J . Phys. Chem. 1986, 90, 953-956
953
Consistency of the Kinetlc Mass Action Law with Thermodynamics L. S. Garcb-Codn,+S. M. T. De la Selva,* and E. Pifia** Department of Physics, Universidad Autbnoma Metropolitana, Iztapalapa. C.P. 09340, MCxico, D.F. (Received: May 7, 1985; I n Final Form: July 29, 1985)
In this paper we place the phenomenological description of systems with nonequilibrium chemical reactions within the frame of extended irreversible thermodynamics. Using the standard method developed for this theory we show that it leads to the prediction that the time derivative of the reaction rate is a general function of all the scalar invariants such as the internal energy, volume, etc. characterizing the system, among them the rate of the reaction itself. In the case of a negligible relaxatkn time for the rate of the reaction, we show that a particular choice for the form of this function leads to the kinetic mass action law for all cases, linear and nonlinear characterized by an integer order. Taking into account the simultaneous Occurrence of heat and diffusive transfer we show also that for fast reactions the divergence of the heat and diffusive fluxes can drive the rate of the reaction in agreement with experiment.
1. Introduction The growing realization since C. L. Berthollet’s times (1799)’ that the rate of a chemical reaction is dependent on the concentrations of the reacting substances, and that chemical reactions can reverse their sense, led to their systemization into rate laws that express this dependency as integral powers of the concentrations. By the years of 1864-1867, Guldberg and Waage’ proposed a statement which synthesizes all the empirical rate laws for reactions occurring in homogeneous fluid phases at constant temperature and pressure, known as the kinetic mass action law (KMAL).1,2 As is well-known, KMAL encompasses all the empirical rate laws of specific one-step reactions and also others which exhibit a definite order. In spite of its success, both its full thermodynamic consequences and its microscopic origid4 are not well understood. The purpose of this paper is to show that KMAL belongs to an extended version of ordinary nonequilibrium thermodynamics in the sense that it may be derived from rather general phenomenological postulate^.^ Together with this derivation, some very interesting features about how the heat and diffusion fluxes are coupled with the chemical reactions are also obtained. To accomplish our goals we have divided this paper into sections. In section 2 we briefly review the up-to-date background of the KMAL. In section 3 we use the formalism of extended irreversible thermodynamics (EIT) to derive the main equations which are the core of this paper. Section 4 is devoted to extracting from our previous results the explicit form for KMAL and giving a physical discussion of its content. Finally, in section 5 we give some general concluding remarks as well as a brief outlook.
2. Review of Known Results To keep the notation to a minimum degree of bulkiness we shall restrict ourselves to the case in which only a single chemical reaction among an unrestricted number of species takes place in the homogeneous fluid under consideration. With this in mind, we recall that, for a one-step representative reaction
where the Q s label the different reacting species and the vk)s the stoichiomeric coefficients, KMAL states that
where Ci is the molar concentration of the ith species at time t , kf and kb are the so-called forward and backward rate constants dependent only on the temperature T and the pressure p , and C, represents a product of the reaction. For irreversible reactions, Also at El Colegio Nacional. *Consultant, Instituto Nacional de Investigaciones Nucleares, Mtxico.
0022-3654/86/2090-0953$01.50/0
kb = 0 in eq 2.2 and for second-order irreversible reactions by example, dependent only on the concentration of one reactant, say A, v A = 2 and all the rest of the v’s are zero, etc. It is thus in this sense that eq 2.2 embodies all the empirical rate laws of specific one-step reactions and also many others which exhibit a definite order. The latter category occurs when the vi are integers although not equal to the stoichiometric coefficients of their overall chemical equation. For the enormous mass of empirical results embodied by eq 2.2, consistency has been shown between the laws of equilibrium thermodynamics and their steady-state consequences where dC,/dt = 0. This consistency is well-known to be summarized in the equilibrium constant K( T , p ) given by6 (2.3) where Ci(ss) is the molar concentration of species i evaluated at the steady state, including the case of equilibrium. An explicit formula for K( T,p) is also obtained when the rate constants are represented with the empirical equation proposed by Hood in 1878 and later justified by Arrhenius and van’t Hoff, now known as the Arrhenius e q ~ a t i o n . ~ Furthermore, a thermodynamic description for chemical reactions was formulated some decades ago by I. Prigogine and c o - w ~ r k e r s . ~It~is~ based on the idea that for a closed system the entropy change between two states of different compositions, constant in time but not necessarily at equilibrium, can be written aslo,ll
T d S = dE
+ p d V + A d.$
(2.4)
(1) Glasstone, S. “Textbooks of Physical Chemistry”; D. Von Nostrand: New York, 1946; Chapters 12 and 13. (2) Jordan, P. C.“Chemical Kinetics and Transport”;Plenum Press: New York, 1979; p 73. (3) Ross, J.; Mazur, P. J . Chem. Phys. 1961, 35, 19. (4) Berrondo, M.; Robles Dominguez, J. A.; Garcia-Colin, L. S. J . Chem. Phys. 1976,-65, 1927. ( 5 ) Garcia-Colin, L.S.; de la Selva, S. M. T.; Piiia, E. Phys. Lett. A 1985, llOA,363. (6) Callen, H.B. “Thermodynamics”; Wiley: New York, 1960; Chapter 12. (7) Eyring, H.;Eyring, E. M. “Modern Chemical Kinetics”; Reinhold: New York, 1963. (8) Prigogine, I.; Defay, R.; “Thermodynamique Chimique”; Desosr: Libge, Belgium, 1950; English translation by: Everett, D.; Longmans Green, London, 1954. (9) Prigogine, I. “Introduction to Thermodynamics of Irreversible Processes”; Interscience: New York, 1961; Chapters 3 and 5. (IO) Kestin, J. “A Course in Thermodynamics”;McGraw Hill: New York, 1979; Vol. 11, Chapter 21. ( 1 1) Haase, R. “Thermodynamics of Irreversible Processes”; AddisonWesley: Reading, MA, 1969; Chapter 2.
0 1986 American Chemical Society
954
The Journal of Physical Chemistry. Vol. 90, No. 5, 1986
where S is the entropy, E the energy, I/ the volume, C; the degree of advancement of the reaction defined as vi dC; = dC, and 34 the affinity defined as k-l
34
the case. On the one hand we have eq 2.2 and on the other LIT postulates the relation lrt
n
= CI V , F , - XkV l ! 4
Garcia-Colin et al.
(2.5)
where y, is the molar chemical potential of the ith species. This point of view that essentially yields the extra term A dC; in the equation of the entropy, coincides with Th. de Donder's account of the Clausius uncompensated heat term when a system is in mechanical and thermal but not in chemical equilibrium. Equation 2.4 implies an extension of thermodynamics in the sense that one is applying equilibrium thermodynamics to systems not in equilibrium albeit in a frozen composition and therefore the term 34 dC; is different from zero and it is only in equilibrium that 34 = 0 is realized with C; = C;(q)(E,V). However, chemical reactions in their evolution toward the equilibrium state are irreversible processes, and for them, it is desirable to show that eq 2.2 is consistent with some type of nonequilibrium thermodynamics. Up to now, this consistency has only been exhibited for the linear case9,11~i2 in the following way. The core of linear irreversible thermodynamics (LIT) is the local equilibrium hypothesis which states that Gibbs' equation is satisfied in the following form
1rc
J(7,t) = -A - - div ii T T
(2.10)
where u' is the hydrodynamic velocity. The consistency between KMAL and this last equation can be achieved under the restrictive conditions fixed by the experimental situations in which (2.2) is valid, namely, first constant temperature and density and uniform composition throughout space, second linear limit, and third perfect solutions whose chemical potentials are of the form = d T ) + R T In
Pl
c,
(2.1 1)
In fact, substitution of eq 2.1 1 into definition 2.5 and calling the equilibrium constant9 n
-Xv,?7,(T) = R T 1% K(T) yields immediately that A / R T = log K ( T , ) n f - ' q ntq. Further substitution of this expression after the term kbIf iC," is taken as common factor in eq 2.2 allows us to write KMAL as follows (2.12)
(2.6) Here Mi is the molecular mass of the ith species and we are taking specific quantities; thus, s is the specific entropy, C,/ the mass fraction of component i, etc. The time rate of change of the C/s and of e are governed by the following balance equations, respectively (2.7) Here (fviMiJ) represents the local production or consumption respectively of mass of species i per total _mass, while there is the simultaneous Occurrence of diffusion flux lP Here, J the chemical flux is a local quantity and therefore more general than J'of eq 2.2, and
It is only under the above-mentioned restrictions and when 34
