J. Phys. Chem. 1996, 100, 11005-11008
11005
Nonequilibrium Flow-Dependent Corrections to Chemical Rate Constants in Gases R. E. Nettleton Department of Physics, UniVersity of the Witwatersrand, Johannesburg 2050, South Africa ReceiVed: January 4, 1996; In Final Form: March 27, 1996X
Second-order corrections to rate constants, quadratic in the traceless shear stress and proportional in a steady state to the gradient of flow velocity, are calculated for decomposition of dilute gases NO2, N2O, and HI. The approach can be applied, with similar conclusions, to a wide variety of reactions of arbitrary order. The method can be extended to heat flow and diffusive flow of particles. The corrections are found negligible in dilute gases under experimentally accessible conditions, contrary to assumptions often made in extended thermodynamics. Similar effects, in a modified theory taking into account collisional transfer, might possibly be seen in computer studies of steady liquid states far from equilibrium.
1. Introduction In the usual1 formulation of kinetics of decomposition of NO2 or N2O, one writes, when no decomposition products are initially present and x is the fraction decomposed:
dx/dt ) k(1 - x)2 - 1/2k′x3
(1.1)
In the decomposition of HI, where the reverse reaction is of second order rather than third as in (1.1), one has
dx/dt ) k(1 - x)2 - k′(x/2)2
(1.2)
where k, and k′ are rate constants which differ for different reactions. It is usually assumed that ν1 ν2 3| k′/k ) K h ) Z|ν 3 /Z1 Z2
(1.3)
where K h is the equilibrium constant, Zi are molecular partition functions with νi the stoichiometric coefficients, and constituent 3 is the one which decomposes in the forward reaction to yield dissociation products 1 and 2. Equation 1.3 reflects the presumption that k and k′ are equilibrium functions of the initial densities and temperature which is well in accord with experiment in the cases mentioned. In nonequilibrium, however, k and k′ may depend on a shear stress, heat flow, or diffusion flow, or on x. Here we examine how to calculate the flow-dependent corrections and verify, for reactions usually observed in gases, that these corrections in the case of a shear stress are small and can be neglected. Later work will consider heat flow and diffusion. We shall conclude that the flow-dependent corrections are most likely to be observed in computer modeling of reactions in liquids, which requires a model which takes into consideration collisional transfer, where computer studies are more practicable than in gases, at least for steady, isothermal Couette flow. The corrections we calculate here are in addition to departures from eq 1.3 arising from the fact that the velocity distributions of two colliding reactant molecules are non-Maxwellian in the presence of a reaction. Calculations of the perturbed velocity distribution and reaction rate have been made2 for a bimoledular reaction by effecting a Chapman-Enskog treatment of a linearized Boltzmann equation or a variational treatment3 of the latter. Additional treatments of this kind have taken into account the heat of reaction4 which can slow the reaction if endothermic X
Abstract published in AdVance ACS Abstracts, May 15, 1996.
S0022-3654(96)00066-4 CCC: $12.00
or speed it up if exothermic, and the effect on the reaction constant k of different compositions and reactant mass ratios.5 An expansion of the perturbed reaction rate in powers of the chemical affinity has been calculated for a single reactant6 and extended to multicomponent mixtures.7 The latter predicts a decrease of less than one percent in the equilibrium rate for the reaction H2 + Cl r HCl + H. A molecular dynamics simulation8 has found a decrease in the reaction rate resulting from a depletion in a fast reaction of the population of particles with relative speeds above the activation threshold. The rate decrease is partially reversed as products accumulate and collisions occur involving these. The nonequilibrium effects are, in general, small and depend on rough models of the reactive cross section. They can be incorporated into the present paper if we assume that k and k′ depend on x and that eq 1.3 holds only very approximately unless x has its equilibrium value. To calculate the further corrections arising from dependence of k and k′ on the fluxes of diffusion, momentum, and energy, we invoke the extended thermodynamics of reaction kinetics, set forth by Garcı´a-Colı´n et al.9 This uses the reaction rate J and the numbers Ni of molecules of the various constituents as independent variables. In the presence of a steady velocity gradient, we have an additional state variable P° , the traceless pressure, and when steady heat or particle flows are present, these enter as further independent variables. The entropy S({Ni},V,T,J,P° ) is postulated10-12 to obey a Gibbs equation
T dS ) dU + P dV + Φ dJ + ∑ΨRβ dP° Rβ - ∑µi dNi Rβ
i
(1.4) The J dependence is not quantitatively important and is introduced only for applications of reciprocity.13 When these have been made in section 2, J will be dropped as an independent variable. It will then be assumed that the reaction rate is given by (1.1) or (1.2). Equation 1.4 plus integrability conditions can be used to determine the P° dependence of the {µi} and thus of the reaction rate which is proportional to the thermodynamic force
A ) -∑µiνi
(ν3 < 0)
(1.5)
i
A is minus the chemical affinity. To calculate the P° dependence of the chemical potentials, we determine ΨRβ by invoking the information-theoretic calculation of S. © 1996 American Chemical Society
11006 J. Phys. Chem., Vol. 100, No. 26, 1996
Nettleton
The thermodynamics will be summarized in section 2, together with the application of reciprocity and integrability. In section 3, we use information theory to calculate ΨRβ which, in turn, is used in section 4 to derive an expression for the effect of shear flow on a steady decomposition reaction, specifically for NO2, N2O, and HI. Numerical estimates of this effect are made in section 5. In section 6, we estimate P° -induced shifts in equilibrium temperature and concentrations from the equation A ) 0. A discussion and conclusions are presented in section 7.
To find A0, we have the dilute-gas relation, with µ0i the P° f 0 limit of µi,
µi0 ) κT[ln Zi - ln Ni]
(2.7a)
Zi ) V(2πmiκT/h2)3/2
(2.7b)
If N1 ) 0 ) N2 and N3 ) N03, with ν3 ) -2 ) -ν1 and ν2 ) 1, before decomposition begins
h N03) + ln[(1 - x)2/x3] A0/κT ) -ln(1/2K
2. Extended Chemical Thermodynamics
(2.8)
In an isotropic fluid, forces of a given tensorial character depend linearly to lowest order on variables of the same tensorial order, so that
The P° dependence of the {µi} is calculated from the integrability condition for the Helmholtz function F ≡ U - TS, viz.
ΨRβ ) -νpP° Rβ + ...
(2.1a)
∂µi/P° Rβ ) -∂ΨRβ/∂Ni ) (∂νp/∂Ni)P° Rβ
Φ ) -νrJ + ...
(2.1b)
To calculate the derivative ∂νp/∂Ni in (2.9) and the P° dependence of τc in (2.5), we need the maximum entropy formalism and its associated information-theoretic entropy which are discussed in the following section.
where νr and νp are functions of T, V, and conserved variables. The ellipsis indicates higher-order terms. The phenomenological equtions are13
J˙ ) LrΦ + MrA
(2.2a)
J ) -(1/νr)Φ
(2.2b)
where Lr and Mr are phenomenological coefficients which may depend on all the variables. Equation 2.2b is the identity J ) J, and the phenomenological coefficient multiplying Φ is determined to make it so. Positive definiteness of the irreversible entropy production requires the antisymmetric Onsager relation
νr-1 ) Mr
(2.3)
Lrνr ) τc-1
(2.4)
where the relaxation time for J is identified with the collision time, τc. In a quasi-steady state, with J negligible, corresponding to conventional kinetics,
J ) τcMrA
(2.5)
In principle, Φ could be calculated via methods outlined below in eqs 3.3 and 3.4 for calculating ΨRβ, if we can find an operator Rˆ such that J ) 〈Rˆ 〉, the indicated average being calculated from information theory. Rˆ depends on motion of molecular subgroups relative to the molecular mass center and should have no appreciable correlation with macroscopic shear rates. In a dilute gas, we neglect momentum transfer during a collision in calculating the macroscopic momentum flux. Accordingly, we take νr in (2.1b) to be independent of P° :P° and likewise Mr from (2.3). We can estimate Mr from the P° f 0 limit of (2.5):
τcMrA0 ) (N03/|ν3|)Υ(x)
3. Statistical Calculation from Maximum Entropy To calculate ΨRβ ) -∂F/∂P° Rβ, we invoke the maximum entropy formalism14 of Jaynes. The operator Pˆ Rβ for the kinetic component of the momentum flux, which is the component we take into account in a dilute gas, is, for a ternary mixture of Ni particles of constituent [i] (1 e i e 3): 3
Pˆ Rβ ) V-1∑mj-1 ∑ pRk pβk j)1
(2.6)
where Υ is the right-hand side of (1.1) or (1.2) and A0 the P° f 0 limit of (1.5). Υ may incorporate the nonequilibrium corrections of refs 2-8. The P° dependence of the reaction rate in (2.5) stems, accordingly, from the stress dependence of τc and A.
(3.1)
k[j]
The maximum entropy formalism14 maximizes the functional
S ) -κ ∫ F ln F dx
to lowest order in A and Φ. In a dilute gas
(2.9)
(3.2)
where the integral is taken over phase space, subject in this case to 0 dx P° Rβ ) ∫FPˆ Rβ
(3.3)
The resulting distribution, F, has the form15 0 ˆ - β∑ΨRβPˆ Rβ ] F(x) ) Z-1 exp[-βH
(3.4)
Rβ
where Z normalizes F to unity and H ˆ is the Hamiltonian. ΨRβ is determined by substituting this F into (2.13) and requiring that the latter be satisfied identically. After a lengthy calculation, we extract (2.1a), with
νp ) V2/[2κT(N1 + N2 + N3)]
(3.5)
This result can be substituted into (2.9) to obtain the P° dependence of the chemical potentials. We do this in the following section. To calculate τc in (2.6), we need to use the distribution F to obtain the rms speed, Vj. We have
〈KE〉 ) 1/2(N1m1 + N2m2 + N3m3)V-2
(3.6)
To O(P° 2), we have, with pi the momentum of a representative particle of constituent i:
Chemical Rate Constants in Gases
J. Phys. Chem., Vol. 100, No. 26, 1996 11007
3
〈KE〉 ) ∫F∑Nipi2/2mi dx ) i)1
(3/2)κT(N1 + N2 + N3)(1 - ∆Z/Zc) + 3
1
/2Vβ2∫exp(-βH ˆ )Zc-1∑(pi2/2mi)(νpPˆ :P° )2 dP (3.7) i)1
where the integration is over the space of all the particle momenta. Zc is the canonical partition function and ∆Z is the O(P° 2) contribution to Z. After extensive algebra we extract
〈KE〉 ) κT(N1 + N2 + N3)[(3/2) + 2(νp/V)2P° :P° ] (3.8) From (3.6) and (3.8):
Vj ) Vj0[1 + (2νp2/3V2)P° :P° ]
(3.9)
where Vj0 is the contribution independent of P° . From this, with l the mean free path, we have
τc ) l/Vj ) τ0c [1 - (2νp2/3V2)P° :P° ]
(3.10)
We are now in a position to calculate the flow-induced corrections to the factors A and τc in eq 2.6 for the reaction rate. 4. Stress-Dependent Corrections to the Rate Constants
normal experimental conditions, in agreement with the fact that these classic experiments have been analyzed successfully without taking such corrections into account. It is of interest to investigate these nevertheless because the density dependence makes it look as though they could be important at low density. We shall point out in section 7 that such corrections might be seen in computer simulations for liquids rather than in gases where simulations of steady Couette flow have not been practicable. In making numerical estimates, we shall take T ) 627 K for dissociation of NO2, T ) 1125 K for decomposition of N2O, and T ) 500 K for dissociation of HI. These are temperatures for which well-known rate measurements have been made.16 In all cases, the initial pressure P0 ) (N03/V)κT is 10-2 atm. Also, initially, x ) 0 ) N1 ) N2. Under these circumstances the gas is sufficiently dilute so that the ideal gas equation of state and the dilute-gas expression for µi should apply, making the correction ∆J/J at t ) 0 proportional to (Pxy/P0)2. ∆J/J increases for t > 0, but the overall reaction rate rapidly decreases and becomes difficult to observe. Any observable corrections are likely to be seen at the start of decomposition. Since A0 becomes large as x f 0, we neglect the term proportional to ∑iνi in eq 4.3 at t ) 0. We have since, P° :P° ) 2Pxy2,
∆J/J = -(1/3)(Pxy/P0)2
(x ) 0)
(5.2)
We calculate ∂νp/∂Ni (1 e i e 3) from (3.5) to obtain via (2.9)
For P0 ) 10-2 atm, we find ∆J/(JPxy2) ) -(3P02)-1. To make ∆J/J ) 10-2, on the threshold of detection, we must have Pxy ) 1.8 × 102 Pa. Since
µi ) µi0 - 1/2P° :P° V2|2κT(N1 + N2 + N3)2|-1
Pxy ) -η0∂ux/∂y
(4.1)
Substitution from (4.1) into (1.5) yields
A ) A0 + (1/4)∑νiP° :P° V2/[κT(N1 + N2 + N3)2] (4.2) i
where A0 is given by (2.9). Combining this result with (3.10), we find, using (2.5), that in a steady state
J ) τ0c MrA0[1 +
{V2/2κT(N1 + N2 + N3)2}P° :P° {(2A0)-1∑νi - (3κT)-1}] i
(4.3) The term in the square brackets which is proportional to P° :P° is ∆J/J, i.e., the fraction change in the reaction rate induced by a stress. The square bracket represents a multiplicative correction to the rate constants k and k′. 5. Numerical Results for Decomposition Reaction Rates We apply eq 4.3 to reactions of the type
|ν3|[A3] T ν2[A2] + ν1[A1]
(ν3 < 0)
(5.1)
Specifically, we take {A3 ) NO2, ν3 ) -2; A1 ) NO, ν1 ) 2; A2 ) O2, ν2 ) 1}, {A3 ) N2O, ν3 ) -2; A1 ) N2, ν1 ) 2; A2 ) O2, ν2 ) 1}, {A3 ) HI, ν3 ) -2; A2 ) H2, ν2 ) 1; A1 ) I2, ν1 ) 1}. In the first two cases, where the reverse reaction is third order, we have ∑iνi ) 1. In the third case, forward and backward reactions are both second order, and ∑iνi ) 0. In all cases, the quadratic stress effect is proportional to (Pxy/n)2, where Pxy is the nonzero component of shear stress and n the total molecular number density. Numerical estimates in the following two sections indicate that the stress effect is negligible under
(5.3)
where η0 is the shear viscosity, we can calculate the steady velocity gradient required for an appreciable correction factor ∆J/J ) 10-2. With the aid of the first Chapman-Enskog approximation for a hard-sphere model, which should be sufficiently accurate for order-of-magnitude estimates, we get η0 ) 4.1 × 10-5 Pa s (NO2); η0 ) 5.4 × 10-5 Pa s (N2O); η0 ) 2.8 × 10-5 Pa s (HI). In all cases |∂ux/∂y| ∼ 106 s-1. This is unlikely to be attained in steady flow of a dilute gas in the laboratory, and so we verify the well-known fact that nonequilibrium corrections to k and k′ are small in empirically-accessible conditions. In principle, an appreciable fraction ∆J/J might be seen at much lower density and smaller |∂ux/∂y|, but then the reaction would be hard to observe in addition to the problem of establishing steady, isothermal flow plus the problem of separating second-order effects from those of higher order. 6. Corrections to Equilibrium Temperature and Concentrations The condition for chemical equilibrium, which determines x ) x(N,V,T,P° ), equates to zero the right-hand side of eq 4.3. The equilibrium value x0 corresponding to P° ) 0 is obtained by setting A0 ) 0. Then if ∆x if the O(P° 2) contribution to x x0, we have on invoking (2.9) for A0 and using A0 (x0) ) 0
(∂A0/∂x)∆x ) -[V2/4κT(N1 + N2 + N3)2]∑νiP° :P° (6.1a) i
(κT)-1 ∂A0/∂x ) (x0 - 3)/[x0(1 - x0)]
(6.1b)
Here (6.1b) has been calculated using stoichiometric coefficients appropriate to dissociation of NO2 and N2O. ∆x ) 0 to this order in HI where ∑iνi ) 0. For dissociation of NO2 at 627 K, we find x0 ) 0.999 98, and for the case of N2O at 1125 K x0 ) 0.999 99. To make
11008 J. Phys. Chem., Vol. 100, No. 26, 1996
Nettleton
∆x/x0 ) 10-3, we need in both cases P ∼ 700 Pa, corresponding to |∂ux/∂y| ∼ 107 s-1. Stress-induced shifts are harder to observe near equilibrium than when x ) 0. We can also ask how the O(P° 2) term in A willl change the chemical equilibrium temperature if x is held constant. This has some theoretical interest because a similar calculation [ref 11, p 265] shows that the shear-rate dependence of chemical potentials can shift the spinodal line in a polymer mixture. Using the fact that, for chemical equilibrium when P° ) 0 (Ni0 ) Ni(x0)):
(N1 + N2 + N3)0 ) N03(1 + 1/2x0)
(6.2)
where N03/V is the initial density of NO2 or N2O at x ) 0. We have
∆T/T ) -(1/3)(3/2)-2(Pxy/P0)2
(6.3)
in both cases. If |∂ux/∂y| ) 1 s-1, this gives ∆T/T ) -7.2 × 10-17. For systems such as these, we cannot expect to observe flow-induced shifts in the chemical equilibrium. 7. Discussion Although it has long been assumed that we can assign equilibrium values to the chemical rate constants, k and k′, whose ratio then gives the equilibrium constant K h , only x-dependent corrections from the reaction itself2-8 have been estimated. We investigate here additional corrections, depending on momentum flux, via the extended thermodynamics10,11 which introduces independent state variables P° , Q, and JD to represent independent flow of momentum, heat carried by phonons and selfdiffusion, and binary diffusive flow, respectively. These appear in the Gibbs equation (1.4), in general, which leads to an equation for dF, the differential of the Helmholtz free energy. Application of the integrability condition, such as (2.10) to this equation permits calculation of the dependence of chemical potentials on P° , Q, JD and any other such variables appearing in an extended description, leading to an estimate of the dependence on these variables of the reaction rate. The calculation invokes Jaynesian information theory14 to calculate ΨRβ in the Gibbs equation (1.4). The calculation has been done for a gas at 10-2 atm because the extended thermodynamics has been developed11 mainly as a theory of dilute gases. The reason for this is that predicted second-order effects depend on (P° /n)2, (Q/n)2, and so on. Therefore, it is assumed that they should show up at low densities. Furthermore, in a dilute gas one can neglect the collisional transfer component of P° . The latter makes the information-theoretic integrals difficult to evaluate. The present investigation is one of a series designed to show that one needs very large fluxes P° , Q, JD which are not readily obtainable in the laboratory in order to observe these nonlinear corrections. In a similar manner, corrections O(Q2) to the internal energy
of a gas have been calculated by Dominguez and Jou.18 An analysis of their results similar to this one19 leads to the conclusion that very large |Q| is needed in order to observe any effect. At very low densities, isothermal steady states with large fluxes are hard to achieve and the unperturbed reaction rates very small. Also, at low density, higher-order terms in (P° /n)4 can enter and are difficult to separate from the secondorder ones. The gaseous phase is probably not the one in which to look for such effects. At x ) 0, when decomposition is just beginning, the order of the reaction affects Mr, but not ∆J/J, in view of the smallness of the A0-1 term, and so these conclusions are independent of the order. This work generalizes to gases an extensive set of estimates made for liquids12 where second-order nonlinear effects involving P, Q, and JD are invariably found to be small under laboratory condition. In liquids, they can show up13,20 in computer simulations which mimic isothermal steady states at high density and high shear rates.21 Such effects can also show up in polymer liquids,11 as we have observed above, through a shear-rate dependence of the spinodal line. A principal difference between the present investigation and corresponding ones for liquids resides in the fact that the operator Pˆ Rβ in (3.1) must include the intermolecular force contributions in the condensed phase. Also, the concept of mean free path is difficult to define for a liquid, and so the Eyring reaction-rate theory22 has been invoked13 in discussing extended thermodynamics of reaction kinetics in liquids. References and Notes (1) Glasstone, S. Textbook of Physical Chemistry; McMillan: London, 1956; Chapter XIII. (2) Present, R. D. J. Chem. Phys. 1959, 31, 747; 1968, 48, 4875. (3) Present, R. D.; Morris, B. M. J. Chem. Phys. 1960, 50, 151. (4) Prigogine, I.; Mahieu, M. Physica 1950, 16, 51. (5) Pyun, C. W.; Ross, J. J. Chem. Phys. 1964, 40, 2572. (6) Shizgal, B.; Karplus, M. J. Chem. Phys. 1970, 52, 4262. (7) Shizgal, B.; Karplus, M. J. Chem. Phys. 1971, 54, 4345, 4357. (8) Gorecki, J.; Popielawski, J.; Cukrowski, A. S. Phys. ReV. 1991, A44, 3791. (9) Garcı´a-Colı´n, L. S.; de la Selva, S. M. T.; Pin˜a, E. Phys. Lett. 1985, A110, 363. (10) Casas-Va´zquez, J.; Jou, D., Lebon, G., Eds. Recent DeVelopments in Nonequilibrium Thermodynamics; Springer: Berlin, 1984. (11) Jou, D.; Casas-Va´zquez, J.; Lebon, G. Extended IrreVersible Thermodynamics; Springer: Berlin, 1993. (12) Nettleton, R. E. AdV. Thermodyn. 1993, 7, 223. (13) Nettleton, R. E. Il NuoVo Cimento 1988, 101B, 53. (14) Jaynes, E. T. Phys. ReV. 1957, 106, 620. (15) Nettleton, R. E.; Freidkin, E. S. Physica 1989, A158, 672. (16) Hinshelwood, C. N. The Kinetics of Chemical Change in Gaseous Systems; Clarendon Press: Oxford, U.K., 1929. (17) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954; p 527. (18) Dominguez, R.; Jou, D. Phys. ReV. 1995, E51, 158. (19) Nettleton, R. E. Phys. ReV. 1996, E53, 1241. (20) Hanley, H. J. M.; Evans, D. J. J. Chem. Phys. 1982, 76, 3225. (21) Nettleton, R. E. J. Non-Equilib. Thermodyn. 1987, 12, 273. (22) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941.
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