statistical mechanical models of chemical reactions iii. solvent effects

MOLECULAR PHYSICS, 1987, VOL. 60, No. 6, 1315-1342

S t a t i s t i c a l m e c h a n i c a l m o d e l s o f c h e m i c a l r e a c t i o n s

III. S o l v e n t e f f e c t s

by P. T. C U M M I N G S

Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, Virginia 22901, U.S.A.

and G. S T E L L

Departments of Chemistry and Mechanical Engineering, State University of New York at Stony Brook,

Stony Brook, New York 11794, U.S.A.

(Received 30 September 1986 ; accepted 14 October 1986)

The statistical mechanical study of chemically reactive fluids via the analytic solution of integral equation approximations for such fluids (Cummings, P. T., and Stell, G., 1984, Molec. Phys., 51, 253; 1985, Ibid., 55, 33) is extended to a consideration of solvent effects. This is achieved by studying mixtures of molecular species some of which can undergo reaction (the solute(s)) and one of which is inert (the solvent). The results obtained near infinite dilution of the reacting solute in the inert solvent are similar in magnitude to those observed in experiments on the dissociation equilibrium of dinitrogen tetroxide.

1. INTRODUCTION

This paper further pursues the analytic study of chemically reacting systems initiated by the authors in [1] (hereafter referred to as I with equation (i.j) of I denoted (I.i.j)). In this paper, the association reaction between species .// and species B molecules,

A + B . -~ -AB, (1.1)

was studied by considering a mixture of A and B atoms in which the pair interactions gave rise to chemical association. In [2], the reaction

2A .-~- A 2 (1.2)

between like species molecules was modelled by using a pair potential between A atoms similar to the potential between A and B species used in I. In this paper, we consider the effect of a solvent on both reacting systems (1.1) and (1.2).

Solvent effects on chemical equilibria are well-known [3] and can be quite dramatic [3, 4]. For example, consider the association equilibrium of nitrogen dioxide with dinitrogen tetroxide

2NO 2 ~--- N 2 0 4 . (1.3)

1316 P . T . Cummings and G. Stell

The ratio of the association constant K (defined in equation (1.4)) to its gas phase value in various liquid solvents at 293'16 K [3].

Solvent K/Kgas

SiCI, 21"5 CS 2 28"7 CC14 47.5

CHC13 69'1 C2H,Br 79-7 C6HsBr 103"2 C6H5C1 103"2

C6H 6 171 '3

As shown by the table, in a liquid solvent it is found experimentally that the equilibrium mass action association constant

K - PN204 (1.4) 2

PNO2

is between 10 and 200 times its value for the same gas phase reaction in the absence of a solvent [33. Chandler and Pratt [4, 53 developed a comprehensive formalism for the study of condensed phase chemical equilibria, and an approx- imate form of their theory (described in more detail in w 2.5 below) yields a shift in K which is of the order of magnitude of the experimentally observed values. They explain this result in terms of the steric effect of the solvent on the reacting solute. In essence, they find that association is dramatically increased in a liquid solvent because the solvent molecules drive pairs of NO 2 groups together in order to save space in the dense fluid medium.

The role of steric effects on reactions (1.1) and (1.2) will be examined in this paper by considering mixtures in which the reacting components (solute) are immersed in a hard sphere solvent. Thus, in the case of the association reaction (1.1) between unlike species A and B molecules, a three component mixture of A and B molecules with a hard sphere solvent C will be considered, the pair interactions (9ij(r) being given by

~ , j ( r ) = o0, r < ~,

= O, r > a ,

ij = AA, BB, AC, BC, CC,

fan(r) = exp [ - - ~an(r)/knT] - 1 (1.5)

12L = - 1 + 3 ( r - L ) , r < a ,

= 0 , r > f f .

Here, fij(r) is the Mayer f-function, k n and T are respectively Boltzmann's constant and the absolute temperature, z is a dimensionless parameter related to the temperature, L is the distance between atom centres in an AB diatomic molecule and 3(x) is the Dirac delta function. The functional form of fan(r) is motivated and justified in I, to which the reader is referred for a detailed explanation. For

Stat is t ical mechanics of chemical reactions 1317

the association reaction (1.2) between like species, a two component mixture is used (solute A and hard sphere solvent C), the pair interactions being given by

q~u(r) = ~ , r < a ,

= O, r > or,

6= AC, C C ,

faa(r) = exp [ - - ~)AA(r)/kBT] -- 1

12L = --1 + 6 ( r - - L) , r < a,

T

= 0 , r > • .

(1.6)

In w the Percus-Yevick approximation [6] for the mixture with interactions (1.5) is solved analytically; results are presented in w T h e modifications of the analysis required to solve the PY approximation for the two-component mixture (1.6) are described in w and selected results are also presented. Section 5 contains our conclusions and discussions.

Since the solvent in the calculations presented in this paper is a hard sphere solvent, we are probing solvent-induced effects on the chemical equil ibrium which are strictly steric in nature. In particular, there is no l iquid-gas phase transition in the hard sphere fluid with the consequence that solvent effects related to, for example, solvent vapour- l iquid critical points cannot be probed in this model. In the next paper in this series, a model applicable to such phenomena is developed by considering reaction (1,1) in a simple fluid solvent which in the integral equation approximation used undergoes a l iquid-gas phase transition at low temperature. This permits the examination of solvent criticality and phase equilibria on the chemical reaction equil ibrium of the solute.

2. ANALYTIC SOLUTION OF THE PY APPROXIMATION FOR THE REACTION A + B ~ A B IN SOLUTION

In this section, we consider the solution of the PY approximation for the three-component mixture with pair interactions given by equations (1.5). Species A, B and C molecules are denoted by the indices 1 to 3 respectively. The mixture Ornste in-Zernike (OZ) equation [7] for an m-component mixture is given by

ho(r) = cij(r) + k~l= Pk Clk( I $ I )hkj( I t -- $ I ) d$ (2.1)

where hij(r ) and cii(r ) are the total and direct correlation functions between species i and j molecules, p~ is the number density of species i molecules, and the integral is per formed over all space. T h e PY approximat ion is given by

cij(r ) = fo ( r ) [1 + hij(r ) - clj(r)], (2.2)

1318 P . T . C u m m i n g s and G. Stell

which, for the interact ions (1.5) yields

hij(r) = - 1 , r < a,

ij = 11, 22, 13, 23, 33,

2L hi2(r ) = - 1 + ~ - J ( r - L ) r < a ,

cij(r) = O, r > o',

l <~i, j<~3.

(2.3)

Since each cij(r) is f ini te-ranged, the Baxter W i e n e r - H o p f factorizat ion [8, 9] for mixtures [10] is immedia te ly applicable, yielding the set of factorized equat ions

rhij(r ) = -q'ij(r) + 27t Pk qik(t)(r -- t)hkj(Jr -- t I) dt, (2.4 a) k = l

i fo '-' rclj(r ) = --q~j(r) + 2~z Pk qki(t)qjk(r + t) dt. (2.4b) k = l

We choose Pl = P2 = Pu, the densi ty of solute atoms, and P3 = Pv, the solvent density. F r o m s y m m e t r y considerat ions, it follows that h13(r ) = hz3(r ) and h l l ( r ) = h22(r); analogous relations hold for the direct correlat ion functions. As a consequence of these identities, we find that q i i ( r ) = q22(r), q l 2 ( r ) = qzl(r), qa3(r) = qz3(r) and q3t(r) = q3/(r). T h e second of these relations is in some sense the least trivial, since in general qij(r) does not equal qji(r) (unlike the cor responding, usual s y m m e t r y in the total and direct correlat ion funct ions) ; it follows in this case since species 1 and 2 a toms are at the same densi ty and have equal diameters.

Tak ing the above symmetr ies into considerat ion, we find that equat ion (2.4 a) yields five dist inct equations,

rhl i (r ) = --q ' l l(r) + 2lrpv[qll | + q12 | hl2(r)] + 2~Pvqia | hi3(r), (2.5 a)

rhl2(r) = -- q'12(r) + 2rcpv[qi i | hi2(r) + q12 | h l i ( r ) ] + 2rCpv qi3 | hi3(r), (2.5 b)

rhi3(r) = -q'13(r) + 27rPv[qll | hla(r) + qi2 | hia(r)] + 27rpv qla | h3a(r), (2.5 c)

rhi3(r) = -- q3i(r) + 2rrpv[qax | hl l ( r ) + q3i | h12(r)] + 2~Zpv q3a | hia(r), (2.5 d)

rh33(r ) = --q~a(r) + 4rrpvq31 | h ia(r ) + 2rcpvq33 | h33(r), (2.5 e)

where f | g(r) symbolizes the convolut ion

f | = f o f ( t ) g ( ] r - t[) dt.

As in the previous two papers in this series [-1, 2], our task is to de termine the funct ional fo rm of the Baxter q-funct ions qij(r), relate the parameter ). to the tempera ture z and the densities Pt~ and Pv, derive an expression for the association constant and to provide an a lgor i thm for c o m p u t i n g the radial dis t r ibut ion funct ions gij(r) = 1 + hq(r). T h e results in this section are given in explicit fo rm for L = o-/2; the cor respond ing results for L = o-/3 are easily derived by apply ing the steps for L = o-/2 out l ined in this section to the cor responding analysis for L = o-/3 given in I.


2.1. The Bax ter q-functions

We begin by cons ider ing the funct ions qlj(r), j = 1, 2, 3. Subs t i tu t ing the closures (2.3 a) and (2.3 b) into equat ions (2.5 a)- (2 .5 c), we obta in the fol lowing different ia l-difference equa t ions :

q'lx(r) + p[q12(r + L) -- qa2(r -- L)] = a i r + bl, (2.6)

2L 2 q]2(r) + p [ q l l ( r + L) -- q l l ( r -- L)] + - ~ - fi(r -- L) = a~r + bl, (2.7)

q'13(r) = a i r + bl, (2.8)

where

p gPu j'L2 - - - - " v = p a = q v J , ( L / a ) 2; q i = g p i a 3 " i = U, V, (2.9)

6 '

a 1 = 1 -- 2repv [q l l ( t ) + q12(t)] dt -- 2rCpv ql3(t) dt, (2 .10a)

fo ;o b 1 = 2zcpv t [q l l ( t ) + q12(t)] dt + 2r~pv tql3(t ) dt. (2.10b)

T h e b o u n d a r y and d iscont inui ty condi t ions on the funct ions qlj(r) are

2L 2 q l t ( a ) = ql2(ff) = q13(0") -~- 0, q l 2 ( L - ) : q~2(Z+) q- 1---2- (2.11)

F r o m equat ions (2.8) and (2.11), the funct ional fo rm of ql 3(r) is given by

q 1 3 ( r ) = � 8 9 2 + h I ( r - a ) , r < a , (2.12)

= 0 , r > ~ a .

T o de te rmine qla(r) and q12(r), a m e t h o d similar to that used in I mus t be employed . T h u s , we define the func t ions

q+(r) = q l l ( r ) + q12(r),'~ (2.13) g

q_(r) qxx(r) -- qx2(r),J

so that

ql l (r ) = �89 + q _ ( r ) ] , ] (2.14)

q12(r) �89 -- q_ ( r ) ] . J s

T h e funct ions q+(r) and q_(r) then satisfy the equat ions

2L 2 q+(r) + p[q+(r + L ) - - q + ( r - - L)-I + - ~ - b ( r - - L ) = a + r + b+, (2 .15a)

)~L 2 q ' ( r ) - - p [ q _ ( r + L) -- q (r -- L)] -- - i2 - fi(r -- L) = O, (2.15 b)

with

a+ = 2a a = 2 -- 4rcpv q+(t) dt - 4rcpv q13(t) dt,

; ;o ~ b+ = 2bl = 4~pv tq+(t) dt + 4rcpv tqa3(t ) dt.

(2 .16a)

(2.16 b)


T h e b o u n d a r y and d iscont inui ty condi t ions on q+(r) and q_(r) are given by

2L 2 q + ( a ) = O , q + ( L - ) = q + ( L + ) + 12 ' (2 .17a)

~,L 2 q_(a) = O, q _ ( L - ) = q_(L + ) - - (2.17b)

12

We now solve equat ions (2.15) specifically for L = a/2; for the reasons noted earlier in this section, explicit resul ts for L = a/3 will not be given in this paper .

T h e solut ion of equat ion (2.15 a) is ob ta ined using the m e t h o d s of I, and we s imply quote the resul t here :

a+ a+ b+ a q + ( r ) = - - - - r + ( 1 - v / 2 ) - ~ + A + c o s ( p r ) + B + s i n ( p r ) , O < r < -

p - ~ p 2

a+ a+ b+ = - - r + (1 -- v/2) + - - + A + cos [p(r -- a /2)]

- - B + sin [ p ( r - - a / 2 ) ] , ~ < r < a . (2.18)

Equa t ion (2.18), when c o m b i n e d with equat ions (2.16) and (2.17 a), yields the fol lowing linear equat ions for the p a r a m e t e r s a + , b+ , A+ and B+ :

1 2 - - 2 - a + + - b'+ - A ' + c - B ' + ( I + s ) - ( 2 . 1 9 ) v v 48 '

1 1 V2 (1 + V/2)a+ + - b'+ -- A'+s -- B'+c = 0, (2.20)

V

24r/u(1 - v/4) ] 24r/v a+ 1 + v2 4qv - O l v b ' + +

V

x [(1 -- c + s)A'+ + (1 -- c -- s)B'+] = 2, (2.21)

V24r/v ~ ] [ 6r/v ] a + [ v---- T- + + b'+ t -- v + 2t/v

24r/v [ v ( v ) ] , v2 - - l + ~ + ( 1 - - v ) c + 1 + ~ s A+

y2 I + 1 -- C + (1 -- V)S B'+ = 0, (2.22)

where

b+ b ' + - , A ' + -

G

A+ B+ a2 , B ' + - a Z , (2.23)

s = sin (v/2), c = cos (v/2). (2.24)

Statist ical mechanics of chemical reactions 1321

Equa t ions (2.19)-(2.22) can be wr i t t en in the fo rm lb [ 48p M + =

A'+

LB'+J

(2.25)

where the e lements of the ma t r ix M are defined obviously . T h e solut ion of equat ion (2.25) is given by

a+ = ~ T l l + 2 T 3 x , (2 .26a)

b'+ = ~ T12 + 2T32 , (2.26b)

1 (-~82 ) A'+ - ~ T13 + 2T33 , (2.26c)

B'+ =

Here , A is the de t e rminan t and Tij the ij cofactor of M given by

12~/v r/---------~v [8 -- 4v + v z - 8c + (8 -- 4v + v2)s] + (@ -- 2~/v)(1 + s), A = A 0 "t- •3

24qvqv y2 [ 4 v - 6 + (6 -- v)c + ( - 6 + 2v)s], T l l = TOn +

12rlvqv T12 = T~ + - - 7 - [8 -- 5v + ( - - 8 + v)c + (8 -- 3v)s],

12qv rlv V 4

- - [ 1 2 - - 6 v + V 2 "~- ( - 1 2 -~- 2V - - ~'2)c T13 = T~ +

+ (12 -- 10v + v 2 ) $ ] --~ (/12 V - - 2r/v)(1 + s)

(2.26 d)

(2.27)

(2.28 a)

(2.28 b)

12rlvq~v [12 - 4v + ( - - 1 2 + 10v -- v2)c T~, = T O + v4

(2.28 c)

+ (12 + 2v - v2)s] + (r/2 - 2r/v)(1 + s), (2.28 d)

T31 = T~ + 2qv(t + s) (2 .29a)

T32 = T~ + -~rlv(1 + s) (2.29 b)

T33 = TO3 + fl_.y_.v [4 -- v -- 2vc + (v -- 4)s] (2.29 c) 2V 2

T34 = T O + fl--E-v [(4 -- v)c -- 2vs]. (2.29 d) 2v 2

T h e supersc r ip t ' 0 ' in equat ions (2.27)-(2.29) is used to denote the value of the co r re spond ing p a r a m e t e r in the l imi t r/v -+ 0, and so A ~ and the ~/j are g iven by equat ions ( I .A I ) - ( I . A 9) wi th f = 2 and r / = qv (or, equivalent ly , ~ = 2r/v ).

1322 P . T . C u m m i n g s a n d G . S t e l l

T h i s c o m p l e t e s t h e s o l u t i o n o f q + ( r ) ; we n o w c o n s i d e r q_(r) . T h e s o l u t i o n is

s t r a i g h t - f o r w a r d a n d w e s i m p l y q u o t e t h e r e s u l t

q_ (r) = A _ cos (pr ) + B _ s in (pr ) , O < r < ~ ,

w h e r e

ff = A _ cos [ p ( r - a / 2 ) ] - B _ s in [ p ( r - a / 2 ) ] , - < r < a , (2 .30)

2

A _ - - ~c A'_ - - - - (2.31 a)

a z 48(1 -- s ) '

B _ - A s B'_ - - . (2.31 b)

a 2 48(1 - s)

C o m b i n i n g e q u a t i o n s (2 .18) a n d (2 .30) w i t h (2 .14) , w e o b t a i n t h e f ina l e x p r e s -

s i o n s fo r qax(r) a n d qaz(r ) ,

qla(r) al al 61 + = - - - - r + (1 -- v/2) -- A s cos (pr) + B o s in (pr ) ,

p p

G O < r < ~ ,

a l a l b l + - r + (1 -- v/2) + A o cos [ p ( r - a / 2 ) ] - - B s s in [ p ( r - - a / 2 ) ]

p p

G - < r < a , ( 2 . 3 2 a ) 2

q l 2 ( r ) a l a l b I + = -- - - r + (1 -- v/2) -- ./1 v cos (pr) + B s s in (p r ) , p p

O < r < ~ ,

a l a l b l = - - r + (1 -- v/2) + - - + A s cos [ p ( r -- a / 2 ) ] -- B o s in [ p ( r -- a / 2 ) ] ,

p p

t7 - < r < a . ( 2 . 3 2 b ) 2

I n e q u a t i o n s (2 .32) ,

a 1 = a+/2, bl = b+/2,

A s = ( A + + A _ ) / 2 , A n = ( A + - - A - ) / 2 ,

B s = ( B + + B _ ) / 2 , B n = ( B + - B _ ) / 2 .

t t ! t r T h e d i m e n s i o n l e s s p r i m e d q u a n t i t i e s bl, A s , AD, B s a n d B o a re d e f i n e d in t h e

u s u a l w a y (el . e q u a t i o n s (2 .23) a n d (2 .31) ) . T h i s c o m p l e t e s t h e a n a l y s i s fo r qxj(r),

j = 1, 2, 3. W e n o w t u r n o u r a t t e n t i o n to t h e f u n c t i o n s q3j(r), j - - I , 3. S u b s t i t u t -

i ng t h e c l o s u r e s (2 .3) i n t o e q u a t i o n s (2.5 d) a n d (2.5 e), w e f i n d

q'31(r) + p[q31(r -4- L ) -- qal(r -- L ) ] = a 3 r + b a , (2 .33)

S t a t i s t i c a l m e c h a n i c s o f c h e m i c a l r e a c t i o n s 1323

q~3(r) = a 3 r + b3, (2.34)

a 3 = 1 - - 4 n p v J i ' q 3 1 ( t ) d t - 2 n p v j i~ (2.35 a)

f; fo b3 = 4 n p v tq31( t ) clt + 2 n p v tq33( t ) d t . (2.35 b)

T h e b o u n d a r y cond i t ions on the func t ions q3i(r) are

q31(0-) = q33(0-) = 0 (2.36)

and we obse rve tha t q31(r) is con t i nuous at r = L. T h e so lu t ion of equa t ion (2.34) is i m m e d i a t e and is g iven b y

q33(r) : �89 - - 0") 2 + b3(r - - t7), r < 0-, (2.37)

= 0 , r >~a.

T h e so lu t ion of equa t ion (2.33) is s t r a igh t fo rward , y i e ld ing

a3 a3 b3 -- - - r - - (1 -- v /2 ) - - - - + A 3 cos (pr) + B 3 sin (pr), q31 (r) = p + p2 p

0- O < r < ~ ,

a3 a3 b3 - r - - (1 - v /2 ) + - - + A 3 cos [ p ( r - a/2)] p + p 2 p

ff - - B 3 sin [ p ( r - - a / 2 ) ] , ~ < r < 0 - . (2.38)

E q u a t i o n s (2.35) and (2.36) and con t i nu i t y of q13(r) at r = L resu l t in four l inear equa t ions for the p a r a m e t e r s a 3 , b 3 , A 3 and B 3 w h i c h can be wr i t t en in the fo rm

where

r a31 [i[ M b~ 0 , (2.39)

[P3J

b3 A3 B3 b~ ' (2.40) -- O" ' A3 - - 0 - 2 ' B ~ - a2

and where M is the same m a t r i x as tha t appea r ing in equa t ion (2.25). Conse - quen t ly , the express ions for the d e t e r m i n a n t and cofac tors g iven in equa t ions (2.27) and (2.29) are app l i cab le to the p r e sen t equa t ion as well . T h u s , we find tha t

T31 (2.41 a) a3 -- A '

732 b ~ - A ' (2 .41b)

T33 (2.41 c) A ~ - A '


T34 B ~ = A '

where T31 , i - - 1 , 2, 3, 4, are given by equations (2.29). This

(2.41 d)

completes the derivation of the functional forms for the Baxter q-functions for L = a/2 .

2.2. D e t e r m i n a t i o n o f the p a r a m e t e r 2

The next step in the analytic solution is to relate 2 to the tempera ture z and the densities qv and t/v via the PY approximation. As in I, we can show that

Yi2(r) = 1 + h12(r) -- c12(r)

is continuous at r = L, so that the PY approximation, equation (2.2), yields

):c = y12(L).

After some tedious but straightforward algebra, for L = ~/2 becomes

(2.42)

this equation

2z 2al . . . . + 2vA's -- F1 -- F 3 ,

Y (2.43)

where F 1 and F s are given by

~ VaZ~ 3a~ alb'l ' F1 = z ' ~ q v L - ~ --4v z - ~ bay (A's + A'D)s -- b--Lv (B's + B~)(1 -- c)

a 1 + ~-2 (A} + A~)(1 -- c + 23 -- vs)

A i 2 + + (v + sin v) +

4

+ Bs2 +4 B~ (v -- sin v)]

F 3 = 24tlv 8v2 2v 2

a l + ~- (B} + B~)(2 -- 2c - s + vs -- v /2)

A r 2 1 ~ t 2 x~r21Qt2 s ~'D + ~D~'S (1 -- COS V)

2

! t

b s A s ' s - - b s B ~ ( 1 - c ) V V

(2.44 a)

a 3 a 3 + ~- A;(1 -- c + 2s -- vs) + - ~ B;(2 -- 2c -- s + vs -- v /2)

A ; 2 . A's2B'3 2 . B's 2 ] + T ( v + s i n v ) + ~ ( 1 - cos v) + T (v -- sin v) . (2.446)

Equation (2.43) combined with equations (2.44) constitutes a single non-linear equation for the parameter 2 which can easily be solved using standard tech- niques.

Statistical mechanics of chemical reactions 1325

2.3. Expression for the association constant

The association constant for the reaction (1.1) is given by

k = PA8 (2.45) p~ p~"

Formally, the expressions given in I for the dimensionless association constant K = k/a 3 and for the expected number ( N ) of B(A) atoms distant L from a given A(B) atom remain valid and are quoted without further comment.

( N ) = 2flu 2(L/a) 3, (2.46)

K = ~z'~(L/~ 3 311 -- (NS] 2" (2.47)

2.4. Calculation of the radial distribution functions

The method used to compute the radial distribution functions qlx(r), g12(r), gt3(r) and g33(r) is an extension of Perram's method [1, 2, 11] for hard sphere mixtures with suitable modification for the presence of the delta function in gx2(r) at r --- L. This modification for the A B binary mixture was described in detail in I and its generalization to the present ternary mixture is straightforward; consequently, the explicit form of the algorithm will not be given, and the reader is referred to I for further details.

2.5. The limit of vanishing solute density

We now consider the important limit of vanishing solute density Pu ~ 0 where the reacting species (A and B) are present at infinite dilution in the hard sphere solvent. (Of course, the other limit of vanishing solvent density, Pv ~ O, corresponds to the model described in I.) In principle, one could simply take the limit Pu ~ 0 in equations derived in w167 2.1 through 2.3 above; however, the limit is very difficult to evaluate directly since the parameter v ~ 0 as Pv--+ 0. Rather, it is much easier to return to the starting equations (2.5), perform the limit at this point in the analysis and derive the Baxter q-functions from there. This is the approach we now take.

For Pu --* O, equations (2.5) become

rhll(r ) = --q'xl(r) + 27rpvqla | h13(r), (2.48 a)

rhl2(r) = - q'l 2(r) + 2gpv ql3 | h13(r),

rh13(r) = --q~3(r) + 2gpvq13 | h3a(r),

(2.48 b)

(2.48 c)

rh13(r) = -q31(r) + 2gpv q33 (~ h13(r), (2.48 d)

rh33(r) = - q33(r) + 2rCpv qsa (~ qss(r) �9 (2.48 e)


Substituting the closure (2.3) into equations (2.48), we obtain

q'ii(r) = a i r + bi

2L 2 q'xa(r) + ~ 6(r -- L ) = a i r + 61,

q'is(r) = axr + bi ,

q~l(r) = a 3r + b3,

q~3(r) = a 3r + b3,

where

a 1 = 1 -- 27~pv ql3(t) dt,

b 1 = 2ZCpv f o t q x 3 ( t ) dt,

a 3 = 1 -- 2~pv q33(t) dt,

b 1 = 2rcpv f j t q 3 3 ( t ) dt.

It is not difficult to see that the q-functions have the form

2 L 2 q12(r) = �89 -- 0) 2 + b(r -- ~) - - - ~ O(r - L ) ,

= O,

q l l ( r ) = q13(r) -- q31(r) = q33(r),

r~o ' , I

r~>ff, i

where

= � 8 9 - - o-) 2 + b ( r - - ~ r ) , r < o , l

= 0 , r >~ a, i

(2.49 a)

(2.49 b)

(2.49 c)

(2.49 d)

(2.49 e)

(2.50 a)

(2.50 b)

(2.50 c)

(2.50d)

(2.51 a)

(2.51 b)

1 + 2rlv b - - 3 q v a - (1 -- qv) 2' a - 2(1 - q v ) 2 (2.52)

and O(x) is the Heaviside step function given by

O(x) = O, x < O,

= 1 , x~>0.

The parameters a and b are simply the coefficients which appear in the Baxter q-function in the analytic solution of the PY approximation for the hard sphere fluid with density t/v. This is not surprising, since in the limit of infinite dilution of the reacting species, the fluid is simply a hard sphere fluid with density qv- Consequently, all of the Baxter q-functions are equal to the hard sphere Baxter q-function except for qx2(r) whose derivative contains a delta function. This observation leads to the conclusion that the correlation functions h i l ( r ) , hi3(r)

Sta t i s t i ca l mechanics of chemical reactions 1327

and h33(r) are simply hard sphere correlation functions at density qv (denoted hHs(Pv; r)) while h12(r) is given by

2L 2 rh12(r) = rhns(Pv; r) + - - ~ ~(r -- L) . (2.53 a)

A similar result holds for q2(r)

2L 2 rc12(r) = rcns(Pv ; r) + - ~ - 6(r -- L) . (2.53 b)

In view of equations (2.53), we find that y l z ( r ) is very simple in the infinite dilution limit, so that 2 is given in this limit by

2z = YHs(Pv; r = L) . (2.54)

More explicitly,

2 = _1 YHs(Pv; r = L) . (2.55) T

This equation is valuable numerically since for a given P v , it yields and starting value for 2 so that 2 can be solved for numerically starting at Pv = 0 and incrementing Pv in small steps to the desired value. It is also valuable in another context: it yields the expression for the association constant K at infinite dilution in the hard sphere solvent. But first we evaluate K in the double limit Pv--* O,

Pv ~ 0 which would correspond to the dilute gas association constant for reaction (1.1) in the absence of a solvent. From equation (2.47) and the fact that in this double limit 2 ~ I/z, we obtain

7t K o = K ( p U = O, P v = O) -=- ~ (L/o') 3. (2.56)

In the limit Pv ~ 0 with Pv non-zero, equation (2.55) combined with equation (2.47) yields

K~~ = K ( p v = O, Pv) = -~z (L /a )3y~s(Pv; r = L) . (2.57)

Equations (2.56) and (2.57) together give the result

K~176 ) _ K ( p v = O, Pv) = Y~s(Pv; r = L) . (2.58)

Ko K0

This equation, which we have obtained in the context of the PY approximation, is in fact exact for our model. This follows from the use of a relation derived by one of us [123 some time ago that holds in the limit Pv ~ 0 limit for all models in which the solute-solute and solute-solvent pair potentials are equal. In both the association reactions (1.1) and (1.2) considered in this paper, this condition is satisfied. In our notation, Stell's relation, equation (2.2) of [12] is

glj(r) = e x p (--~oij(r)/kBT)Ycc(r), ij = A A , A B , B B . (2.59)

Using this in our definition of K , given by (I.1.12), immediately yields equation (2.58). The same result was also derived by Chandler and Pratt [5] from their formal theory of chemical reaction equilibria on the basis of a series of approximations. Chandler and Pratt argued that since YHs(Pv; r = L ) is of the order 1 0 2


at liquid state densities, this explained the order of magnitude of the shift in the association constant characterizing the dissociation equilibrium of nitrogen dioxide and dinitrogen tetroxide described in the w 1 above.

3. R E S U L T S F O R T H E C H E M I C A L E Q U I L I B R I U M A + B ~ A B I N S O L U T I O N

In this section, we report results for the hard sphere solvent effects on the chemical equilibrium A + B ~ A B for the case L = a/2. The results obtained for L = a/3 do not differ qualitatively from those reported here.

As shown in w 2.2, to obtain results for the model defined by the set of interactions (1.5) for given state parameters Po, Pv and z it is necessary to solve the single non-linear equation (2.43) for the single unknown parameter ~. In w it was shown that 2 has a particularly simple form in the limit Pv--* 0 for arbitrary Pv. Thus, numerically it is convenient to start at the point (Pv, Pv) = (0, Pv) and increment Pv in small steps to the desired value by solving (2.43) at each point using a standard numerical technique (such as false position or bisection).

One particularly revealing way of presenting results on the solvent effect is to consider mixtures at constant temperature (z) and at constant dimensionless total density p* = (Pa + PB + Pc) a3. The mole fraction of solute in the solution is then given by

PA + PB (3.1) Pa + P~ + Pc

Xsolute - -

and the solvent mole fraction by

Xsolven t - - Pc (3.2)

PA + PB + Pc"

(Note that Xsolute is the mole fraction of all A and B monomers and counts both A and B free monomers and the monomers bound together into A B dimers.) For Xsolven t --* 1, we have a solution in which the reacting species are present at infinite dilution, while in the opposite limit (Xsolwnt ~ 0) we recover the model described in I.

We begin by considering ( N ) , the expected number of B atoms bound to a given A atom and defined in equation (2.46). In figure 1, ( N ) is shown as a function of solvent mole fraction at temperature z of 0"01 and at five fixed total densities t/, where t /= (n/6)p*. At the lowest density ( t /= 0"1), which corresponds to a fairly dense gas state (since in these reduced units t / ~ 0"15 would represent a typical critical density for systems exhibiting l iquid-vapour coexistence), the complete association line ( ( N ) = 1) is not reached either in the pure reacting species case Xsolve, t = 0 or in solution. Clearly, at fixed temperature, ( N ) is a monotonic decreasing function of Xsolw m since as reacting solute molecules are replaced by solvent molecules the degree of binding into pairs which can take place is increasingly limited.

In figure 2, we consider K / K o , where K0, discussed in w above, is the value which K would have in the ideal gas state in the absence of a solvent (i.e. K 0 equals K in the dual limit Pv ~ 0, Pv --* 0). The temperature is again fixed at z = 0"01 and there are five representative values of t/ spanning the range from moderately dense gas (t/-- 0"1) to very dense liquid (r /= 0"5).


1.0 I I I I I

0-8 o.a ~ \ < ~ >

0-6

0.4

0.2

0"0 I I I I

0 '0 0.2 0-4 0.6 0.8 1.0 Xsolvent

Figure 1. T h e expected n u m b e r ( N ) of species B a toms d i s tan t L = a/2 away f rom a species .4 a tom for the react ion (1.1) in a ha rd sphere solvent at t empe ra tu r e z --- 0'01 and at total reduced densi t ies ~ /= 0"1, 0-2, . . . , 0.5. T h e expected n u m b e r is shown as a func t ion of the mole f ract ion of solvent in the mix tu r e def ined by equa t ion (3.2).

T h e e x p e r i m e n t a l r e s u l t s s h o w n in t h e t a b l e in w 1 w o u l d c o r r e s p o n d , in t h i s

m o d e l , to t h e r e g i m e Xsolven t -~ 1, i .e. t h e r e g i m e i n w h i c h t h e r e a c t a n t s a re p r e s e n t

at n e a r i n f i n i t e d i l u t i o n in t h e h a r d s p h e r e s o l v e n t . M o r e o v e r , t h e s o l v e n t s in t h i s

I O 0 ~ ] r ~

'~ I 0-3 0.4 �9

6(3

4 0

2O

0 0.0 0.2 0-4 0"6 0.8 1.0 Xsolvent

Figure 2. T h e ratio of the equ i l i b r ium ratio K to its ideal gas value K 0 for the react ion (1.1) in a ha rd sphere solvent at t e m p e r a t u r e ~ = 0-01 and at total r educed densi t ies t / = 0.1, 0 '2, . . . , 0-5 for a toms capable of b i n d i n g at L = a/2. T h e ratio K / K o is s h o w n as a func t ion of the mole f ract ion of so lvent in the mix tu re def ined by equa t ion (3.2).

1330 P . T . C u m m i n g s and G. Stel l

table are i n the l iqu id s tate so tha t q- - -0"4-0"5 . W e see then tha t our m o d e l exh ib i t s ve ry s t rong so lvent effects in this reg ime, wi th K shi f ted d r a ma t i c a l l y f rom its ideal gas va lue and b y an a m o u n t which is of the o r d e r of m a g n i t u d e of the e x p e r i m e n t a l resul ts . W e obse rve tha t f igure 2 is cons i s t en t wi th ou r in tu i t ion , n a m e l y that for a g iven total dens i ty , K is la rger (and the ex ten t of reac t ion grea ter ) when the re is m o r e reac tan t p r e sen t (or, in o the r words , Xsolve. t is lower) . F o r i ndus t r i a l p rocesses in wh ich the ex ten ts of r eac t ion are d o m i n a t e d by s ter ic effects (as in the m o d e l here) it is c lear tha t to op t imize y ie ld f rom a chemica l reac t ion concen t r a t i ng the reac tan t s is a m o r e effective s t ra tegy than p e r f o r m i n g the reac t ion in solu t ion . (Th i s conc lus ion assumes , of course , tha t as in ou r m o d e l tha t the pu re so lven t and p u r e reac t ing so lu te are in the same c o n d e n s e d phase at the specif ied t e m p e r a t u r e and dens i ty . In prac t ice , one is m u c h m o r e l ikely to be con f ron t ed wi th a specif ied t e m p e r a t u r e and p re s su re such as a m b i e n t t e m - p e r a t u r e and p r e s s u r e - - a t wh ich the reac t ion is to be p e r f o r m e d and at wh ich the reac tan t s are gaseous and the so lvent l iquid . T h i s is the case for the chosen example of d in i t rogen t e t rox ide equ i l i b r i um. Clear ly , in these c i r cums tance , the cons t an t z and q l ines in f igure 2 w o u l d r ep re sen t p r e s su re changes of m a n y o rde r s of m a g n i t u d e so tha t the s t ra tegy of concen t r a t i ng reac tan t s d e s c r i b e d above becomes imprac t i ca l . )

P e r h a p s the mos t in te res t ing curve in f igure 2 is the m o d e r a t e dens i ty ( t / = 0-1) curve wh ich suggests tha t the effect of so lvent and p resence of reac tan ts are nea r ly compensa t i ng . T h a t is, the value of K o b t a i n e d when on ly reac tan ts are p r e sen t (xsolve, t = 0) is m u c h the same as the va lue o b t a i n e d at inf ini te d i lu t ion .

In f igure 3, we show the rat io K / K o as a func t ion of the so lven t dens i ty for six values of so lu te dens i ty (vary ing f rom 0 to 0"5). A t each so lu te dens i ty as the so lvent dens i ty is inc reased the value of K / K o increases , the r a p i d i t y wi th wh ich this occurs be ing d e p e n d e n t on the dens i ty of the reac t ing solute. F o r each solute dens i ty , except t/v = 0 c o r r e s p o n d i n g to inf in i te ly d i lu te reactants , t he re is a so lvent dens i ty at wh ich the c o m p l e t e associa t ion l imi t is reached . At any g iven solute dens i ty , the e q u i l i b r i u m rat io K can be va r ied over several o rde r s of m a g - n i t ude wi th smal l changes of dens i ty . T h i s sugges t s tha t one m i g h t be able to con t ro l reac t ion equ i l i b r i a (and thus reac t ion rates) b y va ry ing so lvent dens i ty j u s t as one can con t ro l so lub i l i ty over a wide range in a supe rc r i t i ca l so lvent t h r o u g h so lvent dens i ty changes , the basis of superc r i t i ca l ex t rac t ion t echn iques [-13-]. I t is t he re fo re not su rp r i s i ng there have been recent sugges t ions tha t superc r i t i ca l f luid solvents m i g h t be used to con t ro l reac t ion ra tes in i ndus t r i a l l y i m p o r t a n t o rganic syn thes i s reac t ions [14].

W e now focus a t t en t ion on the s t ruc tu re of the f luid m i x t u r e u n d e r g o i n g chemica l react ion . W e begin b y looking at resul ts for the low total dens i ty (~/ = 0"1) for th ree so lvent mole f rac t ions : x~olven t = 0"0, 0"4 and 0"8. ( F o r Xsolven t =

1.0, all the radia l d i s t r i b u t i o n func t ions are equal for r > a as is c lear f rom w In f igure 4. we p o r t r a y the radia l d i s t r i bu t i on func t ions gal(r)= gaA(r)= gBB(r), g12(r) = gA~(r), g13(r) = gAc(r) = gBc(r) and g33(r) = gcc(r) for z ----- 0-01, q = 0"1 and Xsolven t = 0'0. T h u s , the gll(r) and g12(r) are the same as those which w o u l d be o b t a i n e d in the absence of a so lvent and are s imi la r to those d e s c r i b e d in I : i.e. t he re is a cusp at r = a + L = 3a/2 and g l2 (a ) is less than g l l ( a ) i nd i ca t i ng a lower p o p u l a t i o n of un l ike species pa i rs at contact . (No te tha t on ly the i n t e rmo lecu l a r pa r t ofgij(r ) is shown ; the del ta func t ions in g12(r) and gx3(r) at r = L = a/2 have not been shown. ) T h e so lvent species, species 3, wh ich is p re sen t in th is m i x t u r e


100 i ' �9

/ t / >5 0"4 0-3 0-2 H K / K o

6O

40

20

0"0

0 0"0 0.1 0.2 0"3 0-4 0"5

'r/v

Figure 3. The ratio of the equilibrium ratio K to its ideal gas value K 0 for the reaction (1.1) in a hard sphere solvent at temperature z = 0"01 and at reduced solute densities qv = 0-J, 0'2, , . . , 0-5 for atoms capable of binding at L = ~/2. The ratio K / K o is shown as a function of the reduced solvent density ~v �9

at infinite d i lu t ion, exhibi ts considerable short range order. C o m p a r e d to a one c o m p o n e n t hard sphere fluid at vo lume fraction q = 0"1 (equal to the total densi ty of the react ing fluid mix tu re cons idered here), the solvent molecules are shifted preferent ia l ly to shorter separations, perhaps ind ica t ing a t endency for solvent molecules to wan t to aggregate in to a ' solvent r i c h ' phase. (Of course, no phase separat ion takes place bu t the po in t be ing made here is that the steric in teract ions considered in this model migh t conceivably con t r ibu te to phase separat ion in a system with more realistic interact ions.) At X~otv~nt = 0"4, shown in figure 5, we see that g12(a) has not increased percept ibly , bu t that g l l (a )=g22(a) has increased (which is to be expected since the n u m b e r of species 1 and 2 a toms has increased). T h e solvent is still more s t ruc tured at short range than a hard sphere

fluid bu t is less so than at X~olwnt = 0"0. F igure 6 shows xso~ve, t = 0-8. T h e most p r o m i n e n t feature in this figure is the b e g i n n i n g of the approach of each of the correlat ion func t ions to the single hard sphere correlat ion funct ion , a s t ruc tura l reflection of the fact that at inf ini te dilution (X~ol,~, t = 1) no d imers are presen t and a solute molecule canno t d is t inguish be tween another solute molecule and a solvent molecule . T h i s s t ructural feature is seen clearly in figure 7 which shows a reasonably high total dens i ty state (~/= 0-3, ~ = 0"01) with x~o~w,t = 0"993. In this figure it is clear that all the correlat ion func t ions are close to the pure hard sphere fluid at vo lume fraction 0"3.

In contrast , figure 8 shows the s t ruc ture at the same densi ty bu t at x~olv~, t = 0"699, very close to the solvent mole fract ion at which total association ( ( N ) = I) takes place. As in the lower dens i ty case, the solvent s t ruc ture exhibi ts s t ronger shor t range order than the cor responding hard sphere fluid.

At higher densi ty near ( N ) = 1, as in figure 9, we see some similari t ies between the correlat ion func t ions : the so lven t - so lven t correlat ion is close to the

1 3 3 2 P . T . C u m m i n g s a n d G . S t e l l

I 6 ~ T r T - - " T - r - -

1"2

gi j(r)

0 .8

0 .4

0-0 0 I 2 3 4

r/o" F i g u r e 4. T h e radial d i s t r i b u t i o n f u n c t i o n s g12(r) ( ), gll(r) = g22(r) ( - - - ) , g13(r) =

g23(r) ( . . . . . . ), g33(r) ( ) and gns(r) ( - - - ) for t he r eac t ion (1.1) in a h a r d s p h e r e so lven t at t e m p e r a t u r e z = 0 '01, to ta l r e d u c e d d e n s i t y q = 0"1 a n d Xsolve. ~ = 0"0 for a t o m s capab le o f b i n d i n g at L = a/2. T h e sol id a n d d a s h e d cu rves w i t h t he lower c on t a c t va lues are ga2(r) a n d gHs(r) respec t ive ly . T h e h a r d s p h e r e radial d i s t r i b u t i o n f u n c t i o n is ca lcu la ted at r e d u c e d d e n s i t y q. N o t e t ha t c o m p o n e n t s 1, 2 a n d 3 co r r e - s p o n d to spec ies A, B and C.

I" 6 1 X T V T V r v ~

1.2

gij(r)

0 ' 8

0 . 4

k_

0 - 0 ~ L 1 0 2 3 4 r/o"

Figu r e 5. T h e radial d i s t r i b u t i o n f u n c t i o n s gij(r) for t h e r eac t ion (1 . l ) in a h a r d s p h e r e so lven t at t e m p e r a t u r e z = 0"01, tota l r e d u c e d d e n s i t y / / = 0"1 and Xsolve, t = 0 '4 for a t o m s capab le o f b i n d i n g at L = (7/2. T h e l egend is t h e s a m e as for f igure 4.


1.6

1"2

gij(r)

0"8

0.4

"% x:

O'C ~ . . - - . . 1 - - - - L ~ 0 2 3 4 r/o-

Figure 6. T h e radial d is t r ibut ion funct ions gq(r) for the reaction (1 . l ) in a hard sphere solvent at t empera ture ~ = 0"01, total reduced densi ty r / = 0'1 and Xsolv,, t = 0"8 for a toms capable of b ind ing at L = a/2. T h e legend is the same as for figure 4.

3-2

2"4

gij(r)

1"6

0"8

' [ ' I ' ! '

0-0 , , I i I , 0 2 3 4

r/or

Figure 7. T h e radial d is t r ibut ion funct ions gij(r) for the reaction (1.1) in a hard sphere solvent at t empera ture T = 0"01, total reduced densi ty t / = 0"3 and Xso~,en t = 0'993 for a toms capable of b inding at L = a/2. T h e legend is the same as for figure 4.

1 3 3 4 P . T , C u m m i n g s a n d G . S t e l l

3 -2 - - r [ r ~ 1

2 - 4 gij(r)

1"6

0"8

0.0 ~ ~ J 0 2 3 4

r/o" F i g u r e 8. T h e rad ia l d i s t r i b u t i o n f u n c t i o n s gij(r) for t h e r e a c t i o n (1.1) in a h a r d s p h e r e

s o l v e n t at t e m p e r a t u r e �9 = 0"01, to ta l r e d u c e d d e n s i t y t / ~ 0-3 a n d Xsolven I = 0-699 for a t o m s capab l e o f b i n d i n g at L = g /2 . T h e l e g e n d is t h e s a m e as for f igure 4.

4

3

gij(r)

2

I ' I ' I '

0 ' , I , I J 0 2 3 4

r /o-

F i g u r e 9. T h e rad ia l d i s t r i b u t i o n f u n c t i o n s g,j(r) for t h e r e a c t i o n (1.1) in a h a r d s p h e r e s o l v e n t at t e m p e r a t u r e �9 = 0-01, to ta l r e d u c e d d e n s i t y t / = 0-4 a n d Xsomv=, t = 0"901 for a t o m s c a p a b l e o f b i n d i n g at L = a/2. T h e l e g e n d is t h e s a m e as for f igure 4.

Statistical mechanics of chemical reactions 13 3 5

corresponding hard sphere fluid and the solvent-solute correlation is similar to the 1-1 ( = 2-2) solute-solute correlation. This is in contrast to the results at lower density. This suggests that at high density, where steric forces dominate the interactions, one can deduce that the solvent structure is largely unchanged by the presence of the reacting solutes, a deduction which is likely to be general rather than specific to the model considered here. T h e similarity between solvent-solute and 1-1 solute-solute correlations is not general and is at tr ibutable to the equality of the solvent-solute and 1-1 ( = 2-2) interactions.

4. ANALYTIC SOLUTION OF THE P Y APPROXIMATION FOR THE REACTION

2A ~,-~-A 2 In SOLUTION

In this section, we consider the solution of the PY approximation for the two-component mixture with pair interactions given by equations (1.6); species A and C molecules will be denoted by the indices 1 and 2 respectively. The PY approximation (2.2) for the binary mixture with pair interactions (1.6) yields

hit(r ) = - 1 , r < a,

i j= 12, 22,

2L 6 r h l l ( r ) = - 1 + - ~ ( - - L ) , r < a , (4.1)

cu(r ) =0, r > a,

l <~i,j<<.2.

To be consistent with w 2, we shall refer to Pl as Pv, the density of solute atoms, and P2 as Pv, the solvent density. T h e Baxter factorization for mixtures is applicable to the mixture with closures (4.1), yielding the factorized equations (2.4). In the present case, equation (2.4 a) yields five distinct equations,

rhii(r ) = -q ' lx(r ) + 2rcpvql 1 | hl i (r) + 2xpvql 2 | hi2(r), (4.2a)

rh12(r) = --q~2(r) + 2~pvqi 1 Q hlz(r ) + 2Zcpvql 2 ~) h22(r), (4.2 b)

rhl2(r) - -q~ l ( r ) + 2xpvq21 | + 2~pvq22 (~ hl2(r), (4.2c)

rh22(r) = --q22(r) + 2xPvq21 ~) hi2(r) + 2~Pvq22 ~ h22(r), (4.2 d)

where f | g(r) is defined in w 2. As usual, our task is to determine the functional form of the Baxter q-functions qu(r) relate the parameter ). to the temperature z and the densities Pv and Pv, derive an expression for the association constant and to provide an algorithm for comput ing the radial distribution functions gu(r). (As in w 2, the latter is based on a str ightforward extension of Per ram's method [1, 2, 11] and so will not be covered here.) The results in this section are given in explicit form for L = el2; as in w 2, the corresponding results for L = a/3 wilt not be given since they can be derived easily by combining the analysis of this section with the results for L = a/3 contained in [-2].

4.1. The Baxter q-functions

We begin by considering the functions qu(r), j = 1, 2. Substi tut ing the closures (4.1 a) and (4.1 b) into equations (4.2 a) and (4.2b), we obtain the following


differential-difference equations : ).L z

q'l i(r) + p [ q ~ ( r + L) - q l~(r - L)] + - -~ - tS(r - L ) = a l r + b~, (4.3)

q'12(r) = a i r + bl , (4.4)

where p is given by equation (2.9) and

a i = 1 -- 2 ~ p v qil(t) dt - 2rcpv qi2(t) dt, (4.5 a)

bi = 2rcPv fftqll(t) dt + 2rCpv fotqlz(t) dt. (4.56)

The boundary and discontinuity conditions on the functions qlj(r) are :

2L 2 qil(~) = qi2(~) = 0, q l i ( L - ) = qi l (L+) + 1--2- (4.6)

From equations (4.4) and (4.6), the functional form of qa2(r) is given by

q i z ( r ) = � 8 9 = 0 , r < ~,.} r~> (4.7)

We determine qil(r) using the method of ['2] and simply quote the result here for L = a/2.

ai ai bl - - - - r + (1 - v/2) - - - + A i cos (pr) + B1 sin (pr), qll(r) = p p-~ p

O"

O < r < ~ ,

al al bl -- r + (1 -- v/2) -- - - + A1 cos [p(r - a/2)] p ~ p

t7

- B i s i n [ p ( r - - a / 2 ) ] , ~ < r < a . (4.8)

Equation (4.8), when combined with equations (4.5) and (4.6), yields the following linear equations for the parameters al , bl, A 1 and Bi :

1 2 - 2 - a l + - b ' l - A ' i c - - B ] ( 1 + s) - (4 .9) v v 4 8 '

1 1 t v2 (1 + v/2)a x + v b 1 -- A ' l s -- B' ic O, (4.10)

[ 1 2 ~ l v ( 1 - v / 4 ) 1 12qv a i 1 + v2 4qv - - 6 q v b ' l + ~ v

• [(1 -- c + s)A] + (1 -- c - s)B'l] = 1, (4.11)

ai L v---- 5 - + +b'a 1 - v + 2qv

v2 - 1 + 5 + ( 1 - v ) c + 1 + s A'I

~- 1+ 1 - ~ ~+(1-v )~ B'~=o, (4.12)


where

bl A1 B1 (4.13) bl -- 0-, A1 -- 0.2, Btl - 0" 2

and s, c are given by equat ions (2.24). Equa t ions (4.9)-(4.12) can be wr i t ten in the fo rm

M

[a 1 [ - ~ / 4 8 1

i A i '

I_Bi where the e lements of the equat ion (4.14) is g iven by

mat r ix M are

(4.14)

defined obviously . T h e solut ion of

1 ( ~ 8 2 ) a l = ~ T l l + T31 , (4 .15a)

b'l = ~ Tx2 + Ta2 , (4.15b)

A'l = ~ T13 + T33 , (4.15 c)

) B' 1 = A T14 + T34 �9

Here , A is the d e t e r m i n a n t and Tii the ij cofactor of M given by

6 ' /v ' tv [8 - 4v + v 2 A = A ~ + - - 7 - 8c + (8 - 4v + v2)] + ( r / ~ - 2nv)(1 + s).

T i l = TOt + -

Tt2 = T~ +

T13 = T~ +

+ ( 1 2

T14 = T O +

+ ( 1 2

T31 = T~ +

T32 = Z~ +

T33 = T~ +

12tlvtlv [4v -- 6 + (6 -- v)c + ( - - 6 + 2v)s], 1:2

6tlvqv [8 _ 5v + ( _ 8 + v)c + ( 8 _ 3v)s], - 7 -

6qvqv [12 -- 6v + v 2 7 + ( - - 1 2 + 2v -- v2)c

- - 10v + v2)s] + (q2v -- 2qv)(1 + s),

6qvtlv [12 -- 4v + ( - - 1 2 + 10v -- v2)c - V -

+ 2v -- vZ)s] + (q2 v -- 2qv)(1 + s),

2qv(1 + s),

~,/v(1 + s),

fl__.Lv I-4 -- v -- 2vc + (v -- 4)s], 2v 2

(4.15 d)

(4.16)

(4.17 a)

(4.17b)

(4.17c)

(4.17d)

(4.18 a)

(4.18b)

(4.18 c)


Ta4 = T~ + fl-Z-r [(4 -- v)c -- 2vs]. (4.18d) 2v 2

T h e superscr ipt ' 0 ' in equat ions (4.16)-(4.18) is used to denote the value of the cor respond ing parameter in the limit r/v ~ 0, and so A ~ and the ~ are given by equat ions ( I .A.1) - ( I .A.9) with f = 1 and r / = r/t: (or, equivalently, r = r/v ). Th i s completes the analysis for qlj(r), j = 1, 2. We now tu rn our a t tent ion to the funct ions q2~(r), j = 1, 2. Subs t i tu t ing the closures (4.1) into equat ions (4.2 c) and (4.2 d), we find

q'21(r) + p[q21(r + L) - q21(r - L)] = a 2 r + b2,

where

q~2(r) = a2r + b2,

(4.19)

(4.20)

a2 a2 b2 r - - ( 1 - v / 2 ) + - - + . 4 2 cos [ p ( r - a / 2 ) ] - - B 2 sin [ p ( r - - a / 2 ) ] , p + p2 p

G - < r < a. (4.24) 2

T h e parameters a2, b2, `42 and B 2 satisfy four linear equat ions which can be wri t ten in the fo rm

a21 Ii] b~ 0 ,

LB~]

(4.25)

a 2 = 1 -- 2npv q21(t) dt -- 2npv q22(t) dt, (4.21 a)

bs = 2npv I : t q21 ( t ) dt + 2npv fo tq22( t ) dt. (4.21 b)

T h e b o u n d a r y condi t ions on the funct ions qlj(r) are

q21(tT) = q22(0") = 0 (4.22)

and we note that q21(r) is con t inuous at r = L. T h e solution of equat ion (4.20) is immedia te and is given by

q22(r) = �89 -- ff)2 + b2(r - a), r < a,~ (4.23) ~.J = 0 , r~>

T h e solution of equat ion (4.19) with closure (4.22) and cont inui ty at r = L is very similar to the solution of equat ion (2.33) in w 2, and so we s imply quote the results for L = a/2. T h e funct ion q21(r) is given by

a2 q21(r) = - - --p r + ~2 2 (1 - v/2) -- b2p + A 2 c o s (p r ) + B E s i n (p r ) ,

t7 O < r < ~ ,

S ta t i s t i ca l mechanics of chemical reactions 1339

where

b2 A2 B2 !

b E - 0 ' ' A ~ - ~2, B ~ - ~2" (4.26)

Thus, the expressions for the determinant and cofactor given in equations (4.16) and (4.18) are still applicable and we find that

T31 (4.27 a) a2 - - A '

T32 b ~ - A ' (4.27b)

T33 (4.27 c) A ~ - - A '

T34 (4.27 d) B ~ - A '

where T3i , i = 1, 2, 3, 4, are given by equations (4.18). This completes the derivation of the functional forms for the Baxter q-functions for L = o'/2.

4.2. Determina t ion of the parame te r 2

Application of the PY approximation (2.2) yields

,~'g =

where F i , i = 1, 2 are given by

F a, 3a, F i = 24~/i[~-- ~ 8v 2 L Z V ~

2 a 1 + 2vA'x -- F1 -- F 2 , (4.28)

' ' b ' aib~ bi A'is - ~ B'i(1 - c) 2v 2 v v

a i + -~ AI(1 -- c + 2s - vs) + ai R ' t 9 -- v/2) V2 ~i',-- - - 2c -- s + vs

A'i 2 A'i2 B'i 2 B'i 2 ] + T ( v + s i n v ) + T ( 1 - c ~ (4.29)

- - I

and where it is understood that ~/1 = qv and q2 = q v . Equation (4.28) constitutes a single non-linear equation for the parameter 2 which can be solved numerically.

4.3. Express ion f o r the association constant

The association constant for the reaction (1.2) is given by

h - PA2 (4.30) (pA) 2"

The expressions given in [-2] for the dimensionless association constant K = k / a 3

and for the expected number ( N ) of A atoms distant L f rom a given A atom remain valid and are simply quoted here.


( N ) = 2r/v ).(L/tr) 3, (4.31 )

rc~(L/0") 3 K - 6[,1 - ( N ) ] E (4.32)

Note while the expression for ( N ) given in equat ion (4.31) is the same as that for association between unlike species, equat ion (2.46), the 3 in the denomina to r of equat ion (2.47) is replaced by a 6 in equat ion (4.31). Th i s difference arises f rom dist inguishabil i ty considerat ions and is discussed in more detail in [-2].

4.4. The limit of vanishing solute density

T h e analysis for the limit of vanishing solute densi ty Pv ~ O, where the reacting species A is present at infinite di lut ion in the hard sphere solvent is m u c h the same as that presented in w 2.5 above and will no t be repeated. T h e final result for )~(r/v = 0; r/v ) is the same as that given by equat ion (2.5) while the cor responding result for K is

7z K o = K ( p v = O, Pv = 0) = ~zz (L/a)3" (4.33)

T h e expression given in w for the ratio between the association constant at infinite di lut ion and its value in the ideal gas state still holds for the present homogeneous association reaction as an exact result f rom the relation of Stell [-12] given in that section.

5. RESULTS FOR THE CHEMICAL EQUILIBRIUM 2A ~--- A E IN SOLUTION

T h e results obta ined for the chemical equi l ibr ium association ratio K for the model of the reaction (1.2) solved in w do no t differ quali tat ively f rom those descr ibed in w 3 for the model of the reaction (1.1) and so we do not dwell on them at any great length. In figure 10, the ratio K / K o is shown for reaction (5.1) for z = 0"05 as a funct ion of Xsolven t at various total densities t/. As was found in w 3 for reaction (5.2), for l iquid-like systems (r//> 0"4), the ratio K / K o is large ( ~ 20-30) at infinite di lut ion of react ing solute (Xsolven t = 1) and increases rapidly as x,ojven ~ is decreased (cor responding to the rep lacement of solvent molecules with solute molecules capable of reaction). Again, as in w for r / = 0-1, K / K o shows little variat ion with solvent mole fraction.

T h e correlat ion funct ions (g11(r) -- gAA(r), glz(r) ---- gAc(r) and gEE(r) ~-- gcc(r)) exhibit quali tatively the same features as the cor respond ing correlat ion funct ions for reaction (5.2) descr ibed in w In part icular, g22(r) exhibits more s t ructure than the cor respond ing radial dis t r ibut ion funct ion for a hard sphere fluid at densi ty r/, a l though in the case of reaction (5.1) this is evidenced only by the more p rominen t oscillations in gEE(r), the contact value of g22(r) being slightly below the cor responding hard sphere quant i ty (compared to the results in w 3 where the cor respond ing so lvent -so lvent correlat ion funct ion exhibi ted enhanced oscillations and a higher contact value). A typical example is given in figure 11.

Addi t ional interest in our model (5.1) arises f rom the fact that at the complete association limit ( ( N ) = 1) the mode l becomes equivalent to the zero pole approximat ion (ZPA) for a mix ture of hard dumbel l s and hard spheres first


K / K o

I 0 0 - -

80

0":

6O

4O j 0 0"1

D-I 0"2 0-4 0-6 0"8 I'0 Xsolvent

Figure 10. T h e ratio of the equi l ibr ium ratio K to its ideal gas value K0 for the reaction (1.2) in a hard sphere solvent at t empera ture r = 0+05 and at total reduced densit ies t / = 0-1, 0.2 . . . . . 0-5 for a toms capable of b inding at L = ~r/2. T h e ratio K / K o is shown as a funct ion of the mole fraction of solvent in the mixture.

2

gij(r)

i ' I I ' I

IX IX ~ x

\ \

ix ++. \ I \\~: \\

\ ".. \

.~

0 2 3 4 r/o"

Figure 11. T h e radial d is t r ibut ion funct ions g l l ( r ) = gAA(r) ( - - - ) , gl2(r) = gAc(r) (------), g~a(r) = gee(r) ( . . . . . . ) and gns(r) ( - - - ) for the reaction (1.2) in a hard sphere solvent at t empera ture z = 0-05, total reduced densi ty r / = 0'3 and Xsolven t = 0"7 for a toms capable of b ind ing at L = a/2, T h e dashed curve with the higher contact value is gns(r). T h e hard sphere radial d is t r ibut ion funct ion is calculated at reduced densi ty ~/= 0-3.


introduced for the pure dumbell fluid by Morriss and Cummings [15]. In fact, the radial distribution functions shown in figure 11 are at complete association, so that the results given can be regarded as the ZPA prediction for a hard dumbel l / hard sphere mixture with Pd~mb,n~ = Pa/2. Unfortunately, Monte Carlo results are not available for comparison with the theoretical results shown in figure 11.

6. CONCLUSIONS

The study of chemically reactive species using the analytic solution of the Percus-Yevick approximation for models of inter- and intra-molecular interactions which lead to sterically hindered association between molecule pairs has been extended to focus on the effect of a hard sphere solvent on the reaction equilibrium. At liquid-like densities, the order of magni tude of the solvent effect is in accord with experimental results and the trend displayed is consistent with Le Chatelier 's principle. Thus , to the extent which it is possible to be certain without computer simulations to verify the accuracy of the Percus-Yevick approximation, it appears that the present model yields a qualitatively correct picture of solvent effects on chemical reaction equilibria in systems dominated by steric forces.

Acknowledgment is made by P T C to the donors of The Petroleum Research Fund, administered by the American Chemical Society, for support of this research through grant P R F 15589-G5. Computat ional facilities necessary to complete this research were generously provided by the Center for Computer - Aided Engineering and the Academic Comput ing Center at the Universi ty of Virginia.

REFERENCES

[1] CUMMINGS, P. T., and STELE, G., 1984, Molec. Phys., 51,253. [2] CUMMINGS, P. T., and STELE, G., 1985, NIolec. Phys., 55, 33. [3] MOELWYN-HVGHES, E. A., 1964, Physical Chemistry, 2nd revised edition (Pergamon

Press). [4] CHANDLER, D., and PRATT, L., 1976, J. chem. Phys., 65, 2925. [5] CHANDLER, D., 1978, Faraday Discuss. chem. Soc., 66, 184. [6] PERCUS, J. K., and YEVICK, G. J., 1958, Phys. Rev., 110, 1. [7] ORNSTEIN, L. S., and ZERNIKE, F., 1914, Proc. Sect. Sci. K. ned. Akad. Wet., 17, 793. [8] BAXTEr, R. J., 1968, Aust.J. Phys., 21,563. [9] NoBLe, B., 1958, Methods Based on the Wiener-Hopf Technique (Pergamon Press).

[10] BAXTEa, R. J., 1970,J. chem. Phys., 52, 4559. [11] PERaAM, J. W., 1975, Molec. Phys., 30, 1505. [12] STELE, G., 1973,J. chem. Phys., 59, 3926. [13] PAULAITIS, M. E., PENNINGER, J. M. L., GRAY, R. D., JR., and DAVmSON, P., 1983,

Chemical Engineering at Supercritical Fluid Conditions (Ann Arbor Science Publishers).

[14] SUBRAMANIAM, B., and McHUGH, M. A., 1986, Ind. Eng. Process Des. Dev., 25, 1. [15] MoRmss, G. P., and CUMMINGS, P. T., 1983, Molec. Phys., 49, 1103.

statistical mechanical models of chemical reactions iii. solvent effects

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