the evolution of multicomponent systems at high pressures

8/14/2019 The Evolution of Multicomponent Systems at High Pressures

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The evolution of multicomponent systems at high pressures

Part IV. The genesis of optical activity in high-density, abiotic uids

J. F. Kenneya and Ulrich K. Deitersb

a Joint Institute of the Physics of the Earth, Russian Academy of Sciences, Gas Resources

Corporation, 15333 JFK Boulevard, suite 160, Houston, T X 77032, USA.E-mail: JFK=alum.MIT.edu

b Department of Physical Chemistry, University of Cologne, L uxemburger Str. 116, D-50939,Cologne, Germany. E-mail: Deiters=stthd0.pc.uni-koeln.de

Received 25th April 2000, Accepted 19th May 2000

Published on the Web 30th June 2000

A thermodynamic argument has been developed which relates the chirality of the constituents of a mixture of

enantiomers to the system excess volume, and thereby to its Gibbs free enthalpy. A specic connection is

shown between the excess volume and the statistical mechanical partition function. The KiharaSteiner

equations, which describe the geometry of convex hard bodies, have been extended to include also chiral hardbodies. These results have been incorporated into an extension of the equation of PavlcekNezbedaBoublk

state for convex, aspherical, hard-body systems. The Gibbs free enthalpy has been calculated, both for

single-component and racemic mixtures, for a wide variety of hard-body systems of diverse volumes and

degrees of asphericity, prolateness, and chirality. The results show that a system of chiral enantiomers can

evolve to an unbalanced, scalemic mixture, which must manifest optical activity, in many circumstances of

density, particle volume, asphericity, and degree of chirality. The real chiral molecules

uorochloroiodomethane, CHFClI, and 4-vinylcyclohexene, have been investigated by Monte CarloC8

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,

simulation, and observed to both manifest positive excess volumes (in their racemic mixtures) which increase

with pressure, and thereby the racemicscalemic transition to unbalanced distributions of enantiomers. The

racemicscalemic transition, responsible for the evolution of an optically active uid, is shown to be one

particular case of the general, complex phase behavior characteristic of closely-similar molecules (either

chiral or achiral) at high pressures.

1 Introduction

The phenomenon of optical activity in uids, either biotic orabiotic, requires, simultaneously, two distinct phenomena: thepresence of chiral molecules, which lack a center of inversion;and an unequal distribution of those chiral enantiomers. Asystem of chiral molecules characterized by a distribution ofequal abundances of enantiomers is a racemic mixture; onescharacterized by distributions of unequal abundances are sca-lemic mixtures. Only scalemic mixtures manifest optical activ-ity. Certain biological processes, such as natural fermentation,generate chiral molecules of only one enantiomer. Abiologicalprocesses can produce either equal or unequal enantiomer

abundances, depending upon the thermodynamic conditionsof their evolution.

The phenomenon of optical activity in abiotic uids isshown in the following sections to be a direct consequence ofthe chiral geometry of the system particles acting according tothe laws of classical thermodynamics. In the following sectionsa purely thermodynamic argument is developed which relatesthe evolution of optical activity in a system of chiral moleculesto the excess volume of scalemic mixtures. The excess volumeof the scalemic mixture of enantiomers is related to their geo-metric properties using the KiharaSteiner equations, whichhave been extended to describe particles which lack a center ofinversion. The chiral property described by the extension ofthe KiharaSteiner equations is introduced into the

equations for mixtures of hardPavlcekNezbedaBoublk

For Part III see ref. 31.

bodies, with which are calculated the Gibbs free enthalpiesand thermodynamic affinities of hard-body systems. The cal-culated thermodynamic affinities establish that, in accordancewith the dictates of the second law, a system of chiral mol-ecules will often evolve unbalanced (scalemic) abundances ofenantiomers at high densities.

For experimental verication of the theoretical calculationsmade with convex hard-body systems, Monte Carlo simula-tions have been carried out on the real chiral molecules u-orochloroiodomethane (CHFClI), and 4-vinylcyclohexene

Both CHFClI, and have been observed, at(C8

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). C8

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high pressures, to manifest higher densities in their scalemicdistributions, as compared to their racemic ones. Such densitychange drives the racemicscalemic transition.

1.1 Historical background

From its rst demonstration by Pasteur, the phenomenon ofoptical activity in uids has engaged the attention and interestof the scientic community.1 This phenomenon has providedan arena for considerable exercise of imagination and creativefantasy, regrettably almost entirely unleavened by the disci-pline of thermodynamics.

Perhaps for reason of its historical provenance in fermentedwine, the phenomenon of optical activity in uids was forsome time believed to have some intrinsic connection withbiological processes or materials. Such error persisted until

the phenomenon of optical activity was observed in material,some believed previously to be uniquely of biotic origin,extracted from the interiors of meteorites.

DOI: 10.1039/b003265o Phys. Chem. Chem. Phys., 2000, 2, 31633174 3163

This journal is The Owner Societies 2000(


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From the interiors of carbonaceous meteorites have beenextracted the common amino-acid molecules alanine, asparticacid, glutamic acid, glycine, leucine, proline, serine and threo-nine, as well as the unusual ones a-aminoisobutyric acid, iso-valine and pseudoleucine.2h4 At one time, all had beenconsidered to have been solely of biotic origin. The ages of thecarbonaceous meteorites were determined to be 34.5 billionyears, and their origins clearly abiotic. Therefore, those aminoacids had to be recognized as compounds of both biologicaland abiological genesis. Furthermore, solutions of amino-acid

molecules from carbonaceous meteorites were observed tomanifest optical activity. Thus was thoroughly discredited thenotion that the phenomenon of optical activity in uids(particularly those of carbon compounds) might have anyintrinsic connection with biotic matter. Signicantly, theoptical activity observed in the amino acids extracted fromcarbonaceous meteorites has not the characteristics of such ofcommon biotic origin, with only one enantiomer present;instead, it manifests the characteristics observed in naturalpetroleum, with unbalanced, so-called scalemic, abundances ofchiral molecules.

The optical activity commonly observed in natural pet-roleum was for years speciously claimed as a proof of someconnection with biological detritus, albeit one requiring botha willing disregard of the considerable dierences between theoptical activity observed in natural petroleum and that inmaterials of truly biotic origin, such as wine, as well as desue-tude of the dictates of the laws of thermodynamics. Obser-vation of optical activity, typical of such in natural petroleum,in hydrocarbon material extracted from the interiors of car-bonaceous meteorites, discredited those claims.5,6 Nonethe-less, the scientic conundrum remained as to why thehydrocarbons manifest optical activity, in both carbonaceousmeteorites and terrestrial crude oil.

There is common misunderstanding that the molecular pro-perty of chirality, which is responsible for optical activity inuids, is an unusual, complicated property of large, complex,polyatomic molecules. The small, common, single-branchedalkane molecules are usually chiral. Single-branched alkanes

comprise between 715% of the molecular components ofnatural terrestrial petroleum and are also observed in pet-roleum synthesized by such as the FischerTropsch processes.When these chiral molecules are created in low-pressureindustrial processes, they occur always in equal enantiomerabundances, and the resulting synthetic petroleum does notmanifest optical activity. In natural petroleum, these mol-ecules occur in unequal enantiomer abundances, and the uidmanifests optical activity. Molecular chirality results from thehighly directional property of the covalent bond, and is indif-ferent to whether a compound results from a biological or anabiological process.

Previous hypotheses oered to explain optical activity ofthe compounds extracted from carbonaceous meteorites have

invoked such deus ex machina as panspermia, (theseeding of optically active biotic molecules from (literally)the heavens7,8) or the cumulative eects of the chiral weak-interaction involved in beta decay,9h16 with necessary over-sight of the several orders of magnitude of energy dierencecompared to that attributable to the entropy of mixing, whichwould destroy any imbalance responsible for optical activity.

With no recourse to any such artices, the phenomenon ofoptical activity in uids is here shown to be an inevitable con-sequence of the dictates of thermodynamic stability theorymanifested by systems which contain quite ordinary, covalent-bonded molecules, in certain conditions of density.

1.2 The organization of this paper

The topic of optical activity in multicomponent uids is takenup thoroughly, in order that its fundamental thermodynamic

property be set forth explicitly, and that its statistical mecha-nical genesis be demonstrated. This paper is organized intothree parts:

(1) A thermodynamic argument is developed which relatesthe Gibbs free enthalpy, and the phase stability of a mixedsystem, to its excess volume. This argument invokes no spe-cic molecular property, and uses only a strict thermodynamicdenition of a system containing chiral components whichspecies equality of chemical potentials and molar volumes,and a non-vanishing excess volume. The distribution of

species in a multicomponent system is shown to be deter-mined, at low pressures, by its entropy of mixing, and, at highpressures, by its excess volume.

(2) A statistical mechanical argument is developed whichrelates directly the Gibbs free enthalpy and excess volume tospecic molecular geometric properties. The KiharaSteinerequations have been extended to describe hard-body particleswhich do not possess a center of inversion; and the resultshave been applied to the equa-PavlcekNezbedaBoublktions for convex hard-body uids.

(3) Using the equations, thePavlc ekNezbedaBoublkthermodynamic affinity has been calculated formally for adiverse group of hard-body uid systems characterized by dif-ferent molecular volumes, and degrees of asphericity andchirality. These uids are shown to undergo the racemicscalemic transition exactly as required by the general ther-modynamic argument developed in section 2.

Using Monte Carlo simulation, the two real, chiral uids,uorochloroiodomethane (CHFClI), and 4-vinlycyclohexene

have been investigated as pure, chiral components(C8

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)and as racemic mixtures. The latter are shown to developpositive excess volumes at increased densities, which increaseapproximately linearly with pressure, and which thereforeinduce the racemicscalemic transition.

2 The explicit, general prediction of the genesis of

optically active systems by classical thermodynamic

argument

First is shown that the evolution of unbalanced abundances ofenantiomers, scalemic mixtures, often results inevitably fromthe general requirements of thermodynamic stability theory.In keeping with the traditions of classical thermodynamics, noassumptions are made about any detailed properties of thematerial which composes the uid mixture, other than themost basic attributes of their chirality.

From the cross derivatives of the dierential of the Gibbsfree enthalpy, G(p, T, Mn

jN),

dG \ [S dT] V dp ] ;j

kj

dnj

(1)

the dierential equation for the chemical potential, as a func-tion of pressure, at constant temperature is given as:

Adki

dp

BT, KnjL

\AdVdn

i

BT, p, KniEjL

\ Vi

. (2)

With inclusion of the Gibbs mixing factor RT ln thexi

,chemical potential of the ith species is given in terms of itspartial volume as :

ki\ k

ii] RT ln x

i]

APpi

pV

m, idp

BT

. (3)

The volumetric behavior of a multicomponent system can be

described by the intensive variable, excess volume: VE \ VmWhen the formalism developed by[R(j)

xj

Vm, j

.Guggenheim17 and Scatchard18 is applied, the excess volume

3164 Phys. Chem. Chem. Phys., 2000, 2, 31633174


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can be expressed as a series expansion :

VE \ ;i

xi

;j:i

xjMV

0, ijE ] V

1, ijE (x

i[ x

j)

]V2, ijE (x

i[ x

j)2 ] ] V

v, ijE (x

i[ x

j)vN. (4)

For the present analysis, the GuggenheimScatchard expan-sion, eqn. (4), may be truncated at the rst term without lossof generality; when such is done, the molar volume may bewritten as:

Vm \ ;j

xj Vm, j ] ;j

;k:j

xj xk4(Vjk)maxE , (5)

which approximation constitutes application of the Porteransatz. (The excess molar volumes in eqn. (5), 4(V

j, k)maxE ,

contain the factor four in order that it will return itsmaximum value in the case of a binary equimolar mixture atthe racemic molar fractions x

i\ x

j.)

2.1 The thermodynamic chirality function

A thermodynamic system capable of manifesting optical activ-ity may be considered generally as consisting of the two chiralenantiomers, L and D, together with a third achiral com-ponent, A. For such a three-component system, the chemicalpotential of the D-component enantiomer is:

kD

\ kDi] RT ln x

D]

APpi

pV

Ddp

BT

. (6)

When eqn. (5) is applied, together with the property thatthe partial molar volume of the D-(V

DA)maxE \ (V

LA)maxE ,

component enantiomer becomes,

VD

\ VmD

] xD

(1 [ xL

)4(VDL

)maxE ] x

A2 4(V

DA)maxE ; (7)

and its chemical potential is thereby,

kD

\ kDi] RT ln x

D

]AP

pi

p[V

mD] x

L(1 [ x

D)4(V

DL)maxE ] 4x

A2(V

DA)maxE ] dp

BT

.

(8)

Let the thermodynamic chirality functional, Q(p, T; bex)DL

dened as the integral of the excess volume of the system ofchiral molecules over its pressure as:

Qij

(p, T; x) \1

RT

APpi

p4(V

ij(p, T; x))

maxE dp

BT

. (9)

The chemical potential of the D-component enantiomer, eqn.(6) and (8), is then written:

kD

\ kDi ] RT ln x

D]

APpi

pV

m, Ddp

BT

]RT xL(1

[xD)QDL

]RT xA

2QD, A ; (10)

an analogous expression for with the subscript L replacingkL

,that of D in eqn. (10).

2.2 Enantiomeric separation and conversion processes

Three thermodynamic conditions characterize enantiomers:the equality of their reference chemical potentials; the equalityof their molar volumes; and (usually) a non-vanishing excessvolume of their mixtures:

kLi \ k

Di

Vm, L

\ Vm, D

VL, DE D 0

8(11)

The validity of the rst two equations eqn. (11) is intuitivelyobvious. The third, inequality, of eqn. (11) may be consideredto be, at this point, an assertion; in the following sections, it is

proven to hold usually. The simultaneous requirements setforth by eqn. (11) are strictly thermodynamic ones and involveno detailed properties of the molecules themselves. In additionto the equalities of eqn. (11), the excess volume for either enan-tiomer with a non-chiral component, A, is identical ; such thatQ

DA\ Q

LA.

Let it be assumed that the enantiomers can convert into oneanother, as DL. The condition of chemical equilibrium,

together with eqn. (10), (11), and the equalitykD

\ kL

, QDA

\gives:Q

LA,

lnx

Dx

L

[ (xD

[ xL

)QD, L

\ 0. (12)

Eqn. (12) expresses succinctly and rigorously the essentialdetermining factors for the spontaneous, abiotic evolution ofoptical activity in uids: the competition between the entropyof mixing, given by the rst logarithmic term, and theoppositely-directed eects of the inevitable packing ineffi-ciency of mixed enantiomers, given by the thermodynamicchirality function, Eqn. (12) denes the analytic LambertQ

DL.

W function,19,20 which has always one real root and, asshown in Fig. 1, sometimes three.

The molar excess volume of a simple mixture often is rela-tively small, VE B 110 cm3 mol~1.21,22 Therefore, at or near

a pressure of 1 bar, for a system consisting only of the twoenantiomers, kJ. The denomi-Q

DL\ (1/RT)pVE B (10~3/RT)

nator RT is approximately 2.5 kJ at 300 K. Therefore, atmodest pressures and essentially all temperatures, the secondterm in eqn. (12) cannot balance the rst except at the value

which solution describes the racemic mixture. ThusxD

\ xL

,eqn. (12) establishes that, at low pressures, an abiotic systemwill usually evolve into a racemic mixture of enantiomers.

However, the thermodynamic chirality functional, QDL

,depends directly upon the system pressure, for Q

DLB VEp.

For any non-vanishing, positive excess volume, the ther-VDLE ,

modynamic chirality function has no limit as pressureincreases. Therefore, for a system whose thermodynamicchirality function is greater than a certain threshold value,

there will always be a (usually, high) pressure(QDL)threshold,above which the system will evolve unbalanced abundances ofenantiomers, and the resulting system will inevitably be opti-cally active. For a system whose thermodynamic chiralityfunction is less than the threshold value, there(Q

DL)threshold,

Fig. 1 Values of the functions andln(x

D

/(1 [ x

D

)) (2x

D

[ 1)Q

DLwhich satisfy eqn. (12) at dierent values of the fractional abundance,in a binary mixture. Note the one racemic and two scalemic solu-x

D,

tions to eqn. (12) for QDL

\ 4.

Phys. Chem. Chem. Phys., 2000, 2, 31633174 3165


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will be no transition pressure, and such a system will remainracemic. This behavior is shown graphically in Fig. 1 whereare represented the plots of the two functions ln[x

D/(1 [ x

D)]

and (which correspond to a binary system com-(2xD

[ 1)QDL

posed solely of two enantiomers). As seen clearly in Fig. 1, forvalues of the thermodynamic chirality function, less thanQ

DL,

a threshold value, the system cannot evolve an unbalancedsystem. For the value, there is only one solution,Q

DL\ 1.5,

the racemic root, for the value, there is axD

\ 0.5; QDL

\ 4,second solution at the scalemic value, (and a third,x

DB 0.97

symmetrically, at xD B 0.03).For a general system composed of two enantiomers and athird component, A, the threshold value of for the onsetQ

DLof a racemicscalemic transition is that for which the twoterms in eqn. (12) have equal derivatives with respect to x

D;

such that,

2

(1 [ xA

)\ Q

LD. (13)

The foregoing thermodynamic argument has shown thatchiral molecules in an unbalanced, scalemic, distribution ofenantiomers can possess lower chemical potentials, andthereby a lower Gibbs free enthalpy, than a racemic distribu-tion of the same compound, under certain conditions of

density.If is positive and of sufficient magnitude, there is aQ

DLthreshold pressure at which the racemic mixture becomesthermodynamically unstable. If a conversion reaction DLexists, the slightest excess of one enantiomer can cause thesystem to develop a macroscopic excess of this enantiomer,when the pressure is raised further.

2.3 The enantiomeric phase separation in scalemic mixtures

Alternatively, the system can undergo a phase split into twoscalemic phases. In either case, application of sufficient pres-sure always leads to the formation of phases with enantio-meric excess.

The molar Gibbs free enthalpy of the system of chiral enan-tiomers L and D together with an achiral component A is:

Gm

\ ;j

xjkji ] RT ;

j

xj

ln xj

] ;j

xj

Ppl

pVm, j

dP ] RT xL

xD

QLD

] RT xA

(1 [ xA

)QAD

(14)

The limits of stability and critical points of mixtures are deter-mined by the higher derivatives of the molar Gibbs free enth-alpy, which gives for the D enantiomer,G

n, i\ (dnG

m/dx

in)j, p, T

,

G

D

\

AdG

m

dxDBxL, T, p\ k

D

i] RT(ln x

D

[ ln x

A

)

]Ppi

p(V

mD[ V

mA) dp ] RT x

LQ

LD] RT(2x

A[ 1)Q

AD

(15)

The second derivatives of are, respectively:Gm

G2D

\Ad2G

mdx

D2

BxL, T, p

\ RTA 1

xD

]1

xA

B[ 2RT Q

AD

GLD

\A d2G

mdx

Adx

D

BxL, T, p

\ RT1

xA

] RT QLD

] 2RT QADh

G2L

\

Ad2G

m

dxL2 BxL, T, p\ RT

A1

xL

]1

xAB[ 2RT Q

AD

(16)

The conditions for phase stability require that:

G2D

G2L

[ (GDL

)2 P 0 (17)

The examination for phase separation, for which the equalityholds in eqn. (17), introduces the equality

xD

\ xL

\ 12

(1 [ xA

). (18)

The roots of the equality in eqn. (17) then admitted are:

QDL \7

2

(1 [ xA)

4QAD

[2

xA

(1 [ xA

)

. (19)

The rst root of eqn. (19), for the general case of a three com-ponent system, corresponds to that determined by eqn. (12)for a binary system. The molar Gibbs free enthalpy of asystem containing two enantiomers and a third achiral com-ponent, as given by eqn. (14), has been calculated using thevalue for determined by the roots given by eqn. (19), andQ

DLis shown in Fig. 2 as a function of the reduced molar fraction,

which is the D fraction of the total molar fraction of the(xD

)r

enantiomers, (The reduced molar fraction has been(1 [ xA

).used in order that the racemic solution will fall at the value

0.5.) The double minima of the Gibbs free enthalpy as a func-tion of is immediately apparent in Fig. 2, as is the fact(xD

)r

that the racemic mixture, for those values of is at aQDL

,maximum. That the two scalemic minima lie at equal values ofGibbs free enthalpy is shown by their double tangent.

The foregoing analysis has involved determination of theequilibrium states of a multicomponent system, which con-tains a variable distribution of enantiomers together with athird achiral component, without considering the dynamicprocesses by which that system resolves to equilibrium whenthe pressure has changed. The two minima shown in Fig. 2can be reached either by chemical reaction or by phaseseparation.

If the transition time for L ]D conversion of the individualmolecules is much slower than the transport diusion time,then an initially racemic system will undergo a rapid physicalseparation, similar to gasgas demixing; and the system willresolve into two physically separate regions, one enriched in

Fig. 2 Gibbs free enthalpy at the roots of eqn. (19), (in arbitraryunits).



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the D enantiomer, the other in the L. At the opposite extreme,if the transition time for L ]D conversion is much faster thanthe diusion time, the system will undergo a quasichemicaltype of reaction and change its constituent compositionentirely. For cases in between, the system must be expected toundergo often complex chemical and dynamical behavior as itproceeds to equilibrium.

3 Statistical mechanical calculation of the excess

volume using the geometric properties of individualmolecules

The thermodynamic analysis of the previous section estab-lished that a system of chiral particles will possess, in certainconditions of pressure, temperature, and degree of chirality, alower free energy in a scalemic distribution than a racemicone. Consistent with the traditional perspective of classicalthermodynamics, that analysis invoked no detailed propertiesof the chiral system, beyond the minimum functional deni-tion of chiral enantiomers set out in eqn. (11). The thermody-namic analysis identied the systems excess volume, actingthrough the dened thermodynamic chirality parameter, asthe operative physical property responsible for the racemic

scalemic transition. However, the classical thermodynamicanalysis did not address the question of how, or why, a systemof chiral molecules ought, or must, manifest a non-vanishingexcess volume. Nor did the thermodynamic analysis give anyspecic indication how the distribution of the chiral enantio-mers might be determined from properties of those molecules.

In this section, the formalism is developed for direct calcu-lation of the distribution of the chiral particles from thestereochemical properties of the individual molecules. First, aprecise description of the geometry of chiral, convex, hard-body particles is developed by extension of the KiharaSteinerequations. That geometric description is then used in the

equations for convex, hard-bodyPavlcekNezbedaBoublksystems, from which an explicit expression is developed for theHelmholtz free energy of a multicomponent system whichcontains chiral particles.

3.1 The geometric description of a hard-body system :extension of the KiharaSteiner equations

Convex hard-body systems have been described by Kiharaand Steiner in terms of three geometric parameters, andR3

i, S3

iwhich are determined by the support function thatV3

i,

describes the volume and surface generated by the rolling ofone hard body around the surface of another, in all possibleorientations. As shown in the following subsection, theweighted products of these geometric entities, andR3

i, S3

iV3i,

enter the equation of state for systems of convex hard bodiesand determine its thermodynamic properties. The parameter

functional represents the averaged radius of curvature ofR3ithe support function and its averaged surface area; in thisS3

iinstance, does not represent the ith partial volume but theV3

ieective volume determined by the support function. The indi-vidual functionals and are dened by the equations:R3

i, S3

iV3i

R3i\

1

4p

P0

p P0

2pri

Adu

idh]

dui

d/

Bdh d/

S3i\

P0

p P0

2pui

Adr

idh]

dri

d/

Bdh d/ h (20)

V3i\

1

3

P0

p P0

2pri

Adr

idh]

dri

d/

Bdh d/

In eqn. (20), u(h, /) is the unit vector in the direction of thenormal of the supporting plane, and r(h, /) is the vector fromthe origin to the contact point of the convex body with the

Fig. 3 The geometric parameters used by Kihara to determine R, S,and V in eqn. (20).

supporting plane, as indicated in Fig. 3; the angles h and /are polar angles.

In his derivation of the functionals and KiharaR3i

, S3i

V3i,

assumed that the convex hard body possesses a center ofinversion, and used such property to simplify his equations.However, a convex hard body need not be invariant underinversion. In order to represent a system whose components

possess chirality, an appropriate tensorial description must beused. The expressions in the integrands of eqn. (20) are tensorentities, specically pseudo-scalars; strictly each should beexpressed similarly as:

R3i\

1

4n]

1

3]

P0

p P0

2pri(h, /)

Adui(h, /)

dh]

dui(h, /)

d/

Bdh d/]

1

2]

]CP

0

p P0

2pri(h, /)

Adui(h, /)

dh]

dui(h, /)

d/

Bdh d/

Dy/yx?~xz/z ]P

0

p P0

2pri(h, /)

Adui(h, /)

dh]

dui(h, /)

d/

Bdh d/]

]

1

2

[CP

0

p P0

2pri(h, /)

Adui(h, /)

dh]

dui(h, /)

d/

Bdh d/

Dy/yx?~xz/z

] similar terms in

x \ x

y ][y

z \ z

&

x \ x

y \ y

z ] [z

qq R S n ntt t t t ttt t t t ttt t t t ttt t t t ttt t t t ttt T U t ttt t ttt R S t ttt t t t trt t t t o

tt t t t ttt t t t ttt t t t ttt t t t tts T U p tt q n q n tt t t t t tt r o r o tt t t t t ts s p s p p

(21)

Clearly, in cases for which the hard body possesses a center ofinversion, the terms in the second set of square brackets ineqn. (21) cancel, and the terms in the rst set of squarebrackets are identical. Eqn. (21), and the analogous equationsinvolving the Kihara surface and volume functionals, admitthe representation of the geometric functionals, andR3

i, S3

iV3i,

as rst-rank tensors :

R3i\ R3

iSe

S] R3

iAe

AS3i\ S3

iSe

S] S3

iAe

AV3i\ V3

iSe

S] V3

iAe

A

8, (22)

in which the two orthogonal unit vectors, and dene theeS

eA

,two-dimensional vector space for the symmetric and anti-symmetric components, respectively, of the geometric func-tionals. In the rst line of eqn. (22), represents theR3

iS

symmetric terms in the rst set of square brackets of eqn. (21),and the anti-symmetric dierence terms in the second set;R3

iA

there are analogous Kihara functionals and in theS3i, V3

i,

second and third respectively. The symmetric contributions in

eqn. (22) are always positive; the asymmetric contributionscan be either positive or negative, depending upon the partic-ular geometry of the body.



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For example, consider again a tetrahedron, which is aconvex body. Imagine an initially regular tetrahedron, withthe apices of its base at [(a,0,0), ([a,0,0,), (0, and itsaJ3,0)],resulting geometry after that base has been deformed (with thefourth apex, out of the plane, remaining xed) such that thebase apices are at the new positions [(a,0,0), ([a,0,0,), (d,

i.e., by a deformation by which one of the apices isaJ3,0)],moved parallel to its opposite side. The hard body whichresults from such a distortion, while still convex, is chiral; anda collection of molecules of such geometry will be optically

active. The same tetrahedron similarly but oppositelydeformed, with the position of its third apex above the planeof its base remaining xed in both cases, to positions [(a,0,0),([a,0,0,), ([d, is also chiral, but clearly of oppositeaJ3,0)],chiral sense. Clearly, a mixture of regular tetrahedra and onesdeformed as described above cannot occupy the sameminimum volume as would the regular tetrahedra alone,although the volumes of the respective deformed tetrahedraremain unchanged. Equally clearly, the product of the radiusof curvature of the deformed tetrahedron and the surface ofthe regular one would be increased, and, to terms linear in thedeformation, approximately by a factor proportional to d. Itdeserves to be noted that the parameter of distortion, d, neednot be necessarily small; the base of the tetrahedron can bemade very scalene, and the tetrahedron will remain a convexhard body.

3.2 Extension of the equations forPavlcekNezbedaBoublkmixtures of aspherical hard-body systems

In order to develop the most general analysis possible of thethermodynamic stability of systems of chiral particles, suchare described by the statistical mechanical formalism of hardbodies. The equation of state for the pure hard-sphere gasgiven by scaled particle theory (SPT)23 represents one of the(very) few exactly-solvable problems in modern statisticalmechanics. Signicantly, scaled particle theory developed itsmajor componentthe equation for the probability that thecenter of a particle exists within a radial shell at a certain

distance, subject to the condition that the region dened bysuch radius be devoid of particlesfrom analysis of statisticalgeometry. Later, Reiss, Frisch and Lebowitz24 extended SPTto describe mixtures of hard spheres. Twenty years after itsrst enunciation, scaled particle theory was extended furtherby Nezbeda and to describe mixtures of Pavlcek, Boublkconvex, hard-body gases,25 using essentially arguments fromstatistical geometry and the precise descriptive parameters forsuch developed by Kihara and Steiner. Because the

equations represent an extensionPavlcekNezbedaBoublkof scaled particle theory, they share its rigor and precision, aswell as the same of the dierential geometric formalism ofSteiner. For that reason, the PavlcekNezbedaBoublkequations have been used to examine the statistical thermody-namic stability of mixtures of chiral hard bodies.

The equation of state is :PavlcekNezbedaBoublk

p

RT\ o

31 ]

1 g1 [ g

](r8s8)

o(1 [ g)2]

g(12t8[ 5s8)q8s89o(1 [ g)3

]3oq8s82 [ 2(r8s8)2 ] 12(r8s8)(r8t8) [ 18(r8s8)(w8 t8)

9p2(1 [ g)3

24(23)

in which the molecular geometric functionals were originallydened by Nezbeda and as :Pavlcek, Boublk

q8\oq\o ;i

xiR3i2

t8\ot\o ;ixi

V3i

S3i

r8\or\o ;i

xiR3i

w8 \ow\o ;ixi

V3i

S3i

s8\os\o ;i

xiS3i

g\ot\o ;

i

xiV3i8

.

(24)

The shape-dependent parameters q, r, s, t and w are weightedsums of the functionals of the respective support functionswhich describe the volume and surface generated by therolling of one hard body around the surface of another, in allpossible orientations; g represents the weighted packing frac-tion, the molar fractions. The individual functionalsx

iR3i

, S3i

and are dened tensorially by eqn. (22). The tensor contrac-V3i

tions of their products, such as etc., which enter ther8s8, q8s8, r8t8,equation of state, eqn. (23), arePavlcekNezbedaBoublk

dened using the two-dimensional identity matrix and the rst

of the Pauli matrices, A1 00 1

B\ I

2

A0 11 0

B\r

1Pauli

8. (25)

Because the vector space dening the degree of geometricchirality spans two dimensions, only one of the Pauli matricesis required; and the tensor properties of the geometric func-tionals are described using the symmetric and anti-symmetricoperators

rS \ I2

]r1Pauli

rA \ 2r1Pauli H. (26)

The contraction of the geometric vectors, and is then:v1

v2

v1

v2

\ v1v

2\ v

1(rChiral)v

2, (27)

where rChiral\ rs ]rA. For example, the contraction of thegeometric vectors rand s is thereby:

rs \A

;i

xi

R3i

B(rChiral)

A;j

xj

S3j

B. (28)

For the case where there exist only the two enantiomers, Dand L, the product function rs given by eqn. (27) and (28) hasthe form:

(rs)DL

\ (xD

R3D

)rChiral(xL

S3L

)

\ xD2(R3

DS S3

DS ] R3

DA S3

DS ] R3

DS S3

DA ] R3

DA S3

DA)

] xD

xL

(R3DS S3

LS ] R3

LS S3

DS ] R3

DA S3

LA ] R3

LA S3

DA)

] xL2(R3

LS S3

LS ] R3

LA S3

LS ] R3

LS S3

LA ] R3

LA S3

LA) (29)

Plainly, eqn. (27) and (29) return the identical expressions forsymmetric, achiral molecules as do the original KiharaSteiner equations. Equally plainly, eqn. (27) and (29) returnthe correct equations for single-component systems, given byeither the rst or third lines of the expansion of Most(rs)

DL.

importantly, the second line of the expansion of gives an(rs)DL

additional contribution only in mixtures. There are similarcontributions for chirality for the other functionals whichappear in the equation of state,PavlcekNezbedaBoublk

etc. Such cross-terms, which appear only(r8t8)DL

, (w8 t8)DL

, (q8s8)DL

,in mixtures, have all the units of volume and are responsiblefor the excess volume. For two enantiomers, the anti-symmetric components possess identical magnitude andopposite sign :

R3D

\ R3DS e

S] R3

DA e

AR3

L\ R3

DS e

S[ R3

DA e

A

H, (30)

and similarly for both components of and for bothS3D

V3D

enantiomers L and D. When eqn. (30) is applied, the excessparameter (rs)E [the second line of the expansion of (rs)

DL]

takes on the particularly simple form:

((rsDL

)E \ xD

xL

(R3DS S3

DS [ R3

DA S3

DA). (31)



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Dening the terms in the rst (or third) line of the expansionof as the product-functional rs may be written for(rs)

DL(rs)

DL0

a binary system as

(rs)DL

\ (1 [ 2xD

] 2xD2)rs

DL0 ] 2x

D(1 [ x

D)rs

DLE . (32)

When the linear expansion has beenrsDLE \ rs

DL0 (1 ] e

rsChr)

applied, the product-functional has especially simple(rs)DL

forms for the racemic and homochiral systems:

((rs)DL

)homochiral\ rsDL0

((rs)DL

)racemic\ rsDL0 ] rs

DLE \ rs

DL0 A1 ] ersChr

2B8 . (33)

There are similar expressions for the other product-functionals, rt, ts, etc., which will involve similar chiralityparameters, etc. In the following analysis, ae

rtChr, e

tsChr,

minimalist perspective is taken, such that the eects ofchirality are taken into account for only the product-functional rs ; therefore the subscripts will be dropped fromthe parameter of chirality, e

rsChr.

Because this analysis involves the geometry properties ofthe individual molecules, the system density, o, has been fac-tored out of the geometric functionals so as to return thealternate form of the equation of PavlcekNezbedaBoublk

state:

(p)PNB

okB

T\ 1 ]

g(A ] Bg ] Cg2)(1 [ g)3

(34)

in which

A \A

1 ]rs

t

BB \ [

A2]

(rs)

t]

1

t2

A2(rs)tw[

4r2st

3]

(2(rs)2 [ 3qs2)

9

BBhC \11 ] 1t2

A4qst3

]5qs2

9

B2

(35)

When these molecular shape variables, A, B, and C are used,the Helmholtz free energy of the mixed hard-body system iswritten as:

F \ NkB

TC

[lnA V

Nj3B

[ 1 [ C ln(1 [ g) ]g(Dg ] E)

(1 [ g)2D

(36)

in which j is the thermal de Broglie wavelength, and the vari-ables D and E are related by,

D \3C ] B [ A

2

E \ A [ C

8. (37)

It should be noted that the functions r, s, t, w and l, and

thereby also A, B, C, D and E, depend upon the moleculargeometric properties only and are not thermodynamic vari-ables. Eqn. (34), (35) and (36) have been applied to calculatethe Helmholtz free energies, Gibbs free enthalpies and ther-modynamic affinities of a large number of racemic and single-component, chiral hard-body systems of diverse geometricshape and degree of chirality, over a wide range of pressureand temperature.

4 The racemicscalemic transition: the onset of

unequal abundances of enantiomers in a chiral

system at high density

The direction of spontaneous evolution of any system is deter-mined by its thermodynamic affinity, A(p,T, The secondMn

jN).

law of thermodynamics, expressed mathematically by De

Donders inequality,

dQ@ \ A dm P 0

A \ [ ;i, o, a

vi, oa ki, oa

8, (38)

requires that the thermodynamic affinity be positive for anyspontaneous transition, for which the variable of extent ism.26h29

The process of evolution here examined is that of theracemicscalemic transition, which may be expressed as:

12

CD

] 12

CL]x

DC

D] (1 [ x

D)C

L, (39)

where, in the scalemic state [right side of eqn. (39)], the molarfraction The thermodynamic affinity for the onset ofx

DD 1/2.

an unbalanced system of single-component enantiomers, fromone initially racemic, is:

A(p, T; eChr, a)racemic?pure\ (12

kD

] 12

kL

)racemic [ (kD

)pure. (40)

When the thermodynamic affinity, eqn. (40), is negative, thesystem either evolves in the opposite direction, toward aracemic mixture, or remains as such. However, whenever thisthermodynamic affinity becomes positive, a racemic mixturetransforms to a scalemic one with unequal abundances ofenantiomers. For a single-mole system, the thermodynamic

affinity is simply the dierence in the Gibbs free enthalpy ofthe racemic and pure systems.

4.1 Theoretical analysis of a chiral hard-body system

The Gibbs free enthalpy has been calculated over the range ofpressures 1500 kbar, at temperatures of 300 and 1000 K, fora wide collection of convex hard-body systems of dierentmolecular geometries, whose molecular mass and volume, l,were taken to be, respectively, 105 g mol~1 and 65 cm3mol~1, which values correspond roughly to those for single-branched heptane or octane. The thermodynamic affinity wascalculated for general convex hard-body systems characterizedby values of asphericity a \ rs/(3l) \ 1.5, 2.0, 2.5, 3.0, 4.0, 5.0,and for values of the prolateness parameter, b \ r2/s \ 1.75,2.00, 2.25, 2.5, 3.0, 4.0. Identical calculations were performedalso for prolate ellipsoids with ratios of long to short axes of1.75, 2.0, 2.5, 3.0, 4.0.

For each geometry, the respective system was examined forthe following values of the chirality parameter: eChr\ 0.025,0.05, 0.1, 0.5, 1.0. The Gibbs free enthalpy was calculated ineach case for both the racemic and single-component systems.Because a typical experimental value for measured molarexcess volume is approximately 110 cm3 mol~1 for simplesystems,22,30 for several geometries studied, the chiralityparameter, eChr, was specically chosen to produce an excessvolume of 1 cm3 at STP for a hard-body gas with the approx-imate molecular and thermodynamic characteristics of octane.The choice of 1 cm3 mol~1 for molar excess volume was

chosen as most reasonably conservative.Consistent with the intention to take a conservative,

minimalist approach for the analysis, the eects of chiralitywere restricted to the product of the molecular parameters rand s ; such that the product rs which appears in the

equations was applied as given inPavlcekNezbedaBoublkeqn. (29). All other molecular shape variables, q, s, t, w, andtheir powers and products with rs have been left unchanged.Such restriction constitutes a formidable underestimation ofthe eects of chirality upon the thermodynamic functions, par-ticularly upon the Gibbs free enthalpy. An eect of chirality isalways to increase the eective asphericity of the hard body,and thereby also to increase the hard-body components ofboth the pressure and the Gibbs free enthalpy; an addition of

chirality cannot ever decrease the magnitude of those func-tions, at any density or temperature. The property of aspheri-cal hard-body systems has been demonstrated previously.31



8/12

The results of all calculations are qualitatively identical. Atlow pressures, and thereby low densities, the Gibbs free enth-alpy of the racemic mixture is always lower, at all tem-peratures and for every geometry. The contribution to theGibbs free enthalpy attributable to the entropy of mixingdominates the system at low pressures, regardless of geometry.However, at high pressure and for all values of the chiralityparameter greater than a certain threshold value, the contri-bution to the Gibbs free enthalpy attributable to its excessvolume exceeds that of its entropy of mixing, and the affinity

abruptly changes sign, and remains positive for the racemicscalemic transition.An example of the eect of pressure upon the thermodyna-

mic affinity is shown in Fig. 4, on a logarithmic scale of pres-sure through the range 1500 kbar, for typical systems of pureand racemic particles of molecular mass 105 g mol~1 andcharacteristic volume, t\ 65 cm3, with asphericity, a \ 1.5,and degree of prolateness, b \ 2.25, for values of its chiralityparameter, eChr\ 0.025, 0.05, 0.1, 0.5, 1.0. These parameters ofmolecular geometry represent very conservative measures ofasphericity, prolateness, and chirality; the magnitude of eChr isoften substantially greater than one. The asphericity, a \ 1.5,describes a molecule only modestly deformed from spherical,and degree of prolateness, b \ 2.25, characterizes one forwhich the length of its major axis is slightly more than twicethat of its minor axis; these parameters are comparable to thesame, for example, of propane or butane, although assigned toa larger molecule. The degrees of chirality, for which the great-est is eChr\ 1.0, are also modestly taken. Mathematical mod-eling for molecules as simple as 3-methylhexane indicates thatthe dierence in the product of the radius of curvature and thecharacteristic surface of one enantiomer, and of its mirrorimage, can easily be as great as 24. As shown in Fig. 4, anincrease of pressure has no discernable eect upon the systembelow 1 kbar, excepting only for molecules of the greatestdegrees of both asphericity and chirality. However, above 1kbar, the product of the excess volume and pressure begins toshift the relative values of the Gibbs free enthalpy of, respec-tively, the racemic and scalemic system. At pressures higher

than 10 kbar, the additional component to the Gibbs freeenthalpy of the product of the excess volume and pressurebegins to dominate the system; and, at sufficiently high pres-sure, typically of the order of 20 000450 000 atm, the contri-bution to the Gibbs free enthalpy of the racemic mixtureattributable to its excess volume becomes greater than thatattributable to its entropy of mixing. At the density corre-sponding to such pressure, the thermodynamic affinity of the

Fig. 4 The thermodynamic affinity, A(p, T; a) of a chiral hard-bodysystem of asphericity a \ 1.5 and prolateness b \ 2.25 as a function ofpressure for dierent values of the parameter of chirality, eChr.

system changes sign, and at greater densities, the Gibbs freeenthalpy of the scalemic system is lower than that of theracemic one. As shown also in Fig. 4, for the lowest value ofthe parameter of chirality, eChr\ 0.025, the thermodynamicaffinity does not change sign, and the system would remainracemic at all pressures, which behavior is consistent with thepurely thermodynamic argument of section 2.

In Fig. 5 are shown the thermodynamic affinities for therepresentative hard-body system, for each of the test values ofthe parameter of chirality, on a linear pressure scale between

1500 kbar. That the affinity for the lowest value of the chiralparameter, eChr\ 0.025, does not change sign shows clearly.Although the traces in Fig. 5 appear linear, they are not; thetrace for the lowest value of the chiral parameter has a nega-tive second derivative, and, at high pressures, the Gibbs freeenthalpy of the single-component system increases less rapidlythan that for the racemic one. The affinity, for that and lesservalues ofeChr, never crosses the x-axis; and the system remainsracemic.

If the molecular species comprising the enantiomers is gen-erated at high pressure, as for example, hydrocarbon mol-ecules from hydrogen and carbon, the system will evolve,above the racemicscalemic transition pressure, a stronglyunequal distribution of enantiomers for which the abundanceratio will be determined by the thermodynamic affinity andthe general law of mass action. If the system was initially com-posed of both enantiomers, above the racemicscalemic tran-sition pressure, it will either transform into one of an unequaldistribution of enantiomers, or will undergo phase separation;in such case, its specic evolution will depend upon the tem-perature and the transition rate for the processes by whichone enantiomer transforms into its chiral opposite.

4.2 Observation of the evolution of the chiral molecules

CHFClI and by Monte Carlo simulationC8

H12

The previous theoretical analysis was restricted to convex,hard-body molecules. Such restriction allowed formal mathe-matical precision at a cost of generality. In order to investi-

gate the racemicscalemic transition of more realistic chiral

Fig. 5 The thermodynamic affinity, A(p, T; a) of a chiral hard-bodysystem of asphericity a \ 1.5 and prolateness b \ 2.25 as a function ofpressure for dierent values of the parameter of chirality, eChr, on alinear scale.



9/12

molecules, isothermalisobaric Monte Carlo simulations havebeen performed, over wide ranges of density, to calculate theexcess volumes of two uids consisting of the fused-hard-sphere molecules: uorochloroiodomethane (CHFClI), and 4-vinylcyclohexene Fluorochloroiodomethane is both(C

8H

12).

chiral and not convex, although not strongly concave. Themolecule 4-vinylcyclohexene is strongly concave, as shown inits stick diagram representation in Fig. 6.

The atomic coordinates for CHFClI were obtained with theMM3 computer packages of Allinger et al.,32h34 and those of

4-vinylcyclohexene were known from previous work. For bothmolecules each atom was modeled as a hard sphere with theappropriate van der Waals radius, which was taken from thedata compiled by Emsley35 and reduced by approximately10% to agree better with experimental data of their uid den-sities.

The simulation ensembles contained 256 or 512 molecules,either pure enantiomers or a racemic mixture. A simulationcycle consisted of attempted movestranslation androtationof all molecules, followed by a random volumechange. After the system reached equilibrium, such that thevolume remained constant throughout a cycle, 150 000500 000 cycles were executed to obtain the thermodynamicaverages.

The xed property in the simulation of a hard-body systemis the dimensionless pressure/temperature ratio:

pr

\ls3

kB

Tp (41)

where is the length unit of the simulation, taken for thesels

investigations as 1 The calculated property is the systemA .density or the molar volume. Thus a value corre-p

r\ 0.3

sponds roughly to 12 kbar at 300 K. The dierence of themolar volumes of, respectively, the racemic mixture and thepure enantiomer gives directly the excess volume. The simula-tions reported here were limited to a maximum dimensionlesspressure of 0.3. At higher pressures, these samples undergopartial solidication by the AlderWainwright transition.

Therefore, above such pressure, no meaningful density aver-ages can be calculated.The results of the Monte Carlo simulation investigations of

uorochloroiodomethane and 4-vinylcyclohexene are shownin Fig. 7. The negative values and uctuations of the excessvolumes of both CHFClI and at low pressures are notC

8H

12

Fig. 6 4-Vinylcylohexene, C8

H12

.

Fig. 7 Excess volumes of uorochloroiodomethane, CHFClI, and 4-vinylcyclohexene, as functions of reduced pressure.C

8H

12,

important; for the thermodynamic chirality functional, QChr,involves the integral of the excess volume over pressure.Therefore, the high-pressure values of VE determine both QChrand the Gibbs free enthalpy. It can be seen that, for both mol-ecules studied, the pure enantiomer uids have greater densityat high pressures. For interpreting the excess volumes, theabsolute uncertainty of the molar volumes and of the excessvolumes can be estimated as:

*V\dV

dxi

*xi\ [

V

xi

*xi. (42)

Therefore, although the density of a dilute gas at the dimen-sionless pressure of 0.001 can be calculated quite accurately,the excess volume of the halogenated methane has (at that

value of a statistical uncertainty of approximately 1 cm3pr)mol~1 ; and therefore, in that case, a value for the excessvolume of 0.5 cm3 mol~1 must be regarded as insignicant.However, at high pressures, the statistical uncertaintydecreases to approximately 0.1 cm3 mol~1. Therefore thesecalculated excess volumes must be regarded as signicantlypositive.

5 Discussion

The phenomenon of optical activity in abiotic uids has beenshown in the previous sections to be a direct consequence ofthe chiral geometry of the system particles acting according tothe laws of classical thermodynamics.

(1) In section 2, the purely thermodynamic argument was

developed which relates the evolution of optical activity in asystem of chiral molecules to the excess volume of scalemicmixtures. The thermodynamic analysis established that, abovea threshold value of excess volume, the Gibbs free enthalpybecomes, at high densities, lower for an unequal, scalemic dis-tribution of enantiomers.

(2) In section 3, the excess volume of a scalemic mixture ofenantiomers has been related to their geometric propertiesusing the KiharaSteiner equations, which have beenextended to describe particles which lack a center of inversion.The chiral property described by the extension of the KiharaSteiner equations has been introduced into the

equations for mixtures of hardPavlcekNezbedaBoublkbodies, with which are calculated the Gibbs free enthalpies

and thermodynamic affinities of the hard-body systems.(3) In section 4, formal calculations of the thermodynamic

affinities show that, in accordance to the dictates of the second



10/12

law, a system of chiral molecules will often evolve unbalanced,scalemic abundances of enantiomers at high densities. Theresults of Monte Carlo investigations on the chiral (and non-convex) molecules CHClI and demonstrate positive,C

8H

12increasing excess volume at high pressures, the prerequisite forthe racemicscalemic transition.

Although the formal analysis in the preceding section con-sidered model systems composed of hard bodies, the resultsthere described hold true also, without qualication, for realmoleculesi.e., ones which possess additionally a long-range,

attractive, van der Waals-type component to their intermolec-ular potential. The racemicscalemic transition occurs at adensity at which the free energy contribution attributable tothe product of the systems excess volume and pressureexceeds that attributable to its entropy of mixing. The formeris usually positive, the latter always negative. As indicated ineqn. (12) and Fig. 1, and as demonstrated using eqns. (34), (35)and (36), and as shown explicitly in Fig. 4, the onset of theracemicscalemic transition occurs at increased densities. Thedependence of the racemicscalemic transition upon pressureresults from its control of the density. The attractive, long-range, van der Waals-type component to the potential canonly act to increase the density, at all temperatures. Therefore,the presence of a van der Waals-type component to the poten-tial could not eliminate, or cause somehow to vanish, theonset of the racemicscalemic transition in any uid of realmolecules the geometry of whose short-range, repulsivei.e.,hard-bodycomponent of potential imposes an increase inthe excess-volume component of the Gibbs free enthalpy themagnitude of which exceeds its mixing-entropy component.

Many of the qualitative features of the racemicscalemictransition can be understood by considering the virial expan-sion of the chemical potentials which enter the thermodyna-mic affinity. The chemical potential, when written in terms ofthe virial coefficients, has the form:

ki(p, T; a) \ k

iI.G.(p, T) ] (B

2)ipi] , (43)

in which represents the second virial coefficient in theB2

density expansion of the compression factor Z \pV/nRT,

and equals, to the rst-order approximation of the virialexpansion, the natural logarithm of the fugacity coefficient.For a mixed, hard-body gas, the second virial coefficient is,exactly,

B2

\ t] rs \A

;i

xiV3i

B]

A;i

xi

R3i

BA;j

xj

S3j

B; (44)

and for the contribution of the ith component,

(B2

)i\ V3

i] R3

iS3i] R3

i

A;jEi

xj

S3j

B] S3

i

A;jEi

xj

R3j

B.

\ ti0 ] (rs)

i0 ] (rs)

iE (45)

For a binary mixture of enantiomers, the excess contribution

to the second virial coefficient is, when the approximation eqn.(22) is applied:

(rs)DLE \ (rs)

DL0 e

DLChr. (46)

Comparing eqns. (46) and (9), one may note that

QL, D

(p, T)D 4rseDLChr

p

RT, at low pressures. (47)

When the Kihara asphericity parameter, a \ rs/3t, is intro-duced into eqn. (47), the thermodynamic chirality functionmay be approximated as,

QL, D

(p, T)D12taeDLChr

p

RT. (48)

The approximation eqn. (48) indicates qualitatively many ofthe principle characteristics of the racemicscalemic transition,and, together with eqns. (12) and (13), provides a direct inter-

pretation of the limitations of the thermodynamic chiralityfunction described in section 2:

(1) For a combination of hard-body molecular volume, t,degree of asphericity, a, and parameter of molecular chirality,eChr, there will exist a (usually high) pressure at which theadditional free energy attributable to the excess volume of aracemic mixture will exceed that attributable to its entropy ofmixing.

(2) For a given parameter of chirality, eChr, the greater is thedegree of asphericity, a, the lower will be the density and pres-

sure of the racemicscalemic transition; similarly, for a givendegree of asphericity, a, the larger is the parameter of chirality,eChr, the lower will be the density and pressure of the racemicscalemic transition.

(3) For any combination of the degree of asphericity, a, andthe parameter of chirality, tChr, the larger is the hard-bodymolecular volume, t, the lower will be the density and pres-sure of the racemicscalemic transition.

(4) If the product of the hard-body molecular volume, t, thedegree of asphericity, a, and the parameter of chirality, eChr, isless than a threshold value, the system will never undergo theracemicscalemic transition no matter how high its pressure.As noted in the previous section, these easily-understoodproperties are exactly those demonstrated by many detailedcalculations of binary chiral, hard-body systems of diversedegrees of asphericity, chirality and particle volumes.

The phenomena of expansion upon mixing, and segregation(or phase separation) at high pressures, are interrelated. Thepressure dependence of the complex phase behavior of binaryliquid systems has been described generally by Rebelo.36 Thetype of phase separation characteristic of the racemicscalemictransition corresponds to liquidliquid equilibrium (LLE)behavior described by Rebelo as either type 1 (or a) or type 3(or c) as shown in Fig. 3 on page 4279 in the work cited. Bothtypes of uid behavior manifest phase separation withincreased pressure and a lower critical solution pressure tran-sition (LCSP) point. Therefore, a mixture of enantiomers withfavorable physical parameters should be expected to manifestan LCSP. When a liquid mixture of enantiomers coexists with

its vapor under the constraints of eqn. (11), the system willinevitably present an azeotrope, for the equal pressures ofboth L and D enantiomers, combined with their non-idealbehavior, assure that the pressure must be at an extremum.

As noted in section 2, for a system to manifest optical activ-ity, the onset of the racemicscalemic transition must occurbefore the system undergoes the high-pressure, AlderWainwright, uidsolid transition. The prediction of theAlderWainwright transition by scaled particle theory hasalready been discussed generally37 and further investigatedspecically for aspherical, hard-body systems.31 The absolutelimits of the uid phase of any system are determined by thevalues of density at which its entropy vanishes. The

equations give such values by thePavlc ekNezbedaBoublk

roots of the equation :

5

2] ln

A VNj3

B\ [

(Shc)PNB

NkB

, (49)

in which the hard-core com-PavlcekNezbedaBoublkponent of the entropy, (Shc)PNB, is:

(Shc)PNB\[ NkB

CC ln(1 [ g) ]

g(Dg ] E)(1 [ g)2

D. (50)

The geometric functionals, C, D and E, in eqn. (50) are denedin eqn. (35) and (37); and from those equations, one can per-ceive quickly that the complexity of the relationships between

the KiharaSteiner functionals, and do not permit aR3i , S3i V3isimple estimate for predicting the particular geometries forwhich the onset of the racemicscalemic transition will



11/12


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this present analysis for the previous ones of meteoritematerial.

Analysis of the material in carbonaceous meteorites hasestablished that their minerals, rocks, and agglomeratedmaterial went through various stages of fractionation in thecourse of formation. From their chemical composition andcalculated velocity of cooling, the carbonaceous chondriteshave been ascertained to be high-pressure remnants, originallyformed at depths between 70150 km in parent celestialbodies whose diameters were more than 800 km, and perhaps

as great as 2500 km.44 The observations not only of heavyhydrocarbon molecules but also of diamond minerals, in theircubic (diamond), hexagonal (lonsdaleite), and mixed (chaoite)forms, in such as the Novo Urei, Goalpara, and North Haigmeteorites, attest to their high-pressure history.45

Thus the present thermodynamic analysis of optical activ-ity, which has invoked no specic material properties exceptthe necessary one of molecular chirality, is consistent with theastrophysical and chemical analysis of materials extractedfrom meteorites. The results here reported both conrm andsupport the previous analyses of the material from meteorites.

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the evolution of multicomponent systems at high pressures

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