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Thermodynamic consistency of reaction rate expressions
SHORTERCOMMUNICATIONS
Thermodynamic consistency of reaction rate expressions (Received 10 June 1963; in revised form 4 October 1963) The proof will show that equation (6) necessarily implies the ‘following form for (5):
WE PRESENTa rigorous proof that DENMGH’Srelation for the thermodynamic consistency of reversible reaction rates is both necessary and sufficient. DENBIGH[l] has shown that it suffices for the ratio of the forward and the reverse reaction rate expressions to be a simple power of the equilibrium constant expression. However, that this relation is also necessary for thermodynamic consistency does not seem to be generally appreciated. As recently as 1961, FROST and PEARSONin a widely used text [2] surmised that the relation might be “any function for that matter, and still be consistent”. We have proved that the relation is in fact more restricted, that Denbigh’s power function is the only one mathematically in accord with thermodynamics, The proof is valid for the entire class of reversible reactions for which the over-all reaction rate is written conventionally [3] as the difference of two separable rate terms. Consider the general reversible reaction
k/k’
f (K) = p,
(j = 1, . . . , n?). (7)
n = +
The exponent n can be any positive number. PROOF In equation (6) set the activity of every species except the j-th species equal to unity. Equation (6) then reduces to f [(Ajy~] = (Al)@‘i-“j).
(8)
With the substitution (A,) = z~‘~J,
m 0=c
=
equation (8) becomes viAi
(1)
f(Z) =
i=l
(At)%.
E
Z*.
(9)
The arbitrary selection of the j-th species implies that n = (a’g - aj)/vj for allj. From (9), the definition of a function forces f(K) to equal Kn, and establishes that the simple power function is necessary. The substitution of (7) into (6) yields
where the vi are the stoichiometric coefficients, positive for products and negative for reactants, and the AI represent the chemical species involved. From thermodynamics, we know the equilibrium constant expression
K(T) = fi
Z(b~-aj)/~
(2)
i=l
In kinetics, for the class of reactions which follow the above convention, we can write the over-all reaction rate for a product species AI,
+c;
II
1 GM = vj
fj
(A#(
k’+f!I
(Ai)a’r.
i=l
(3)
i=l
i=l
Here k and k’ are the individual reaction rate constants, functions of temperature alone, and 4 is in general an arbitrary function of all the species AS. The exponents CL{and a’( may differ from the stoichiometric coefficients. At equilibrium, since the net reaction rate of every species falls to zero, k/k’ = fi
(,&)M’i-ar).
’ d&l _ dx v,
(4)
i=l Both the ratio k/k’ from kinetics
and the equilibrium constant Kfrom thermodynamics are independent of composition and depend only on temperature. Thus k/k’ = f (K), or, from equations f
fi [ i=l
(5)
(2) and (4), (A~)‘,
1
= n
(Ag)(“‘i-af).
which proves that the relation also is sufficient. To complete the proof n must be proven positive. To this end, consider the over-all reaction rate (3) for non-equilibrium compositions. Define for the chemical reaction an extent of reaction X such that dt
dt’
and denote by G(X) the product of activities in equation (4). By convention, products in a given chemical reaction are those species whose concentrations increase with decreasing free energy, while reactants are those which spontaneously decline. Applying this convention and the stoichiometric sign convention for either products or reactants to equation (lo), we find dX/dt > 0. Since the equality holds only at equilibrium, where the reaction rates are zero, Xep is a maximum. Away from equilibrium, X < Xcq and G(X) < k/k’, while at equilibrium X = XeQ and G(X) = k/k’. Since k/k’ is independent of composition,
(6)
i=l
322
(II)
SHORTERCOMMUNICATIONS From the definition of G(X) and from equation (7),
which is equivalent to a = ), p = 0. The reverse rate law then must take the form
dG(X)
-=
dX i=l
i=l
;w
APPLICATION The working formula for thermo-
a2
=
V2
alrn - an -
V?Ti
>
0.
(13)
For any species As in a given reaction we can rewrite (13) as a’$ = at $ me
n > 0,
(14),
which limits the form of the reverse. rate expression given the forward rate law, and vice versa. Since the power n differs among various reactions and must be experimentally determined for each, this constraint does not define the relation uniquely. As an example, consider the dissociation of hydrogen on a hot tungsten surface. The over-all reaction is
Ha + 2H.
d$ = k~(HaHHY-
thermodynamic that
(21)
Substituting
(15)
k’~(H3%)~‘,
(22)
which corresponds to n = 4. WOODand Wrs~ [6] determined the kinetics of hydrogen atom recombination on a number of surfaces, including tungsten, and found these kinetics to be first order with respect to the gaseous atom concentration over the measured range. Since both direct and indirect observations agree, we may conclude that n = 4, and that the reverse rate expression is given by (22). Had the observations suggested an inconsistent rate form, we should have demanded further evidence before attempting to select a final rate expression. In this example, however, with no conflict, we can write (16) as
If we assume the over-all rate law to have the form
f
B’=2n
r’ = klp~,
a’1 - al a’a ~=----_~~----_~~~ Vl
(20)
where the only a priori information about n is that it must be positive. Normally n is a small integer or a simple fraction. In considering the reverse reaction, we may gain additional information by directly measuring hydrogen atom recombination rates on an appropriate surface, or by measuring adsorption rates, adsorption energies, surface coverage, etc., which give information about particular steps in a theoretical recombination mechanism. After examiniig adsorption data and possible theoretical mechanisms related to reaction (15), B~UDART[5] concluded that the reverse rate expression is
which can be true if and only if n 2 0, since the concentrations and activities are certainly all positive. If n = 0, however, no reaction takes place because. the reaction rates for all species vanish identically. Thus for all reactions where some change occurs, the power n must be positive, and the proof is complete.
Consider an application. dynamic consistency is
a’=+--n
(23) into equations (2) and (4), we obtain K = pas-‘p~= k/k’ = pz~~-~/~p~~
(16)
or k/k’ = K’/s = K”,
(14),
which confirms for this reaction what we have #roved in general.
a’=a--n
(17)
B’ = j9 + 2n.
(18)
Department of Chemical Engineering Princeton University Princeton, New Jersey
consistency
requires,
from equation
BRENNANand FLETCHER[4] correlated their data for the production of hydrogen atoms at temperatures below 1400°K and at pressures exceeding 1O-s mm with the expression r = kpRslr2,
(19)
(A‘) [At] IT
E. H. BLUM R. Luus
NOTATION Activity of species At Volumetric concentration of species At Product over the m indexed quantities following
REFERENCES
HI DENB~GHK. G., The Principles of Chemical Equilibrium, pp. 442445.
Cambridge University Press, 1961. FROSTA. A. and PEARSONR. G., Kinetics and Mechanism, p. 191. Wiley, New York 1961. 131 The conventional form is that given in equation (3). The proof does not in general apply to other forms. GRAVENand LONG, J. Amer. Chem. Sot. 1954 76 2602, for example, suggest on mechanistic grounds rate expressions for the water gas reaction which contain additive as well as multiplicative terms. Their resulting rate expressions then need not be and, indeed, are not consistent with Denbigh’s relation. 141 BRENNAND. and FLETCHERP. C., Proc. Roy. Sot. 1959 A250 389. PI B~UDARTtvi., Znd. Chim. Belge 1958 23 383. WI WOOD B. J. and WISE H., J. Phys. Chem. 1961 65 1976.
PI
323
Thermodynamic consistency of reaction rate expressions (Received 10 June 1963; in revised form 4 October 1963) The proof will show that equation (6) necessarily implies the ‘following form for (5):
WE PRESENTa rigorous proof that DENMGH’Srelation for the thermodynamic consistency of reversible reaction rates is both necessary and sufficient. DENBIGH[l] has shown that it suffices for the ratio of the forward and the reverse reaction rate expressions to be a simple power of the equilibrium constant expression. However, that this relation is also necessary for thermodynamic consistency does not seem to be generally appreciated. As recently as 1961, FROST and PEARSONin a widely used text [2] surmised that the relation might be “any function for that matter, and still be consistent”. We have proved that the relation is in fact more restricted, that Denbigh’s power function is the only one mathematically in accord with thermodynamics, The proof is valid for the entire class of reversible reactions for which the over-all reaction rate is written conventionally [3] as the difference of two separable rate terms. Consider the general reversible reaction
k/k’
f (K) = p,
(j = 1, . . . , n?). (7)
n = +
The exponent n can be any positive number. PROOF In equation (6) set the activity of every species except the j-th species equal to unity. Equation (6) then reduces to f [(Ajy~] = (Al)@‘i-“j).
(8)
With the substitution (A,) = z~‘~J,
m 0=c
=
equation (8) becomes viAi
(1)
f(Z) =
i=l
(At)%.
E
Z*.
(9)
The arbitrary selection of the j-th species implies that n = (a’g - aj)/vj for allj. From (9), the definition of a function forces f(K) to equal Kn, and establishes that the simple power function is necessary. The substitution of (7) into (6) yields
where the vi are the stoichiometric coefficients, positive for products and negative for reactants, and the AI represent the chemical species involved. From thermodynamics, we know the equilibrium constant expression
K(T) = fi
Z(b~-aj)/~
(2)
i=l
In kinetics, for the class of reactions which follow the above convention, we can write the over-all reaction rate for a product species AI,
+c;
II
1 GM = vj
fj
(A#(
k’+f!I
(Ai)a’r.
i=l
(3)
i=l
i=l
Here k and k’ are the individual reaction rate constants, functions of temperature alone, and 4 is in general an arbitrary function of all the species AS. The exponents CL{and a’( may differ from the stoichiometric coefficients. At equilibrium, since the net reaction rate of every species falls to zero, k/k’ = fi
(,&)M’i-ar).
’ d&l _ dx v,
(4)
i=l Both the ratio k/k’ from kinetics
and the equilibrium constant Kfrom thermodynamics are independent of composition and depend only on temperature. Thus k/k’ = f (K), or, from equations f
fi [ i=l
(5)
(2) and (4), (A~)‘,
1
= n
(Ag)(“‘i-af).
which proves that the relation also is sufficient. To complete the proof n must be proven positive. To this end, consider the over-all reaction rate (3) for non-equilibrium compositions. Define for the chemical reaction an extent of reaction X such that dt
dt’
and denote by G(X) the product of activities in equation (4). By convention, products in a given chemical reaction are those species whose concentrations increase with decreasing free energy, while reactants are those which spontaneously decline. Applying this convention and the stoichiometric sign convention for either products or reactants to equation (lo), we find dX/dt > 0. Since the equality holds only at equilibrium, where the reaction rates are zero, Xep is a maximum. Away from equilibrium, X < Xcq and G(X) < k/k’, while at equilibrium X = XeQ and G(X) = k/k’. Since k/k’ is independent of composition,
(6)
i=l
322
(II)
SHORTERCOMMUNICATIONS From the definition of G(X) and from equation (7),
which is equivalent to a = ), p = 0. The reverse rate law then must take the form
dG(X)
-=
dX i=l
i=l
;w
APPLICATION The working formula for thermo-
a2
=
V2
alrn - an -
V?Ti
>
0.
(13)
For any species As in a given reaction we can rewrite (13) as a’$ = at $ me
n > 0,
(14),
which limits the form of the reverse. rate expression given the forward rate law, and vice versa. Since the power n differs among various reactions and must be experimentally determined for each, this constraint does not define the relation uniquely. As an example, consider the dissociation of hydrogen on a hot tungsten surface. The over-all reaction is
Ha + 2H.
d$ = k~(HaHHY-
thermodynamic that
(21)
Substituting
(15)
k’~(H3%)~‘,
(22)
which corresponds to n = 4. WOODand Wrs~ [6] determined the kinetics of hydrogen atom recombination on a number of surfaces, including tungsten, and found these kinetics to be first order with respect to the gaseous atom concentration over the measured range. Since both direct and indirect observations agree, we may conclude that n = 4, and that the reverse rate expression is given by (22). Had the observations suggested an inconsistent rate form, we should have demanded further evidence before attempting to select a final rate expression. In this example, however, with no conflict, we can write (16) as
If we assume the over-all rate law to have the form
f
B’=2n
r’ = klp~,
a’1 - al a’a ~=----_~~----_~~~ Vl
(20)
where the only a priori information about n is that it must be positive. Normally n is a small integer or a simple fraction. In considering the reverse reaction, we may gain additional information by directly measuring hydrogen atom recombination rates on an appropriate surface, or by measuring adsorption rates, adsorption energies, surface coverage, etc., which give information about particular steps in a theoretical recombination mechanism. After examiniig adsorption data and possible theoretical mechanisms related to reaction (15), B~UDART[5] concluded that the reverse rate expression is
which can be true if and only if n 2 0, since the concentrations and activities are certainly all positive. If n = 0, however, no reaction takes place because. the reaction rates for all species vanish identically. Thus for all reactions where some change occurs, the power n must be positive, and the proof is complete.
Consider an application. dynamic consistency is
a’=+--n
(23) into equations (2) and (4), we obtain K = pas-‘p~= k/k’ = pz~~-~/~p~~
(16)
or k/k’ = K’/s = K”,
(14),
which confirms for this reaction what we have #roved in general.
a’=a--n
(17)
B’ = j9 + 2n.
(18)
Department of Chemical Engineering Princeton University Princeton, New Jersey
consistency
requires,
from equation
BRENNANand FLETCHER[4] correlated their data for the production of hydrogen atoms at temperatures below 1400°K and at pressures exceeding 1O-s mm with the expression r = kpRslr2,
(19)
(A‘) [At] IT
E. H. BLUM R. Luus
NOTATION Activity of species At Volumetric concentration of species At Product over the m indexed quantities following
REFERENCES
HI DENB~GHK. G., The Principles of Chemical Equilibrium, pp. 442445.
Cambridge University Press, 1961. FROSTA. A. and PEARSONR. G., Kinetics and Mechanism, p. 191. Wiley, New York 1961. 131 The conventional form is that given in equation (3). The proof does not in general apply to other forms. GRAVENand LONG, J. Amer. Chem. Sot. 1954 76 2602, for example, suggest on mechanistic grounds rate expressions for the water gas reaction which contain additive as well as multiplicative terms. Their resulting rate expressions then need not be and, indeed, are not consistent with Denbigh’s relation. 141 BRENNAND. and FLETCHERP. C., Proc. Roy. Sot. 1959 A250 389. PI B~UDARTtvi., Znd. Chim. Belge 1958 23 383. WI WOOD B. J. and WISE H., J. Phys. Chem. 1961 65 1976.
PI
323