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The relationship between the thermodynamic and kinetic reaction diffusion parameters
Shorter Communications ChemicalEngineering Science, 1972, Vol. 27, pp. I190- 1192.
in GreatBritain Pergamon Press.Printed
The relationship between the thermodynamic and kinetic reaction diffusion parameters (First received
4 August
197 1; accepted
THE SOLUTION to the isothermal
steady-state reaction diiTusion problem, which arises both in the study of catalyst performance and membrane transport, has been shown to depend upon a single dimensionless parameter. In catalysis, this parameter is called the Thiele modulus-h [2,7-93, while in the study of membrane transport involving a near equilibrium reaction, a thermodynamic parameter called the “relaxation length”[4,5] has been introduced. It is the purpose of this note to show that these parameters are equivalent in describing the reaction-d8usion problem under near equilibrium conditions; both reduce to the ratio of the relaxation time of the reaction to the average relaxation time for diffusion. Friedlander and Keller [4] showed that in the steady state, when a single chemical reaction near equilibrium is considered, assuming the diffusional flows to be uncoupled, the characteristic reaction-diffusion length A is given by the relation
where
9 September
197 1)
The affinity may be expanded in a Taylor series around its equilibrium value, resulting in: $=L’(K++J+...).
(5)
Retaining only the first two terms,_and recognizing that at equilibrium the tinity is zero, (A = 0), we obtained by comparing Eqs. (4) and (5)
7,-' = -L,z.
aA
(6)
Since A is defined by 71 A=-2
i=1
vipi
we find by using the chain rule, and noting that d[ = (dci/vr) that its derivative with respect to 6 is
Hence, Eq. (6) can be rewritten in the form: (7)
and the where are Onsager phenomenological coefficients evaluated at equilibrium, V,;s the forward reaction velocity at equilibrium and pii denotes a&& evaluated at equilibrium. For ideal soluiidns, Li = (Di&/RT). Blumenthal and Katchalsky [ 11 applied this result to facilitated diffusion through a membrane, and Frazier and Ulanowicz [3] to a model for chemical pumping, based on the asymmetric properties of the nonlinear chemical affinity field equation. Katchalsky and Oster[S] have generalized this result to include coupled diffisional flows. The relaxation length A can be expressed as the ratio of two relaxation times. Meixner [6] has defined the relaxation time of a chemical reaction, TV.by
Solving Eq. (7) for L, and substituting Eq. (3), into Eq. (l), yields:
The Einstein equation determines T$, for a diffusional process
Li, as defined by
the relaxation
(8 time,
2D,rk = 12
(4) where 5 is the degree of advancement of the chemical reaction and dt/dt the rate of the reaction. Near equilibrium, the reaction rate is linearly related to the free energy change, or affinity, A, of the chemical reaction. Thus, as is well known
$=
(9) where I2 is the mean square distance that a nonreacting species i diffuses in the time r,$. For the case of transport through a membrane, 1 is taken to be the thickness of the membrane, while for the catalyst pellet it can be taken to be the characteristic dimension VP/S, where VP is the volume and S the surface area of the pellet. Replacing DI in Eq. (8) by its value in Eq. (9), we obtain: A2 7, 0T =- 2(7d)
L,A.
1190
(10)
Shorter Communications
where (rd) = F Td'YI/~YI is an average ditfusional relaxation time. For non-reactive species y, = 0, and thus only species that both ditfuse and react affect the value of (TV) and hence A. For A < I, 7r < (To) and the reaction will come to equilibrium near the surface of the catalyst (or membrane). This dX%sion-controlled case corresponds to a large value of the Thiele modulus h. On the other hand, when A % I, (TV) Q 7, and the components diffuse through the system without coming to equilibrium. This is the reaction-controlled case and corresponds to small values of h. In the calculation that follows, it will be demonstrated that for a chemical reaction close to equilibrium the Thiele modulus is equal to the dimensionless number l/A. It is worth noting that A is in some sense a measure of the distance a reactant must penetrate the system before coming to equilibrium; hence it is called a relaxation length. THIELE MODULUS FOR THE GENERAL NEAR EQUILIBRIUM REVERSIBLE REACTION The rate of a reversible reaction, r, can be expressed in the general form
which can be rearranged into the standard form
6
(a) -h%
=f(l),
aj, B,)
(17)
where
and
THE
~=~~(i~c~-“i)--(*=~+,c~~l)
(11)
The homogeneous part of Eq. (17) illustrates that h is the Thiele modulus and from the definitions of A, L, and L( [Eqs. (l)-(3)] we see that h=l/A.
where q is the number of reactants and n - q the number of products. Near equilibrium, this rate expression may be linearized by expanding the concentrations around their equilibrium values, i.e.
where
Thus for a single near-equilibrium chemical reaction, the kinetic parameter h and the thermodynamic parameter provide equivalent descriptions of the steady state reactiondiffusion problem, and in their dimensionless form characterize the ratio of chemical to diffisional relaxation times. Donner Laboratory University of California Berkely, Calif.. U.S.A.
Substituting Eq. (12) into Eq. (1 l), using the binomial theorem to expand each term and neglecting the second order terms, yields the linearized expression r=-G$~8c,.
tAnd Weizmann Institute of Science, Rehovot, Israel.
NOTATION
(13) A
The steady state mass balance equation for species i with constant diffusivity has the familiar form *.a=_
’ dx2
w;
i = 1,2,. . . n.
(14)
Since Eq. (14) holds for all species i, dividing by vi we obtain DJ d’cj V$ dx=
Dj d% _ 0 v, dx= ’
D”:
f h k,, kr 1 Li
(15) L,
Replacing the concentrations by their values from Eq. (12), Eq. (15) may be solved for 6c, in terms of 6c,: gc, =2 n,X+B+, DI (
)
(16)
where (Y$and pj (j # i) are constants determined by the boundary conditions. Combining Eqs. (12), (14) and (16)
1191
ALAN S. PERELSON tAHARON KATCHALSKY
n 4 R’ s T
chemical affinity, A = -T v++, ergs/mole concentration of the ith species, moles/cm3 diffusivity of the ith species, cm*/sec defined by Eq. (17), moles/cm3 Thiele modulus, dimensionless forward and reverse reaction rate constants, respectively, dimensions vary characteristic dimension, cm phenomenological coefficients relating the chemical potential gradient of species i to the molar flux of species i; Lj = Di/pic, moleG/erg cm set phenomenological coefficient relating the affinity to the rate of a chemical reaction; r = L,A, moles2/ erg cm%ec number of species present, dimensionless number of reactants, dimensionless chemical reaction rate per unit volume, moles/cm%ec gas constant, ergs/mole “C surface area of catalyst pellet, cm* temperature, “C
Shorter Communications vF forward reaction velocity, v, = kF jfi, ci-“f, moles/ cm%ec V, volume of catalyst pellet, cm3 x rectilinear coordinate, cm Greek symbols a,, p, constants determined by boundary conditions yc weighting factor, yi = r~$&,cm3/mole Bc( deviation of concentration of species i from its equilibrium value, mole/cm3 n dimensionless coordinate, r) = x/l, dimensionless
A characteristic
length, A = L, s (v~/L~)-~‘*, cm
pi pit
chemical potential of species i, ergs/mole partial derivative of pf with respect to ci, pil = (&/&), ergs/cm3/molez Y, stoichiometric coefficient for species i, dimensionless 6 molar extent of reaction, moles/cm3 rdi diffisional relaxation times for the ith species, set (rd) average diffusion relaxation time, set rr relaxation time of a chemical reaction, set Overbars on quantities such as A, c,, vFand pii indicate their equilibrium value.
REFERENCES 111 BLUMENTHAL R. and KATCHALSKY A., Biochim. Biophys. Acta. 1969 173 357. [Zl DAMKOHLER G., Chemie-lng. 1937 3 430. 131 FRAZIER G. C. and ULANOWICZ R. E., Chem. Engng Sci. 1970 25 249. 141 FRIEDLANDER S. K. and KELLER K. H.. Chem. Enena Sci. 1965 20 121. [51 KATCHALSKY A. and OSTER G., The kfolecularkks of Membrane Function, (Ed. D. C. TOSTESON), Prentice-Hall 1969. [61 MEIXNER J. and REIK H. G., Encyclopedia ofPhysics, Vol. 11l/2 1959. [71 THIELE E. W., Ind. Engng Chem. 1939 31916. 181WEISZ P. B. and PRATER C. D., Adv. Caral. 1954 6 143. [93 WHEELER A., Adv. Catal. 195 13 250.
1192
p. 1.
in GreatBritain Pergamon Press.Printed
The relationship between the thermodynamic and kinetic reaction diffusion parameters (First received
4 August
197 1; accepted
THE SOLUTION to the isothermal
steady-state reaction diiTusion problem, which arises both in the study of catalyst performance and membrane transport, has been shown to depend upon a single dimensionless parameter. In catalysis, this parameter is called the Thiele modulus-h [2,7-93, while in the study of membrane transport involving a near equilibrium reaction, a thermodynamic parameter called the “relaxation length”[4,5] has been introduced. It is the purpose of this note to show that these parameters are equivalent in describing the reaction-d8usion problem under near equilibrium conditions; both reduce to the ratio of the relaxation time of the reaction to the average relaxation time for diffusion. Friedlander and Keller [4] showed that in the steady state, when a single chemical reaction near equilibrium is considered, assuming the diffusional flows to be uncoupled, the characteristic reaction-diffusion length A is given by the relation
where
9 September
197 1)
The affinity may be expanded in a Taylor series around its equilibrium value, resulting in: $=L’(K++J+...).
(5)
Retaining only the first two terms,_and recognizing that at equilibrium the tinity is zero, (A = 0), we obtained by comparing Eqs. (4) and (5)
7,-' = -L,z.
aA
(6)
Since A is defined by 71 A=-2
i=1
vipi
we find by using the chain rule, and noting that d[ = (dci/vr) that its derivative with respect to 6 is
Hence, Eq. (6) can be rewritten in the form: (7)
and the where are Onsager phenomenological coefficients evaluated at equilibrium, V,;s the forward reaction velocity at equilibrium and pii denotes a&& evaluated at equilibrium. For ideal soluiidns, Li = (Di&/RT). Blumenthal and Katchalsky [ 11 applied this result to facilitated diffusion through a membrane, and Frazier and Ulanowicz [3] to a model for chemical pumping, based on the asymmetric properties of the nonlinear chemical affinity field equation. Katchalsky and Oster[S] have generalized this result to include coupled diffisional flows. The relaxation length A can be expressed as the ratio of two relaxation times. Meixner [6] has defined the relaxation time of a chemical reaction, TV.by
Solving Eq. (7) for L, and substituting Eq. (3), into Eq. (l), yields:
The Einstein equation determines T$, for a diffusional process
Li, as defined by
the relaxation
(8 time,
2D,rk = 12
(4) where 5 is the degree of advancement of the chemical reaction and dt/dt the rate of the reaction. Near equilibrium, the reaction rate is linearly related to the free energy change, or affinity, A, of the chemical reaction. Thus, as is well known
$=
(9) where I2 is the mean square distance that a nonreacting species i diffuses in the time r,$. For the case of transport through a membrane, 1 is taken to be the thickness of the membrane, while for the catalyst pellet it can be taken to be the characteristic dimension VP/S, where VP is the volume and S the surface area of the pellet. Replacing DI in Eq. (8) by its value in Eq. (9), we obtain: A2 7, 0T =- 2(7d)
L,A.
1190
(10)
Shorter Communications
where (rd) = F Td'YI/~YI is an average ditfusional relaxation time. For non-reactive species y, = 0, and thus only species that both ditfuse and react affect the value of (TV) and hence A. For A < I, 7r < (To) and the reaction will come to equilibrium near the surface of the catalyst (or membrane). This dX%sion-controlled case corresponds to a large value of the Thiele modulus h. On the other hand, when A % I, (TV) Q 7, and the components diffuse through the system without coming to equilibrium. This is the reaction-controlled case and corresponds to small values of h. In the calculation that follows, it will be demonstrated that for a chemical reaction close to equilibrium the Thiele modulus is equal to the dimensionless number l/A. It is worth noting that A is in some sense a measure of the distance a reactant must penetrate the system before coming to equilibrium; hence it is called a relaxation length. THIELE MODULUS FOR THE GENERAL NEAR EQUILIBRIUM REVERSIBLE REACTION The rate of a reversible reaction, r, can be expressed in the general form
which can be rearranged into the standard form
6
(a) -h%
=f(l),
aj, B,)
(17)
where
and
THE
~=~~(i~c~-“i)--(*=~+,c~~l)
(11)
The homogeneous part of Eq. (17) illustrates that h is the Thiele modulus and from the definitions of A, L, and L( [Eqs. (l)-(3)] we see that h=l/A.
where q is the number of reactants and n - q the number of products. Near equilibrium, this rate expression may be linearized by expanding the concentrations around their equilibrium values, i.e.
where
Thus for a single near-equilibrium chemical reaction, the kinetic parameter h and the thermodynamic parameter provide equivalent descriptions of the steady state reactiondiffusion problem, and in their dimensionless form characterize the ratio of chemical to diffisional relaxation times. Donner Laboratory University of California Berkely, Calif.. U.S.A.
Substituting Eq. (12) into Eq. (1 l), using the binomial theorem to expand each term and neglecting the second order terms, yields the linearized expression r=-G$~8c,.
tAnd Weizmann Institute of Science, Rehovot, Israel.
NOTATION
(13) A
The steady state mass balance equation for species i with constant diffusivity has the familiar form *.a=_
’ dx2
w;
i = 1,2,. . . n.
(14)
Since Eq. (14) holds for all species i, dividing by vi we obtain DJ d’cj V$ dx=
Dj d% _ 0 v, dx= ’
D”:
f h k,, kr 1 Li
(15) L,
Replacing the concentrations by their values from Eq. (12), Eq. (15) may be solved for 6c, in terms of 6c,: gc, =2 n,X+B+, DI (
)
(16)
where (Y$and pj (j # i) are constants determined by the boundary conditions. Combining Eqs. (12), (14) and (16)
1191
ALAN S. PERELSON tAHARON KATCHALSKY
n 4 R’ s T
chemical affinity, A = -T v++, ergs/mole concentration of the ith species, moles/cm3 diffusivity of the ith species, cm*/sec defined by Eq. (17), moles/cm3 Thiele modulus, dimensionless forward and reverse reaction rate constants, respectively, dimensions vary characteristic dimension, cm phenomenological coefficients relating the chemical potential gradient of species i to the molar flux of species i; Lj = Di/pic, moleG/erg cm set phenomenological coefficient relating the affinity to the rate of a chemical reaction; r = L,A, moles2/ erg cm%ec number of species present, dimensionless number of reactants, dimensionless chemical reaction rate per unit volume, moles/cm%ec gas constant, ergs/mole “C surface area of catalyst pellet, cm* temperature, “C
Shorter Communications vF forward reaction velocity, v, = kF jfi, ci-“f, moles/ cm%ec V, volume of catalyst pellet, cm3 x rectilinear coordinate, cm Greek symbols a,, p, constants determined by boundary conditions yc weighting factor, yi = r~$&,cm3/mole Bc( deviation of concentration of species i from its equilibrium value, mole/cm3 n dimensionless coordinate, r) = x/l, dimensionless
A characteristic
length, A = L, s (v~/L~)-~‘*, cm
pi pit
chemical potential of species i, ergs/mole partial derivative of pf with respect to ci, pil = (&/&), ergs/cm3/molez Y, stoichiometric coefficient for species i, dimensionless 6 molar extent of reaction, moles/cm3 rdi diffisional relaxation times for the ith species, set (rd) average diffusion relaxation time, set rr relaxation time of a chemical reaction, set Overbars on quantities such as A, c,, vFand pii indicate their equilibrium value.
REFERENCES 111 BLUMENTHAL R. and KATCHALSKY A., Biochim. Biophys. Acta. 1969 173 357. [Zl DAMKOHLER G., Chemie-lng. 1937 3 430. 131 FRAZIER G. C. and ULANOWICZ R. E., Chem. Engng Sci. 1970 25 249. 141 FRIEDLANDER S. K. and KELLER K. H.. Chem. Enena Sci. 1965 20 121. [51 KATCHALSKY A. and OSTER G., The kfolecularkks of Membrane Function, (Ed. D. C. TOSTESON), Prentice-Hall 1969. [61 MEIXNER J. and REIK H. G., Encyclopedia ofPhysics, Vol. 11l/2 1959. [71 THIELE E. W., Ind. Engng Chem. 1939 31916. 181WEISZ P. B. and PRATER C. D., Adv. Caral. 1954 6 143. [93 WHEELER A., Adv. Catal. 195 13 250.
1192
p. 1.