Statistical Thermodynamic Model Research Articles

Pt3Ga: Thermodynamics and Nonstoichiometry

Thermodynamic activities of gallium were measured between about 1000 and 1300 K in the nonstoichiometric intermetallic compound Pt3Ga using an emf-method based on an oxygen conducting solid electrolyte. The variation of the lattice parameter with composition was determined by powder X-ray diffraction. The results of the activity measurements are interpreted in terms of a statisticalthermodynamic model for L12-phases considering four types of point defects, i. e. anti-structure atoms and vacancies on the platinum and gallium substructures. The energies of formation of the point defects at the stoichiometric composition are estimated from a curve fitting procedure yielding Ef(PtGa) = Ef(GaPt) = 1.25 eV, assuming that Ef(VPt) = Ef(VGa) = 2.0 eV. This results in a disorder parameter α’ = 3 · 10−6 at 1173 K which means that at the stoichiometric composition 0.0012% of the gallium substructure sites are occupied by platinum atoms and 0.0004% of the platinum sites by gallium atoms at this temperature.

Thermodynamic prediction of clathrate hydrate dissociation conditions in mesoporous media

We present modifications to a previously reported statistical thermodynamics model that facilitates the prediction of capillary pressure effects on hydrate equilibria in narrow pores. The model uses the Valderrama modification of the Patel and Teja Equation of State (VPT EoS) for fugacity calculations in fluid phases, while the hydrate phase is modeled using the solid solution theory of van der Waals and Platteeuw (1959), as implemented by Cole and Goodwin (1990). The Kihara model for spherical molecules is applied to calculate the potential function for hydrate-forming gases. To account for capillary pressure effects on phase fugacities, we apply a correction similar to the Poynting correction for saturated liquids. This correction can be applied to any model capable of predicting bulk (unconfined) hydrate phase equilibria. The only new parameter required is hydrate-liquid interfacial tension—values for which we have derived previously from experimental data. The model assumes cylindrical pores, although differs from the majority of existing literature models in how the curvature of the solid-liquid interface is considered; we assume a curvature of 2/ r for growth and 1/ r for dissociation, in accordance with accepted capillary theory. Model predictions are validated against previously published experimental hydrate dissociation data for binary CO2-H2O, CH4-H2O and ternary CH4-CO2-H2O systems, and newly reported data for CH4-H2O-CH4O (3.5 mass% methanol aqueous solutions representing the salinity of seawater) systems, confined to mesoporous silica glass. Good agreement between predictions and experimental data is observed.

Equilibrium hydrate formation conditions for hydrotrope–water–natural gas systems

The thermodynamic influence of hydrotrope para-toluene sulfonic acid (pTSA), when present in water–natural gas–hydrate system, is reported here for the first time. Hydrotropes are aromatic compounds amphiphilic in nature, yet differ from classical surfactants with their small hydrophobic part and comparatively larger hydrophilic part. The presence of surfactants in hydrate systems were reported not to influence the equilibrium thermodynamics of hydrate formation, i.e., the presence of surfactants did not shift the pure water–gas–hydrate equilibrium curves. This work finds pTSA, a hydrotrope and a surface-active agent, to influence the hydrate forming thermodynamics. The water–gas–hydrate equilibrium curves for three different additive concentrations in water were obtained from isochoric ‘temperature cycling’ experiments in a state-of-the-art PVT cell. The hydrotrope aggregation behaviour in water was incorporated, for the first time, into the classical statistical thermodynamic hydrate equilibrium prediction model proposed by van der Waals and Platteeuw [Am. Inst. Chem. Eng. 32 (1986) 1321]. The micelle size, or aggregation number, of hydrotropes was taken as the only adjustable parameter in the model, which was adjusted according to the hydrate equilibrium experiment data to give a minimum average absolute deviation (AAD) fit between the model and the experiment values. A relationship was subsequently developed for the micelle size and pTSA concentration, making the scheme a priori prediction model for concentrations within the tested region.

A Statistical-Thermodynamic Model of Viral Budding

We present a simple statistical thermodynamic model for budding of viral nucleocapsids at the cell membrane. The membrane is modeled as a flexible lipid bilayer embedding linker (spike) proteins, which serve to anchor and thus wrap the membrane around the viral capsids. The free energy of a single bud is expressed as a sum of the bending energy of its membrane coat, the spike-mediated capsid-membrane adhesion energy, and the line energy associated with the bud's rim, all depending on the extent of wrapping (i.e., bud size), and density of spikes in the curved membrane. This self-energy is incorporated into a simple free energy functional for the many-bud system, allowing for different spike densities, and hence entropy, in the curved (budding) and planar membrane regions, as well as for the configurational entropy of the polydisperse bud population. The equilibrium spike densities in the coexisting, curved and planar, membrane regions are calculated as a function of the membrane bending energy and the spike-mediated adhesion energy, for different spike and nucleocapsid concentrations in the membrane plane, as well as for several values of the bud's rim energy. We show that complete budding (full wrapping of nucleocapsids) can only take place if the adhesion energy exceeds a certain, critical, bending free energy. Whenever budding takes place, the spike density in the mature virions is saturated, i.e., all spike adhesion sites are occupied. The rim energy plays an important role in determining the size distribution of buds. The fraction of fully wrapped buds increases as this energy increases, resulting eventually in an all-or-nothing mechanism, whereby nucleocapsids at the plasma membrane are either fully enveloped or completely naked (just touching the membrane). We also find that at low concentrations all capsids arriving at the membrane get tightly and fully enveloped. Beyond a certain concentration, corresponding approximately to a stoichiometric spike/capsid ratio, newly arriving capsids cannot be fully wrapped; i.e., the budding yield decreases.

Sedimentation equilibrium in a solution containing an arbitrary number of solute species at arbitrary concentrations: theory and application to concentrated solutions of ribonuclease

Simple expressions are derived describing the equilibrium concentration gradient of each species in a solution containing an arbitrary number of solute species at arbitrary concentration, as a function of the concentration of all species. Quantitative relationships between the species gradients and experimentally observable signal gradients are presented. The expressions are model-free and take into account both attractive and repulsive interactions between all species. In order to analyze data obtained from strongly nonideal solutions, a statistical thermodynamic model for repulsive solute–solute interactions is required. The relations obtained are utilized to analyze the dependence of the equilibrium gradient of ribonuclease A in phosphate-buffered saline, pH 7.4, upon total protein concentration. Experimental results are interpreted in the context of a model for weak self-association leading to the formation of significant amounts of oligomers at total protein concentrations exceeding 25 g/l.

Thermodynamics and nonstoichiometry in the intermetallic compound Pt3In

Thermodynamic activities of indium were measured between 973 and 1273 K in the nonstoichiometric intermetallic compound Pt3In using an emf-method based on an oxygen conducting solid electrolyte. The variation of the lattice parameter with composition was determined by powder X-ray diffraction, and the corresponding results point to a considerably wider homogeneity range on the indium-rich side than previously reported. The results of the activity measurements are interpreted in terms of a statistical-thermodynamic model for L12-phases considering four types of point defects, i.e. anti-structure atoms and vacancies on the platinum and indium sublattices. The energies of formation of the point defects at the stoichiometric composition are estimated from a curve fitting procedure yielding Ef(PtIn)=Ef(InPt)=1.15 eV, assuming that EfV(Pt)=EfV(In)=2.0 eV. This results in a disorder parameter α′=5·10−6 at 1173 K which means that at the stoichiometric composition 0.002% of the indium sublattice sites are occupied by platinum atoms and 0.0007% of the platinum sites by indium atoms at this temperature.

First-Principles Study of Structural and Defect Properties in FeCo Intermetallics

ABSTRACTEmploying ab-initio calculations and statistical thermodynamic modeling, we investigated the structural stability, defect energies, and ordering of B2 FeCo intermetallics. We find that FeCo in the B2 structure is a marginally stable and weakly ordered system, with a high density of antisite defects on both sublattices and low APB energies for the <111> slip on both {110} and {112} planes. The structural stability of B2 FeCo is very sensitive to the change in local atomic environment, as the system transforms to a lower-symmetry L10 phase under the effects of reduced dimensionality or applied shear stress. We suggest that internal stresses near dislocation cores might be closely connected with the intrinsic brittleness of ordered FeCo, as it is likely to induce a local structural transformation from the B2 structure to the L10 structure.

Proton nuclear magnetic resonanse studies on hydration of oxaalkanes in benzene solutions

Hydrogen bonding between water and oxaalkanes (monoethers, diethers and acetals) in [ 2H] 6-benzene solutions was studied by 1H-NMR spectroscopy. It has been found that the stability constants of 1:1 hydrates of monoethers and acetals are similar, whereas the stability constants of diether hydrates are about twice as large. These differences indicate that the hydrophilic properties of the etheral oxygen atoms in both mono- and diethers are nearly the same, and the differences in the stability constants can be rationalised with a simple statistical thermodynamic model. On the contrary, the acetal oxygen atoms, separated by a single methylene group, are considerably less hydrophilic, probably due to their mutual inductive effects.

Viral self-assembly as a thermodynamic process.

The protein shells, or capsids, of nearly all spherelike viruses adopt icosahedral symmetry. In the present Letter, we propose a statistical thermodynamic model for viral self-assembly. We find that icosahedral symmetry is not expected for viral capsids constructed from structurally identical protein subunits and that this symmetry requires (at least) two internal "switching" configurations of the protein. Our results indicate that icosahedral symmetry is not a generic consequence of free energy minimization but requires optimization of internal structural parameters of the capsid proteins.

Interactions of the Escherichia coli DnaB Helicase Hexamer with the Replication Factor the DnaC Protein. Effect of Nucleotide Cofactors and the ssDNA on Protein–Protein Interactions and the Topology of the Complex

Quantitative studies of interactions between the Escherichia coli replication factor DnaC protein and the DnaB helicase have been performed using sedimentation velocity and fluorescence energy transfer techniques. The applied novel analysis of the sedimentation data allows us to construct thermodynamic rigorous binding isotherms without any assumption as to the relationship between the observed molecular property of the complexes formed, the average sedimentation coefficient, or the degree of binding. Experiments have been performed with the fluorescein-modified DnaB helicase, which allows an exclusive monitoring of the DnaB–DnaC complex formation. The DnaC binding to the unmodified helicase has been characterized in competition experiments. The data establish that, in the presence of the ATP analog AMP-PNP, or ADP, a maximum of six DnaC monomers bind cooperatively to the DnaB hexamer. The positive cooperative interactions are limited to the two neighboring DnaC molecules. Analyses using a statistical thermodynamic hexagon model indicate that, under the solution conditions examined, the affinity is characterized by the intrinsic binding constant K=1.4(±0.5)×10 5 M −1 and cooperativity parameter σ=21±5. These data suggest strongly that the DnaC–DnaB complex exists in vivo as a mixture of complexes with a different number of bound DnaC molecules, although the complex with six DnaC molecules bound dominates the distribution. The DnaC nucleotide-binding site is not involved in the stabilization of the complex. Moreover, the hydrolysis of NTP bound to the helicase or the DnaC is not required for the release of the DnaC protein from the complex. The single-stranded DNA (ssDNA) bound to the helicase does not affect the DnaC protein binding. However, in the presence of the DNA, there is a significant difference in the energetics and structure of the ternary complex, DnaC–DnaB–ssDNA, formed in the presence of AMP-PNP as compared to ADP. The topology of the ternary complex DnaC–DnaB–ssDNA has been determined using the fluorescence energy transfer method. In solution, the DnaC protein-binding site is located on the large 33 kDa domain of the DnaB helicase. The significance of the results in the functioning of the DnaB helicase–DnaC protein complex is discussed.

A Compact and Lipophilic Enantiomer Showing a Bilayer Crystal Structure − Free Energy of Spontaneous Resolution of the Racemic Compound into the Crystalline Enantiomers

AbstractThe melting points and the enthalpies and entropies of fusion of (R)‐1 and rac‐1, with the racemate both as the conglomerate and as the true racemic compound, were determined by differential scanning calorimetry: the melting points were 113.2 ± 0.3, 90.7 ± 0.4, and 80.5 ± 0.6 °C, respectively, the ΔHfus values 8.82 ± 0.03, 7.94 ± 0.27, and 5.98 ± 0.59 kcal·mol−1, respectively, and the ΔSfus values 22.8 ± 0.1, 21.8 ± 0.7, and 16.9 ± 1.7 cal·deg−1·mol−1, respectively. The specific heat of fusion (ΔCfus) of the pure enantiomer 1 calculated from these data is 39.1 cal·deg−1·mol−1. The ΔHfus, ΔSfus, and ΔCfus values of these enantiomers are consistently high relative to average values for organic compounds. The enthalpy, entropy, and free energy of formation of the racemic compound from the crystalline enantiomers are 1.56 kcal·mol−1, 3.8 cal·deg−1·mol−1, and 0.22 kcal·mol−1, as calculated from the above data. The positive sign of the free energy of formation bears out the observed spontaneous resolution of the racemic compound into the crystalline enantiomers. A statistical thermodynamic model of melting, weighing the different mechanical contributions to the crystalline and the liquid states, is described in conjunction with the interpretation of the data. It is concluded that both the interaction potential energy of the crystal of the pure enantiomer 1 and the vibrational mobility of the crystal structure are low, features pertinent to a low enthalpy and entropy of the crystal, respectively. X‐ray crystallography of (R)‐1 revealed a simple bilayer arrangement of the roughly ellipsoidal molecules: a characteristic bilayer consists of two monomolecular layers with interactions between the lipophilic groups in between, featuring a methyl−phenyl interaction, while the cyano groups project towards the outsides of the bilayer, which stacks by means of dipolar interactions to a constant distance. (© Wiley‐VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2003)

Energetics and Self-Assembly of Amphipathic Peptide Pores in Lipid Membranes

We present a theoretical study of the energetics, equilibrium size, and size distribution of membrane pores composed of electrically charged amphipathic peptides. The peptides are modeled as cylinders (mimicking α-helices) carrying different amounts of charge, with the charge being uniformly distributed over a hydrophilic face, defined by the angle subtended by polar amino acid residues. The free energy of a pore of a given radius, R, and a given number of peptides, s, is expressed as a sum of the peptides’ electrostatic charging energy (calculated using Poisson-Boltzmann theory), and the lipid-perturbation energy associated with the formation of a membrane rim (which we model as being semitoroidal) in the gap between neighboring peptides. A simple phenomenological model is used to calculate the membrane perturbation energy. The balance between the opposing forces (namely, the radial free energy derivatives) associated with the electrostatic free energy that favors large R, and the membrane perturbation term that favors small R, dictates the equilibrium properties of the pore. Systematic calculations are reported for circular pores composed of various numbers of peptides, carrying different amounts of charge (1–6 elementary, positive charges) and characterized by different polar angles. We find that the optimal R's, for all (except, possibly, very weakly) charged peptides conform to the “toroidal” pore model, whereby a membrane rim larger than ∼1 nm intervenes between neighboring peptides. Only weakly charged peptides are likely to form “barrel-stave” pores where the peptides essentially touch one another. Treating pore formation as a two-dimensional self-assembly phenomenon, a simple statistical thermodynamic model is formulated and used to calculate pore size distributions. We find that the average pore size and size polydispersity increase with peptide charge and with the amphipathic polar angle. We also argue that the transition of peptides from the adsorbed to the inserted (membrane pore) state is cooperative and thus occurs rather abruptly upon a change in ambient conditions.

Sites occupation and thermodynamic properties of the TiCr 2− xMn x–H 2 (0≤ x≤1) system: statistical thermodynamics analysis

Pressure–composition isotherms of the ternary (pseudobinary) Laves phases TiCr 2− x Mn x –H 2(D 2) (0≤ x≤1) systems were measured over a wide range of temperatures and pressures (up to 1000 atm H 2). The lower composition hydride phase, TiB 2H ∼3 (where B=Cr+Mn), is formed for all these compound series (i.e. for 0≤ x≤1) with a similar stability (i.e. independent of x). On the other hand, the high-composition hydride phase, TiB 2H ∼4, is apparent only for lower concentrations of manganese substitution ( x<0.75). A generalized statistical thermodynamics treatment, which considers the possible simultaneous occupation of two interstitial sites (namely the Ti 2B 2 and the TiB 3 tetrahedral sites) was applied, leading to calculated isotherms with two plateau regions. The fit of the calculated isotherms to the experimental ones yielded the microscopic energy-related parameters utilized in the statistical thermodynamics models (i.e. the H-lattice and the H–H nearest neighbors interaction parameters) as well as the two sites occupation fraction and its hydrogen composition variation. It turned out that an almost step-like sequential occupation of the above two sites is indicated by the above analysis, consistent with experimental structural data in the literature. A correlation between the interstitial sites volume and the pairwise H–H interaction parameter was found, with an opposite trend displayed for the two types of sites. The disappearance of the higher composition hydride phase at higher manganese composition is accounted for by the above correlations.

Molecular thermodynamics for some applications in biotechnology

As biotechnology sweeps the world, it is appropriate to remember that the great virtue of thermodynamics is its broad range of applicability. As a result, there is a growing literature describing how chemical thermodynamics can be used to inform processes for old and new biochemical products for industry and medicine. A particular application of molecular thermodynamics concerns separation of aqueous proteins by selective precipitation. For this purpose, we need phase diagrams; for constructing such diagrams, we need to understand not only the qualitative nature of phase equilibria of aqueous proteins but also the quantitative intermolecular forces between proteins in solution. Some examples are given to show how aqueous protein–protein forces can be calculated or measured to yield a potential of mean force and how that potential is then used along with a statistical thermodynamic model to establish liquid–liquid and liquid–crystal equilibria. Such equilibria are useful not only for separation processes but also for understanding diseases like Alzheimer’s, cataracts and sickle-cell anemia that appear to be caused by protein agglomeration. ▪

Molecular thermodynamics for some applications in biotechnology

Abstract As biotechnology sweeps the world, it is appropriate to remember that the great virtue of thermodynamics is its broad range of applicability. As a result, there is a growing literature describing how chemical thermodynamics can be used to inform processes for old and new biochemical products for industry and medicine.A particular application of molecular thermodynamics concerns separation of aqueous proteins by selective precipitation. For this purpose, we need phase diagrams; for constructing such diagrams, we need to understand not only the qualitative nature of phase equilibria of aqueous proteins, but also the quantitative intermolecular forces between proteins in solution. Some examples are given to show how aqueous protein-protein forces can be calculated or measured to yield a potential of mean force and how that potential is then used along with a statistical-thermodynamic model to establish liquid -liquid and liquid -crystal equilibria. Such equilibria are useful not only for separation processes, but also for understanding diseases like Alzheimer’s, eye cataracts, and sickle-cell anemia that appear to be caused by protein agglomeration.

Hydrogen solubility and diffusivity in amorphous La 14Ni 86 films

Hydrogen solubility and diffusivity in Pd-coated amorphous La 14Ni 86 films have been studied by thermal desorption spectroscopy (TDS). The TDS spectra were analyzed according to the statistical Harris and thermodynamic Kirchheim models. For H-loading pressures up to 100 mbar, hydrogen essentially occupies La 2Ni 2 and LaNi 3 tetrahedral sites at room temperature. Only hydrogen in LaNi 3 sites can be thermally desorbed before crystallization of the films takes place. Progressive filling of LaNi 3 sites with increasing hydrogen pressure occurs in agreement with the thermodynamic model. The density of LaNi 3 sites is well characterized by a Gaussian distribution with a mean energy of −0.44 eV. As regards diffusivity, hydrogen atoms jump from mean energy LaNi 3 sites with an activation enthalpy of 0.47±0.03 eV. Nonetheless, discrepancies in the width of the site energy distribution were found when it was independently evaluated from solubility and diffusivity analyses.

Estimation of point defect formation energies in the D0 19-type intermetallic compound Ti 3Al

A statistical-thermodynamic model for binary nonstoichiometric intermetallic A 3B compounds with D0 19-structure was developed based on a mean-field approximation. Vacancies and anti-structure atoms are allowed on the two sublattices as possible point defects. Due to the identical stoichiometry and the analogous coordination around A and B atoms it turned out that the same approach is valid as for A 3B compounds with L1 2-structure, and identical expressions were obtained, both for the concentrations of the different point defects and for the thermodynamic activities. The energies of formation of the four types of point defects were used as parameters. The model equations were applied to the intermetallic compound Ti 3Al using experimental aluminum activities from the literature. By a simple curve fitting procedure the following defect formation energies were obtained: E f (V Ti)= E f(V Al)=1.5 eV, and E f (Ti Al)= E f(Al Ti)=0.6 eV. This results in very low vacancy concentrations which means that the thermal disorder and the deviation from stoichiometry in Ti 3Al is caused almost entirely by anti-structure atoms. Their concentrations (referred to the total number of lattice sites) are found to be about 0.0009 at 1123 K, i.e., 0.12% of the Ti sites are occupied by Al atoms and 0.36% of the Al sites by Ti atoms.

Stability of an ion channel in lipid bilayers: implicit solvent model calculations with gramicidin.

Gramicidin is a helical peptide, 15 residues in length, which dimerizes to form ion-conducting channels in lipid bilayers. Here we report calculations of its free energy of transfer from the aqueous phase into bilayers of different widths. The electrostatic and nonpolar contributions to the desolvation free energy were calculated using implicit solvent models, in which gramicidin was described in atomic detail and the hydrocarbon region of the membrane was described as a slab of hydrophobic medium embedded in water. The free energy penalties from the lipid perturbation and membrane deformation effects, and the entropy loss associated with gramicidin immobilization in the bilayer, were estimated from a statistical thermodynamic model of the bilayer. The calculations were carried out using two classes of experimentally observed conformations: a head-to-head dimer of two single-stranded (SS) beta-helices and a double-stranded (DS) intertwined double helix. The calculations showed that gramicidin is likely to partition into the bilayer in all of these conformations. However, the SS conformation was found to be significantly more stable than the DS in the bilayer, in agreement with most of the experimental data. We tested numerous transmembrane and surface orientations of gramicidin in bilayers of various widths. Our calculations indicate that the most favorable orientation is transmembrane, which is indeed to be expected from a channel-forming peptide. The calculations demonstrate that gramicidin insertion into the membrane is likely to involve a significant deformation of the bilayer to match the hydrophobic width of the peptide (22 A), again in good agreement with experimental data. Interestingly, deformation of the bilayer was induced by all of the gramicidin conformations.

Dependencies of Clathrate Hydrate Dissociation Fugacities on the Inverse Temperature and Inverse Pore Radius

Presented are simple explicit relations for the equilibrium fugacity of single-component (single-guest) gas hydrates as a function of temperature. It is shown how relations of the form ln f = α + β/T can be derived starting with a standard statistical-thermodynamic model based on the van der Waals−Platteeuw equation. In addition to this demonstration of the validity of empirical relations that have been previously presented in the literature for bulk hydrates, explicit relations of the form ln f = α + β/T + δ/rT (where r is the pore radius) are derived for hydrate formation in porous media. The validity of these explicit relations is established for both structure I and structure II hydrates involving hydrocarbons that have negligible water solubilities by comparison of predictions for bulk hydrates with experimental data. In addition to their ease of use, these relations explicitly show the role played by the standard model parameters in the prediction of hydrate equilibrium conditions.

Intermetallic phases with D0 3-structure: a statistical-thermodynamic model

A statistical-thermodynamic model for binary non-stoichiometric D0 3-phases has been developed based on a mean-field approximation. The corresponding D0 3 crystal lattice is divided into three sublattices, the α-, β- and γ-sublattices, where A-atoms occupy the α- and γ-sublattices, and B-atoms occupy the β-sublattice in the perfectly ordered case. Neglecting interstitial defects, all other possible point defects, i.e. anti-structure atoms and vacancies on all three sublattices are considered. Based on a grand-canonical approach the expressions for the defect concentrations are derived as functions of composition and temperature, and from these the thermodynamic activities of the two components can be calculated using energies of formation of the six types of point defects as parameters. The model equations are applied to the two intermetallic phases Fe 3Al (at 500 K) and Ni 3Sb (at 1200 K).