Ind. Eng. Chem. Res. 1997, 36, 4779-4783
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Kinetics of Ammonia Synthesis Close to Equilibrium Dmitry Yu. Murzin* and Alexander K. Avetisov Department of Chemical Kinetics and Catalysis, Karpov Institute of Physical Chemistry, Vorontsovo pole 10, Moscow 103064, Russia
Ammonia synthesis kinetics near equilibrium is considered for an induced inhomogeneous surface using the electronic gas model. At medium nitrogen surface coverage, the resultant kinetic equation matches the Temkin-Pyshev equation. The rate model is then evaluated at high nitrogen pressures assuming an irreversible reaction, finding that the predicted rate is different from first order in nitrogen. Compared to an alternative rate equation, derived from the use of biographical nonuniformity, which predicts first-order kinetics in nitrogen, this model is superior because it matches the observed behavior better. Introduction The kinetics of ammonia synthesis has been a subject of research over the years. A large number of kinetic equations have been suggested based either on certain mechanistic considerations or on empirical evaluations (Aparicio and Dumesic, 1994; Appl, 1992, 1997; Boudart, 1981; ICI, 1970; Nielsen, 1981; Temkin, 1979; Ullmann’s, 1985). This is certainly due to the immense industrial importance of this reaction, as tens of millions of tons of ammonia are produced annually worldwide (Appl, 1992). Based on experimental data which cover the pressure range from below 1 atm up to 500 atm, it was proposed that near the equilibrium, the reaction rate is described by the following equation, which is often referred to in the literature as the Temkin-Pyshev equation (Temkin, 1979):
r ) k+PN2(PH23/PNH32)m - k-(PNH32/PH23)1-m
(1)
where m is a constant (0 < m < 1). Under equilibrium, the reaction rate equals 0; therefore, k+/k- ) K, where K is the equilibrium constant. Hence, only one of the constants, either k+ or k-, together with m should be determined from the experimental data. Equation 1 was supported by numerous studies on various types of catalysts (Ullmann’s, 1985). This equation was based on the supposition that nitrogen chemisorption on a energetically nonuniform surface determines the rate of the overall reaction. The second step is, in fact, an equilibrium between the adsorbed nitrogen and the gasphase concentrations between hydrogen and ammonia. The concept of nonuniform surfaces originated from the fact that the differential heat of adsorption is not constant as a rule but decreases with surface coverage, and adsorption rates and adsorption equilibrium are not described by the Langmuir equation or the Langmuir isotherm correspondingly (Temkin, 1979). The physical nature of the nonuniformity advanced originally by Langmuir and Taylor is not yet well-understood, but to a certain extent can be attributed to the difference in properties of the different crystal faces, the importance of dislocations, other disturbances, etc. This idea forms the basis of a concept of biographical (intrinsic or a priori) nonuniformity which can be either chaotic, when the adsorption energy on a given site is independent of * Correspondence to Dr. D. Murzin, BASF Moscow, B. Gnezdnikovskij per 7, Moscow 103009, Russia. Phone: +7 095 956 91 70. Fax: +7 095 956 91 74. E-mail:
[email protected]. S0888-5885(97)00173-5 CCC: $14.00
the neighbor site, or discrete. However, if in an elementary surface reaction only one adsorbed particle is involved, the difference in these distributions cannot be observed. On biographical nonuniform surfaces, a certain distribution of properties is considered, and a differential distribution function is given by (Temkin, 1979)
φ(λ) ) eγλγL/(eγf - 1)
(2)
where L is the total number of sites, λ is the value of the desorbability exponent, φ(λ) is the number of sites, which corresponds to a variation in the desorbability exponent from λ to λ + dλ, f is the variation interval of the desorbability exponent, and γ is the distribution parameter. Different functions determining the fraction of total number of surface sites characterized by the definite adsorption energy of a given substance were proposed, and one, which accounts for Temkin adsorption isotherm, corresponds evenly to the nonuniform surface. At high pressures, eq 1 should be modified (Temkin, 1979) so as to include the deviations from the laws of ideal gases and to incorporate the effect of pressure on the reaction rate depending on the volume change at activation. Therefore, eq 1 at high pressures contains not partial pressures, but fugacities, and besides that, the right-hand side of it includes a factor (Shishkova et al., 1957), exp[-(VN2# - RVZN2)P/RT], where VN2# is the partial molar volume of the activated complex of nitrogen adsorption, VZN2 is the partial molar volume of the adsorbed nitrogen and P is the total pressure. Although in the original derivation it was supposed that nitrogen is adsorbed in the molecular form, an assumption on the dissociation nitrogen adsorption also leads to the same equation (Temkin, 1979). In the region far from equilibrium, it was suggested that the reaction rate is determined by two slow irreversible steps. The first step is nitrogen chemisorption, and the second one is the addition of hydrogen to molecular adsorbed nitrogen (Temkin, 1979). The reaction rate is expressed as
r)
k(PN21-m(1 - PNH32/KPN22PH23) (l/PH2 + PNH32/KPN22PH23)m(l/PH2 + 1)1-m
(3)
This more general equation was successfully tested (ICI, 1970) over a wide range of operating conditions. Recently, eq 3 was derived with the supposition of dis© 1997 American Chemical Society
4780 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997
sociative nitrogen adsorption (Kuchaev et al., 1995). At relatively high ammonia pressures, eq 3 is transformed into eq 1. In fact, there are no physical reasons for a surface to be evenly nonuniform. Therefore, some of the kinetic dependencies in the ammonia synthesis were explained by Temkin (1990) within the framework of the surface electronic gas model, although without derivation. It was suggested (Aparicio and Dumesic, 1994) that adsorption isotherms based on lateral interactions should lead to microkinetic models (e.g., models which incorporate available knowledge from extensive surface science studies) that describe the observed reaction kinetics better than microkinetic models based on the uniform surface concept. Recently, the model was extensively used (Murzin, 1992, 1993, 1994, 1995a,b, 1996a-c; Murzin and Salmi, 1996) for the discussion of several aspects of heterogeneous catalytic reactions over metals. Among other subjects analyzed was a two-step sequence, initially proposed by Temkin (1979) and treated in detail by Boudart (1972). Such a sequence corresponds to the situation when among the reaction intermediates, one is the most abundant and all others are present at the surface at much inferior concentration levels. It was shown (Murzin, 1996c) that for induced nonuniform surfaces in the region of medium coverage, the concentration dependence is more complicated, according to the derivation for biographical nonuniform surfaces. In the present paper, we treat the kinetics of ammonia synthesis quantitatively by applying the surface electronic gas model and address the possible differences which could arise in the treatments based on induced and biographically nonuniform surfaces.
In the case of a generalized form of an elementary reaction
A + ZI + Z w S
(4)
the reaction rate within the framework of the surface electronic gas model is expressed as (Murzin, 1995a)
r)k
fA fI PAθ0θI fq
(5)
where k is the rate constant with no interaction case, θ is the surface coverage, and the activity coefficients are given by J
fi ) exp(ωiiηi2θiC/T)
∏
exp(ωijηiηjθjC/T)
(6)
j)j1, j*i
and J
fiq ) exp(ηqiηiθiC/T)
exp(ηqiηjθjC/T) ∏ j)j
(7)
Here fi and fiq are the activity coefficients of substrate i in the adsorbed condition and transition state, ηq is the effective charge of the transition state, ωii (ωij) can be either +1 (repulsive interactions) or -1 (attractive), and
C ) h2/4kπm*
(8)
where h is the Planck constant, k the Boltzmann constant, and m* the effective electron mass. Model Formulation
Surface Electronic Gas Model Usually in the models, based on the induced nonuniformity concept, like the surface electronic gas model, it is assumed that on one site, the equations for ideal adsorbed layers hold and that the overall value is a sum of all local values. The surface electronic gas model explains mutual interactions between adsorbed particles and their interaction with the catalyst by changes in the position of the Fermi level. The model is based on the assumption that a complete or partial ionization of the adsorbed particles takes place during chemisorption, with electrons being transferred to the surface layer. This means that chemisorbed particles feed their electrons into the surface layer of the solid or take electrons from it, forming at the surface a kind of two-dimensional electronic gas. The changes in electron concentration in the solid proceed only in the subsurface layer. The model can be used only when the surface coverage is not small. A peculiar characteristic of the model is that the energy of the adsorbed layer is determined only by the total number of adsorbed particles and does not depend on their arrangement (Temkin, 1972). Another essential feature of the model is the statement that electrons in the subsurface layer can be treated as isolated from other metal electrons, as if they are put in a narrow and deep bath. The model gives the possibilities for predicting (or estimating) the values of the interaction parameter, as based on the Sommerefeld theory of metals . Contrary to that, the interaction parameter within the framework of the conventional Bragg-Williams lattice gas remains unclear.
Schematically in the most simplest treatment, the ammonia synthesis mechanism near the equilibrium can be pictured in the following way:
(1) N2 + Z S ZN2 (2) ZN2 +3H2 ≡ Z + 2NH3 N2 + 3H2 ) 2NH3
(9)
Here ZN2 is an adsorbed intermediate in the form of dinitrogen, step 1 is a reversible one, and step 2 is an equilibrium one. In mechanism 9, it is supposed that surface species others than chemisorbed nitrogen are presented on the surface in inferior quantities (Aparicio and Dumesic, 1994, Temkin, 1979). This assumption is backed by experimental evidence showing that nitrogen adsorption on iron catalysts proceeds at a rate approximately equal to that of ammonia synthesis (Aparicio and Dumesic, 1994). The equilibrium constant of step 2 in eq 9 can be expressed, following the general treatment (Temkin, 1972; Murzin, 1995a)
K2 ) (PNH32/PH23)(θ0/θN2) exp(-η2θN2C/T)
(10)
The equilibrium constant of the overall reaction heterogeneous catalytic reaction can be expressed by multiplication of the equilibrium constants of the elementary steps, which constitute the overall reaction. In the particular case of ammonia synthesis (eq 9), the equilibrium constant is given as
Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4781
K ) k1/k-1K2 ) PNH32/PH23PN2
(11)
From eqs 10 and 11, it follows that
θN2/θ0 ) (k1/k-1)PN2 exp(-η2θN2C/T)
(12)
The reaction rate is expressed as
r ) r1 - r-1 ) k1PN2θ0 exp(-Rη2θN2C/T) k-1θN2 exp((1 - R)η2θN2C/T) (13) where R is the Polanyi parameter
ηq ) Rη
(14)
The balance equation for surface coverage is given by
θN2 + θ0 ) 1
(15)
Figure 1. Dependence of the relative reaction rate r+/r+b on nitrogen pressure at k1/k-1 ) 1 and (a) η2CT-1 ) 20, (b) η2CT-1 ) 40, (c) η2CT-1 ) 100, and (d) η2CT-1 ) 200.
θN2(1 + [(k1/k-1)PN2 exp(-η2θN2C/T)]-1) ) 1 (16)
(which is sometimes assumed, for instance, in the consideration of transient ammonia synthesis kinetics (Kuchaev et al., 1991)) and the overall reaction rate is determined by the ammonia synthesis rate in the forward direction. According to the conventional derivation based on the model of biographically nonuniform surfaces, this rate in the forward direction, r+b, is given by
leading to
The estimation of surface coverage (eq 16) and reaction rate (eq 13) should be performed numerically. For instance, the value of the surface coverage could be solved using the Newton-Raphson procedure. An analytical expression (although not strictly correct) can be obtained for the region of medium coverage when it is usually assumed that
θN2/θ0 ≈ 1
(17)
Hence, it follows from eqs 12 and 17 that
θN2 ) ln U/(η2CT-1)
(18)
where, taking eq 11 into account,
U ) (k1/k-1)PN2 ) (k1/k-1K)(PNH32/PH23)
(19)
From the balance equation (15), one arrives at
θ0 ) 1 - ln U/(η2CT-1)
(20)
Equation 13 can be transformed (not strictly, we should emphasize) therefore into
r ) k1PN2(1 - ln U/(η2CT-1))U-R k-1 ln UU1-R/(η2CT-1) (21) Then we finally arrive at an equation for ammonia synthesis in the region of medium coverage
r ) (1 - ln U/(η2CT-1))k+PN2(PH23/PNH32)R ln U/(η2CT-1)k+/K(PNH32/PH23)1-R (22) which is in fact very similar to eq 1. Practically in the region of medium coverage, the difference between (1) and (22) is very small and, thus, can be neglected. Conditions of Irreversibility In certain conditions, the rate of the ammonia synthesis reaction in the reverse direction is very small
r+b ) k+PN2(PH23/PNH32)R
(23)
By looking at eq 22 (and the system of equations (13) and (16)), one would expect intuitively certain deviations from the first-order kinetics toward nitrogen, contrary to eq 23. Such experimentally observed deviations (Shiskova et al., 1957) within the framework of the model of biographically nonuniform surfaces were attributed to changes in the molar volume of the activated complex at high pressures. Note that the values of the molar volumes of the activated complexes cannot be determined experimentally; therefore, the pressure correction factor was an adjusted parameter during the parameter estimation (Shiskova et al., 1957). As a first step in our analysis, it was reasonable just to estimate what error can be introduced in the calculations, if one uses instead of (23) the equation for ammonia synthesis in the forward direction in the region of medium coverage
r+ ) (1 - ln((k1/k-1)PN2)/(η2CT-1))k+PN2(PH23/PNH32)R (24) The dependence of the ratio r+/r+b on nitrogen pressure at the different values of the parameters is presented in Figure 1. For simplification, it was assumed that the ratio between k1 and k-1 is equal to unity. It follows from Figure 1 that at high values of nitrogen pressures, if the value of the parameter η2CT-1 is around 20, then big differences between eqs 23 and 24 are expected. Such values were in fact reported for several heterogeneous catalytic reactions over metals (Temkin, 1972; Snagovskii et al., 1978). A more detailed numerical analysis was performed for the following system of equations:
4782 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997
are not solely due to the influence of high pressure on the kinetic constant but can be explained within the framework of the surface electronic gas model without any additional assumption about the partial molar volumes of adsorbed nitrogen and activated complex. Conclusions
Figure 2. Dependence of the reaction rate/nitrogen pressure as a function of nitrogen pressure at R ) 0.5 and (1) k1 ) 1, K2/K ) 1, and η2CT-1 ) 0.1, (2) k1 ) 0.65, K2/K ) 1, and η2CT-1 ) 1.
Ammonia synthesis kinetics near equilibrium is usually described by the famous Temkin-Pyshev equation. The underlying concept is the assumption of nonuniformity, which is based on the fact that the differential heat of adsorption decreases with coverage. The original derivation of ammonia synthesis kinetics was based on the concept of a priori nonuniform surfaces. However, in recent years, there is more and more evidence that lateral interactions (e.g., the interactions between the adsorbed species) play a decisive role. One of the physically reasonable, although simple, models accounting for such induced nonuniformity is the surface electronic gas model. In the present paper, the kinetics of ammonia synthesis near equilibrium was derived based on this model. It was demonstrated that although the concentration dependencies for induced nonuniform surfaces are somewhat complicated compared to the conventional Temkin-Pyshev equation for ammonia synthesis, for engineering purposes, the differences between the two approaches are of minor importance. However, if the reverse reaction can be neglected, at high nitrogen pressures, the reaction rate according to the model of induced nonuniformity differs from that for the model, based on biographical nonuniformity. This could be an explanation for the experimentally observed deviations from the first order in nitrogen at high nitrogen pressures. Literature Cited
Figure 3. Dependence of the reaction rate/nitrogen pressure as a function of nitrogen pressure at R ) 0.5 and (1) k1 ) 0.1, K2/K ) 10, and η2CT-1 ) 0.1, (2) k1 ) 0.05; K2/K ) 100, and η2CT-1 ) 0.1, (3) k1 ) 0.1, K2/K ) 10, and η2CT-1 ) 1, (4) k1 ) 0.15, K2/K ) 1, and η2CT-1 ) 10.
θN2(1 + (K2/KPN2) exp(η2θN2C/T)) ) 1
(25)
r+/(PN2) ) k1(1 - θN2) exp(-Rη2θN2C/T)
(26)
In Figures 2 and 3, the dependence of the reaction rate divided by nitrogen pressure is presented as functions of the reaction parameters. The value of the Polanyi parameter was taken in the calculations to be equal to 0.5, as usually obtained for ammonia synthesis. One can state that when surface nonuniformity is not manifested (e.g., nonuniformity factor η2CT-1 is much less than unity), then the reaction rate divided by nitrogen pressure is almost independent of N2 pressure. Note that at high values of the parameter K2/K, even at relatively low values of the nonuniformity factor, it obviously follows from eq 25 and from Figure 3 that deviations from the first order are prominent. Naturally, drastic deviations from first order with respect to nitrogen follow from Figure 3 at high values of the nonuniformity factor, which gives us a possibility to speculate that experimentally observed deviations from the first order in nitrogen at high nitrogen pressures
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Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4783 Murzin, D. Yu. Three-step Catalytic Reaction Mechanism with Coverage Dependent Parameters. Chem. Eng. Technol. 1996b, 19, 113. Murzin, D. Yu. On Reaction Kinetics over Inhomogeneous Surfaces. React. Kinet. Catal. Lett. 1997, in press. Murzin, D. Yu.; Salmi, T. Isothermal Multicplicity in Catalytic Surface Reactions with Coverage Dependent Parameters: Case of Polyatomic Species. Chem. Eng. Sci. 1996, 51, 55. Nielsen, A. Ammonia Synthesis: Exploratory and Applied Research. Catal. Rev. Sci. Eng. 1981, 23, 17. Shishkova, V. N.; Sidorov, I. N.; Temkin, M. I. Investigation of Ammonia Synthesis Kinetics by FlowsCirculation Method at High Pressures. Trudy GIAP 1957, 7, 62. Snagovskii, Yu. S.; Khomenko, A. A.; Ivanova, R. A.; Shub, F. S.; Temkin, M. I. Comparison of Model Catalysts with Homogeneous and Heterogeneous Surfaces with Respect to the Kinetics of Methane Conversion by Water Vapor on Nickel. Kinet. Katal. 1978, 19, 942.
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Received for review February 24, 1997 Revised manuscript received July 8, 1997 Accepted July 14, 1997X IE9701736
X Abstract published in Advance ACS Abstracts, September 1, 1997.
