The Evolution of Multicomponent Systems at High Pressures:  IV.  The Genesis of Optical Activity in High-density, Abiotic Fluids.

J. F. Kenney

Joint Institute of the Physics of the Earth, Russian Academy of Sciences;

Gas Resources Corporation,

11811 North Freeway, fl. 5, Houston, TX 77060, U.S.A.; JFK@alum.MIT.edu   

Ulrich K. Deiters

Department of Physical Chemistry, University of Cologne

Luxemburger Strasse 116, D-50939, Cologne, GERMANY

 

 

Abstract:

 

          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 specific connection is shown between the excess volume and the statistical mechanical partition function.  The Kihara-Steiner equations, which describe the geometry of convex hard bodies, have been extended to include also chiral hard bodies.  These results have been incorporated into an extension of the Pavlíek-Nezbeda-Boublík equation of 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 fluorochloroiodomethane, CHFClI, and 4-vinylcyclohexene, C₈H₁₂, have been investigated by Monte Carlo simulation, and observed to manifest, both, positive excess volumes (in their racemic mixtures) which increase with pressure, and thereby the racemic-scalemic transition to unbalanced distributions of enantiomers.  The racemic-scalemic transition, responsible for the evolution of an optically active fluid, 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.

 

[Keywords:  optical activity, chirality, hard-body systems, thermodynamics.]

 

1.       Introduction.

          The phenomenon of optical activity in fluids, either biotic or abiotic, requires, simultaneously, two distinct phenomena:  the presence of chiral molecules, which lack a center of inversion;  and an unequal distribution of those chiral enantiomers.  A system of chiral molecules characterized by a distribution of equal abundances of enantiomers is a racemic mixture;  ones characterized by distributions of unequal abundances are scalemic mixtures.  Only scalemic mixtures manifest optical activity.  Certain biological processes, such as natural fermentation, generate chiral molecules of only one enantiomer.  Abiological processes can produce either equal or unequal enantiomer abundances, depending upon the thermodynamic conditions of their evolution.

          The phenomenon of optical activity in abiotic fluids is shown in the following sections to be a direct consequence of the chiral geometry of the system particles acting according to the laws of classical thermodynamics.  In the following sections a purely thermodynamic argument is developed which relates the evolution of optical activity in a system of chiral molecules to the excess volume of scalemic mixtures.  The excess volume of the scalemic mixture of enantiomers is related to their geometric properties using the Kihara-Steiner equations, which have been extended to describe particles which lack a center of inversion.  The chiral property described by the extension of the Kihara-Steiner equations is introduced into the Pavlíek-Nezbeda-Boublík equations for mixtures of hard bodies, with which are calculated the Gibbs free enthalpies and thermodynamic Affinities of hard-body systems.  The calculated thermodynamic Affinities establish that, in accordance with the dictates of the second law, a system of chiral molecules will often evolve unbalanced (scalemic) abundances of enantiomers at high densities.

          For experimental verification of the theoretical calculations made with convex hard-body systems, Monte Carlo simulations have been carried out on the real chiral molecules fluorochloroiodomethane, CHFClI, and 4-vinlycyclohexene, C₈H₁₂.  Both CHFClI, and C₈H₁₂ have been observed, at high pressures, to manifest higher densities in their scalemic distributions, as compared to their racemic ones.  Such density change drives the racemic-scalemic transition.

 

          1.1     Historical background.

          From its first demonstration by Pasteur, the phenomenon of optical activity in fluids has engaged the attention and interest of the scientific community.¹  This phenomenon has provided an arena for considerable exercise of imagination and creative fantasy, regrettably almost entirely unleavened by the discipline of thermodynamics.

          Perhaps for reason of its historical provenance in fermented wine, the phenomenon of optical activity in fluids was for some time believed to have some intrinsic connection with biological 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.

          From the interiors of carbonaceous meteorites have been extracted the common amino-acid molecules alanine, aspartic acid, glutamic acid, glycine, leusine, proline, serine, threonine, as well as the unusual ones α-aminoisobutyric acid, isovaline, pseudoleucine.^2-4  At one time, all had been considered to have be solely of biotic origin.  The ages of the carbonaceous meteorites were determined to be 3-4.5 billion years, and their origins clearly abiotic.  Therefore, those amino acids had to be recognized as compounds of both biological and abiological genesis.  Furthermore, solutions of amino acid molecules from carbonaceous meteorites were observed to manifest optical activity. Thus was thoroughly discredited the notion that the phenomenon of optical activity in fluids (particularly those of carbon compounds) might have any intrinsic connection with biotic matter.  Significantly, the optical activity observed in the amino acids extracted from carbonaceous meteorites has not the characteristics of such of common biotic origin, with only one enantiomer present; instead, it manifests the characteristics observed in natural petroleum, with unbalanced, so-called scalemic, abundances of chiral molecules.

          The optical activity commonly observed in natural petroleum was for years speciously claimed as a "proof" of some connection with biological detritus, - albeit one requiring both a willing disregard of the considerable differences between the optical activity observed in natural petroleum and that in materials of truly biotic origin, such as wine, as well as desuetude of the dictates of the laws of thermodynamics.  Observation of optical activity, typical of such in natural petroleum, in hydrocarbon material extracted from the interiors of carbonaceous meteorites, discredited those claims.^{5, 6}  Nonetheless, the scientific conundrum remained as to why the hydrocarbons manifest optical activity, in both carbonaceous meteorites and terrestrial crude oil.

          There is common misunderstanding that the molecular property of chirality, which is responsible for optical activity in fluids, is an unusual, complicated property of large, complex, multi-atomic molecules.   The small, common, single-branched alkane molecules are usually chiral.  Single-branched alkanes comprise between 7-15% of the molecular components of natural terrestrial petroleum and are also observed in petroleum synthesized by such as the Fischer-Tropsch processes.  When these chiral molecules are created in low-pressure industrial processes, they occur always in equal enantiomer abundances, and the resulting synthetic petroleum does not manifest optical activity.  In natural petroleum, these molecules occur in unequal enantiomer abundances, and the fluid manifests optical activity.  Molecular chirality results from the highly directional property of the covalent bond, and is indifferent to whether a compound results from a biological or an abiological process.

          Previous hypotheses offered to explain optical activity of the compounds extracted from carbonaceous meteorites have invoked such deus ex machina as "panspermia,"  the "seeding" of optically active biotic molecules from (literally) the heavens,^{7, 8} - or cumulative effects of the chiral weak-interaction involved in beta decay,^9-16 - with necessary oversight of the several orders of magnitude of energy difference compared to that attributable to the entropy of mixing, which would destroy any imbalance responsible for optical activity.

          With no recourse to any such artifices, the phenomenon of optical activity in fluids is here shown to be an inevitable consequence of the dictates of thermodynamic stability theory manifested 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 fluids is taken up thoroughly, in order that its fundamental thermodynamic property be set forth explicitly, and that its statistical mechanical genesis be demonstrated.  This paper is organized into three parts:

1.                 A thermodynamic argument is developed which relates the Gibbs free enthalpy, and the phase stability of a mixed system, to its excess volume.  This argument invokes no specific molecular property, and uses only a strict thermodynamic definition of a system containing chiral components which specifies 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 determined, at low pressures, by its entropy of mixing,  and, at high pressures, by its excess volume.

2.                 A statistical mechanical argument is developed which relates directly the Gibbs free enthalpy and excess volume to specific molecular geometric properties.  The Kihara-Steiner equations have been extended to describe hard-body particles which do not possess a center of inversion;  and the results have been applied to the Pavlíek-Nezbeda-Boublík equations for convex hard-body fluids.

3.                 Using the Pavlíek-Nezbeda-Boublík equations, the thermodynamic Affinity has been calculated formally for a diverse group of hard-body fluid systems characterized by different molecular volumes, and degrees of asphericity and chirality.  These fluids are shown to undergo the racemic-scalemic transition exactly as required by the general thermodynamic argument developed in section 2.

                    Using Monte Carlo simulation, the two real, chiral fluids, fluorochloroiodomethane, CHFClI, and 4-vinlycyclohexene, C₈H₁₂, have been investigated as pure chiral components and as racemic mixtures.  The latter are shown to develop positive excess volumes at increased densities, which increase approximately linearly with pressure, and which therefore induce the racemic-scalemic transition.

 

2        Molecular chirality.

          There is common misunderstanding that the molecular property of chirality, which is responsible for optical activity in fluids, is an unusual, complicated property of large, complex molecules, themselves probably of biotic origin and comprised of numerous different atomic species.  This misunderstanding appears supported by such (otherwise erudite) treatises on the subject as that by Jacques et al.¹⁷, which discuss such complex molecules as D-glucopyranose, and L-hydroxy-2-hydridamine-d-α-bromocamphor-π-sulfonate, but fail even to mention the chirality of the banal, simple molecules described below.  The purpose of this short section is to correct such misperception with a simple, clear, elementary description of molecular chirality, demonstrated by small, common molecules, comprised of only two atomic species.  

          Chiral geometry, characterized by many compounds, results from the highly directional property of the covalent bond, itself the characteristic responsible for the wealth and diversity of stereochemistry.  The directional property of the covalent bond, including particularly that of carbon, is indifferent to whether a compound results from either a biological or abiological process.

          Consider, for example, the tetrahedral methane molecule, altered so that the hydrogen atom at its uppermost apex remains while those at the 2-, 6-, and 10-o’clock positions on the base of the tetrahedron are replaced, respectively, by a propyl radical (-C₃H₇), a methyl radical (-CH₃), and an ethyl radical (-C₂H₅).  The molecule which results, represented schematically in Fig. 1, is 3-methylhexane, a common, single-branched alkane, observed both in natural petroleum and also in petroleum industrially synthesized by Fischer-Tropsch processes.  Plainly, 3-methylhexane is a chiral molecule.  Equally plainly, the isomer which would result by exchanging the ethyl and methyl radicals, is another, distinct chiral molecule.  [These isomers are often designated (R)-3-methylhexane and (S)-3-methylhexane, respectively.]  Furthermore, if the propyl radical at the 2-o’clock position of the base of the tetrahedron were substituted by a butyl group (-C₄H₉), or any n-alkyl group for which n > 3 (-C_nH_2n+1), the resulting single-branched alkane would also be chiral.  Furthermore still, the distinct, single-branched, alkane isomers formed by changing the position of the methyl group from the 3-position to the 2-, or 4-, or any other position on the alkane chain (except the center one), will also be chiral.

          These simple considerations demonstrate that chirality is an inevitable and banally common molecular property, particularly among carbon compounds.  Single-branched, chiral alkanes comprise between 7-15% of the molecular components of natural terrestrial petroleum and are observed in petroleum synthesized by such as the Fischer-Tropsch processes.  When these chiral molecules are created in low-pressure industrial processes, they occur always in equal enantiomer abundances, and the resulting synthetic petroleum does not manifest optical activity.  In natural petroleum, these molecules occur in unequal enantiomer abundances, and the fluid manifests optical activity.

 

3.       The explicit, general prediction of the genesis of optically active systems by classical thermodynamic argument.

          First is shown that the evolution of unbalanced abundances of enantiomers, scalemic mixtures, often results inevitably from the general requirements of thermodynamic stability theory.  In keeping with the traditions of classical thermodynamics, no assumptions are made about any detailed properties of the material which composes the fluid mixture, other than the most basic attributes of their chirality.

          From the cross derivatives of the differential of the Gibbs free enthalpy, G(p,T,{n_j}),

                                                                             (1)

the differential equation for the chemical potential, as a function of pressure, at constant temperature is given as:

.                                                                           (2)

With inclusion of the Gibbs mixing factor RTlnx_i, the chemical potential of the i-th species is given in terms of its partial volume as:

.                                                                        (3)

The volumetric behavior of a multicomponent system can be described by the intensive variable, excess volume:  V^E = V - Σ_(j)x_jV_m,j,.  When the formalism developed by Guggenheim¹⁸ and Scatchard¹⁹ is applied, the excess volume can be expressed as a series expansion:

.              (4)

For the present analysis, the Guggenheim-Scatchard expansion, (4), may be truncated at the first term without loss of generality;  when such is done, the molar volume may be written as:

,                                                                    (5)

which approximation constitutes application of the Porter Ansatz.  (The excess molar volumes in equation (5), , contain the factor four in order that the it will return its maximum value in the case of a binary equimolar mixture at the racemic molar fractions x_i = x_j.)

 

          3.1     The thermodynamic chirality function.

          A thermodynamic system capable of manifesting optical activity may be considered generally as consisting of the two chiral enantiomers, L and D, together with a third achiral component, A.  For such a three-component system, the chemical potential of the D-component enantiomer is:

.                                                                     (6)

When equation (5) is applied, together with the property that , the partial molar volume of the D-component enantiomer becomes,

;                                                  (7)

and its chemical potential is thereby,

.                               (8)

Let the thermodynamic chirality functional, Q(p,T;  )_DL be defined as the integral of the excess volume of the system of chiral molecules over its pressure as:

.                                                     (9)

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

;                 (10)

an analogous expression for μ_L, with the subscript L replacing that of D in equation (10).

 

          3.2     Enantiomeric separation and conversion processes.

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

                                                                                                  (11)

[The validity of the first two of equations (11) is intuitively obvious.  The third of equations (11) may be considered to be (at this point) an assertion;  it is proven in the following sections to hold usually.]  The simultaneous requirements set forth by equations (11) are strictly thermodynamic ones and involve no detailed properties of the molecules themselves.  In addition to the equalities of equations (11), the excess volume for either enantiomer with a non-chiral component, A, is identical, such that Q_DA =  Q_LA.

          Let it be assumed that the system is a reactive one, for the enantiomers can convert into one another, .  The condition of chemical equilibrium, μ_D = μ_L, together with equations (10) and (11) and the equality Q_DA =  Q_LA, gives:

.                                                                                (12)

Equation (12) expresses succinctly and rigorously the essential determining factors for the spontaneous, abiotic, evolution of optical activity in fluids:  the competition between the entropy of mixing, given by the first logarithmic term, and the oppositely-directed effects of the inevitable packing inefficiency of mixed enantiomers, given by the thermodynamic chirality function, Q_DL.  Equation (12) defines the analytic Lambert W function,^{20, 21} which has always one real root and, as shown in Fig. 2., sometimes three.

 

          The molar excess volume of a simple mixture often is relatively small, V^E ~ 1-10 cm³/mol.^{22, 23}  Therefore, at or near a pressure of one bar, for a system consisting only of the two enantiomers, Q_DL = (1/RT)pV^E  (10^-3/RT) kJ.  The denominator RT is approximately 2.5 kJ at 300 K.  Therefore, at modest pressures and essentially all temperatures, the second term in equation (12) cannot balance the first except at the value x_D = x_L, which solution describes the racemic mixture.  Thus equation (12) establishes that, at low pressures, an abiotic system will usually evolve into a racemic mixture of enantiomers.

          However, the thermodynamic chirality functional, Q_DL, depends directly upon the system pressure, for Q_DL ~ V^Ep.  For any non-vanishing, positive excess volume, V_DL^E, the thermodynamic chirality function has no limit as pressure increases.  Therefore, for a system whose thermodynamic chirality function is greater than a certain threshold value, (Q_DL)^threshold, there will always be a (usually, high) pressure above which the system will evolve unbalanced abundances of enantiomers, and the resulting system will inevitably be optically active.  For a system whose thermodynamic chirality function is less than the threshold value, (Q_DL)^threshold, there will be no transition pressure, and such system will remain racemic.  This behavior is shown graphically in Fig. 2 where are represented the plots of the two functions ln(x_D/(1-x_D)) and (2x_D-1)Q_DL (which correspond to a binary system composed solely of two enantiomers).  As seen clearly in  Fig. 2, for values of the thermodynamic chirality function, Q_DL, less than a threshold value, the system cannot evolve an unbalanced system.  For the value, Q_DL = 1.5, there is only one solution, the racemic root, x_D = 0.5;  for the value, Q_DL = 4, there is a second solution at the scalemic value, x_D  0.97 (and a third, symmetrically, at x_D  0.03).

          For a general system composed of two enantiomers and a third component, A,  the threshold value of Q_DL for the onset of a racemic-scalemic transition is that for which the two terms in equation (12) have equal derivatives with respect to x_D;  such that,

.                                                                                                 (13)

          The foregoing thermodynamic argument has shown that chiral molecules in an unbalanced, scalemic, distribution of enantiomers can possess lower chemical potentials, and thereby lower Gibbs free enthalpy, than a racemic distribution of the same compound, under certain conditions of density.   However, that argument gave no indication how a given distribution of enantiomers might, in new conditions of temperature or pressure,  convert into a different one, such that the system could assume a lower free enthalpy.

          For a chemically-reactive system, in which the achiral constituent molecules A and B are present, and in which the chiral molecules C evolve as,

,                                                                                         (14)

the system will always transform into the distribution of reagents and products which possesses the lowest possible Gibbs free enthalpy.  A chemically-reactive system will similarly evolve always that distribution of enantiomers which renders it the lowest free enthalpy.

          A system which is not considered chemically-reactive, composed solely of the enantiomers C_L and C_D will also transform itself always into that distribution of enantiomers which renders it the lowest Gibbs free enthalpy.  As a simple example, if either C_D or C_L enantiomer might result from a single-step reaction such as (14), the principle of detailed balance requires that there exists at equilibrium always a distribution of components,

,                                                                                     (15)

of which the abundances are determined by the general law of mass action.  With a change of temperature or pressure, the equilibrium coefficient will change;  and the abundance distribution of enantiomers will also change, so as to conform to the law of mass action and effect the minimum Gibbs free enthalpy.  The constant, simultaneous production of enantiomers C and dissociation into constituent reagents A  and B, in accordance with (15) and the principle of detailed balance, assure that the system always will evolve into the distribution of lowest free enthalpy.

 

          3.3     The enantiomeric phase separation in scalemic mixtures.

    If Q_DL is positive and of sufficient magnitude, there is a threshold pressure at which the racemic mixture becomes thermodynamically unstable.  If a conversion reaction D  L exists, the slightest excess of one enantiomer can make the system to develop a macroscopic excess of  this enantiomer, when the pressure is raised further.  Alternatively, the system can undergo a phase split into two scalemic phases.  In either case, application of sufficient pressure always leads to the formation of phases with enantiomeric excess.

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

                                   (16)

The limits of stability and critical points of mixtures are determined by the higher derivatives of the molar Gibbs free enthalpy, G_n,i = (∂ⁿG_m/∂x_iⁿ)_j.p,T , which gives for the D-enantiomer,

.                                 (17)

The second derivatives of G_m are, respectively:

                                    (18)

 

The conditions for phase stability require that:

                                                                                   (19)

The examination for phase separation, for which the equality holds in (19), introduces the equality

.                                                                                        (20)

The roots of the equality (19) then admitted are:

.                                                                   (21)

The first root of equation (21), for the general case of a three component system, corresponds to that determined by equation (12) for a binary system.

 The molar Gibbs free enthalpy of a system containing two enantiomers and a third achiral component, as given by equation (16), has been calculated using the value for Q_DL determined by the roots given by equation (21), and is shown in Fig. 3 as a function of the reduced molar fraction, (x_D)_r which is the D fraction of the total molar fraction of the enantiomers, (1-x_A).    (The reduced molar fraction has been used in order that the racemic solution will fall at the value 0.5.)  The double minima of the Gibbs free enthalpy as a function of (x_D)_r is immediately apparent in Fig. 3, as is the fact that the racemic mixture, for those values of Q_DL, is at a maximum.  That the two scalemic minima lie at equal values of Gibbs free enthalpy is shown by their double tangent.

          The foregoing analysis has involved determination of the equilibrium states of a multicomponent system, which contains a variable distribution of enantiomers together with a third achiral component, without considering the dynamic processes by which that system resolves to equilibrium when the pressure has changed.  The two minima shown in Fig. 3 can be reached either by chemical reaction or by phase separation.

          If the transition time for L → D conversion of the individual molecules is much slower than the transport diffusion time, then an initially racemic system will undergo a rapid physical separation, similar to gas-gas demixing;  and the system will resolve into two physically separate regions, one enriched in the D enantiomer, the other in the L.  At the opposite extreme, if the transition time for L → D conversion is much faster than the diffusion time, the system will undergo a quasi-chemical type of reaction and change its constituent composition entirely.    For cases in between, the system must be expected to undergo often complex chemical and dynamical behavior as it proceeds to equilibrium.

 

4.       Statistical mechanical calculation of the excess volume using the geometric properties of individual molecules.

          The thermodynamic analysis of the previous section established that a system of chiral particles will possess, in certain conditions of pressure, temperature, and degree of chirality, a lower free energy in a scalemic distribution than a racemic one.  Consistent with the traditional perspective of classical thermodynamics, that analysis invoked no detailed properties of the chiral system, beyond the minimum functional definition of chiral enantiomers set out in equations (11).  The thermodynamic analysis identified the system’s excess volume, acting through the defined thermodynamic chirality parameter, as the operative physical property responsible for the racemic-scalemic transition.  However, the classical thermodynamic analysis did not address the question how, or why, a system of chiral molecules ought, or must, manifest a non-vanishing excess volume.  Nor did the thermodynamic analysis give any specific indication how the distribution of the chiral enantiomers might be determined from properties of those molecules.

          In this section, the formalism is developed for direct calculation of the distribution of the chiral particles from the stereochemical properties of the individual molecules.  First, a precise description of the geometry of chiral, convex, hard-body particles is developed by extension of the Kihara-Steiner equations.   That geometric description is then used in the Pavlíek-Nezbeda-Boublík equations for convex, hard-body systems, from which an explicit expression is developed for the Helmholtz free energy of a multicomponent system which contains chiral particles.

 

          4.1.    The geometric description of a hard-body system:  Extension of the Kihara-Steiner equations.

          Convex hard-body systems have been described by Kihara and Steiner in terms of three geometric parameters, , , and , which are determined by the support function that describes the volume and surface generated by the rolling of one hard body around the surface of another, in all possible orientations.  As shown in the following subsection, the weighted products of these geometric entities, , , and , enter the equation of state for systems of convex hard bodies and determine its thermodynamic properties.  The parameter functional  represents the averaged radius of curvature of the support function and  its averaged surface area;  in this instance,  does not represent the i-th partial volume but the effective volume determined by the support function.  The individual functionals , , and  are defined by the equations:

                                                              (22)