefficients of the various molecular types in the chemical equation. On rearranging, this equation becomes: k K=-=+n.bs=E b' *',-*A6 hmnB
.
-".-
(5)
This is the classical form of the mass action ." equilibrium a"bP equation, in which the ratio of constants - has been c'd6 combined with the constants 4 and 4'. Had Guldberg and Waage thrown the original equation into the form:
in which the product of the constants, 4ab, is replaced by k. ~ , = k(cr)-i(ds)s = The above argument of Guldberg and Waage is given (6) ' (aP)DL(bd@ in detail in order to show conclusively that they had the they would have had the modem equation involving 1 GULDBERG AND WAAGB. I. prakt. chem., [2], 19.69 (1879). 328
activities in which ap is the activity of the p molecules, bg the mtivity of the q molecules, etc. In modern terminology a , b, c, and d are called activity coefficients. At the time of Guldberg's and Waage's work no methods existed for determining these activity coefficients. It was soon recognized that the value of K was not a true constant, but nothiig was done about it until G. N. Lewis2 in 1901 and 1907 introduced the terms fugacity and activity, which he defined in such a way that 'the equilibrium constant is a true constant. Let us now consider certain conventions in writing equations and in obtaining values of equilibrium constants. In the reaction: Nz +3Hs 2NHs, AFZ.. = -7820cal. (7) we place the activities a of the substances appearing on the right-hand side of the equation in the numerator, and those on the left-hand side in the denominator, thus:
When we write the equilibrium equation: we obtain in a similar manner:
Obviously, the first constant is the reciprocal of the second. In order t o calculate the standard free energy change given in equation 7, the value of K from equation 8 would have to be used in the general equation: A c , . = -RTlnK = -7820cal.
(11)
Had the reaction been written as in 9, we should have obtained:
Moreover, had we written the equation involving just one molecule of ammonia:
the equilibrium constant would have been:
the same, but the sign changes. If the equation is written for double the number of mols, the value of AFo is doubled, etc. The question naturally arises as to the reason for an equilibrium constant having d i e r e n t values depending on the way the equation has been written. The obvious answer is that equation 9 indicates a given mechanism of the reaction and equation 13 indicates another mechanism. The first is the equation of a bimolecular reaction and the second is the equation of a monomolecular reaction. Since equation 14, which is based on 13, contains fractional exponents, we frequently find the statement that "these fractional exponents are usually avoided through multiplication by an appropriate integer." When, however, equation 13 is multiplied by the integer two, equation 9 results, which, as stated before, involves a different mechanism. At a given temperature only one of these two equations represents the facts, but since we do not always know the actual mechanism of the reaction, we sometimes see one constant, sometimes another in the literature. All of this makes no difference until we wish to make use of the constant in the evaluation of some thennodynamic function as, for example, the free energy change AFo accompanying a given reaction. Thermodynamics takes no account of any kinetic mechanism of a reaction, so here again the value of the constant may be taken as that given by either equation 10 or 14. We are not even restricted to these two values, but may use values obtained from equations similar to 9 and 13 in which we have multiplied through by any number whatsoever. The value of AFo obtained will in each case represent the true free energy change when the number of mols of ammonia indicated on the lefthand side of the equation is transformed into the number of mols of nitrogen and hydrogen on the righthand side of the equation a t the temperature indicated. Another matter which is a frequent source of confusion is the way in which we raise the activities of the various molecular species to the power corresponding to the coefficient of the particular compound in the chemical equation. Thus in equation 8 the activity of hydrogen is cubed in accordance with the fact that in equation 7 the hydrogen has the coefficient 3. We usually justify this by writing equation 7 as follows: N*
+ Hz + HP+ HI St NHs + NHr
(16)
from which we obtain the equation:
and A z s . = -RTlnK"/z
=
-RF
= 3910cal.
=
(15)
These equations have been given in detail in order to emphasize the fact that.the free energy change calculated with the aid of a ginen equilibrium constant always yields a oalue for the reaction of the number of mols of the substance indicated by the equation used in obtaining the oalue of the constant. If the equation is written for the reverse reaction, the numerical value of AFo remains
l G . N. LEWIS.Proc. Am. Acad., 37, 49 (1901); 2. phyrik. Chcm.. 38, 205 (1901); Proc. Am. Acod., 43 259 (1907); 2. physik. Chem., 61,129 (1907).
a ~ m ' a t i m = Q'NR~ a.waa..a.a..aa, awzaa,
(17)
In doing this we say nothing of the fact that, although we write NH, twice and Hz three times, we proceed to assign the total values of the ammonia and hydrogen activities to each. term. It is quite natural t o raise the question as to the justification of this procedure, and to argue that if Hz is to be written three times and NHJ twice, why should not one-third of the hydrogen activity be assigned to each H2 term and one-half of the ammonia activity to each NHs term. This suggests that the true equation should be:
Thus, for example, Soddys considers the case involving the isotopes of chlorine in the reaction: suggested by Chapman4in connection with a theoretical discussion of the possibility of separating isotopes. On the basis of probability considerations, he arrives at the conclusion that k, = 4& when he applies the mass action law in the conventional manner. Continuing, he says: This, t o say the least, is unexpected, because if cw5cients of velocity of reaction have any physical significance a t all, one would expect them t o be the same for substances assumed t o be chemically identical. The result is clearly due t o a loose method of choosing the concentrations, for if we rewrite the reversible equation: Cl.Cl' CI1 Cl'zz== CI.CI' (20)
+
+
i t transpires that we have chosen for the concentration of the resultants, because they are the same, the sum of their individual concentrations. althoueh for the reactants. which also are chemi&ly the same', the in;dividual concentrations have been taken. ~t is clear t h a t i t is the individual concentrations in both cases t h a t have t o be taken, and therefore that one-half of the CI.CI' concentration is involved. Then kl kn. So with any reaction of this type, involving two molecules, apart from the question of isotopes altogether, the 4 that always appears in the conventional textbook examples is merely a consequence of a loose and physically unjustifiable mode of representing the concentrations. Writers of future textbooks might ponder a little over this.. I do not imagine I have exhausted the physical possibilities, hut, so far as I can see, my distribution relation covers the physically conceivable cases, and therefore the half and not the whole, concentration of the substance undergoing a bimolecular reaction with itself ought t o enter into the equilibrium equation.
-
.. . ..
The above quotation from Soddy is given a t length in order to emphasize the fact that the confusion really exists. Actuallv, we have proved nothing - by. using equations such & 16 and 20: We can, however, show by a study of free energy changes which take place in a chemical reaction that the original method used in writing equation 8 was correct. To do this we proceed as follows, studying a reaction between perfect -w e s A, B, G, H, etc., according - to the equation:
There will be no free energy change in the passage through the semipermeable membranes since this is accomplished under equilibrium conditions. If a mols of A are expanded or compressed reversibly from P'A to PA,and b mols of B from PIBto PB,the changes in free energy will be:
Similarly if g mols of G and h mols of H are expanded or compressed from PC and PHto P'c and P'n, the changes in free energy will be: P'o AF. = gRTln -; Po
P'H AF4 = hRTlu Pn
(23)
The total free energy change will be:
If P'A,P'B, P'c, and P'H are taken as one atmosphere, we have :
in which K+is the equilibrium constant when concentrations of the perfect gases are expressed as partial pressures. the above rmoning we have made use of the fact that the pressure of A in the reaction chamher was PA. Had we followed the line of reasoning which led us to equation 18 we would have made use not of the pressure Pa, but rather PA/a, stating that Of the of A was to be assigned to each of the e mols. However* it 's obvious that the pressure against which A must be introduced is PA and not Pnla. If a is one, then: AF, = l R n n
PA P A
~f
a is two, then: AF, = 2RTln
PA P A
=RTh
P ' A etc. P'z~'
(27)
This reasoning, therefore, shows conclusively that it is the total activity or partial pressure of each constituent that should be raised to a power corresponding to the d + b B + ......c ; g G + h H + ...... (21) coefficient of the particular substance appearing in the in which the partial pressures a t equilibrium are PA, chemical equilibrium equation. It also shows that the PB,PG,Pa, etc. The different gases undergo pressure value of a, which becomes the exponent in the change changes and react with each other in an imaginary of free energy equation, merely takes account of the equilibrium box equipped with cylinders, pistons, and fact that the free energy change involved in the reachypothetical semipermeable membranes. The in- tion of two mols is twice the free energy change in the dividual gases in the separate chambers may have any reaction of one mol. pressures represented by P'A, P'B, PIG, and P'H. We The above has been presented not with the thought shall proceed to calculate the free energy change in- of adding any new facts to the theory of equilibrium volved in changing the pressure of a mols of A from constants, but rather to show, first, that Guldberg and P k to PA,of b mols of B from P'B to PB,of g mols of G W a g e had thought in t m s of activities, and second, from PC to PIG, and of h mols of H from PH to P'g. to clear UD certain ooints in the theom of eauilibrium constantstvhich arebften not clearly understdod by the * FREDERICICS~DDY, Nature, 105,516,642 (1920). student. ' D. L.C n ~ ~ m ~ , i b i105,487 d., (1920).
