J. Phys. Chem. 1992,96, 9048-9052
9048
(26) Floris, F.; Tomasi, J.; Pascual-Ahuir, J. L. J. Comput. Chem. 1991,
12, 784.
(27) Pierotti, R. A. Chem. Rev. 1976, 76, 717. (28) Alagona, G.; Ghio, C.; Igual, J.; Tomasi, J. J . Mol. Struct. (THEOCHEM) 1990, 204, 253. (29) Sinchez Marcos, E.; Pappalardo, R. R.; Rinaldi, D. J. Phys. Chem. 1991, 95, 8928. (30) Alagona, G.; Ghio, C.; Kollman, P. J . Am. Chem. SOC.1986, 108, 185. (31) Boudon, S.;Wipff, G. J . Compur. Chem. 1991, 12, 42. (32) Dunning, T. H.; Hay, P. J. In Modern Theoretical Chemistry.
Methods of Electronic Structure Theory; Schaeffer, H. F., 111, Ed.; Plenum Press: New York, 1977; Chapter 1. (33) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213. (34) Alagona, G.; Ghio, C.; Igual, J.; Tomasi, J. J . Am. Chem. Soc. 1989, 111, 3417. (35) (a) Jackman, M. L.; Kelly, D. P. J. Chem. SOC.( E ) 1970, 102. (b) Laurie, V. W.; Muenter, J. S.J. Am. Chem. Soc. 1966, 58, 2883. (36) (a) Murray, J. S.; Politzer, P. Chem. Phys. Lett. 1988, 152, 364. (b) Bawagan, A. 0.;Brion, C. E. Chem. Phys. Lori. 1987, 137, 573. (37) Ruiz, M.; Oliva, A.; Fernandez-Alonso, J. I.; Bertran, J. An. Quim. 1982, 78, 202.
Thermodynamics of Open, Nonisothermal Chemical Systems Far from Equilibrium Nobuo Yoshidat Department of Applied Chemistry, Faculty of Science and Engineering, Kinki University, Kowakae. Higashi-Osaka 577, Japan (Received: April 27, 1992; In Final Form: July 22, 1992)
The thermodynamic behavior of kinetic models based on a continuously stirred tank reactor (CSTR) is studied in an attempt to seek general trends in the thermodynamic properties of open nonlinear systems. The models consist of two reversible reactions, A + nB (n + l)B (n = 0, 1, or 2) and B * C, taking place in an adiabatic CSTR. The heat of reaction is incorporated, and the rate constants are assumed to follow an Arrhenius temperature dependence. The models give rise to multiple stationary states and sustained oscillations (limit cycles). The entropy difference between stationary or oscillatory states and equilibrium and the rate of entropy production in these states are calculated as a function of the residence time in the reactor. The entropy difference and entropy production may be taken, to some extent, as indicative of the influence of irreversible processes, which disappears at equilibrium. The results of the calculations reveal the following systematic trends: (i) The entropy difference or entropy production for stable states or both always increase as the residence time is shortened, namely, as the system is displaced further from equilibrium. (ii) If stable and unstable states (stationary or oscillatory) coexist under identical conditions, then the stable state invariably has a smaller value of the entropy difference or entropy production or both than the corresponding unstable state.
Introduction The behavior of open chemical systems has been the subject of extensive research in recent years. When such systems are maintained sufficiently far from thermodynamic equilibrium by flows of matter or energy across their boundaries, they may exhibit a wide variety of dynamical phenomena, including multiple stationary states, sustained oscillations (stable limit cycles), and spatial pattern~.I-~ The thermodynamic formulation of these phenomena is important but is in the process of development. Some theoretical approaches have been proposed for systems far from equilibrium.They mostly contain new thermodynamic quantities introduced by extension of classical thermodynamics for equilibrium systems. Attempts have been made, on the other hand, to characterize the nonequilibrium behavior in terms of conventional thermodynamic functions such as entropy and the rate of entropy production.6J*-20 The previous paper^'^^*^ have dealt specifically with the cubic autocatalator model (a cubic autocatalysis coupled with a firstorder reaction) in an isothermal, continuously stirred tank reactor (CSTR). Calculations were made of the entropy production due to chemical reaction in stationary and oscillatory states and of the entropy difference between these states and corresponding equilibrium states. The entropy difference and entropy production both vanish at equilibrium and may be regarded, to a certain degree, as indicating how intensely irreversible processes function. In the CSTR, the residence time may also be considered a measure of the effects of irreversible processes; one may think that the shorter the residence time, the stronger they become. It has been found that as the residence time is shortened, either the enttropy difference or the entropy production or both always increase. In the model in question, there are ranges of residence times at which stable stationary or oscillatory states and unstable stationary states 'Present address: 4-48, Mikuriyaminami 2-chome, Higashi-Osaka 577, Japan.
exist simultaneously. If small perturbations are given to a system lying at an unstable state, it evolves spontaneously toward a corresponding stable state. If one can assume that the effects of irreversible processes relax during spontaneous evolutions, then one might expect that the transition from an unstable to a stable state is accompanied by a decrease in entropy difference or entropy production or both, in analogy with the finding stated above. Although it is not obvious whether this line of argument is valid or not, the final result has always been shown to be true. There exist similar tendencies for a different type of open system, viz., Poiseuille flows in a tube, where the flow is laminar or turbulent depending on the magnitude of the Reynolds number.2'~22At the present stage of theoretical developments, these systematic trends seem difficult to prove by a priori reasoning; rather they may need to be illustrated by many specific models. The purpose of this article is to investigate the thermodynamics of nonisothermal chemical systems in an adiabatic CSTR, taking the heat of reaction into consideration. Much work has been done on the dynamical behavior of the nonisothermal CSTR,4J3-25but its thermodynamics has received little attenti0n.I' It will be demonstrated below that the present nonisothermal models conform to the same general trends as described above.
Model Systems and Their Kinetics and Thermodynamics The kinetic schemes to be studied comprise the two reversible steps A + nB * (n + l ) B (1) B+C proceeding in an adiabatic CSTR (Le., through whose walls no heat is lost), where n = 0, 1, or 2. When n = 1 or 2, step 1 becomes an autocatalytic process. There is a continuous flow of matter at a volumetric rate v through the reactor of volume V. The reactor and its outflow, both at temperature T,contain the species A, B, and C at concentrations a, b, and c, respectively, while its
0022-3654/92/2096-9048f03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 9049
Thermodynamics of Open, Nonisothermal Systems inflow has the corresponding concentrations uo, bo, and co at temperature To. The reactions are assumed to obey mass-action kinetics and to take place only in the reactor. Mass and heat balance yield v-da = u(a0 - a ) Tq-k,(T)ub" + k-l(T)b"+l] (3) dr db V - = bo - b) +
These new variables, parameters, and abbreviations transform eqs 3-6 into
+
dt
V[kl(nab'' - k-l( T)b"+' - k2( T)b + k-2( T)c] (4) dc V - = U ( C ~- C) + V[k2(T)b - k-2( T)c] (5) dt
dT
VC - = uC(TO- T ) - VAHI[kI(T)ab"- k-I(T)b"+'] dr VPH,[k2(77b- k-2(T)cI (6) where tis time, k l ( T ) ,and k2(T) [k-,(T) and k2(T)] are the rate constants in the forward (reverse) direction of reactions 1 and 2, respectively, C is the heat capacity per unit volume of the fluid in the reactor, and -AHl and -AH2 are the heats of reaction, Le., the enthalpy changes for reactions 1 and 2, respectively. The quantities C, AHl,and AH2 are regarded as constants, independent of concentrations and temperature. The rate constants are assumed to follow the Arrhenius law: kl(T) = AI exp[-E,/(RT)I (7a) k-,(T) = A-l ~XP[-E-~/(RT)I
(7b)
k2(T) = A2 exp[-Ez/(RT)I
(7c)
k-2(T) = A-2 exp[-E-2/(RT)l (7d) where AI, A2,and A-2 are the "pre-exponential factors", E l , E-,, E2,and E-2are the activation energies, and R is the universal gas constant. The rate constants are related to thermodynamic functions by
k2( T )
-= exp( k-203
-%)
= exp(
-AH2
+ TAS2
RT
)
(8b)
where AGI and MI(AG2 and AS2) are the standard free energy and entropy changes for reaction 1 (2), respectively. The quantities AS1 and AS2 are considered constants, as are AH1 and AH2. The following dimensionless variables are now introduced:
The evaluation from eqs 12 to 15 of the time evolution of the variables and their stationary-state values will be facilitated if the number of the variables is reduced. This can be done as follows. Combination of eqs 12-15 yields d(a + B + 7 ) - (1 + Po + Yo) - ( f f + B+ 7 ) (16) dr Trcs d(Ba - Bpy + 8 ) ( B - Bpyo) - ( B a - Bpy 8 ) (17) dT ?res The general solutions of eqs 16 and 17 are, respectively, a B + y = K I exp(-r/r,,) 1 Bo yo (18) Ba - Bpy + 8 = K2 exp(-?/rra) + B - Bpyo (19) with K 1and K2 constants. Equations 16 and 17 (or 18 and 19) show that if the relationships a B + y = 1 60 yo (20) BCY- Bpy 6 = B - Bpyo (21)
+
+
+ + +
+
+
+ +
are chosen as initial conditions in the reactor, Le., if the reactor is initially filled with the same fluid as that forming the inflow stream, then eqs 20 and 21 remain valid at all times afterward and may be utilized to eliminate two of the four variables a,8, y, and 8. In view of eqs 18 and 19, however, even if arbitrary initial conditions other than eqs 20 and 21 are adopted, eqs 20 and 21 still hold in the limit of sufficiently long times, i.e., when the system sits at a stationary state or displays limit-cycle oscillations, a situation with which the present paper is mainly concerned. Stationary-state values of a,8, y, and 8 may be obtained by integrating two of eqs 12-15 numerically with the aid of eqs 20 and 21. A more straightforward way is also possible. It follows from eqs 20 and 21 that = PYSS - Oss/B + 1 - PYO (22) f f ,
(9) The first, next three, and last of eq 9 are, respectively, the dimensionless time, concentrations, and excess temperature. Dimensionless parameter groups are also defined:
+ P ) Y , ~+ Oss/B + Bo + YO + PYO (23) B, = Here and in what ensues the subscript ss refers to a stationary state. Setting d-yldr = 0 in eq 14, substituting eq 23, and solving for yss,one has Yss
=
Bo + 'Yo + PYol 1 + K T r e s e s ~ ( l+ P +32,d
Yo + KTr,%,Y[(8,/B)
(24)
When eq 24 is inserted into eqs 22 and 23, and @, are expressed in terms of 0,. These assand in turn, are substituted into eq 12 in which d a l d r = 0:
os,,
1 - ffss --
The first of eq 10 is a dimensionless form of the mean residence time t,,, which is given by Ires= V/u. Further, three functions of 8 are abbreviated as
Tres
ess(a,Bs," -XdL"+') =0
(25)
Solving eq 25 numerically, one may find the value of e,, which, together with eqs 22-24, produces the values of a,, p,, and y,. The local stability of the stationary states represented by the values of a,, B,, yss,and OSs thus obtained may be examined by a standard linear-stability a n a l y ~ i s . ~ Let us now p r d to a thermodynamic analysis of the present models. For evaluation of the entropy of the mixture inside the reactor is stationary and oscillatory states, an appropriate reference
Yoshida
9050 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
state needs to be specified. Let this state be an equilibrium state corresponding to a stationary or oscillatory state under consideration. This equilibrium state is a stationary state that would be established in the limit as the residence time T,, approaches infinity, with the other parameters maintained constant. It may also be viewed as a state that would be reached if the contents of the reactor were isolated from their surroundings, with their volume held fured. In the limit as T ~ , a,eqs 24 and 25 reduce to @ , / B + Bo + Yo + Pro 7, = (26) 1 + p + 92,,
-
a,
- El,,B,
-- 0
(27)
where the subscript eq stands for equilibrium. The equilibrium values aq, yq, and may be obtained from eqs 22,23,26, and 27, with the subscript ss replaced by eq in the first two equations, in a manner analogous to that described before for the stationary-state values CY,, B,, y,, and Oss. To calculate the entropy difference between a stationary state (as, A, ys, 0,) and the corresponding equilibrium state (aq,Bq, yq, O,), the path between the two states is divided in two as (a,, B,,, Yss?)e, (a,, B,, r s s , ),0 (a,, B q l Y q , 0),
Pq,
-
-
Let S,, S, and S , be the entropies per unit volume in these three states, respectively. When the first path is followed, the entropy change is
S-S,= S"fdT=Cln(:)
From this, the dimensionless entropy production PS is defined as
In an oscillatory state where the system executes limit-cycle oscillations, it follows from eq 20 that a + b + c = a, + be, + c, (37) which is similar in form to eq 33. Hence, indicating a time average over one period of a limit cycle by the angular brackets ( ), one has, in analogy with eq 34, an expression for the dimensionless difference % between the average entropy over a cycle and the corresponding equilibrium entropy:
tlb, - a) + t2(r, - 7)- a, ln aeq - a, In B q y, In y,
+ a In a + @ In
The average entropy production E is, corresponding to eq 36,
(28)
T.
Taking the concentrations a, b, and c as the molar concentrations, one has for the second path
s,-s= (sA,qaq
+ SB,qb, + sC,qcq) - (SAOS + SBbss + sCcs)
(29)
where si,q and si are the partial molar entropies of species i in the states (ae9,B , y e,) and (ass, A, Y,. eq), respectively, with i = A, B, or%. the fluid in the reactor is assumed to be an ideal mixture, the partial molar entropies may be written as si = si*- R In [i] (30)
2
where si*and [i] are the standard partial molar entropy and molar concentration of species i, respectively, Le., [A] = a, [B] = b, and [C] = c. Substitution of eq 30 into eq 29 leads to S, - S = -PSl(a, - a,,) + M 2 ( c q - c,) - R(a, In a, + b, In b, + c, In c, - a,, In a,, - b,, In b,, - c,, In c,,) (31) where PSI and ASz are defined by MI sB* - SA*, = sc* - sB* (32) and are identical with those appearing in eq 8. In the derivation of eq 31, it has been noted from eq 20 that a, b,, + c,, = a, b, + c, (33)
+
+
Adding eqs 28 and 31, dividing the result by Rao, and using the dimensionless quantities introduced earlier, one obtains the desired entropy difference in dimensionless form, to be denoted by DS:
lib,- ass) + Si(r, - Yss) - a, In *q - B q In B, 7, In y, + a,,In ass+ A, In B, + -is In ,Y,, (34) On the other hand, the rate of entropy production u, per unit volume due to reactions 1 and 2 in a stationary state is given by
It is apparent that PS and are always positive. That this is also the case with DS and %,as required, can be shown from eqs 34 and 38 with the help of eqs 20 and 21.
Results and Discussion The models considered give birth to a variety of patterns for the dependence of stationary and oscillatory states on the residence time rrtswhen the values of the ten parameters Bo, yo,K , t, Y, q l , q2, tI,lz, and B are suitably chosen. The entropy difference and entropy production were calculated as a function of T= for various combinations of these parameters. The entropy difference DS between a stationary state and the corresponding equilibrium state and the stationary-stateentropy production PS were obtained first by evaluating the values of a,, B,, y,, and 0, from eqs 22-25 and those of aq,Bq, yq, and ,0 from eqs 22,23,26, and 27 (with the stationary-state quantities replaced by the equilibrium ones in eqs 22 and 23) and then by substituting these values into eqs 34 and 36. The average entropy difference % between an oscillatory state and the corresponding equilibrium state and the average entropy production E for oscillatory behavior were computed from eqs 38 and 39 by averaging the relevant quantities over one oscillatory period while solving eqs 12 and 14 numerically by a fourth-order Runge-Kutta method, with B and 8 in eqs 12 and 14 eliminated by the use of eqs 20 and 21. A large number of systems with different combinations of the parameters were examined to verify the general trends stated in the Introduction for the entropy difference and entropy production. Although only a few representative systems are discussed below because of limitations of space, the conclusions to be drawn from them are quite general for the present models. The simplest chemical model based on the CSTR permitting thermodynamic analysis is one in which only a first-order reversible reaction A F= B occurs. This situation is realized by putting n = 0 and K = 0 in the previous treatment. This model gives either unique stationary states (always stable) over the whole range of T,, or multiple (viz., three) stationary states over some range of depending on the values of the five parameters Bo, t, til, ti, and B.26 Results for a typical system displaying multiple stationary states are shown in Figure 1, where DS and PS are depicted as a function
Thermodynamics of Open, Nonisothermal Systems I. I ,
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 9051 140
4r--720 22 -
j P S L
I\
--30
-4
100 I56 4es Figure 3. Entroy difference DS between stationary (oscillatory) states and equilibrium and the entropy production PS in stationary (oscillatory) states as a function of the residence time T~~ for a system with the two-reaction scheme A 2B 3B, B C. The symbol 0 indicates a point of Hopf bifurcation. Between the two bifurcation points the solid lines represent and for stable oscillatory states and the dashed lines DS and PS for unstable stationary states. The rest of the solid lines stand for DS and PS for stable stationary states. Parameter values: B = 0.5, K = 0.05, e = 0,Y = 1, qI = 2.3, tl = 0, q2 = 2.3, f2 = 2.3, 00 = 0.15,yo = 0.
50
(m)
(s)
+ * E
or both than the corresponding unstable state. References and Notes (1) Epstein, I. R. J . Phys. Chem. 1984, 88, 187. (2) Gray, P.; Scott, S. K. J . Phys. Chem. 1985, 89, 22. (3) Oscillations and Traveling Waves in Chemical Systems; Field, R.J., Burger, M., Eds.; Wiley: New York, 1985. (4) Gray, P.; Scott, S . K. Chemical Oscillations and Instabilities; Clarendon Press: Oxford, U.K., 1990. ( 5 ) Glansdorff, P.; Prigogine, I. Thermodynamic Theory of Structure, Stability, and Fluctuations; Wiley: New York, 1971. (6) Nicolis, G.; Prigogine, 1. Self-Organization in Nonequilibrium Systems; Wiley: New York, 1977. (7) Eu, B. C. Acc. Chem. Res. 1986, 19, 153. ( 8 ) Keizer, J. Sfatistical Thermodynamics of Nonequilibrium Processes; Springer: New York, 1987. (9) Hunt, K. L. C.; Hunt, P. M.; Ross,J. Annu. Rev. Phys. Chem. 1990, 41,409. Ross,J.; Hunt, K.L. C.; Hunt, P. M.J . Chem. Phys. 1992, 96,618. Zheng, Q.;Ross, J.; Hunt, K.L. C.; Hunt, P. M. J . Chem. Phys. 1992,96, 630. (10) Lefever, R.; Nicolis, G. J . Theor. Biol. 1971, 30, 267. Nicolis, G. Adu. Chem. Phys. 1971, 19, 209. (11) McNeil, K. J.; Walls, D. F. J . Stat. Phys. 1974, 10, 439. (1 2) Ishida, K.; Matsumoto, S . J . Theor. Biol. 1975, 52, 343. (13) Richter, P. H.; Rehmus, P.; Ross,J. Prog. Theor. Phys. 1981,66, 385. (14) Termonia, Y . ;Ross,J. Proc. N a d Acad. Sci. U S A . 1981,78,2952. (1 5 ) Mansson, B. A. G. Z . Naturforsch. Teil A 1985, 40, 877. (16) Dutt, A. K. J. Chem. Phys. 1987,86, 3959. (17) Roelofs, M. G. J . Chem. Phys. 1988,88, 5516. (18) Irvin, B. R.;Ross, J. J . Chem. Phys. 1988,89, 1064, 7039.
9052
J . Phys. Chem. 1992, 96, 9052-9058
(19) Yoshida, N. J. Chem. Phys. 1990, 92, 2593. (20) Yoshida, N. J . Phys. Chem. 1991, 95, 6666. (21) Klimontovich, Yu. L.; Engel-Herbert, H. Zh. Tekh. Fiz. 1984, 54, 440. (22) Ebeling, W.; Engel-Herbert, H.; Herzel, H. Ann. Phys. (Leibzig) 1986, 43, 187. Ebeling, W.; Engel-Herbert, H.; Herzel, H. In SelfOrganiration by Nonlinear Irreuersible Processes; Ebeling, W., Ulbricht, H., Eds.; Springer: Berlin, 1986; pp 2-16. Engel-Herbert, H.; Ebeling, W. Physico A 1988, 149, 195.
(23) Gavalas, G. R. Nonlinear Differential Equarions of Chemically Reocting Sysrems; Springer: Berlin, 1968. (24) Uppal, A.; Ray, W. H.; Poore, A. B. Chem. Eng. Sci. 1974,29,967; 1976, 31, 205. (25) Griffiths, J. F. Adu. Chem. Phys. 1986,64, 203. (26) The simple reaction scheme A + B can exhibit limit-cycle oscillations when it is operated in a CSTR allowing heat transport via its walls (see, e.g., ref 24). The thermodynamics of such a nonadiabatic CSTR will be treated in a later paper.
Thermochemistry of Alkyl Free Radicals N.Cohen Space and Environment Technology Center, The Aerospace Corporation, P.O.Box 92957, Los Angeles, California 90009-2957 (Received: March 9, 1992; In Final Form: May 21, 1992)
The group additivity difference method is used to estimate the thermochemical properties (Cpo(T), 298 < T/K < 1O00, of alkyl free radicals from ethyl through the C6radicals. Revised values are calculated for the alkyl radical groups. Recent experimental and theoretical work on hindered rotor barrier heights necessitating these revised calculations is discussed. All Cp's are also expressed in polynomial form, Cpo(r ) = a, + a2T + u3p a 4 P . Polynomial coefficients are given for the calculation of the equilibrium constants for formation of the radicals from the elements: nC + 1/2(2n HZ CnHZn+,. S0298, ArHO298)
+
-
Introduction In recent years, the JANAF Thermochemical Tables' have become the standard source (and have established the standard presentation format) for thermochemical d a t a s o much so that inexperienced users may be stymied when they require data for a compound for which JANAF-type tables are not available. This limitation is due both to the constraints of time and labor required for the preparation of the tables and also to the paucity of experimental data for many species of interest. The latter factor will always (in the foreseeable future) pose a limitation to the compilation of thermochemical data, because the number of compounds of practical interest will continue to exceed greatly the resources required for properties to be measured in the l a b oratory. Two decades ago, when thermochemical or kinetic calculations were confmed to models that could be solved manually with pencil and paper, the unavailability of thermochemical data presented a conceptual but not practical limitation. Now, the ready accessibility of fast supercomputers allows engineers and modelers to construct detailed chemical models for reacting systems that contain upward of several hundred individual chemical reactions and their reverse processes. These large models have contributed greatly to our understanding of the gross nature of complex chemical processes, such as oxidation and combustion or atmospheric pollution chemistry. Since forward and reverse rate coefficients for elementary reactions are always related to the equilibrium constant by kr/kb = K the availability of thermochemical data halves the number rate coefficients that have to be known or measured. Consequently, thiscore of practitioners has become a primary group of users of thermochemical data-nd the limitations on the body of available data have an immediate impact on their efforts. One solution to this dilemma has been to develop techniques for the reliable (but rapid and inexpensive) estimation of thermochemical data when measurements are not available. We have recently reviewed the extensive literature in the relatively narrow field of alkane and cycloalkane thermochemistry.2 Of closely related interest is the family of alkyl free radicals. Over 2 decades ago, h m e l l and Quinn3 provided the first detailed calculations of the entropies for the four smallest alkyl
3
TABLE I: RZeume of Asrrumcd Internal Barrier Heights barrier in
radical/rotor
radical"
ethyl: 'CH2CHj
propyl: 'CH,C,H, isopropyl: CH,'CHCH, sec-butyl: CH3-'CHCH2CH3 sec-butyl: CH3'CHCH2CH, sec-butyl: CH,'CHCH2-CH3 isobutyl: CH2'CH(CHJ2 ten-butyl: CH,'C(CH,), neopentyl: CH;C(CH,), large radical: 'CH2R large radical: CH,'CHR large radical: C2HS'CHR " Barrier heights given in kcal/mol.
-
-
