11835
J. Phys. Chem. 1993,97, 11835-11842
Reaction Volumes in Model Fluid Systems. 2. Diatomic Dissociation in Lennard-Jones Solvents R. Ravi, Luis E. S. de Souza, and Dor Ben-Amotz' Department of Chemistry, Purdue University, West kfayette, Indiana 47907 Received: June 30, 1993'
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A model diatomic dissociation reaction representing Brz 2Br dissolved in rare gas solvents (Ne, Ar, and Xe) is examined using a Lennard-Jones perturbed hard sphere fluid theory. Properties of the appropriate hard sphere reference system are derived from the 'hard fluid" two-cavity distribution function, and attractive solvation energies are calculated using Percus-Yevick reference-interaction-sitemodel (PY-RISM) solvent distribution functions (about the bound and dissociated diatomic). Small changes (10-3095) in the solvent atom-solute atom attractive interaction well depth upon dissociation are found to qualitatively change the corresponding reaction free energies and volumes. Calculations are performed as a function of solvent temperature, density (pressure), and solute-solvent interaction parameters.
1. Introduction
Chemical reaction volumes, which are defined as the pressure derivativesof the corresponding reaction free energies, are often found to depend in a complexway on solvent density, temperature, and molecular structure.'J This sensitivity to solute-solvent interactions makes it impossible to assign a unique numerical value to the volume of a particular chemical reaction, without regard to its detailed coupling to the solvent. In order to better understand the relationship between experimental reaction volumes and molecular interactions, we have carried out an analysis of a model dissociation reaction (Brz 2Br) dissolved in Lennard-Jones (LJ) solvents (Ne, Ar,and Xe). In a previous study' (which will from here on be referred to as paper l), we focused on reaction volumes in purely repulsive (hard sphere) fluids. The results of that study indicate that, even in such relatively simple systems, reactionvolumes are never simply equal to the correspondingisolated reactant volume changes but in fact reflect subtle changesin solvent structure (packing) around the reactants and products. Repulsive fluids do not, however, display all of the interesting complexity of real fluid systems. For example, reaction volumes in repulsive fluids invariably have the same sign as the isolated reactant volume changes,' which is not always the case for experimental systems.2-5 Qualitatively, this more complicated behavior of real systems is, at least in part, due to the influence of attractive solvent-solute interactions. For instance, dissociation reactions (for which the isolated solute volume change is positive) may display negative reaction volumes if the productshave a sufficiently large attractive solvation energy, resulting in a constriction of the solvent around the product species. Such effects have traditionally been interpreted using dielectric continuum solvent models: which clearly do not offer a molecularly detailed picture of solvation. In this paper, we carry out a molecular analysis of solvent (excess) contributions to chemical reaction free energies and volumes by exploring the effects of both short-range repulsive and long-range attractive solute-solvent interactions on a model chemical reaction. The calculations are based on expressions closely related to those of the Weeks-Chandler-Andersen (WCA) perturbation theory,6.' in which long-range attractive interactions are treated as perturbations to the properties of an appropriately chosen hard body reference system. The results are used to illustrate the way in which the delicate balance of attractive and repulsive solvation forces may produce large changes in the magnitude (and even the sign) of chemical reaction free energies and volumes.
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*Abstract published in Advance ACS Absrrocrs, October 15, 1993.
This paper is organized as follows. Section 2 contains a summary of the procedures used to calculate repulsive and attractive contributionsto diatomicdissociation free energies and volumes. Numerical results for the dissociation of Brz in rare gas solvents, as a function of pressure, temperature, and solute-solvent interaction well depth, are presented and discussed in section 3, and general conclusions are drawn in section 4. Details of the calculations are collected in three appendices. 2. Theoretical and Computational Procedures 2.1. Thermodynamic Relations. For a general chemical reaction, the change in the Gibbs free energy, AG,at constant P and T, and the change in Helmholz free energy, AA, at constant V and T,can be expressed as
- prcactantr
(1) where pi represents the corresponding chemical potential. For notational brevity, we shall from here on omit the subscripts on AG and AA. The chemical potential of a solute pi at infinite dilution in a solvent can be written as
AGP,T= ~
V , =Tppmducts
= A(N+l,T,V) -A(N,T,V) (2) where Vis thesystemvolume,Nis the number of solvent molecules, and N + 1 is the total number of molecules in the system, including the solute. If it is further assumed that theintermolecularpotential energy is independent of the internal coordinates of the solute (reactant and product) molecules, then the following separation of the chemical potential is obtained by expressing A in terms of the canonical partition function:
+
+
pi = p]ic"s py (3) where pidcB'is the chemical potential of the solute in a system of noninteracting ("ideal gas") particles, p? results from the confinement of the solute in the system volume, and pf" represents the remainder (the direct solvent contribution to the chemical potential).' The above separation of the chemical potential leads to a corresponding separation of the dissociation free energy
+
+
AG = AP A G ~ " ~ and the corresponding reaction volume is
(F)~ = P&.(Y)~ = Avd"* + ATxc
A V r dAG
0022-3654/93/2097-11835%04.00/0 0 1993 American Chemical Society
(4)
(5)
Ravi et al.
11836 The Journal of Physical Chemistry, Vol. 97, No. 45, 1993
where
and
and
where "=($)T
is the isothermal compressibilityof the solvent, k is Boltmnn's constant, and An is the difference between the number of product and reactant molecules in solution (for a diatomic dissociation An= +l).l Noticethat the'idea1"contribution tothefreeenergy is pressure independent and therefore does not contribute to the reaction volume. Also, the 'density" contribution to the free energy depends only on the equation of state of the pure solvent (through &) and thus is independentof solute-solventinteraction parameters (see paper 1 for further discussion of AVeM).In this paper, we focus on the yexcess" contribution to the free energy and volume of reaction. This is the most chemically interesting contribution, since it is the only one that depends on interactions between the solvent and solute. 2.2. Excess Cbemical Potential C p l ~ u l n t i ~Now ~ . consider the following specific dissociation reaction, carried out in a monatomic Lennard-Jones (LJ) solvent: Br,
-
(8)
2Br
The solvent-solvent LJ interaction potential is
uLJ(r)= 4cs
[):(
l2
-
(91
(9)
where cs and US represent the solvent LJ well depth and diameter, respectively, and r is the separation between solvent atoms. The interaction between a solvent atom (S) and a single solute atom (Br) is also described by eq 9, with LJ parameters, QSB~, and US.B~ (and r equal to the solute atom-solvent atom separation). The intramolecular interaction between the two bromine atoms in Br2 is excluded from our analysis since it is contained in the ideal part of the chemical potential and does not contribute to the excess free energy or reaction volume. In order to calculate the effects of solvation on Br2 dissociation, we use expressions closely related to the WCA perturbation theory.6 In particular, the solvent-solute LJ potential is separated into a repulsive, do)(&and attractive, d l ) ( r ) ,contribution,
u'O'(r)
ESB,
+ fdLJ(r), r < 21/6af&B,
= 0, r 2 21/6af&B,
(loa)
d l ) ( r ) = t(Lj(r) - u ( O ) ( ~ )
( 1Ob)
and
It is customary to introduce a coupling parameter A (0 S A I1) to represent a continuous transition of the pair potential from its reference value to the fully coupled value.
uLJ(r)= U
y r )
+ Au(')(r)
(1 1)
Using the above WCA splitting of the potential, one can write the excess chemical potential of Br and Br2 at infinite dilution in a monatomic solvent as9
where p?' and p z o represent repulsive contributions to the chemical potential for an infinitely dilute Br atom and Br2 molecule, respectively; g&,(f) is the radial distribution function of the solvent about a dissociated Br atom (for a given value of the coupling parameter, A); ~ ~ ~is the~attractive ( ~ interacr ~ tion potential between a solvent atom and a bound Br atom; ggLr($J is the spherically averaged distribution of solvent atoms about a bound Br atom; and re is the equilibrium bond length of Br2. A more detailed discussion of the calculations is presented in the following subsections. Notice that the integrals in eqs 12 and 13 would yield exact results if the exact g'A)(r)were used (at all values of A). Since this is generally not available, the WCA theory invokes the fact that the structure of liquids at high densities is determined primarily by repulsive forces, and thus gcBr(r)= g?B,(r) is assumed. In this work, we use a slightly different approximation, assumingthat the true @(r) is equal to that of the full LJ system (A = 1)
gg;w
(14) g&) = &(r) and = g&dd for an isolated and bound Br atom, respectively. This approximation may bejustified by noting that the integrals in eqs 12 and 13 represent the exact excess average attractive internal energy, Ucl), of Br and Br2. This implies that our approximation to the attractive chemical potential is exact, except for the neglect of the entropy contribution to the attractive free energy change (since A = U - TS). The standard WCA approximation,on the other hand, not only neglects the entropy contribution to the perturbation free energy but, in addition, assumes a less accurate (reference system) distribution function in calculating the attractive internal energy. On the other hand, our calculations make use of the PY-RISM approximation to determine ggB1(r)and g2BI(+c) (see section 2.4), which may introduce additional errors (see section 2.6). Finally, the chemical potentials of Br and Brl are related to each other through the distribution function, y(r),for two cavities representing the two noninteracting Br atoms dissolved in the solvent of interest at an intercavity separation equal tor. At the equilibrium bond length of Br2, r = re, this is closely related to the excess chemical potential change upon dissociation of the diatomic in solution. p g ( r e ) = 2 p r - kT In y(rJ
(15)
Thus, kT In y(re)represents the mean force potential exerted on the Brz by the solvent. Furthermore, the cavity distribution functioncan be factored into a product of repulsive and attractive contributions.10The repulsive contributionto y(r) (and therefore to Apcxc= A B x c )is calculated directly from the 'hard fluid" cavity distribution function,' AM^^^,^ kT In y ~ ~ ( r while ~ o ) , the attractive contribution, ApcxcJ,is calculated using the integrals on the right-hand side of eqs 12 and 13. In other words,
ApeXc = ApcXC$ + Apcxc,l
(16) where ApcXc= 2 p r - p g . The details of these calculations are described below (and in Appendixes I and 11). 23. Repulsive Excess Chemical Potentid. Calculation of repulsive contributions to AGcxCand A P x crequires hard sphere diameters for the Br atom (&,) and the solvent atom (ds).l These are calculated from a generalization of the WCA prescription to mixtures, as first suggested by Perram," and shown to be reasonably good for predicting thermodynamic properties of mixtures.11J2
)
The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 11837
Diatomic Dissociation in Lennard-Jones Solvents In the particular case of an atom (Br) at infinite dilution in an atomic solvent (S),the hard sphere diameters of the solvent and solute atoms are defined by
TABLE I: Atom-Atom U Parameters solvent atom4olvent atomu Br atom-solventatom* solvent US (A) e3 (K) US-Br (A) S-Br (K) Ne 2.749 35.6 3.324 54.6 3.405
Ar
119.8
221 Xe 4.100 From ref 22. Derived as d e ”
where yHS(r) is the cavity distribution function in the hard sphere reference fluid (which depends parametrically on the densities and hard sphere diameters of all the system components), u s ( r ) is the repulsive (reference) part of the LJ potential for the solvent-solvent interaction, uFBr(r)is the repulsive part of the LJ potential for the Br atom-solvent interaction, and uZiI(r) is the corresponding hard sphere potential.
HS
UsBr(r)
= +=, I (dBr + 4 ) / 2 = 0, r > (dBr+ ds)/2
(20)
In practice, ds is obtained by minimizing the integral on the left-hand side of eq 17. This value of ds is substituted into eq 18, which is in turn solved for der. In these calculations, the “hard fluid” mode11J3is used to approximateyf;(r) in the region where the hard sphere potential is infinite. Outside this region, the Verlet-Weis mixed hard spheredistribution functionis used.14 As in previous work,15analytical representations of the dependence of ds and dBron T and p have been generated for each set of LJ parameters (see Appendix I1 for details). Analytical expressions derived by Nicolas et a1.I6 for the equation of state of a LJ fluid are used to determine pressure, P, and / 3 values ~ needed in the calculation of the pressure dependenceof AGXcand APxc (eq 7). 2.4. Attractive Excess Chemical Potential. To evaluate the integrals in eqs 12 and 13, which represent the attractive contribution to AGexc,we must compute the solute atom-solvent atom pair correlation functions for two cases: (i) an isolated Br atom in solution and (ii) a Br atom in a Brz molecule in solution. In both cases, the solute is assumed to be at infinite dilution. For case (i), the set of Ornstein-Zernike (OZ) integral equations for atomic fluids is solved with a Percus-Yevick (PY) closure using the algorithm of Balk.” Balk‘s method is an extension, to infinitely dilute solutions, of Gillan’s a1gorithml8 for neat monatomic fluids. To model the structure of molecular fluids, Chandler and AndersenI9proposed a matrix of site-site Omstein-&mike (SSOZ) equations along with a hard sphere closure, collectively known as the RISM equations. For case (ii), we used a closely related approximation, termed the PYRISM approximation, in which the SSOZ equations are solved with PY closure, for the assumed LJ potentials, using the program written by Muralidhar et al.zo This program is based on the algorithm of Morriss and MacGowan,Z1which is an extension of Gillan’s algorithm to mixtures of molecular f l ~ i d s . l ~ J ~ . ~ ~ The following additional details of the case (ii) calculations are noteworthy. First, the infinite dilution solvent distribution function is approximated by using a very low mole fraction of Brl (O.OOO1) in the algorithm that is meant for mixtures at finite concentrations. Second, there was no problem in obtaining convergence at the lowest density for each set LJ parameters and temperatures. For each successive higher density, the results from the previous lower densityare used as initial guesses. Third, the LJ well depth for the interaction of the two bromine atoms on different molecules is set to zero. For both cases (i) and (ii), a grid point spacing of dr = 0.040s and a total number of 320 grid points are used, yielding an upper
3.441 3.709
161 25 1
in Appendix I.
TABLE II: Pohrizrbilitia nod Ionization Energiesn SpCCiCS
Brz Br
Ne Ar
Xe
a (44’) 7.02 3.05 0.3956 1.641 1 4.044
I (eV) 10.54 11.814 21.564 15.759 12.130
integration limit of 12.760s. Finally, to evaluate the integrals in eqs 12 and 13 (with eq 14), an upper integration limit of 6.40s is chosen. To faciIitate integration, a piecewise quadratic polynomial fit to the corresponding distribution function is employed. These values and procedures are found to yield accurate results and efficient convergence. 2.5. SoluteSolvent Interaction Parameters. In order to evaluate the solute-solventinteraction parameters needed in the calculation of A?lGcxcand A P C , two self-consistency criteria are applied. These relate the propertiesof Br2 modeled as a spherical LJ particle with those of Br2 modeled as a diatomic composed of two LJ atoms. The details of these calculations are given in Appendix 1. Briefly, the LJ diameter, OS+, representing the interaction of a single Br atom (bound in Br2) and the rare gas of interest is obtained by equating the excluded volume of the spherical and diatomic models for Br2. The corresponding well depth, ES-B~, is then obtained by equating the excess internal energies of sphericaland diatomic models for Brz dissolved in the same rare gas (in the zero density limit). The input solvent LJ parameters, us and 6,and the resulting interaction parameters, and fS-Br,are listed in Table I. The above solute atom-solvent interaction parameters pertain to the interaction of a bound Br atom with a rare gas solvent. In general, these parameters are not expected to be the same as those for the interaction of a dissociated Br atom with the same solvent. Althoughthe interaction diameter, U s B r , may not change significantly upon dissociation, this is not expected to be the case for the well depth, cS.Br, as illustrated by the following argument, based on the difference in the polarizabilities of Brz and Br. The dispersion interaction energy, 4, between two particles i and j separated by a distance r is approximatelyZZ
where It or j and ai or j represent the ionization energies and the polarizabilities, respectively, of the two particles. If, in addition, it is assumed that the negative term of the LJ potential, 4q-ju1-,6, is proportional to then it follows that
where t$errUSB~, and 7 % refer ~ ~to the interaction of a dissociated Br atom and a rare gas atom. The right-hand side of this expression may be evaluated directly using eq 21 along with the parameters in Table 11, and all parameters on the left-hand side, except CS-Brr are also given (see above). Thus, eq 22 can be used to estimate % B ~ for the dissociated Br atom-rare gas atom interaction. Such calculations (combined with €S-& values for the bound Br atom given in Table I) suggest that the well depth for the interaction of a Br atom and a rare gas atom increases by about 30% upon dissociation ( A E S B r = 30%). In the following
The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 5
Ravi et al.
.......................
c
1
AP+A+A
-5001 _--
Repulsive Attractive Total
-0004
10
0.0
0.2
0.4
0.6
0.8
P 0;
Figure 1. Comprisonofthecalculatedfreeenergyofdis.dation (curves) with MC computer simulationmeasurements (points) for a system at 7" = T ( k / e ) = 1.5, in which the LJ parameters of the diatomic atoms are the same as those of the solvent atoms. The open points correspond to simulation data24z6for a system in which there is no change in solutesolvent attraction upon dissociation, and the solid points reprmnt simulation r e s ~ l t s ~for ' . ~a~ system in which there is a 50% increase in solute-solventattraction upon dissociation (see Appendix 111for details). The reaction free energy values are reduced to the solvent-solvent LJ well depth.
r-7 ,.......'_.........
,..
.-
I....
.
-2001
1
.
.I"
Figure.2. Calculatedpressuredependenceoftherepulsive(dashedcurves), attractive (dotted curves), and total (solid curves) excess reaction free energies and volumes for the dissociation of Brz in Ar at Ts = T(k/c+) = 2. The two left-hand subpanels (a and c) correspond to a system in which the solvent-solute LJ itlteraction strength, t k - ~does ~ , not change upon dissociation, and the two right-hand subpanels (b and d) contain results for a system in which the Br atom-solvent atom interactionstrength is assumed to increase by 20% upon dissociation.
section, calculations will be performed assuming values of 0% 5 A s ~I B 30% ~ in order to explore the influence of changes of this magnitude on reaction free energies and volumes. A value of A t s =~20% ~ will be taken as a reasonable conservativeestimate of the true A t s in ~ the ~ temperature- and solvent-dependent calculation. In all these calculations, U ~ will B be ~ assumed to be the same for a bound and dissociated Br. not change upon dissociation, AEA,B, = 0%. The two right-hand 2.6. Comparison with Simulation Results. Although no panels represent results obtained when the Br atom-solvent experimental or computer simulation results are available to interaction is assumed to increase by 20% upon dissociation. In directly test the accuracy of our predictions for the effects of rare each subpanel, the repulsive (dashed curves), attractive (dotted gas solvents on the dissociation thermodynamics of Br2, simucurves), and total (solid curves) excess free energy and volume lations of closely related model systems have been performed. changes are indicated. Thesearesystems in which the LJdiametersof thesolutediatomic With A c ~ - B = 0%, ~ the attractivecontribution to the freeenergy atoms are the same as those of the solvent. Results for the cavity of reaction (Figure 2a) is relatively small, and the total free energy distributionfunction in this combined with simulations change tracks the pressure dependence of the repulsive contriof the chemical potential of a LJ sphere in a LJ fluid as a function bution. The increase in AGexcwith pressure in this case is of solvent-solute interaction well depth, t ~ - ~may ~ , be 2 ~ used to dominated by the work performed in pushing solvent atoms out derive the dissociation free energy of a diatomic as a function of of the way to make room for the larger volume product A~s-B~. (dissociated)solute species. Thisbehavior issimilar to that found In particular, we compare our calculations with simulation for diatomic dissociations in hard sphere fluids, as discussed in results for the dissociation of a diatomic composed of two atoms paper 1. with the same LJ parameters as the solvent, where the solventA moderate increase in solvent-solute coupling upon dissosolute atom interaction well depth either stays the same ( A t s ~ ~ ciation, A t k - ~=~ 20%, produces a dramatic change in the = 0%)or increases (Atser = 50%)upon dissociation. The results attractive contribution to AGcxc. In this case, the attractive of these calculations are illustrated in Figure 1 (see Appendix I11 contribution is comparable in magnitude and opposite in sign to for details). Thelargest deviationsbetween thecalculated (curves) the repulsive contribution. In fact, it is slightly larger, and as a and simulated (points) results occur at low density, where the result the total AGmCis negative rather than p i t i v e a t all densities. PY-RISM approximation is known to be i n a c c ~ r a t e . Overall, ~~*~~ Therepulsive,attractive,andtotalcontributions t o A P c ,shown the predicted dependence of AGexcon both density and AES.B~is in the lower subpanels of Figure 2, are equal to the slopes of the quite similar for the calculations and the simulations. The AGxccurvesin the upper subpanels. Physically, the total reaction discrepancies between the two predictions can be taken as a volume represents the volume change of the system upon practical measure of the quantitative accuracy of our perturbed dissociationof the solute, and it clearly depends in a complicated hard sphere fluid calculations. way on solvent-solute coupling ( A ~ A ~ -and B ~ )solvent pressure. At high pressure, the repulsive contribution to A P x c decreases 3. Results and Discussion slowly with pressure, much as it does for the dissociation of hard diatomics in hard sphere fluids, as discussed in paper 1. At lower Figure 2 shows the repulsive, attractive, and total contributions pressure the repulsive contribution to A P x c rises sharply to a to AGCxcand A P x c for the dissociation of Brz dissolved in Ar. maximum, in a way that is qualitatively different from that in These results pertain to infinite dilution, and T* = kT/tA, = 2, hard sphere solvents. This maximum reflects the compressibility and are plotted as a function of reduced external pressure, P* = PUA,~/C&. The two left-hand subpanels in this figure refer to a maximum of the solvent in the vicinity of the critical point. (The system in which the solvent-Br atom interaction strength does close connection between AVXC and BT is evident in eq 7). Similar
Diatomic Dissociation in Lennard-Jones Solvents
The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 11839
in Argon
500
E
--________----.-..._....._._.-.-.-.-::
-5-
t.
......
-I ....................
0
_............. ......................................... -.
-501
T'
-2 3
............. 4
.look
.--_ 9
t -150
P
QA:
Figure3. Calculated density dependenceof the total exccss reaction free energy and volume for the dissociation of Br2 in Ar and four different values for the increase in the solute atom-solvent atom interaction well depth upon dissociation.
maxima in the partial molar volumes of solutes dissolved in supercritical fluids have been discussed in detail by Debenedetti and co-worker~.~~ When AQA~-B~ = 20%. both the attractive and the total excess reaction volume undergo large negative excursions at near critical densities. The negative sign of A P x c reflects the constriction of the more tightly bound solvent atoms around the dissociated Br atoms. In other words, the increase in solute-solvent attraction upon dissociation produces a decrease in the total system volume, and thus a negative APxc. The large negative maximum again tracks the compressibilitymaximum of the solvent near its critical point. At high pressure, the total excess reaction free energy increases with pressureeven when AQ&-B~ = 20% (Figure 2b). This reflects the sharper increase in repulsive interactions than in attractive interactions at high pressure and results in a positive A P x c at high pressure for both ACAr-Br = 0% (Figure 2c) and AEk-Br = 20% (Figure 2d). Nevertheless, the magnitudeof the total A P x c at high pressure is significantly smaller when A ~ A ~=- B20%, ~ as a consequence of solvent contraction around the more strongly attracting products. The effects of changing solvent-solute coupling on dissociation thermodynamics are also shown in Figure 3, which illustrates the effect of increasing A t k - ~from ~ 0% to 30% on the total AGCXc and APXC for the dissociation of Brz in Ar (at T* = 2). In this case, the results are plotted as a function of the reduced solvent density, p* = pku3 rather than pressure. Thedifferences between the density- and pressure-dependent behaviors are evident in eq I, which relates the pressure and density derivatives of the excess free energy of reaction. The effect of changing Atk-er at a fixed temperature is qualitatively the same as the effects of changing temperature at a fixed value of A ~ A ~ - B as~ illustrated , in Figure 4. These calculations were carried out for the dissociation of Brz in Ar over a reduced temperature range of 2 I T* I 9, assuming A c A ~ - B ~
0.0
0.2
0.4
0.6
0.8
p OAl'
Figure 4. Calculated temperature (and density) dependenceof the total excess reaction free energy and volume for the dissociation of Br2 in Ar.
= 20%. The qualitative similarity of the results in Figures 3 and 4 reflects the fact that at high temperature the effects of attractive
solvent-solute interactions become less significant. At the lowest temperature, which is just above the critical temperature of the LJ solvent ( Tc* = 1.32),30the excess free energy change strongly favors the products. As the temperature of the system increases, so does the excess free energy of reaction, ultimately producing a positive AGCxc at all densities. Notice also that since the compressibility maximum of the solvent is reduced at higher temperatures (farther away from thecritical point), the magnitude of the excursions of A P x c from zero is similarly reduced at the higher temperatures. Figure 5 illustrates the effects of changing the rare gas solvent on the dissociation of Brz at a reduced temperature of T* = 2. The percent change in E ~ assumed B ~ in these calculations is A t s ~ ~ = 20%. This is solvent independent, in keeping with the assumption that the solvent-solvent well depth, em, is the same in the reactant and product system (and the Lorentz-Berthelot combining rules, = ( Q ~ Q B ~ - B ~ ) ' / ~The ) . rough solvent independence of A t s is~ also ~ consistent with more detailed estimatesobtained using the KohlerSmith-Kong combiningrules for 6 % and ~ ~ solvent and solute polarizabilities to estimate A E ~ B ~ (as described in Appendix I and section 2.5). This latter approach suggests a slight decrease in A E ~with B ~increasing rare gas size (AQN-B~E l . l A t h - ~and ~ AQX-B~ O . ~ ~ A Q A ~ - B ~ ) . Notice that, although the percent change in ACS_B,is nearly solvent independent, the absolute value of A c ~ scales B ~ with Q ~ B ~ and thus increases by a factor of 5 in going from Ne to Xe (see Table I). Nevertheless, the predicted absolute value of AGexcis found to be nearly solventindependent. In fact, if a slight decrease in the percent A-Br with increasing rare gas sizehad been assumed (in keeping with the Kohler-Smith-Kong combining rule estimates), then the AGexC results in the three solvents would have moved even closer together. The reason for the near solvent
11840 The Journal Df Physical Chemistry, Vol. 97, No. 45, 1993
_...-. _I._..._....._...._..... .
t 4
3 8
b 9
-200 0.0
0.2
0.4
0.6
0.8
P 0,”
Figure 5. Calculated solvent dependence of the total excess reaction free energy and volume for the dissociation of Br2 dissolved in Lennard-Jones Ne, Ar, and Xe.
independence of AGCXC can again be traced to the competing influence of repulsive (solvent size) and attractive (dispersion) interactions. In order to understand how this occurs, it is important to recognize that LJ attractive solute-solvent interactions extend over a length scale roughly equal to the thickness of the first solvation shell around a solute. Thus, attractive solvation energies scale roughly as the product of the number of solvent atoms in the first solvation shell times the depth of the attractive interaction well. These two numbers tend to be inversely proportional in a series of similar solvents such as the rare gases. In particular, the larger rare gas solvents have a deeper attractive interaction well depth but smaller coordination numbers around the solutes (both Br2 and Br). In other words, the effects of increasing the LJ well depth are almost exactly cancelled by the associated decrease in number of solvent atoms in the solvation shell around the solute. Furthermore, the purely repulsive contribution to the excess free energy change is also rather similar in each of the threesolvents at the samereduceddensity.’ Thus, bothattractive and repulsive forces scale with solvent size in such a way as to produce similar reduced density dependence for the excess reaction free energy. The resulting A P C values are remarkably similar in the three rare gas solvents at the same reduced density. They are in fact so similar that any small differences that may actually exist in these solvents are obscured by uncertainties in the values of the solute-solvent interaction parameters used in the calculation. In particular, thevariation in LJ parameters obtained from different experimental sources22931 is sufficient to change the ordering of the reaction volumes in the lower panel of Figure 5. Thus, the reaction volumes for dissociation of Br2 appear to scale with reduced solvent density in this series of rare gas solvents.
Ravi et al. small increases in solute-solvent attraction upon dissociation, large negative excursions in APxC (of the order of -100 A3) are found to occur at moderate (near critical) densities and temperatures. Such apparently anomalous negative APCvalues for dissociation reactions are reminiscent of those observed experimentally in more complex molecular supercritical fluid system.29J-s The complex dependence of APxcon solvent density, temperature, and small changes in solvent-solute attraction clearly illustrates the prominent role which the solvent plays in determining the chemical reaction volumes, even for nonpolar homonuclear diatomic dissociation reactions. Repulsive solvent-solute interactions tend to dominate the behavior of both AGOxcand APXC whenever the change in solventsolute attractive coupling upon reaction is small or the temperature of the system is high, resulting in positive AGOxcand A P x cvalues. Even in this limit, however, reaction volumes are predicted to depend significantly on solvent pressure. Attractive solvent-solute interactions may, however, compete effectively with repulsion if thereis even a relatively slight (110%) increase in attractive solvent-solute coupling upon dissociation. This may give rise to a complicated dependence of AGOXCand APCon solvent density and temperature. In general, both AGCXC and APxctend to be negative at low (and positive at high) densities and temperatures. Furthermore, although the magnitude of the total reaction volume at high densities may be similar to that of the intrinsic volume change of the reactants upon dissociation ( ~ V-BVB,, ~ = 4 A3),its strong dependence on solvent density, temperature, and solvent-solute coupling clearly indicates that it cannot be interpreted as a property of the isolated solute species. Although our calculations have been carried out for a specific reaction representing the dissociation of Br2 in rare gas solvents, the general way in which the results are found to scale with solvent density, temperature, and solvent-solute coupling parameters may be expected to hold for other chemical reactions. For example, AGcxcand A P x c are found to scale with the reduced density of the solvent in the series of rare gas solvents, Ne, Ar, and Xe. This may be traced to a correlation between repulsive (hard core size) and theattractive (polarizability) solvation forces in thesesolvents, leading to universal behavior strikingly reminiscent of the wellknown “corresponding states“ scaling of the equations of state of fluids with similar interaction potential shapese22 As a consequence, it is intriguing to speculate that AGCXC and A P E values for other chemical reactions in other series of similar solvents may display the same sort of universal scaling behavior.
Acknowledgment. Support for this work from the National Science Foundation (CHE-9157535) and the Office of Naval Research (N00014-92-1559) is gratefully acknowledged. We also thank S.A. Adelman for supplying the computer program20 which we used in our PY-RISM calculations and D. Henderson for the Verlet-Weis mixed hard sphere distribution program.14 Appendix I. Estimation of U Interaction Parameters The LJ parameters, U S - B ~ and ~ ~ s - B ~ , , for the spherical Br2 moleculs-solvent atom interaction are evaluated using the KohlerSmith-Kong combining rules32
4. Conclusions Analysis of a model diatomic dissociation reaction in a monatomic solvent offers insights into the influence of intermolecular interactions on excess chemical reaction free energies, AGCXC, and volumes, APC. When proper account is taken of
where CS,US and tell, UB~,, are the LJ parameters far the solventsolvent and spherical Br2-Br2 interaction, respectively. The a’s are the corresponding polarizabilities (see Table 11). The B r r Br2 parameters used in these calculations, t~~~= 520 K and uer2
The Journal of Physicul Chemistry, Vol. 97, No. 45, 1993 11841
Diatomic Dissociation in Lennard-Jones Solvents
TABLE W. Parameters for Calculating the Br Atom Hard Sphere Diameter br(over the Temperature and Density Range 1 IP I4,O Ip* S 1.1) parameter Ne Ar Xe
TABLE IIk Solvent Atom-Spherical Br2 Molecule U InterPCtion Parametersa solvent a-erl (K) ‘JS-BrZ (A) Ne 71.3 3.80 Ar 211 3.92 Xe 333 4.195 a Derived as described in Appendix I. ~~
= 4.268 A, are derived fromvapor viscosity dataz2and the solventsolvent parameters from the same source as given in Table I. The resulting spherical BrTsolvent atom interaction parameters are given in Table 111. The LJ diametersfor thesphericalBrmlvent atom interaction, u ~ B ~define , , the volume excluded to the solvent by a spherical Brz vxcludcd sphere
= 4a
3
T‘S-Br,
On the other hand, the excluded volume of a diatomic Brz with atom LJ diameters user,whose centers are separatedby a distance re = 2.281 A,33is
The values of ai-, given in Table I are obtained by equating two excluded volume expressions, = G?&:. The well depth parameter for a bound Br atom-solvent atom interaction, w B r r is calculated from fS-Br2, and US-&,, by applying another self-consistency criterion. This requires the excess internal energy of Brz insolutionobtained from thediatomic model (using a~~ and U s B r ) to be equal to that obtained from the spherical model (using BS-Br, and u ~ B ~ , In ) . particular, we chose the low-density infinite dilution limit and T+ = 2 for these calculations (as described below). In the low-density limit, the radial distribution function of the solvent about a spherical Brz is given by the following function of solvent-Brz separation, r.$-Br2:ZZ
F$tz“
where
and thus the solvation internal energy of a spherical Brz in the low-density limit, Gty, can be written explicitly as uphere Br,
=
p , f gS-Br,(‘S-Br,)
uS-Br2(rS-Br2)
&kBrZ
4.328 4.9148 0.049076 -0.0013019 -0.15827 0.040857 -0.091695
a0 a1
(A*7)
Using eq A S and spherical coordinates, this becomes
9 a3 a4 a3 a6
3.8916 2.919 -0.027306 0.00068604 -0.089224 0.002406 -0.0291 55
3.6263 5.6917 0.17836 -0.0044442 -0.050016 -0.065997 -0.041 367
distribution function is not spherical as a result of the presence of the two bound Br atoms.
Appendix II. Determination of Atomic Hard Sphere Diameters The solvent hard sphere diameters are calculated using the following equation, which has been shown15 to accurately reproduce the results of minimizing eq 17 over a temperature and density range 0 S T* I5 and 0 Ip* I 1.1, respectively:
H
dj = 2‘I6uj 1 +
( p+ U
a,(l
z p 2
+ U 3 p 4,
+ a@* + asp* + a@* ’)
1”’r
(A.lO) where a1 = 1.5001, a2 = -0.033 67, a3 = 0.000 393 5, a4 = -0.098 35, UJ = 0.049 37, and U6 = 4.1415.” Bromine atom hard sphere diameters, d ~ are ~ defined , by minimizing the integral in eq 18, as described in section 2.3. These calculations have been performed over the temperature range 1 IT+ I4 and density range 0.0 I p* I 1.1, again only within the fluid phase region of the solvent. The results have again been fit to an equation of the form dBr
+
I[
= 00 1
+ U3T* a,(l + a g * + u5p* + (T* + U z T *
4,
U@* 3)
I’Y
(A.11) Table IV lists the ui parameters obtained for Br in the three rare gas solvents, Ne, Ar, and Xe. The standard deviations of the hard sphere diameters obtained from these fits are all within i0.03% of the results obtained directly from eq 18. The hard sphere diameter of a Br atom, d ~is ~assumed , to be the same for the bound and dissociated Br atom. In particular, the LJ parameters for the bound Br atom are used to calculate the hard sphere contact diameters for both the bound and dissociated Br atom. This approximation is justified by representative calculations using d~~values derived from LJ parameters for a dissociated Br atom, with a well depth increased by up to 30%. These indicate that the resulting dBr changes do not have a significant qualitative effect on the thermodynamic results. The argon atom and the Br atom hard sphere diameters used in the calculation for T+ = 9 (Figure 4) were obtained directly from eqs 17 and 18, since the solute and the solvent parameters obtained from eqs A. 10 and A. 1 1 yield an error of about 1 % in the diameter estimates at T* = 9.
(A.8) On the other hand, an infinitely dilute diatomic Brz molecule has the following excess internal energy (at p = 0):
Appendix III. Comparison with Simulation Results
64.9) The integration is performed over the system volume. The factor of 2 outside the integral accounts for the fact that there are two equivalent Br atoms whose distances from each solvent atom are r s B r and r’sBr. The exponential term inside the integral is the distribution function of solvent atoms around the Brz molecule. In other words, it is the exact low-density (superposition) result,34 which resembles that in eq AS, except that in this case the solvent
Monte-Carlo (MC) simulation results reported for the dissociation of a diatomic (Az) in a LJ spherical solvent (S)for a system in which the LJ diameters of the solute atoms and the solventatomsarethesamemaybeusedtoevaluate thequantitative accuracy of our thermodynamic calculations. The following conditions are chosen to make optimum use of available simulation results: 2“ = kT/es = 1.5, re*= r p s = 0.5, UA = u ~ In order . to probe the effects of variation in the attractive well depth, our calculation results are compared with the MC simulation results for two cases: (i) a system in which the solute atom-solvent atom
~
~
11842 The Journal of Physical Chemistry, Vol. 97, No. 45, 1993
well depth does not change upon dissociation (CAS/ES = 1 or A a ~ a= 0%) and (ii) one in which it increases by 50% upon dissociation (CAS/CS = 1.5 or A C A=~ 50%), where CAS is the interaction well depth for a dissociated solute atom and the solvent. The corresponding well depth in the bound state is in both cases the same as the solvent-solvent well depth. For case (i), the excess chemical potential of dissociation Apexc is calculated using eqs 12-14. The repulsivecontribution,ApeXc,O, is calculated using the WCA hard sphere diameters for the pure solvent.Is In this case, simulationvaluesfor thechemical potential of the isolated A atom, p y , 2 6and the diatomic A2, are related to ApUCby eq 15. Apexc= lny(re) has also been simulated directly.25 For case (ii), Apex, is calculated by combining the case (i) p r results24with simulation results for p r , with A C A ~ = 50%.z6 +he correspondingsimulation results are represented by the points in Figure 1,24,26 along with curves calculated by the procedure described in section 2. Our results overestimate the magnitude of free energy change by about 20% for case (ii). The agreement is somewhat worse for case (i) at the lower densities, though it progressively improves as the density increases. Nevertheless, our results clearly reproduce the qualitative behavior of the simulation data and in particular the sign change in the excess reaction free energy for going from case (i) to case (ii).
pr2c,24
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Ravi et al. (6) Chandler, D.; Weeks, J. D.; Andersen, H. C. Science 1983,220,787. (7) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986 and references therein. (8) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 55,5422. Weeks, J. D.; Chandler, D.; Andersen, H. C. J . Chem. Phys. 1971, 54, 5237. Weeks, J. D.; Chandler, D. Phys. Reo. Lett. 1970, 25, 149. (9) Chandler, D. The Liquid State of Matter: Fluids, Simple and Complex; Montroll, E. W., Lebowitz, J. L.,Eds.;NorthHolland: New York, 1982. (IO) Pratt, L.R.; Chandler, D. J. Chem. Phys. 1980, 72, 4045. (1 1) Fischer, J.; Lotfi, A.; Lucas, K. Fluid Phase Equilib. 1989,47,239. (12) Lotfi, A.; Fisher, J. Mol. Phys. 1989, 66, 199. (13) Ben-Amotz, D.; Herschbach, D. R. J. Phys. Chem. 1993,97,2295. (14) Leonard, P. J.; Henderson, D.; Barker, J. A. Mol. Phys. 1971, 21, 107. Grundke, E. W.; Henderson, D. Mol. Phys. 1972, 24,269. (15) de Souza, L. E. S.;Ben-Amotz, D. Mol. Phys. 1993, 78, 137. (16) Nicolas, J. J.; Gubbins, K. E.; Street, W. B.; Tildwley, D. J. Mol. Phys. 1979,37, 1429. (17) Balk, M. W. Mol. Phys. 1982, 46, 577. (18) Gillan, M. J. Mol. Phys. 1979, 38, 1781. (19) Chandler, D.; Andersen, H. C. J . Chem. Phys. 1972.57, 1930. (20) Muralidhar,R.;Stote,R. H., Adelman,S. A.,privatecommunication. (21) Morriss, G. P.; MacGowan, D. Mol. Phys. 1986,58,745. (22) Hirschfelder, J. 0.;Curtiss, C.F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley: New York, 1964. (23) Weast, R. C., Astle, M. J., Beyers, W. H., Eds. CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL,1986. (24) Ghonasgi, D.; Llano-Restrepo,M.; Chapman, W. G. J. Chem. Phys. 1993, 98, 5667. (25) Llano-Restrepo,M.;Chapman, W.G.J. Chem. Phys. 1992,97,2046. (26) Shing, K. S.;Gubbins, K. E.; Lucas, K. Mol. Phys. 1988,65, 1235. (27) Chandler, D. Mol. Phys. 1976.31, 1213. (28) Monson, P. A. Mol. Phys. 1984, 53, 1209. (29) Debenedetti, P. G. Chem. Eng. Sci. 1987,42,2203. Debenedetti,P. G.; Mohamed, R. S.J. Chem. Phys. 1989,90,4528. (30) Panagiotopoulos, A. Z . Molec. Simulation 1992, 9, 1. (31) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw Hill: New York, 1987. (32) Shukla, K. P.; Lucas,K. Fluid Phase Equilib. 1986, 28, 21 1 and references therein. (33) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure. IV. ConstantsofDiatomicMolecules;VanNostrand New York, 1979. (34) de Souza, L. E. S.;Guerin, C. B. E.; Ben-Amotz, D.; Szleifer, I., J . Chem. Phys., in press.
