Non-equilibrium thermodynamics of closed-system reactions

Matertals Chemistry 7 (1982) 3 5 9 - 394

NON-EQUILIBRIUM THERMODYNAMICS OF CLOSED-SYSTEM REACTIONS

M GARFINKLE D e p a r t m e n t o f Materials E n g z n e e r m g - D r e x e l U m v e r s t t y - P H I L A D E L P H I A 19104-

- PA

USA

Received 11 December 1981, revised 5 March 1982 Abstract - Stolchiometrlc chemical reactions m a closed isothermal system are studied m terms of classtcal and statistical thermodynamics. It is demonstrated that the affinity decay rate m such a system Is independent of reaction mechamsm On this basts a general thermodynamic description of the reactions is provtded, valid for reacUons with diverse mechanisms In contrast to earher approaches dealing only with systems close to equfllbrmm, the present formalism is apphcable to systems arbitrarily far from equfllbrmm The agreement between computaUon and expertment is distinctly better than m calculations based on absolute rate theory

INTRODUCTION The thermodynamic behavlour o f chemical reactions proceeding tn file closed system has been largely neglected by investigators In contrast, b o t h the eqmh b r m m and the stationary states have been extensively examined This is understandable, because all o f the tools o f classical thermodynamics are avazlable m examining the equihbrmm state, and the stationary state can be described by linear phenomenological equations Moreover, both the e q u ~ b r m m and the stationary states are ttme-mvarlant, so that the thermodynarmc functions have fixed values Tlus is not so m the closed system, a comphcation that has resisted resolution. 0390-6035/82/030359-3552 00/0 Copyr,ght © 1982 by CENFOR S R L All nghts of reproducuon m any form reserved

360 Just how neglected has been the closed system can be gauged by the cursory examination afforded chemical reactions by both de Groot i and PrBogme 2 m thetr treatises on the thermodynamics of non-equdlbrium processes Both agree that the sole region of vahd~ty of irreversible thermodynamics, as far as chemical reactions are concerned, ~s inmted to the mu'nedmte wcmlty of equdlbrmm. That is, there exists no body of theory that deals with the thermodynamics of chermcal reactions proceeding m the closed system Consequently, whale the mmal state of a chemical reaction (before products appear) can be described explicitly m terms of thermodynamic functions, as can the terminal state (equdlbnum), no analytical descnpnon Is avadable that can represent the trine-dependency of the thermodynamic functions as the reaction proceeds from its m~tial to its terminal state Thus, it is not even possMe at present to compute as fundamental a quantity as the rate of entropy production for chem. lcal reactions m the closed system In large measure, this lack of a thermodynamic approach to the study of chemical reacnon kmencs arises because only the classical m~croscop~c approach is generally recogmzed, according to which a chemical reacnon is viewed as a system of reacting molecules m which kinetic behavior is described in terms of the trine-dependency of the concentranon of reacting components The great unpetus to this approach was the development of absolute rate theory by Eyrmg 3 The principal feature of this theory is the transition state the energy state that the reactants must attain as a condition for their transformarion into products Th~s requirement constitutes a classical zero-permeabdlty potential-energy barrier In the translnon state the reactants assume a singular and sp.eclfic molecular configuration (activated complex) that is m equfllbrmm with the reactants m the mlnal state Wtule generally successful m interpreting reaction mechamsms, tranm~on state theory predicts reaction rates slower than observed at low temperatures Tlus discrepancy has been attributed to bamer tunneling, which presently appears as a correction factor in the classical rate theory 4 Because tunneling ~s a quantum-mechanical concept, several formulations of a quantum-mechamcal translnon-state rate-theory have appeared s, in which tunneling' is an integral element The notable feature of this hybrid rate theory Is that the energy states of the complex are fully quannzed However, because of the very short hfettme of the complex 6, there exasts some doubt whether the required equ,hbrmm energy-partition is ever attained, and consequently whether full quannzat~on occurs

361 Chnstov 7 deftly avoids tlus difficulty m his quantum-mechamcal model by dispensing entirely with the activated complex as. a singular and specific molecu. lar configuration. The partition functlons m Ins rate equation refer solely to the reactants, and h~s rate equation includes a factor that accounts for all quantum effects, including barrier tunneling Whale ChnstovS has referred to the activated complex as an actual rather than a "'virtual" state when the reaction coordinate is dynamically completely separable from the other nuclear coordinates, his theory treats the actwated complex as a fictlt~ous concept9 Because the translt~on state is so widely accepted as an objective reality, Chnstov's quantum approach ~s being strongly resisted by traditional kmetlclsts 1° Another difficulty that has arisen m the formulation of a rxgorous mecharustic description o f reaction kinetics concerns the determination o f nucleoslde conformations A knowledge o f conformations can be essential m choosing between alternate and doubtful mechanisms m describing specific chemical reactions The Karplus-Vorontsova-Jankowskl controversy concerning the role o f an emptrlcal "electronegatwlty" term m calculating victual coupling constants has introduced uncertainty m the rehabfllty o f such deterrmnatlons Karplus 11 hamself had suggested such a term to improve data coirelatlon, and Vorontsova e a 12 have proposed such a modification o f the Karplus equation Now Jankowskl and Rabczenko l 3 find that such modifications are unnecessary Apparently, once a good set o f empirical coupling constants are obtained, an electronegatiwty term Is superfluous Chmsson and Jankowskl 14 have found that conformation variations can be explained by population changes alone Because these mechanistic complexities are intrinsic to any m~croscop~c approach to chemical reaction kinetics, what is now proposed is an mdependent approach that is essentlaUy macroscopw m that it dispenses entirely with molecular considerations A chemical reaction is viewed simply as an isothermal smk or source o f energy m which kinetic behavior is described m terms of the tune-dependency o f a thermodynamic function Unfortunately, such a tune-dependency is not generally accepted by classical thermodynamlclsts Brostow'sl s admomshment that thermodynaunclsts are mistaken in assuming that the scope and contents o f thermodynauncs are tune-rodependent has fallen on deaf ears Even non-equdlbnum thermodynamlc~sts tgnore the closed system and lumt thetr exaa-nmat~ons to systems that are tune-mvanant Because there ~s no theoretical foundation on which to base a thermodynamic description of closed-system reactions, thas study was undertaken to determine whether it was possible to base such a description on a heuristic approach mvolv-

362 m g a numerical analysis of empirical data The tune-dependency of the chemical affinity, a thermodynamic function introduced by De Donder 16, was chosen for exammahon The reacUons examined were restricted to those proceeding under conditions of constant volume (no work is done on or by the system) and constant temperature (reaction rates suffictently slow that macroscopic thermal equlhbrmm is maintained). Under these condRxons, the chemical afFunty can be defined as a function of state The value of this function at measured tune intervals can be readily calculated from chemical kinetic data

THERMODYNAMIC CONSIDERATIONS Consider the homogeneous stoichiometnc chemical reaction ]~ viM l = 0

(1)

1

:'roceedmg m a closed isothermal system where vl is the stoichiometnc coefficient and M1 is the molecular mass of component ~ The stoichiometnc coefficients have negative values when the components are reactants The driving force for reaction ( 1 ) is the chemical potential difference between the products and the reactants, and the reaction proceeds m the d~rection that mmun~zes this difference The reaction path can be viewed as a chemical potential gradient whose slope is determined sole ly by the potential difference at successive tune intervals. Pngogme e a. 17 defined the chemical affinity A as a generahzed force m terms of this potential difference as A = - Y~~1 ~1

(2)

1

where/a I is the chemical potential of component 1 According to this definition, at equthbrmm A = 0, and m any state m which the reaction proceeds spontaneously A > 0, To deterunne the specific value of A m the non-equdlbrmm states, consider the reaction described by equation (1) Classically, the chemical potentml #1 of component i m any arhitrary state can be rdated to its chermcal potential/z ° m some standard state by an equation of the form O

VI~.£ l : V1~d I ~"

RT ln(al) vl

(3)

363 where R is the gas constant and a~ is the activity o f component I m the chosen arbitrary state Equation (3) can be expressed m terms o f affinities by substituting from equation (2) to yield A = A ° - RT Z l n ( a l ) vl

(4)

where A ° is the affinity o f the reacting system when the components are m their standard state and it IS a function o f temperature only If an activity ratio Q is deFreed as

Q = II (al) vl

(5)

1

then equation (4) can be rewritten as A = A ° - RT In(Q)

(6)

At reaction initiation Q = 0, so that A = ~ , whale at equdlbrmm A = 0, so that A ° = RT In(K), where K is the thermodynamic equlhbrmm constant Substituting for A ° in equation (6) yields

a = - RT ln(¢ o )

(7)

where ~Q = (Q/K) By definition, the value o f ~Q IS limited to the range 0 < ~Q < O

1, and its standard state value ~Q is equal to 1/K Hence, ~'Q IS a dimensionless measure o f the extent o f reaction from A = ~' to A = 0. Accordingly, for any chemical reaction proceeding spontaneously m a closed system o f fixed volume V and at constant temperature T, the afftmty decays towards zero, so that

Ar,v < 0

(8)

where A is the affinity decay rate a A / a t Expression (8) ostensibly represents the furthest extent to which thermodynamic considerations alone can be earned m describing chemical reactions m a closed system Beyond this point a kinetic description is requtred, and because it is dependent on reaction mechamsm, it ~s necessary to stipulate specific classes o f chemical reactions Because kinetic behavior is de-

364 scribed m terms of the reaction velocity v, whach ~s dependent on reaction mochamsm, It ~s logical to assume that the affinity decay rate A is hkew~se so dependent

AFFINITY DECAY RATE To determine whether this assumption of mechamstlc dependence is vahd, consider the hypothetical stolcluometnc reacUon AY + B2

' A + B2Y

(9)

which can proceed by several different reaction mechamsms, two of which are illustrated m Table 1 Each possible mechamsm revolves several mtermedmte steps, and associated with each step ~s an intermediate reaction velocity vj and an mtermedmte affunty Aj, each with a specific value at any specified elapsed tune between reaction unttat~on and equdibnum Table 1 - Hypothetical Reaction Steps Postulated Reaction Mechamsm 1 Step

Process

vj

Aj

B2 + X -~ 2 B + X

0

0

A Y + B -~ A + BY

V1

AI

B Y + B --~ B2Y

V2

A2

Postulated Reaction Mechamsm 2 Step

Process

AY+B2 ~AYB+B AYB --> A + BY B Y + B --> B2Y

v3

Aj

VI

A2

V2 V3

c

A2 A3

365 The first mechanism revolves a dtssocmhon that attains equ~brlum at reaction m~tlation, where X ~s any molecular species m the system The second mechamsm revolves the decomposition of the intermediate molecule AYB It is not possible to determine absolutely which of several postulated mechamsms is correct, but by conducting critical experunents the kraetlclst can elunmate many of the alternatives I 8 Consider the reaction velocity for this hypothetical reaction Whichever mechamsm is chosen, the overall reaction velocity v wall be a function of the mtermedmte reactJon velocities vj This choice ultimately determines the form of the reactjon rate equauon selected by the klnetlclSt to represent the reaction m question Consequently, the rate equation can only be represented m the most general form V----'V(V1, V2,V 3

Vn)

(10)

w~thout knowledge of reaction mechanlsrn Consider now the affinity decay rate for this same hypothetical reaction From the definition of chemical affunty (2), the affunty Aj for any mtermedmte step j is equal to the difference m eheuncal potential between the products and the reactants for that particular step Because of the stolchiometry criterion, on summing the intermediate reaction steps for each of the mechanisms m Table 1, all of the tranmory chermcal species cancel out Consequently, on sumrnmg the intermediate affinities Aj, all of the chemical potentials of the transitory species must also cancel out, resulting m the overall chemical affinity for the stolch~ometnc reaction, regardless of which reaction mechanism ~s chosen That is, the value of the overall chemical aff'mlty at any elapsed tune between reaction m~tmt~on and equthbnum must be equal to the sum of the intermediate chemical affimtles, rodependent of reaction mechanism. This observation is m accordance with thermodynaunc considerations because by defmit~on the chemical affinity is a function of state, and therefore could not be dependent on reaction mechanism Because the aff'unty is independent of mechanistic considerations, so must be the time.differential of the affinity the affinity decay rate A Consequently, for any stoichlometnc reaction m a closed system AT, v = • A j J

(Ii)

where Aj is the value of the afffmlty decay rate for mtermedmte step j at any specified elapsed tune Expresslon (l i) extends (8) one step further, and suggests that

366 a general thermodynamic description of closed-system reactions ~s posstble because there is no mechanistic contribution to the affinity decay rate

APPROACH TO EQUILIBRIUM Having determined the value of the chemical aff'mlty m the initial and terminal states of a closed-system reaction, and that the affinity decays as the reaction proceeds from the former state to the latter independent of reaction mechanism, only knowledge concerning the elapsed time required for the affinity to decay is necessary for th~s thermodynamic description to be complete Unfortunately, thermodynamic considerations alone reveal nothing of the time required for a reacting system displaced from equdlbrlum to attain equlhbnum Equthbrium can be defined as a state whose macroscopic properties are time.mvarlant, but this defimtxon doesn't preclude microscopic fluctuations about the mean value of these properties Boltzmann 19 argued that the energy probability distribution in an isolated system is itself an almost periodic function of time that effectively never repeats itself. That is, the probablhty distribution itself fluctuates with a period that approaches infinity. Hence, some d~splacement must always be present between a system approaching equihbrmm and the state defined as equthbrmm by vn-tue of its canomcal distribution In contrast, Pngogme e a 2o argued that as long as the system has a Finite volume encompassing a f'mite number of particles, ~ts energy distribution must m time attain the canomcal form From either of these arguments ~t can be concluded that the displaced system can attain macroscopic equthbnum within any desired degree of precls~on without having to attain m~croscop~c equthbrmm. That is, macroscopic time.mvariance is possible m some Finite length of time depending on how that elapsed time ~s represented Because this study pertains to the time-dependence of a thermodynamic function, it is only macroscopic time-mvanance that concerns us. Evidently, the crucml consideration m ttus analysts of the approach to equihbrlum is that it is concerned with the probabdlty of an event occurring m accordance with some statistical distribution function Under these circumstances, the 100 percent certainty of any specified event occurring must require an mfintte length of time, whether the event is the time to attain equdibrlum or to attain any other state. Fortunately, this absolute tmae-span can be dispensed with, as it has no empirical significance Instead, the most-probable time can be constdered, as it must have a fimte value according to any distribution function Accordmgly~ when

367 Bmerlem 21 asserts that the nuclear spins in LiF attain equdlbrlum m a tune on the order o f 10 "s seconds, or that it requtres as long as five minutes for spin and vibrational equd]brmm to be attained, these values can only be interpreted at most-probable times There exists an exceedingly small but finite probabthty that equd]brmm might instantaneously be attained, or not at all That is, insufficient mformat~on Is avatlable to specify the time any closer 2 2

REACTION COORDINATES In examining a chemical reaction, we are likewise hmlted to speclfymg only a most-probable ttme for some event to occur Of course the most tmportant event is the elapsed tune required for the macroscopic properties o f the system to become t u n e - m v a n a n t chermcal equthbrmm Fig 1 dlustrates a schematic representahon o f a chemmal reaction proceeding from its m~tlal to ~ts terminal state

I

00 I

>Itl la. ¢Z

..J (.) tlJ ar t.) A o O -

0

)1

)K

ELAPSED TIME, t Ftg 1 - S c h e m a t w representatzon o f the reactzon p a t h f o l l o w e d b y a h o m o g e n e o u s s t o z c h t o m e t r l c c h e m t e a l reactton tn a closed zsothermal s y s t e m

368 Though physically an mfunte number of possible reaction paths exist depending on reaction mechamsm, thermodynamically only a single path is followed because the affunty decay rate is independent of reachon mechanism Consequently, a most-probable tune t K requtred for a chemical reaction to attain xts equdlbrlum state (Q = K) can be defined Accordingly, the progress of a reaction along this smgular path can be specified exphcltly by the reaction coordinates (A,t) Thus, the initial and terminal states of the system have the specific coordinates ( ~ , 0) and (0, tK) respectxvely, and as the reaction proceeds from the former state to the latter, ~ts coordinates wall change from their mmal value to thetr terminal In addition to the untlal and terminal states of the system, this scheme also permits any thermodynauncally-defmed intermediate state to be represented Accordingly, a most-probable tune t I required for a cheuncal reacUon to attain its standard state (Q = 1) can be defined Consequently, the reaction coordinates of the standard state are (A °, tl), as shown m Fig 1 However, m regards to any of the states described, because of the uncertainties inherent to the reacting system, the more closely an affinity coordinate is specified, the greater is the uncertainty in the elapsed tune. Ltkewlse, at any closely-specified tune-coordinate, there ~s an uncertainty tn the value of the affinity These uncertainties m the energy and time coordinates do not arise from experimental error, but are tntrmslc to any system described by statistical probabtlitles By referring to classical and statistical thermodynaunc considerations alone, two general propositions can be made concerning the tune-dependence of a thermodynamic function the afflmty decay rate m a closed system IS independent of reaction mechanism, the most-probable time required for the chemical affimty to attain any specified state has a fimte value. However, thennodynaunc eons~deratlons alone reveal nothing concerning the specific analytical expremon that describes the affinity decay rate as a chemical reaction proceeds from Its mltml to its terminal state To fred such an expression requtres a numerical analysis of kinetic data

369 EMPIRICAL ANALYSIS To ascertaan the form of the reqmred analytical expression necessitates that actw~ty ratxos be computed from experunental kinetic data at the observed tune intervals, and that A then be calculated from the ratios Because Q Is defined an terms of the sto~chlometry of a particular reaction, the reactions evaluated were restricted to those that can be represented by a sangle stolchlometrlc equation The actual reaction may proceed m multiple steps and revolve transitory chermcal species as an the case of the hypothetical reaction, but all the products must appear and all the reactants disappear according to the stolcluometry of the overall reaction Though actlvales are experimentally measurable, virtually all kinetic data are reported an terms of partml pressures or concentrations Consequently, the reactions evaluated must not only meet the stolchlometry requirement, but must occur at low pressures or m ddute solutions so that partaal pressures or concentra. t~ons of reacting components approxunate their actwmes In accordance w~th accepted kinetic practice, all expertmental data are presented as e~ther molant~es or atmospheres In addition, thermodynamic equdlbrlum data must be avadable for the reactions examined To unplement ttus study, approxunately one-hundred chemical reactions were examined These were restricted to homogeneous stolchlometnc reactions m a closed isothermal system The kinetic data were gathered from tabulations that had been pubhshed over the last half-century, or were sohclted from investigators whose results appeared m the recent literature Thermodynamic equilibrium data were found an pubhshed tabulations of standard free energies or calculated from appropriate half-cell potentmls All of these data were reduced to a common format cheuncal affinity versus elapsed tune To formulate an analytical expression that properly describes the affinity decay rate as a function of elapsed tune requtres that the tune-derivative of the calculated affmlt~es be determined at each measured elapsed-tune anterval Although methods of numerical dffferentmtaon that are well stated to electromc data-processing are avadable, they are admittedly error-enhancing procedures 23 Consequently, the chord-area method of dffferentmtlon was employed 24 This procedure was not used to generate analytical reformation but only to survey a large number of tune functions The calculated values of .~ were correlated with various functions of elapsed tune by a regression analysis to determine the best data fit However, as soon as

370 the reciprocal-trine function was considered, it was apparent that the affinity decay rate was inversely proportional to the elapsed ttme t AT,V ~ - 1/t

(12)

Using this reciprocal-ttme function, four representative reactmns wall be disTable 2 - Untmoleculax Reaction Rate Data Isomenzatlon of Ethyhdenecyclopropane to 2- Methylm ethylenecyclopropane T = 507 K [Eth]o = 0 329 atm A ° = - 992 J/tool

371 815 1393 3495

[Eth]' atm

A J/tool

0 297 0 269 0 244 0.197

8340 5340 3460 715

I s o m e n z a h o n of 2-Methylrnethylenecyclopropane to Ethyhdenecyclopropane T= 507K [Meth]o = 0 329 atm A ° = 992 J/mol [Meth] atm

371 815 1393 3495

0 0 0 0

291 256 220 166

A J/tool 9600 6280 3980 1060

371 cussed The reaction mechamsms postulated and the notation employed are those of the mvestzgators, and wall not be critiqued* Because one standard atmosphere is the reference state used m the thermodynamic computations, atmosphere units are retained, where one a r m = 0 1013 MPa. Table 2 lists the kinetic data for the forward and reverse isomerazatlon of ethyhdenecyclopropane to 2-methylmethylenecyclopropane2s The postulated reaction steps for this unnnolecular reaction

are hsted m Table 3 It IS apparent that the overall reaction velocity equals v I Table 3 - Untmolecular Reaction Steps Isomenzat~on of Ethyhdenecyclopropane to 2-Methylmethylenecyclopropane Step

Process

vj

Aj

0

o

vl

A1

the reaction velocity of step 1, where D* denotes an actwated complex as specified by the mvestgator, but whose configuration is notpreclsely k n o w n From these mechamstic considerations, the process is assumed to follow the unmaolecular reaction rate equation v = - k [Eth] + k-[Meth]

(14)

where ~ and ~- are the forward and reverse rate constants and Eth and Meth de* Many of the mechamsms cRed revolve concepts and configurations that are no longer generally accepted, and alternatives and revisions have been proposed However, the author does not presume that thB paper is a proper vehicle for a comprehensive analysis of reactaon mechanisms

372 note the reactant and product respecUvely. F~g 2 illustrates the reciprocal-trine dependence of the aff'mtty decay rate for this unnnolecular reaction.

0 -2 -4 J.

oE

p

6

.,~>.- 8 -I0 I

-12 -14

I

0

I

1

I

I

05 I 0 15 2 0 2 5 3xlO -3 RECIPROCAL TIME, s -1

F~g. 2 - I s o m e r l z a t z o n o f E t h y l i d e n e c y c l o p r o p a n e to 2 - M e t h y l m e t h y l e n e c y c l o propane at 5 0 7 K. F o r w a r d reactzon 0, reverse reaction •

Table 4 hsts the kinetic data for the reduction of Pu(IV) by Fe(II) m perchlonc acid soluUon 26 The most probable reaction steps for this blmolecular reaction Pu(IV) + Fe(II)

) Pu(III) + F e ( I I I )

(15)

are hsted m Table 5. Because the overall reacUon rate has been found to be reversely proportional to hydrogen 1on concentration and directly proportional to chlo ride ion concentration, the investigators have assumed that the reaction probably revolves the activated complex (Pu • X Fe'S)~, where X denotes either the hydroxyl or chloride ion. Because variations m product concentration had an ms~gmficant effect on reaction rates, the reactson ts assumed to follow the b~molecu. lar reaction rate equation

373 Table 4 - Blmolecular Reaction Rate Data Reduction of Pu(IV) by F e ( I I ) m 0 5M Perchlonc Acid Solution T=289K [Pu(IV)]o = 0 00115M [Fe(II)]o =000117M A ° = 18,600 J/tool t s

[Pu(IV)]o M

A J]mol

20 25 30 35 40 50 60 70 80 90 100 105 110 120 130

6 90 x 10 -4 6 25 5 66 519 480 416 369 328 2 96 2 74 251 2.40 233 214 201

20,700 19,600 18,600 17,800 17,200 16,000 15,200 14,400 13,700 13,200 12,700 12,500 12,300 11,800 11,400

Table 5 - Blmolecular Reachon Steps Reduction o f P u ( I V ) by F e ( I I ) m 0 5M Perchlonc Acid Solution

Step

vj

Aj

Fe*S) *

0

0

X Fe+S) * --~-Pu(III) + F e ( I I I ) + X"

VI

Al

Process Pu(IV)+Fe(II)+X'~-(Pu (Pu

X

374 v = - k [ P u ( I V ) ] [Fe(II) ]

(16)

where k is the forward reactmn rate constant Fig 3 dlustrates the reciprocal-trine dependence of the affinity decay rate for ttus btmolecular reaction O

-

50-

ql

o - I00' --j

,

i~_.- 150' ,<~ - 200

I

-250

001

1

002

RECIPROCAL

003

I 004

TIME,

005

s- I

Fzg 3 - Reductzon o f Pu(lK} by Fe(IIJ m 0 5 M Perchlortc Aczd Solutton at 289 K.

The kdnetlc data for the formation of nltrosyl chloride 27 are hsted m Table 6 Wlule this trtmolecular reaenon 2NO + C12

) 2NOCI

(17)

ostensibly depends on ternary colhs~ons, which are esttmated to be about 100 times as rare as binary colhsxons 2 s, this process can also be explained by the two-stage bimolecular process 29 which is illustrated m Table 7 Nevertheless, this protess ~s known to follow the trtmolecular rate equation v = - ~ ' [ N O I 2 [Cla] + k-tNOCl] 2

(18)

Fig 4 dlustrates the rectprocal-ttme depe,,dence of the affinity decay rate

375 Table 6 - Trunolecular Reaction Rate Data. Formation of Nxtrosyl Chloride T=333K [NO]o = 0 168 arm [C12]o = 0 0791 arm A ° = 36,600 J/mol

60 105 165 230 325 380 435 540 600 690 795 945 1050 1230 1360 1515 1650 1915 2165

[NO] atm

A J/tool

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

46,900 44,400 42,300 40,600 38,100 37,500 36,700 35,100 34,600 34,000 33,100 32,300 31,200 30,600 30,500 29,800 29,100 28,400 27,700

161 158 153 149 141 139 135 128 125 122 118 113 106 103 102 0983 0938 0899 0859

for tins tnmolecular reaction Table 8 hsts the kmetw data for the dehydrogenation o f ~sobutane b y iodine vapor 3° Because the reaction lsobutane + I~

> lsobutene + 2HI

(19)

revolves a free-radical cham-mechamsm, the postulated reaction steps listed m Ta-

376 Table 7 - Tnmolecular Reachon Steps Formation of Nltrosyl Chloride Step

Process NO+CI2

~

NOCI 2 + NO

vj

Aj

NOC12

V1

A!

> 2NOC1

V2

A2

t%

-50 -6~

I

0

I 0004 RECIPROCAl..

I

I 0008 TIME, $-I

Fzg. 4 - F o r m a t z o n o f N z t r o s y l Chlorzde at 3 3 3 K

ble 9 are quite revolved Consequently, the overall reaction velocity is a complex function of the mtermedmte reaction velocaies, and the reaetmn rate equation takes the form v = - kl

K* 1/2

[I2] 1/2 [ x - B u l l ] (1 - ~ ' q ) ~ *k, [HI] kL[I ] 1 + -, [l+ ] k2 [ I2 ] ka

(20)

377

Table 8 - Chain Reaction Rate Data Dehydrogenation of Isobutane by Iodine T=525K (1 - Bull) o = 0 264 atm (I2)o = 0 0127 atm A ° = - 26,800 J / m o l t s

(I2) atm

A J/ m ol

1200 1800 2400 3000 3600 4800 6000 7200 9000 10800 14400 18000 21600 25200

0 0119 0 0114 0 0111 0 0107 0 0103 0 00963 0 00905 0 00855 0 00790 0 00728 0 00632 0 00562 0 00516 0,00484

34,500 29,000 25,600 22,700 20,300 16,700 14,100 12,200 9,960 7,990 5,230 3,360 2,150 1,340

Table 9 - Chain Reaction Steps Dehydrogenation of Isobutane by Iodine Vapor Step

Process 12 + M

~

2I+M

vj

Aj

0

0

1-Bull + I ~ t-Bu + HI

VI

A1

t-Bu+I 2 ~ t-BuI+I

V2

A2

t-BuI ~ t-butene + HI

V3

A3

378 where K* is the equdlbnum constant for Step 0 and ~'Q is the extent of reaction The rate-constant subscripts denote the mtermedlate reachon step w~th which each constant ~s assocmted Fig 5 glustrates the reciprocal-tune dependence of the at'fauty decay rate for ttus chain reachon 0

~

i

_

I

l

-2

m

-4 0

l= -6 >. p•4 - 8

-lO

-12

0

02

04

RECIPROCAL

06 TIME,

08

1 IO"a s -I

Fzg 5 - Dehydrogenatmn o f Isoautane by Iodine vapor at 525 K.

The four reactions discussed represent diverse mechanisms, yet the affmlty decay rate was mversely proportional to the elapsed tune for all of them Tlus observahon suggests that Proposition 1 is essentmlly valid, and consequently, that expression (12) can describe the affnuty decay rate for a homogeneous stolcluometnc reaction m a closed isothermal system independent of reachon mechanism Expression (12) constitutes a thermodynanuc descnptlon of the reactions discussed, as Is suggested by the examples shown. However, as these examples of the tune-dependence of the affinity decay rate also show, the lines of regression do not pass through the origin, and do not do so for any of ~ e reactions examined for which experimental data were available near equdlbrmm Wtule tlns observation can be readily dlsnussed as no more than an artifact of the dtfferentmtlon

379 method 3 i, it Is m direct agreement wtth Proposition 2 that the most-probable tune to attain equilibrium ~s ftrute Consequently, an intercept term must be represented m expression (12) to obtain agreement with both thermodynarmc considerations and the empirical data AT,V , x - 1/t + I

(21)

The intercept term I denotes the elapsed tune at which the lines of regression intercept the abscissa, and therefore can be interpreted m terms of the most-probable tune required for the reactions discussed to attain equd~brmm From the equdlbrlum condmons A = 0 at t = tK, the intercept term of expression (21) can be evaluated to yield AT,V = A r [ 1 / t - 1/tK]

(22)

where Ar is the proportionality constant This affinay rate equation is the basis for the following analysis of experunental kinetic data to determine the extent of validity of an analytical expression m describing the tune-dependence of a thermodynamic function

DATA C O R R E L A ~ O N Th~s comprehensive examination of homogeneous closed-system reactions meluded those with mechanisms sufficiently diverse so as to preclude any posslbdlty that the observed correlations might be anomohes The reactions examined included thermal decomposmon, electron and proton transfer, lsomerlzation, and dtrect combination m the gas phase and m aqueous and orgame solvents In order to directly correlate the calculated values of the chemical affunty wah the measured values of the elapsed tune, equation (22) must be integrated, which yields A = Ar ln[~'t exp(1 -~'t)]

(23)

where ~'t = (t/tK) Sunflarly to ~'Q, the extent of reaction ~'t is lunlted to the range 0 ~< ~'t ~< 1 With ~'t so defined, all of the thermodynaunc and kinetic parameters relevant to this study have been introduced, and are summarized m Table 10

T a b l e l 0 - T h e r m o d y n a m i c states d e f i n i n g t h e r e a c t i o n p a t h m a closed s y s t e m

Elapsed Time

Activity Ratio

Extent of Reaction

Extent of Reaction

Chemical Affimty

Affinity Rate

t

Q

rt

~'Q.

A

A

lnltml

0

0

0

0

+=

-=

Standard

tt

1

~'t

~'~

A°

Equlhbrmm

tK

K

1

1

0

System State

400

0

I

I

Reaction Velocity

VO

0

I

:390 \ I ~ 580 - - ~ ~ _ ~

AFFINI" Y RATE E " JATION A° 304,000 J/tool A r - 19,500 J/tool

-

570

360 sso 340 330 520

31C -20

I -,9

I -,s In [~t'exp(I-~t)

I -,7

-,6

]

Fig. 6 - Decomposttzon o f Hydrogen Peroxide containing 66.7 tool percent Hehum at 705 K [ H 2 0 2 ] o = 9 6 torr

381 To correlate the experimental data requtres that the value of t K for each reaction examined be generated by an lteratwe subroutine Th~s revolves a multiple regression analysis that is satisfied by the requuement of equation (23) that the intercept vamshes Tlus procedure ~s outlined in Appendix I To tllustrate the degree of data correlation possible with equation (23), the ttme-dependence of the chemical affinity for four more reactions with dwerse mechamsms will be discussed Fig 6 illustrates this tune-dependency for the thermal decomposition of hydrogen peroxide vapor 32 2H202

> 2H20 + 02

(24)

Because the decomposition rate depends on reaction-vessel surface area, this ummolecular reaction is normally heterogeneous However, when an inert gas zs

I00,

I i

99

~'~ \ k

I

AFFINITY RATEEQUATION A° I00,000 J/tool

Ar

98 _

- 3,4600J/mo/ )4 s_

97 E

96

.1¢

95

94 93 92 -29

I

-28

I

-27'

-26

Fzg 7 - Oxldatzon o f F e ( I I ) by Co (IIIJ m 0 5 M Perchlorw Acid Solut,on at 298 K [ F e ( I I ) ] o = 1 46 × l O ' 4 M , [Co(III)]o = 1.46 x l O ' 4 M

382

present, as preferenual adsorption on reactor surfaces reduces the number of saes avadable for reaction Tlus effectwely inhibits the heterogeneous process, the conditions under wtuch the decomposatlon was examined m this study The oxidation of Fe(lI) by Co(III) m perchlorlc acid solution Fe(II) + Co(III)

(2s)

, Fe(III) + Co(II)

probably revolves an activated intermediate molecule m wluch Is included the hydroxyl 1on, according to the mveshgators 3a . The trine-dependency o f the affLmty for this bunolecular electron-transfer reaction lS shown m Fig 7 The rate o f affinity decay for the reduchon of m t n c oxide by hydrogen 34 Is shown m Fig 8 33(

I

I

I

I

AFFI N ITY RATE EQUATION , A° 2 8 5 , O O O J Imol

310

_

300

"~

290

"~

Ar

-17,3OOJlmol

tK

2 56x101

s

_

2~

27C -

260 -19

~

I -18

I -17

I -16

-

-15

-14

Fzg. 8 - R e d u c t l o n o f Nztrzc O x i d e b y H y d r o g e n at 5 5 9 K. [NO]o = 4 0 6 tort, [H2 ]o = 2 8 9 torr

383 2NO + 2H 2

, N 2 + 2H20

(26)

The trtmolecular reduction mechamsm is not precisely known, but possibly involves the HNO radical, which has been detected 35 The reaction of styrene with iodine m carbon tetrachlonde

(27)

C6H s -CH-CH 2 + 12 ------~ C6H 5 -CHI-CH2I

to form styrene dttod]de apparently revolves both a free-radical chain mechamsm and a concurring non-chain reaction that has a first-order dependency on ~odme concentration and a 3/2 order dependency on styrene concentration 36 Because of the pronounced effect of trace contaminants on reaction rates, reproduclbd]ty of results Is a major difficulty m following this process Fig 9 illustrates the tune-dependency of the affinity for this chain reaction

"6 E

5

OJ \ \

,9,

~

AFFINITY I I RATE EOUATION Ao 234 J/tool Ar - 2,370 J / r r ~

3-

,-j "'¢

2-

<~ I-

0

-2s

I

-20

i

).[

-15

I

I

-)o

-o5

o

Fig 9 - Reactzon o f Styrene and Iodine in Carbon Tetrachlorlde at 295 K = 2 03 x 10"4111, [ C 6 H s C H C H 2 ] o = 0 3 4 7 M

[/2 ]o

These several figures illustrate the high degree of data correlation possible with equation (23), and consequently, that the tune-dependency of a thermodynamic function can be described by an analytical function independent of mech-

384

amstlc comphcations Moreover, this description appears to be not only valid for reactions on close approach to equilibrium (Styrene-Iodine reaction. K ~ 1, t K < < 103 s), but equally valid for reactions greatly dxsplaced from equdlbrmm (Fe ( I I ) - C o ( I I l ) reaction K > 1017, t K > 1014 s)

DISCUSSION Tlus study of the kineUc behavlour of stolchlometnc chemical reactions m the closed isothermal system suggests that there are two separate, distract, and ostensibly independent approaches to the examination of such behavlour The classical approach 18 essentially microscopic m which a chemacal reaction xs viewed as a system of reacting molecules m which kinetic behavior 18 described m terms of the tmae-dependency of the concentration of reacting components the reactton velocity v The alternate approach 1s essentially macroscopic m which a chemxcal reaction ls viewed as simply an isothermal sink or source of energy m wtuch kinetic behavior is described in terms of the trine-dependency of a thermodynamic function the affinity decay rate A Correlations of the empirical data by these two approaches are compared m Appendix II, wtuch illustrates the superior data correlation possible by the single affmaty rate equation over the enttre range of expertmental observations independent of reaction mechamsm Whale these separate approaches appear to be independent of one another, the reaction velocity and the affinity decay rate can be related Dlfferentmtmg equation (4) with respect to the elapsed time yields 2 dc 1 AT,V = - RT ~ v-'-L( - ) el vl dt

(28)

where cl is the concentration of component l, and can be subsUtuted for the activity of component 1 under the observed eond~tlons The term m parentheses is the reaction velocity v Rearranging terms yields v~Ic, = -~,/RT 1

(29)

V

Equation (29) relates the concentration of reacting components at any specified time between reaction initiation and equdlbrium to a macroscopic thermodynamic term (numerator) and a rmcroscoplc kmeUc term (denominator) Consequently, these terms are complementary Presumably, the macroscopic approach must ul-

385 ttrnately rest on a macroscopic foundation based on statistical mechanics • Equation (29) goes as far as purely thermodynamic considerations can be extended m relating these two approaches to reaction kinetics Beyond this point recourse m generally made to a phenomenologlcal approach m wluch flux terms are linearly related to force terms, an approach particularly apphcable to thermal and mass diffusion processes When applied to chemical reactions, the reaction velocv ty (flux term) is lmeafly related to the chermcal affinity (force term) 37 Hence vj : l_j A j / R I

(30)

where L 1 is the linear phenomenologlcal coefficient for step j From experimental observations, the validity of equation (30) is limited to chermcal reactions m the stationary state not greatly d~splaced from equilibrium as FlttS 39 demonstrated that this hnntatlon extends to any chemacal reaction greatly disolaced flora equilibrium Going one step further, Manes e a 4o cast suspicion on the validity of the linear equation even on close approach to equlhbrmm by showmg that t.j = vj

(31)

--+

where vj Is the forward reaction velocity,. . ~and<....is related to the reaction velocity vj of equation (30) by the relatlonstup vj = vj - vj Thus, equation (29) must stand as the closest connection available m relating the classical microscopic approach to chemical kinetics to the thermodynamic macroscopic approach It can be readdy demonstrated that the singular reaction path of the macroscoplc approach, as dlustrated m Fig 1, is related to the standard affmity A ° By definition, A ° has the same absolute value as does the standard free energy Sqbstltutmg the reaction coordinates of the standard state (A °, t l) Into equation (23) yields A ° = A r l n [ t t ° exp(1 - t o ) ]

(32)

where ~o = ( h / t K ) Hence, the reaction parameters Ar and ~.o are lmuted to values consastant with equation (32) T/us equation shows a direct link between chemical kmetacs and chemical thermodynamics That is, for stolcluometnc reactions m a dosed system, the chemical-potential gradient defines the reaction path, demon. stratmg the essential validity of Proposition 1

386 As for Proposlhon 2, it can also be demonstrated that a Finite value for the most-probable tune to attain equilibrium ~s not a condit,on unposed on the afFtmty rate equation Instead th~s finite value for the most-probable time aries dtrectly from the form assumed by the affinity rate equation, much m the same manner that an mfmlte value for the absolute time requtred to attain equ~hbrmm ar,ses d~rectly from the form assumed by the tradmonal rate equations, both without era. ptr~cal verification If the elapsed-ttrne value tK = oo is substituted into equation (22) and this equation Is then integrated, the value of the most-probable ttme tK calculated at A = 0 is not mfmlte but finite, which contradtcts the ongmal specification that tK = ~ . That ~s, for the integrated form of the affinity rate "equation to sahsfy the thermodynamic cond,hon at equ,librium that A = 0 requrres that the value o f t K be finite This requirement was met when it was observed that the lines of regression did not pass through the or~,in when the form of the aff'mlty rate equation was being determined Insufficient data were available to determine exphc]tly the temperature de. pendency of the aff'm~ty rate constant, but apparently Ar/T ,s reversely proportional to the reaction temperature, the same temperature dependency as the free energy of the equdlbrmm system

CONCLUDING

REMARKS

This study suggests that m addihon to the classlcalor microscoplc view of chemical kmehcs in which the reaction is consdered as a system of reacting molecules, a th'ermodynamlc or macroscopic view is possible m which the reaction is considered as a sink or source of energy. According to the nucroscoplc approach kinetic behawor is described m terms of the tune-dependency of the concentration of reacting components; wh,le according to the macroscopxc approach kinetic behavior ,s described m terms of the time-dependency of a thermodynamic function. For homogeneous sto~cluomemc chemical reactions in a closed ~otherreal system, the macroscopic approach as represented by the affinity rate equation extub~ted superior data correlation over the enttre range of experimental observations. As predicted from thermodynanuc conmd~rattons, the observed correlations axe independent of reaction mechanism.

387 Appendix I Fzg l0 outhnes the procedure reqmred to determine the value of the reaction parameters Ar and t K from experimental kinetic data The data input zs m the form of several parameters assocmted wzth each reaction examined, and an array of t - Q parrs associated with each tune interval surveyed

READ PARAMETERS T,K~z tl ,t2,t3"" trl QI ,O2 ,Oa..On

DECREASEoFVALUE ~_ z INCREMENTALLY

(;e (O/K), A -RTIn((;Q) II,x . , o ' , ¢, • (,/,K) . AI'AI'All An IL~l. CtS.(;tll "'" (;In

1

l

REGRESSION ANALYSIS [

A VERSUS In[~;texp(I-(;t)] j

1

I INTERCEPT T
Because the value of the intercept I increases towards zero from large negatzve values wzth decreasing values of tK, the value of tK as mztmlly chosen must he larger than zts final computed value To insure that this cond~tzon is met, the

388

mRlal value of the exponent z in the relationship t K = 10z is arbJtrartiy placed at 7O The condRlon that the intercept has vamshed reqmred to satisfy the Rerat~on loop ~s assumed to be met when [II~< An/106, where An IS the last and hence smallest value of the chemical affinity computed from the activRy-rat~o array. Tlus condition assures that errors arising from the Reratlve procedure are smaller than any expertmental errors Table 11 hsts the values of the synthetic data used to test the computation procedure This data input should yield the data output shown The value of t 1 is determined after Ar and tK are computed by calculating the value of ~o from equation (32) by the method of successive approxunatlons Table I I - Synthetlc Reaction Rate Data A2

> 2A

T=550K [A2]o = 0 10000 arm A ° = 6,473 J/tool [A2] atm 1000 2000 3000 4000 5000 6000 7000 8000 9000 A r = - 8,000 J/tool t K = 20,000 s

tl

= 4,000 s

0 05887 0 04028 0 02984 0 02344 0 01926 0 01639 0 01434 001283 0 01169

389 Appendix 11 The reactant concentratzons at the measured elapsed.ttme intervals were computed from both the affinity rate equation and from the appropriate reaction-velocity rate equation and compared with the expertmental values Fig 11 illustrates the kinetic data for the lsomenzahon of ethyhdenecyclopropane and the concentratlon-tmae curves calculated from the affinity rate

260[

=

I

I

I

I

I

[Eth] o UNIMOLEClJI-tld RATE E ' ~ T I O N

240 220

o

t-

\ ~.~

K

301111 O - 4IS

K"

381X I0" 4/S

RATEAIF~oN/~TION -- -Ar - 4 Z 6 0 J/rnel % Ix 6840 s _

2O0 180

LIJ

160

-

140 -

0

[Eth]e q

I I

I 2

--

I 3

ELAPSED

I 4

I 5

I 6

7xlO 3

TIME, s

Ftg 11 - Isomerzzatzon o f E t h y h d e n e c y c l o p r o p a n e to 2 - M e t h y l m e t h y l e n e c y c l o propane at 5 0 7 K [Eth ]o = 250 0 tort, [Eth ]eq = 139 7 tort

equation and from the unanolecular rate equation (14) after integration The f~,ure suggests that the two computed curves correlate w~th the experimental data points equally well The coefficients of correlation for the curves are both 0 9 9 9 + , and even at t = tK, where the greatest differences in calculated concentrattons mBht be expected, the difference Is less than one percent, well within expermaental error

390 Fig 12 dlustrates a stmdar presentation for the reduction of Pu(IV) m perehlone acid solution, w]th the reachon velocity curve calculated from equation (16) after mtegratlGn As m the previous example, any differences were well within expertmental error 12x10 -4

I

I

I

I

I

I

I

I

[Pu (IV)] o

2

BIMOLECULAR RATE EQUATION "K 512 $ mob ~" 00237 ~/mobs AFFINITY RATE EAC~Jr ATION080 J/tool -

II

10 -~ 9 -- \ 8 7

\\

h< \\

187o s

-

-

\

-

-

4 3

_

2

I0

I

I

20

I

ELAPSED

I

40

I

I

60

I

80

TIME, s

Ftg 12 - Reductzon o f P u ( I V ) by F e ( H ) m 0.5M Perchlomc Acid Solution at 293 K. [Pu(IV)]o = 11 5 x 10"4M, [Pu(IV)]eq = 1 66 x lO-S M

The concentrations as computed from the affinity rate equation and from the integrated trh-nolecular rate equation (18) for the formation of mtrosyl chloride showed vu-tually tmpercelvable differences over the enttre range of experimental observations, so that only a single computed line ~s shown m F~g 13 A single computed line is also shown m Fig 14, but solely because the

_

C 0

2C-

4C

6C

80

I

I

I

I

- 548o J/mj 9.27xlOSl

1000 2000 ELAPSED TIME, s

[ N O]eq - I I

Ar tK

I

37 3 ,I ! / m o l l , I

AFFINITY RATE EQUATION

K

TRIMOLECULAR RATE EQU/JTI0 N

I [NOlo

F2g. 13 - Formatzon o f N~trosyl Chloride at 333 K [NO]o = 128 torT, [NO ]eq = 8 13 tort

z

o

0 4--

100

120 ~ t ~

140~

860

87-

88-

89-

90

91

92

•

Ar lK

2

ELAPSED

1

4

T

J/n 4 |

TIME, s

3

-13,500 404x|O

AFFINITY RATE EQUATION

ix,},

l i t ] eq--

1

5xlO 4

F~g 14 - Dehydrogenatzon o[ Isobutane by Iodine Vapor at 525 K [/210=967 torr, [I~]eq=C 69 torr

ll

0 e-

93

94

95

96

97

Oo xD

392 chain-reaction rate equation (20) xs too complex to be integrated The curve shown was computed from the affinity rate equation

Acknowledgements The autho/" thanks the Lewis Research Center of the Nattonal Aeronautics and Space Admmtstratton, Cleveland, Ohto, USA, [or supportmg this work The helpful suggesttons and constructtve cmttcisms of Professors I¢ Brostow (Department o[ Materials Engmeermg, Drexel Untverstty, Phtladelphta, Pennsylvama, USA}, S G Chnstov (Instttute for Phystcal Chemistry, Bulgarian Academy o[ Sciences, So[ta, Bulgarta) and K Jankowskt (Department o f Chemistry, Untverszty o[ Moncton, Moncton, New Brunswzck, Canada) are most appreciated Special tb.m~ks are clue to the many investigators who made thetr expertmental data available for thts study

REFERENCES 1 2 3 4

5. 6 7 8 9 10 11

S R DE GROOT - T h e r m o d y n a m t c s of Irreverszble Processes, North-Holland, Amsterdam, 1951, p 166 I PRIGOGINE - Thermodynamzcs of Irreversible Processes, Intersctence, New York, 1955, p 59 H EYRING - J . Chem Phys, 3, 107, 1935,S GLASSTONE, K J LAIDLER, H EYRING - Theory o f Rate Processes, McGraw-Hall, New York, 1941 J O HIRSCHFELDER, E WIGNER - 3" Chem Phys, 7, 616, 1939, H EYRING, J WALTER, G E. KIMBALL - Quantum Chemtstry, Wiley, New York, 1946 A KUPPERMANN - 3" Phys Chem, 83, 171, 1979, W H. MILLER - 3" Chem Phys., 65, 2216, 1976 L S K A S S E L - J. Chem Phys., 3, 339, 1935 S G CHRISTOV - Bet. Bunsenges Phys Chem, 76, 507, 1972, 78, 537, 1974,3. Res Inst. for Catalysts, 28, 1t9, 1980 S G CHRISTO v - B e r Bunsenges Phys. Chem, 79, 358, 1975 S G CHRISTOV - Colhswn Theory and Statzsttcal Theory of Chemtcal Reactions, Sprmger-Verlag, Heidelberg, 1980 R P BELL - The Tunnel Effect m Chemistry, Chapman and Hall, London, 1980, p 168 M. KARPLUS - J Chem Phys., 30, 11, 1959, 3" Am. Chem. Soc, 85, 2870, 1963

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19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

L.G V O R O N T S O V A , A F B O C H K O V - Org Magn Reson, 6, 654, 1974 K J A N K O W S K I , A R A B C Z E N K O - Org Magn Reson, 9, 480, 1977 J B C H I A S S O N , K J A N K O W S K I - I Am Chem Soc, 104, (m press),

1982 W BROSTOW - Science, 178, 123, 1972, Proe Soc Gen Systems, 21, 5t~O, 1977 T DE D O N D E R - Acad Roy Belg Bull Classe Sct, 5, 197, 1922 I PRIGOGINE, R DEFAY - Chemtcai Thermodynamtcs, Wiley, New Yo~k, 1962, p 1 J H E S P E N S O N - C h e m t c a l Kmettcs and React,on Mechanzsms, McGraw-Hill, New York, 1981, p 1 I C P E R C I V A L - J Math Physzcs, 2, 235, 1961 I P R I G O G I N E , R B A L E S C U , F H E N I N , P R E S I B O I S - Phvstca, 27, 541, 1961 R BAIERLEIN '-- Atoms and Information Theory, Freeman, San Francxsco, 1971, p 92 W BROSTOW - S c t e n c e of Materlals, Wiley, New York, 1979, p 20 R L LA F A R A - Computer Methods for Sctence and Engmccrmg, Hayden, Rochelle Park, 1973, p 202 I M K L O T Z - Chemwal Thermodynamics, Prentice-Hall, Englewood Cliffs, 1957, p 14 J P CHESICK - J Am Chem Soc, 85, 2720, 1963 T W NEWTON, H D C O W A N - J Phys Chem, 64, 244, 196t~ I W E L I N S K Y , H A T A Y L O R - J Chem Phys, 6, 469, J938 S W BENSON - Fundamentals o f Chemtcal Kmettc~, McGraw-HtU, New York, 1960, p 305 A G SYKES - Kmettcs oflnorgamc Reactzons, Pergamon, Oxford, 1966, p 86 H TERANISHI, S W B E N S O N - J Am Chem Soc, 85, 2887, 1963 B R O G D E N - General Dynamics Res R e p o r t E R R FW-853, 19o8, p 5 W F O R S T - Can J Chem, 36, 1308, 1958 L E BENNET, J C SHEPPARD - J Phys ('hem, 66, 1275, 1962 C N HINSHELWOOD, T E GREEN - J Chem Soc, 128, 730, 1926 H A TAYLOR, H A TANFORD - J Chem Phys, 6, 466, 1938 G FRAENKEL, P D BARTLETT-J Am ('hem Soc, 81, 5582, 1959 M G A R F I N K L E - NASA Techmcal Not~ D-2875, July 1965, p 2 I PRIGOGINE, P OUTER, C HERBO -- J Phys and Collotd Chem, 52 321, 1948 D D FITTS - Nonequzhbnum Thermodynam,cs, McGraw-Hill, New York 1962, p 134 M M A N E S , L J E H O F E R , S W E I , L E R - J Chem Phys, 18 1355, 1950