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35 A Theorem on the Relation between Rate Constants and Equilibrium Constant
35
A Theorem on the Relation between Rate Constants and Equilibrium Constant JURO HORIUTI Research Institute for Catalysis, Hokkaido University, Sapporo, Japan -4 theorem k/- k = Kl/~(r) is shown valid for any thermal reaction having a rate-determing step r , where ~ ( ris) the stoichiometric number of r , k or - k the rate constant of the forward or the backward reaction, and K the equilibrium constant. The theorem includes the classical one, k/-k = K as a special case when Y(T) = 1 and states that a catalyst varies with k/- k, according to whether shifts the part of r from one step to the other of different v ( r ) or not, and that the difference of the activation energies of the forward and the backward reactions equals I / Y ( T ) times the negative heat of reaction.
I. INTRODUCTION It is a well-known classical theorem that
k/-k
K (1) where k or -k is the forward or backward rate constant and K the equilibrium constant, deduced from a particular picture of a reaction consisting in a single step on the basis of the mass action law. Attempts have been made to verify the theorem in general ( I ) , but Manes, Hofer, and Weller have demonstrated it invalid with special reference to an example of reaction consisting of two steps ( 2 ) .To cover the general case, the latter authors put forward a sufficient assumption,
Vs/-V,
=
=
(k/ -k) (u”/u”)’
k/-k
=
K”
(24 (2b)
where V. or -V, is the forward or backward rate of the over-all reaction in the steady state and a” or uR the activity product* of the left or the right of the appropriate chemical equation ; z has been anticipated by these authors to be a small integer or its reciprocal from the fact that chemical reaction takes place between discrete particles.
* The corresponding concentration products are used in the original presentation (2). 339
340
JURO HORIUTI
Horiuti and Enomoto have shown ( 3 ) in extension of the theory of the stoichiometric number (4, 5 ) that V,/-V, = (KuL/uR)””(‘), i.e., that the exponent t o the equation obtained by eliminating k/-k from (2)* must be the reciprocal of the stoichiometric number v ( r ) of r, i.e., the number of r occurringfor every act of the over-all reaction in the steady state. This conclusion would lead along with the above sufficient assumption (2b) to the relation, k/-k = K””(‘) (3)
It is reported herewith that (3) is verified generally to hold, irrespective of the validity of the mass action law, for any reaction having a rate-determining step, homogeneous or heterogeneous. 11. VERIFICATION OF THE THEOREM Let L = R be the over-all reaction going on with a rate-determining step r in an assembly A . Areaction with a rate-determining step means one such that its steadyrate, V, = V, - (-V8)is practicallydetermined by the condition that all appropriate steps except r , denoted by j’s, are in partial equilibrium, i.e., Pli = PFi
(4)
where p r j or p F j is the chemical potential of initial complex Ij or final complex F j of j ; chemical potential of any set of chemical species 6 will be denoted by p6 in what follows. Since the free energy change of A is effected by v ( r ) acts of r for every over-all reaction but by none of the j’s, according to (4),we have PL
- PR
=
v W G ’ - PF)
(5)
where I or F is the initial or the final complex of r . The conversion of L into v(r)I through j’s may be written as L = v(r)I B, where B is the remainder of the resultant, algebraically inclusive of the deficit. We have now
+
v(r)Pr
+
PB
P R = v(r)CcP
+
PB
PL
=
or according t o ( 5 ) ,
(6.1)
The state of B is not necessarily unique, although the value of p B is, it being converted from one to another state through some of the j’s without affecting p B .
* (2) stands for (2a) and (2b); this manner of reference will be followed below throughout.
35.
RATE CONSTANTS AND EQUILIBRIUM CONSTANT
341
It has been shown (5-7), on the other hand, in extension of the transitionstate method of Eyring (8) and of Evans and Polanyi (9),that forward and backward rates, v and -v of any thermal elementary reaction and hence of r, are given as
where K is the transmission coefficient and p*, etc., are the Boltzmann factor of p*, etc., p* in particular referring t o a single * existing in A . t Developing p’ in terms of standard free energy p:, activity a’, activity coefficient f’ and concentration N s according t o the relation, p6 = p:
+ RT log a6
a’
=
j’N6
(8)
we have from (7)
v
=
fE exp (p:/RT)a’/f*,
-v
=
L exp (pIF/RT)uF/f*
where = K(kT/h) exp (-pl*/RT)/N*, the reciprocal of the concentration N*, formally of a single activated complex * in A , giving the volume of the phase or the area of the boundary surface where the elementary reaction r occurs. The general statistical mechanical expression for the activity coefficient j* of the activated complex has been explicitly given for the homogeneous (5) as well as for the heterogeneous (10) elementary reaction. It was shown with special reference t o the hydrogen electrode reaction that the current neglect off*, i.e., the approximation, f* = 1 , is responsible for the conclusion, a = 2 invariably deduced as done by Tafel (11) half a century ago for the Tafel’s constant a of the catalytic mechanism. The correct inclusion of j* leads t o a 0.5 for a certain range of cathodic polarization and to the constant saturation current a t the extreme polarization in agreement with experiments (10). We have now, from (6), (S), and (9),
+
V
=
-v=
v / v ( r ) = k(aL/aB)l’v(r)/f*
-v/ v(r) = -k(uR/uB)l’”(‘)/f*
(10)
which give the rate laws for V, and -V, , where
Equation (3) follows immediately from (11),remembering that the thermodynamical equilibrium constant K = exp [(PI” - plR)/RT].
t It is not meant that there exists physically a single * alone in A , but t h a t rates are given adequately by (7) using the relevant statistical-mechanical functions (6,r).
342
JURO HORIUTI
It is inferred from (3) that (a) a catalyst changes k and -k individually but not the ratio k/-k so long as v(r) remains the same and (b) a catalyst does change the ratio, k/-k, as it shifts the part of r from one step to the other of different v(r) and (c) the excess of the activation energy
RT2 d (log k)/ dT of the forward reaction over that of the backward one equals l/v(r) times the negative heat of reaction.
111. EXAMPLES If the catalyzed synthesis of ammonia, Nz 3H2 = 2NH3, proceeds through steps, Nz---t 2N(a), HZ-+2H(a), N(a) H(a) NH(a), NH(a) H(a) -+ NHZ(a), NHz(a) H(a) + NH3, where (a) refers to the adsorbed state on the catalyst, the exponent l / v ( r ) to K of (3) is 1/1, 1/3, or 1/2, according as the first, second, or any of the last three steps determines the rate; if the first step does, B = 3H2, V, a aN2/f*and -V, a (aNH3)>”/ (aH2)?, and so on. The B in the above examples is reducible by j,s to chemical species implied in L or R , but this is not generally the case. Let a homogeneous reaction consist of steps, L L’ -l- m,L‘ -+ R’, and m R’ -+ R , the second one determining the rate; the intermediate m = B is not similarly reducible, although as is determined as aB = ( K , aL K3aR)1’2 by the condition of the partial equilibria, KlaL = uL‘u*, a”aR’ = &aR ( K 1, K a , equilibrium constants) and the stoichiometric relation, am = uL’ a R ‘ where , activities are identified respectively with the concentrations.
+ +
+
--f
+
+
+
+
Received February 24, 1956
REFERENCES 1. Gadsby, J., Hinshelwood, C. N., and Sykes, K. W., Proc. Roy. SOC.A187, 129
(1946).
8. Manes, M., Hofer, L. J. E., and Weller, S., J . Chem. Phys. 18, 1355 (1950); 22,
1612 (1954). 3. Horiuti, J., and Enomoto, S., Proc. Japan Acad. 29, 160, 164 (1953). 4 . Horiuti, J., and Ikusima, M., Proc. Imp. Acad. (Tokyo) 16, 39 (1939). 6. Horiuti, J., J . Research Inst. Catalysis Hokkaido Univ. 1, 8 (1948). 6. Horiuti, J., Bull. Chem. SOC.Japan 13, 210 (1938). 7. Hirota, K., and Horiuti, J., Sci. Papers Inst. Phys. Chem. Research Tokyo 34,1174 (1938). 8. Eyring, H., J . Chem. Phys. 3, 107 (1935). 9 . Evans, M. G., and Polanyi, M., Trans. Faraday SOC.31, 875 (1935). 10. Horiuti, J., J. Research Inst. Catalysis Hokkaido Univ. 4, 56 (1956). 1 1 . Tafel, J., 2. physik. Chem. 60, 641 (1905).
A Theorem on the Relation between Rate Constants and Equilibrium Constant JURO HORIUTI Research Institute for Catalysis, Hokkaido University, Sapporo, Japan -4 theorem k/- k = Kl/~(r) is shown valid for any thermal reaction having a rate-determing step r , where ~ ( ris) the stoichiometric number of r , k or - k the rate constant of the forward or the backward reaction, and K the equilibrium constant. The theorem includes the classical one, k/-k = K as a special case when Y(T) = 1 and states that a catalyst varies with k/- k, according to whether shifts the part of r from one step to the other of different v ( r ) or not, and that the difference of the activation energies of the forward and the backward reactions equals I / Y ( T ) times the negative heat of reaction.
I. INTRODUCTION It is a well-known classical theorem that
k/-k
K (1) where k or -k is the forward or backward rate constant and K the equilibrium constant, deduced from a particular picture of a reaction consisting in a single step on the basis of the mass action law. Attempts have been made to verify the theorem in general ( I ) , but Manes, Hofer, and Weller have demonstrated it invalid with special reference to an example of reaction consisting of two steps ( 2 ) .To cover the general case, the latter authors put forward a sufficient assumption,
Vs/-V,
=
=
(k/ -k) (u”/u”)’
k/-k
=
K”
(24 (2b)
where V. or -V, is the forward or backward rate of the over-all reaction in the steady state and a” or uR the activity product* of the left or the right of the appropriate chemical equation ; z has been anticipated by these authors to be a small integer or its reciprocal from the fact that chemical reaction takes place between discrete particles.
* The corresponding concentration products are used in the original presentation (2). 339
340
JURO HORIUTI
Horiuti and Enomoto have shown ( 3 ) in extension of the theory of the stoichiometric number (4, 5 ) that V,/-V, = (KuL/uR)””(‘), i.e., that the exponent t o the equation obtained by eliminating k/-k from (2)* must be the reciprocal of the stoichiometric number v ( r ) of r, i.e., the number of r occurringfor every act of the over-all reaction in the steady state. This conclusion would lead along with the above sufficient assumption (2b) to the relation, k/-k = K””(‘) (3)
It is reported herewith that (3) is verified generally to hold, irrespective of the validity of the mass action law, for any reaction having a rate-determining step, homogeneous or heterogeneous. 11. VERIFICATION OF THE THEOREM Let L = R be the over-all reaction going on with a rate-determining step r in an assembly A . Areaction with a rate-determining step means one such that its steadyrate, V, = V, - (-V8)is practicallydetermined by the condition that all appropriate steps except r , denoted by j’s, are in partial equilibrium, i.e., Pli = PFi
(4)
where p r j or p F j is the chemical potential of initial complex Ij or final complex F j of j ; chemical potential of any set of chemical species 6 will be denoted by p6 in what follows. Since the free energy change of A is effected by v ( r ) acts of r for every over-all reaction but by none of the j’s, according to (4),we have PL
- PR
=
v W G ’ - PF)
(5)
where I or F is the initial or the final complex of r . The conversion of L into v(r)I through j’s may be written as L = v(r)I B, where B is the remainder of the resultant, algebraically inclusive of the deficit. We have now
+
v(r)Pr
+
PB
P R = v(r)CcP
+
PB
PL
=
or according t o ( 5 ) ,
(6.1)
The state of B is not necessarily unique, although the value of p B is, it being converted from one to another state through some of the j’s without affecting p B .
* (2) stands for (2a) and (2b); this manner of reference will be followed below throughout.
35.
RATE CONSTANTS AND EQUILIBRIUM CONSTANT
341
It has been shown (5-7), on the other hand, in extension of the transitionstate method of Eyring (8) and of Evans and Polanyi (9),that forward and backward rates, v and -v of any thermal elementary reaction and hence of r, are given as
where K is the transmission coefficient and p*, etc., are the Boltzmann factor of p*, etc., p* in particular referring t o a single * existing in A . t Developing p’ in terms of standard free energy p:, activity a’, activity coefficient f’ and concentration N s according t o the relation, p6 = p:
+ RT log a6
a’
=
j’N6
(8)
we have from (7)
v
=
fE exp (p:/RT)a’/f*,
-v
=
L exp (pIF/RT)uF/f*
where = K(kT/h) exp (-pl*/RT)/N*, the reciprocal of the concentration N*, formally of a single activated complex * in A , giving the volume of the phase or the area of the boundary surface where the elementary reaction r occurs. The general statistical mechanical expression for the activity coefficient j* of the activated complex has been explicitly given for the homogeneous (5) as well as for the heterogeneous (10) elementary reaction. It was shown with special reference t o the hydrogen electrode reaction that the current neglect off*, i.e., the approximation, f* = 1 , is responsible for the conclusion, a = 2 invariably deduced as done by Tafel (11) half a century ago for the Tafel’s constant a of the catalytic mechanism. The correct inclusion of j* leads t o a 0.5 for a certain range of cathodic polarization and to the constant saturation current a t the extreme polarization in agreement with experiments (10). We have now, from (6), (S), and (9),
+
V
=
-v=
v / v ( r ) = k(aL/aB)l’v(r)/f*
-v/ v(r) = -k(uR/uB)l’”(‘)/f*
(10)
which give the rate laws for V, and -V, , where
Equation (3) follows immediately from (11),remembering that the thermodynamical equilibrium constant K = exp [(PI” - plR)/RT].
t It is not meant that there exists physically a single * alone in A , but t h a t rates are given adequately by (7) using the relevant statistical-mechanical functions (6,r).
342
JURO HORIUTI
It is inferred from (3) that (a) a catalyst changes k and -k individually but not the ratio k/-k so long as v(r) remains the same and (b) a catalyst does change the ratio, k/-k, as it shifts the part of r from one step to the other of different v(r) and (c) the excess of the activation energy
RT2 d (log k)/ dT of the forward reaction over that of the backward one equals l/v(r) times the negative heat of reaction.
111. EXAMPLES If the catalyzed synthesis of ammonia, Nz 3H2 = 2NH3, proceeds through steps, Nz---t 2N(a), HZ-+2H(a), N(a) H(a) NH(a), NH(a) H(a) -+ NHZ(a), NHz(a) H(a) + NH3, where (a) refers to the adsorbed state on the catalyst, the exponent l / v ( r ) to K of (3) is 1/1, 1/3, or 1/2, according as the first, second, or any of the last three steps determines the rate; if the first step does, B = 3H2, V, a aN2/f*and -V, a (aNH3)>”/ (aH2)?, and so on. The B in the above examples is reducible by j,s to chemical species implied in L or R , but this is not generally the case. Let a homogeneous reaction consist of steps, L L’ -l- m,L‘ -+ R’, and m R’ -+ R , the second one determining the rate; the intermediate m = B is not similarly reducible, although as is determined as aB = ( K , aL K3aR)1’2 by the condition of the partial equilibria, KlaL = uL‘u*, a”aR’ = &aR ( K 1, K a , equilibrium constants) and the stoichiometric relation, am = uL’ a R ‘ where , activities are identified respectively with the concentrations.
+ +
+
--f
+
+
+
+
Received February 24, 1956
REFERENCES 1. Gadsby, J., Hinshelwood, C. N., and Sykes, K. W., Proc. Roy. SOC.A187, 129
(1946).
8. Manes, M., Hofer, L. J. E., and Weller, S., J . Chem. Phys. 18, 1355 (1950); 22,
1612 (1954). 3. Horiuti, J., and Enomoto, S., Proc. Japan Acad. 29, 160, 164 (1953). 4 . Horiuti, J., and Ikusima, M., Proc. Imp. Acad. (Tokyo) 16, 39 (1939). 6. Horiuti, J., J . Research Inst. Catalysis Hokkaido Univ. 1, 8 (1948). 6. Horiuti, J., Bull. Chem. SOC.Japan 13, 210 (1938). 7. Hirota, K., and Horiuti, J., Sci. Papers Inst. Phys. Chem. Research Tokyo 34,1174 (1938). 8. Eyring, H., J . Chem. Phys. 3, 107 (1935). 9 . Evans, M. G., and Polanyi, M., Trans. Faraday SOC.31, 875 (1935). 10. Horiuti, J., J. Research Inst. Catalysis Hokkaido Univ. 4, 56 (1956). 1 1 . Tafel, J., 2. physik. Chem. 60, 641 (1905).