REACTION VELOCITY AND EQUILIBRIUM
BY WILDER D. BANCROFT
'
In the last dozen years a number of reaction velocities have been studied where the order of the reaction was lower than that to be deduced from the formula describing the reaction. T h e hypothesis, which has been advanced to explain this phenomenon, is that the reactions in question take place in one or more stages, and that if we consider the reaction taking place in some one of these stages only, we shall be able to deduce a formula which will correspond with that found experimentally. I t so happens that none of these abnormal reactions have been reversible, and it is therefore not surprising to find irreversibility postulated as a necessary pre-requisite to an abnormal reaction. Ostwald has discussed the matter from this point of view.' ( ( W emust now ask how these hypotheses are to be reconciled with the relations, previously discussed, between the formula for reaction velocity and that for chemical equilibrium. T h e answer is that these relations are not applicable in these cases, and that therefore there is no difficulty on this point. This must be so because the assumption underlying the explanation was that the products of the primary reaction were immediately further changed by a secondary reaction. Under these circumstances the essential condition of chemical equilibrium is lacking, that the reaction products react to form the original substances ; consequently the laws of equilibrium are not applicable to these cases. In this last consideration we can probably find a criterion for the correctness of these hypotheses, because differences between the order of reaction and the number of molecules taking ('
Lehrbuch,
2,
11,243.
706
WiZder D. Balzcvoft
part should be possible only when the process is irreversible ; the number of molecules and the order of reaction should agree in processes where the original substances can be formed from the reaction products. Too few reactions have been studied as yet from this point of view to enable one to state finally whether this requirement is satisfied experimentally or not." This is entirely clear. Abnormal reaction velocities cannot occur when the reactions are reversible. I t will be desirable to examine this a little more closely. I n order to deal with concrete substances, let us take the equilibrium between ammonium carbamate and its dissociation products, all in the gaseous phase and we will first make the assumption that the reaction takes place in one stage according to the equation
+ CO, '",NH,CO,NH,,
2NH3
and then the assumption that the reaction takes place in two stages according to the equations NH,
NH,
+ CO,
NH,CO,H
+ NH,CO,H "z NH,CO,NH,,
the final concentration of NHZCOZHto be practically negligible. Under the first assumption, we will let 2A be the initial concentration of NH3, B of CO,, and we will let x be the concentration of NH4COINH2at any moment. We may then write dX
dt
=R,(A-Z)'(B-Z)
-R,x.
For equilibrium we shall have KX = ( A - x)'(B
-
x).
Under the second assumption we will let y = the concentration of NHZCO,H at any moment, the signification of A, B, and x remaining as before. We may then write dY -dt - h, (A -- x -J/Z
) ( B - x - y) - k , ~ ,
dX xK,(A-x--J/~)J-~~~.
Reactioiz Velocity aizd Equilib~ium
707
For equilibrium dx/dt= o and dyidt = 0,therefore K,y= ( A - x - ~ / ~ ) ( B - x - J ) K,x = ( A - x -yY/2)y.
Multiplying these last two expressions, we have K,x= ( A - x - ~ Y / ~ ) ' . ( B - x - ~ ) .
Since y is by definition practically zero, this is the theoretical equation for equilibrium. We thus see that if the NH2C0,H were used up as fast as formed we should have a reaction of the second order if we studied reaction velocity, though the chemical equation would call for a reaction of the third order. From this, it is clear that abnormal reaction velocities may and probably do occur even in the case of reversible reactions.' That none have yet been found experimentally is not surprising when one recalls how few reversible reactions have been studied. I t is now possible to drop one limitation. In the discussion, it has been assumed explicitly that the final amount of NH2COZH is zero. Even supposing this were not the case, the concentration of the ammonia is ( 2 A - 2x -y), of the carbonic acid (B -- x - y ) , and of the ammonium carbamate x , so that the last equation is the theoretical equation for equilibrium between ammonium carbamate, ammonia, and carbonic acid. If we drop the further explicit assumption that the NHZCOzHremains constantly at its equilibrium concentration, we shall no longer get a perfect constant for a reaction of the second order ; but, during the first stages, at any rate, the reaction may be more nearly of the second order than anything else. T h e lack of necessary connection between the order of a reaction and the final equilibrium offersa strong argument against the common pedagogical practice of treating reaction velocities first and deducing equilibrium relations from them. T h e equilibrium phenomena are the fundamental ones and are based solely on analytical data. If we know the equation representing the reaction, we can predict the form of the equilibrium Cf. Nernst. Theor. Chem. 3rd. Ed., 523.
708
Reactioiz Velocity a d Equilibrium
equation. We cannot predict the form of the reaction velocity equation from a knowledge of the initial and final concentrations ; we cannot predict it from anything. If we know the order of the reaction and the analytical data, we can then say that the reaction takes place in one or more stages - even then, only in case we have not overlooked some catalytic action. T h e only sound method is to teach the equilibrium phenomena as based on the stoichiometric relations, and then to consider reaction velocities as far as possible with relation to the expressions for equilibrium. Cornell University,
