Molecular Structure and Statistical Thermodynamics

The following section are included:

• FOREWORD

• CONTENTS

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The partition function is considered for molecules having restricted internal rotations, and expressions are obtained which are reasonably accurate for most actual molecules. Tables are presented which list the contributions of a single restricted rotational degree of freedom to the entropy, energy, free energy, and heat capacity.

It is pointed out that the assumption of completely free internal rotation in the simpler hydrocarbon molecules is probably responsible for the discrepancies between the results of previous statistical mechanical calculations and the experimental data. Using the formulas and tables of the preceding paper, calculations are presented which show that, for reasonable values of rotation restricting potentials, complete agreement can be obtained with all experimental results. The uncertainty as to the exact height and shape of these potential barriers, together with the possible errors in estimated vibration frequencies make highly precise calculations of thermodynamic functions out of the question at present. Nevertheless the general agreement with experiment indicates that the potentials and frequencies selected must be approximately correct. These molecular structure data together with the available values of heats of combustion and hydrogenation are then employed in calculations which yield thermodynamic constants and the free energy of formation as a function of the temperature in the range from 300 to 1500°K. The various calculations have been made for all of the hydrocarbons listed in the title.

A method is developed for calculating thermodynamic functions for long chain molecules with particular attention to normal paraffins. In this connection the infinite chain approximation method for calculating vibration frequencies is considered. Starting with the results of Kirkwood, a modification is made which improves the agreement with the exact values for the simpler cases. In addition this method of attack is extended to out of plane motions. This vibrational analysis shows that all skeletal frequencies for molecules of the normal paraffin type can be put into two groups, one fairly narrow band near 1000 cm⁻¹, and a broader band extending from 0 to 460 cm⁻¹. The partition function is then set up on the assumption that motions in the low frequency group can be treated classically, and that the high frequency band can be replaced by a suitable number of 1000 cm⁻¹ frequencies. Contributions from hydrogen atom vibrations are added on later. A formula is finally obtained which is quite simple, considering the complexity of the molecules. The calculated entropies can be brought into agreement with experimental values on the basis of very reasonable internal rotation restricting barriers.

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A general treatment of internal rotation is given for molecules whose moments of inertia for over-all rotation are independent of internal rotational coordinates. Tables are presented for the various thermodynamic functions which are accurate for molecules with one internal rotation and for the potential energy (V/2) (1−Cos nφ). The tables are shown to be a good approximation for molecules with several internal rotational coordinates, provided the potential energy can be expressed as a sum of terms of this type. Methods are suggested for treating problems where cross terms involving more than one internal coordinate are present in the potential energy. The energy level expressions are developed for the more general case with the potential energy expressed by a Fourier series. Although a few specific cases were worked out with different shape potential barriers, it appears that the simple form assumed above will be satisfactory for many purposes.

The infrared spectra of the four isotopic species of formic acid were measured in the vapor phase as well as in the solid nitrogen matrix in the region 400–800 cm⁻¹. Absorption bands in the vapor phase were analyzed by considerations of band contours and comparisons with the frequencies distinctly observed in the matrix, and the previous assignment was revised. The torsional vibrations of the terminal OH groups of short chain polymers of HCOOH were observed in the matrix at 685 and 694 cm⁻¹. The OH torsional frequencies of the trans isomer were located in the vapor phase at 582 (HCOOH), 576 (DCOOH), 450 (HCOOD), and 448 cm⁻¹ (DCOOD). The energy difference between the two isomers was determined to be 2.0±0.3 kcal/mole. The potential hindering internal rotation was calculated to be 2V (kcal/mole) = 2_.1 (1 − cosθ) + 9_.9 (1 − cos2θ) − 0_.1(1 − cos3θ) from the energy difference and the fundamental torsional frequencies. The entropy of mixing of the two isomers was found to be 0.3±0.1 cal deg⁻¹ mole⁻¹ at 25°C. A normal coordinate treatment was made of the cis and trans isomers of monomer and of the dimer and polymer. The double bond characters calculated from the torsional potential constant agree with those calculated from bond lengths.

The theoretical relationships between the potential energy function for internal rotation and the energy levels and thermodynamic properties are summarized briefly. The height and shape of the potential barrier in ethane are re-examined in relation to all pertinent data. It is concluded that the barrier is very close to the cosine function in shape and from 2750 to 3000 cal./mole in height. The barrier heights for a number of other hydrocarbons are summarized and a few comments are made concerning the cause of these barriers.

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Infrared and Raman measurements on cyclobutane are reported. The spectroscopic data, when used in conjunction with electron diffraction measurements and the measured entropy, suggest that the equilibrium configuration for cyclobutane is one with D_2d symmetry, but that the barrier hindering inversion of the molecule is sufficiently low so that even at ordinary temperatures an appreciable number of the molecules obey D_4h selection rules. A normal coördinate analysis for cyclobutane has been performed, and force constants compared with those of related molecules. A possible potential function for the out-of-plane bending motion of the ring has been suggested, and some of the thermodynamic properties of cyclobutane calculated.

Earlier calculations of the conformation and strain energy of cyclopentane are refined by the addition of terms for the election correlation energy of non-bonded but nearby atoms and for the zero point vibrational energy. The agreement with observed properties of cyclopentane is substantially improved. It is shown that substitution of the cyclopentane will ordinarily yield a barrier to the pseudo-rotation of ring puckering and that the energy minimum will in some cases correspond to C₂, symmetry and in other cases to C_s symmetry. It is shown that certain of the parameters evaluated for cyclopentane may be transferred to derivatives and that strain energies may be calculated for the derivatives if potential barrier values are available for the appropriate open chain model compounds.

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Relativistic quantum mechanics as applied to radon (or element 118) fluoride structures indicates that ionic crystalline forms are probably more stable for the fluorides of these elements in contrast to the molecular form of xenon fluorides.

An effective core potential system has been developed for heavy atoms in which relativistic effects are included in the effective potentials (EP). The EP's are based on numerical Dirac-Hartree-Fock calculations for atoms and on the Phillips-Kleinman transformation with other aspects similar to the treatments of Goddard and Melius and Kahn, Baybutt, and Truhlar. The EP's may be written

where |lim > is a two-component angular basis function that is a product of a two-component Pauli spinor and spherical harmonics. The numerical functions U_ij^EP(r) are approximated as expansions in terms of Gaussian or exponential functions. The use of these EP's enables one to use the jj-coupling scheme for subsequent applications in all-valence-electron calculations on heavy atoms and their molecules. A standard atomic SCF program has been modified to accommodate these EP's and Gaussian and exponential basis sets having the proper j angular dependence. Energy levels for many atomic states of Xe and An were calculated. The study of Xe excited states indicates that the spin-orbit splittings are reasonably approximated and that the numerical DHF calculations are adequately reproduced. Au has been treated as an atom with 1, 11, 17, 19, or 33 valence electrons to investigate the effects of redefinition of the core.

Potential energy curves for the ground state of Xe₂, the first four states of the ions, and the eight Xe*₂ eacimer states corresponding to the addition of a 6s σ_g, Rydberg electron to these ion cores have been computed using averaged relativistic effective core potentials (AREP) and the self-consistent field approximation for the valence electrons. The calculations were carried out using the IS-coupling scheme with the effects of spin–orbit coupling included in the resulting potential energy curves using an empirical procedure. A comparison of nonrelativistic and averaged relativistic EP's and subsequent molecular calculations indicates that relativistic effects arising from the mass -velocity and Darwin terms are not important for these properties of Xe₂ molecules. Spectroscopic constants for are in good agreement with all electron CI calculations suggesting that the computed values for Xe₂ excimers should be reliable. The lifetime for the O_u⁺ state of the Xe₂* is computed to be 5.6 nsec which is in the range of the experimentally determined values.

SCF calculations have been carried out for the ground state of Au₂ using a variety of ab initio effective core potentials (EP). The effective core potentials studied both include a two-component relativistic EP (REP) that includes spin-orbit effects and also averaged relativistic EP (AREP) and a nonrelativistic EP (NREP). All-electron nonrelativistic calculations were also performed. The values of spectroscopic constants obtained from these calculations indicate that relativistic effects account for a decrease in R_e of over 0.3 Å and an increase in the bond energy of the order of 1 eV. Various intercomparisons indicate the general validity of effective potential methods, properly applied, but also show certain limitations. In particular, the NREP results agree well with the all-electron, nonrelativistic calculations. Also, various relativistic effective-potential methods agree for SCF calculations provided both the basis sets and the EP are carried to sufficiently high order in angular quantum number. The bond distance calculated relativistically agrees very well with experiment.

The ground and excited states of Au² are studied using ab initio averaged relativistic effective core potentials (AREP) and MCSCP-CI procedures. Spin-orbit effects are included in the excited states derived from ²S_1/2+²D_3/2 and ²S_1/2+²D_5/2 atomic states using an empirical procedure. The ground state dissociation energy is calculated to be 2.27 eV as compared to the experimental value of 2.31 eV. The calculated energies for the two spectroscopically allowed to transitions and other molecular parameters also agree reasonably well with experiment.

We have investigated the sources of error in bond lengths and dissociation energies computed from ab initio effective potential derived from Phillips-Kleinman type pseudo-orbitals. We propose an alternate pseudo-orbital, effective potential treatment with the primary objective of agreement with all-electron molecular calculations. This new treatment forces the pseudo-orbitals to match precisely the Hartree–Fock orbitals in the valence region and thereby eliminates the major cause of error in the earlier calculations. Effective core potentials derived from these revised pseudo-orbitals were used to compute potential energy curves for the ground states of F₂, Cl₂ and LiCl and the results are compared with previous all-electron and effective potential calculations. Our effective potentials yield dissociation energies and bond lengths which are in excellent agreement with the all-electron values. Furthermore, in contrast to other procedures, our revised effective potentials result in an excellent description of the inner repulsive walls of the dissociation curves.

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Ab initio self-consistent field calculations are reported for a series of diatomic molecules using relativistic effective core potentials (REP) and basis sets appropriate for ω–ω coupling. The molecular orbitals are expressed as linear combinations of two-component analogs of Dirac spinors. The unique feature of the present approach is the retention of the spin-orbit operator in the generation of the REP's and the propagation of its effects into the molecular wave functions in a totally consistent fashion. The nature of bonding in the molecules Au₂⁺, TIH, PbSe, and PbS is discussed with consideration of the orbital energies, spectroscopic constants, and population analyses. Comparisons with recently obtained photoelectron spectra of PbSe and PbS are made. It is noted that 6p_1/2 and 6p_3/2 orbitals exhibits bonding characteristics that are different from the nonrelativistic pσ and pπ molecular orbitals.

The dissociation curve for the ground state of TIH was computed using a relativistic ω–ω coupling formalism. The relativistic effects represented by the Dirac equation were introduced using effective potentials generated from atomic Dirao-Fock wave functions using a generalization of the improved effective potential formulation of Christiansen , Lee, and Pitzer. The multiconfiguration SCF treatment used is a generalization of the two-component molecular spinor formalism of Lee, Ermler, and Pitzer. Using a five configuration wave function we were able to obtain approximately 85% of the experimental dissociation energy. Our computations indicate that the bond is principally sigma in form, despite the large spin–orbit splitting in atomic thallium. Furthermore the bond appears to be slightly ionic (TI⁺H⁻) with about 0.3 extra electron charge on the hydrogen.

The dissociation curves for the ground states of Tl₂ and were computed using a generalization of the molecular relativistic ω–ω coupling formalism of Lee, Ermler, and Pitzer. Relativistic effects, as represented by the Dirac equation, were introduced using effective potentials generated from atomic Dirac-Fock wave functions using a generalization of the improved effective potential formulation of Christiansen, Lee, and Pitzer. Our calculations show that the ground state of is 1/2_g with computed D_e and R_e values of 0.58 eV and 3.84 Å. For Tl₂ we find that the ground state is 0_u⁻ but the 0_g⁺, and the 1_u, states are only slightly higher in energy; the potential curves for these states are repulsive to about 3.5 Å and then essentially flat beyond that radius. While corrections for correlation will increase D_e somewhat, Tl₂ is only weakly bound in any of these states which dissociate to normal atoms. The cause is undoubtedly related to the large spin-orbit splitting between the 6p_1/2 and 6P_3/2 thallium spinors.

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The first ab initio procedure for the treatment of spin–orbit coupling in molecules based on the use of relativistic effective potentials derived from Dirac–Fock atomic wavefunctions is presented. A rigorous definition for the spin–orbit operator is given and its use in molecular calculations discussed.

A system for the inclusion of spin-orbit coupling along with moderate scale CI in calculations for molecules containing very heavy elements is demonstrated. In this effective potential procedure rigorous ab initio spinorbit integrals are computed and added to the conventional integral set after the SCF and integral transformation steps of the calculation. This avoids the use of complex coefficients in the integral transformation and yet includes spin-orbit corrections on an equal footing with electron correlation. The diagonalization of the resulting complex CI plus SO matrix requires only about twice the time of a real CI diagonalization. Our present calculations on the two lowest 0⁺ and 1 states and the lowest 0⁻ and 2 states of TIH indicate that this procedure allows adequate flexibility in the electronic coupling, resulting in bonding curves which are in good agreement with the experimentally established curves. The results also help to understand and to confirm previously conjectural interpretations of other spectral data.

The energy levels and thermodynamic functions for a double minimum vibrational degree of freedom are presented, and their application illustrated for the case of ammonia. A simple formula is given for the range of validity of the classical expressions for rotational heat capacity and entropy. Also the symmetry number, which is troublesome in examples of double minimum vibration, is discussed briefly.

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The heat capacity of gaseous methanol was measured in a flow calorimeter over the range 345 to 521 °K. at pressures of roughly 1/3, 2/3 and 1 atm. The results at 345 ° K. show a very large and distinctly non-linear trend with pressure. This is interpreted as indicating the presence of methanol tetramers in the gas, somewhat analogous to the polymers of hydrogen fluoride. The heat of dissociation of this tetramer is found to be 24,200 cal./mole which corresponds to a hydrogen bond energy of 6050 cal./mole if the tetramer is a ring. Experimental values from the literature and from this research for the heat capacity and entropy of methanol are compared with values calculated from spectroscopic data, including particularly a potential barrier to internal rotation of 932 cal./mole. Agreement is obtained, but there remain uncertain elements in the spectroscopic data which could affect this result. The apparent variance of this conclusion from that of earlier investigators is due to the unsuspected type of gas imperfection. In addition values are reported for the beat capacity of benzene vapor at several temperatures and for the heat of vaporization of benzene and methyl alcohol.

The molecular orbital theory is used in appropriate semi -empirical forms to predict the properties of carbon vapor. The results indicate that linear polyatomic molecules: C=C…C—C=C: are the important species. Experimental results from the literature for C₃ are combined with the calculated conjugation or resonance energies and with the heats of formation of allene and ethylene to predict heats of formation for all larger carbon molecules. It is found that the odd species have closed shell structures and lower energies than the even species but that the even species should show greater electron affinity. Both of these, results are consistent with the mass spectrometric results of Honig and of Chupka and Inghram. Molecular spectroscopic data on C₃O₂ are used to estimate the free energy function increments for the species above C₃ The calculated partial vapor pressures predict C₅ to be the most abundant species in the saturated vapor even at 2000°K. with C₇ becoming comparably abundant in the 2500 to 3000°K. range. At higher temperatures even larger molecules should become important. The results are shown to be generally consistent with all reliable vaporization data provided the evaporation coefficients decrease rapidly for increasing molecular size and vary for different crystal surfaces of graphite. The calculated electronic energy levels for C₂ and C₃ agree satisfactorily with the observed spectra, and trends are predicted for both even and odd larger species. It is proposed that liquid carbon consists of essentially infinite linear chains of this type. Both entropy and energy considerations lead to predicted heats of fusion of about 10 kcal./g. atom at 4000°K.; the agreement between the two values indicates at least the absence of any serious inconsistency.

The following sections are included:

• Introduction

• Theory for Many-Electron Systems
  • Polarizability
  • London Forces

• Experimental Intermolecular Energies

• Intramolecular Applications
  • Saturated Hydrocarbons
  • Unsaturated Hydrocarbons
  • Halogens

• Anisotropic Effects
  • Polarizability
  • London Forces

• References

The full array of structure-related experimental data for XeF₆ is considered simultaneously. Electron diffraction, infrared, Raman and ultraviolet spectra, electric field deflection of molecular beam, and calorimetrically determined entropy data are all found to be consistent with a structural model involving substantial distortion in the T_1u bending mode from octahedral symmetry and a pseudorotation in two dimensions of this distortion. Several other vibration frequencies are assigned, and the mean value of three unobserved frequencies is determined from the entropy. No quantitative value could be given for the potential restricting pseudorotation , but various arguments agree in indicating that it is small. There is no reason to question previous proposals that this peculiar structure arises from a pseudo-Jahn–Teller effect associated with the relatively low energy of excitation of a xenon 5s electron to the 5p level.

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On the basis of certain assumptions concerning the properties of the molecules, and by using classical statistics, the van der Waals theory of corresponding states is derived. No mathematical approximations are involved. Argon, krypton and xenon, which conform closely to the assumptions, show the expected correspondence in behavior and are taken as standards. The name perfect liquid is suggested for this behavior. In a second part, the deviations from perfection are discussed for a wide variety of substances. Particularly considered are heat capacities and entropies of vaporization which should be compared at points where the vapor to liquid volume ratio has the same value. The observed deviations are explained in a general way.

The basic theory of the volumetric and thermodynamic properties of fluids is examined in relation to the statistical mechanical proof of the theory of corresponding states. It is shown that the general range of non-polar or slightly polar “normal” liquids cannot be expected to conform to the hypothesis of corresponding states but may well conform to a slightly more complex equation involving just one additional parameter. Rigorous theoretical test can be made only for the second virial coefficient. There it is found that linear and globular non-polar molecules fit this theory very exactly and that slightly polar molecules also conform in reasonable approximation.

The theoretical considerations of Part I suggested that the compressibility factor of a normal liquid in either gas or liquid state should be expressible as a function of just one parameter in addition to the reduced temperature and reduced pressure. The additional parameter is defined in terms of the vapor pressure at T_r = 0.7. This third parameter is required because the intermolecular force in complex molecules is a sum of interactions between various parts of the molecules—not just their centers—hence the name acentric factor is suggested. The theory requires that any group of substances with equal values of the acentric factor should conform among themselves to the principle of corresponding states. This result is verified with relatively high accuracy. While a completely analytical expression for the compressibility factor was not obtained, power series expressions in the acentric factor proved satisfactory and the coefficients are tabulated for a wide range of reduced temperature and pressure. The reduced vapor pressure and the entropy of vaporization are also treated similarly. Agreement is obtained to 0.5% over most regions with maximum deviations of about 2%.

An equation has been developed to represent the second virial coefficient of a normal fluid: BP_c./RT_c. = (0.1445 + 0.073ω) − (0.330 − 0.46ω)T_r⁻¹ − (0.1385 + 0.50ω)T_r⁻¹ − (0.0121 + 0.097ω)T_r⁻³ − 0.0073ωT_r⁻⁸ where ω is the acentric factor (defined by ω = −log (P/P_c.)sat − 1.000 at T_r = 0.7) which was discussed in an earlier paper. This equation not only fits the volumetric data with considerable accuracy but its second derivative also yields agreement with measured values of the pressure derivative of the gas heat capacity.

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A method is developed for the derivation of improved intermolecular potentials from empirical second virial coefficient data together with the acentric factors of the fluids. The Kihara core model is considered first, and relationships are derived between the acentric factor and the core size for several shapes. The resulting cores for CH₄, CF₄, C (CH₃)₄, C₆H₆, and N₂ are reasonable in view of semiquantitative theoretical expectations, but the core for CO₂ is somewhat larger than expected. An approximate treatment is given which combines the Kihara core model with an electric quadrupole interaction. The core calculated for CO₂ on this model including the quadrupole moment is of a reasonable size.

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For nonlinear molecules with equivalent identical nuclei an important pathway leading to equilibration of nuclear spin statistics isomers is provided by wavefunction mixing induced by the spin-rotation interaction and in some cases the spin–spin interaction. For CH₄ this mixing is very large and provides for very rapid equilibration of the three spin statistics isomers. For asymmetric rotor molecules such as H₂O and CH₂O the rapidity of the equilibration is sensitive to the exact rotational energy level pattern. The spin-rotation mixing may be very important if there is an accidental near degeneracy of the right sort. Then most of the isomerization “funnels” through the near-degenerate levels.

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Studies of the time evolution of intensities in the fine structure of infrared bands of CH₃D, H₂O, and NH₃ isolated in rare-gas matrices in the temperature region 6.5°–20°K have been made. The molecule CH₃D exhibits a fine structure at 6.5°K with time-dependent intensities. The data are interpreted in terms of nuclear-spin-species conversion. Because of rapid reconversion at 20°K it is concluded that CH₃D cannot be obtained at room temperature with a nonequilibrium distribution of spin isomers by low-temperature equilibration. The time dependence of the fine structure of the H₂O bending mode at 6.5°K has been reinvestigated. The essence of the previous work is confirmed. The rate of nuclear-spin conversion is found to be rapid at 30°K. The fine structure of the NH₃ umbrella motion exhibits a time dependence which indicates that most of the fine structure observed is not due to rotation.

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The heat capacity of methane from 0.4 to 28 K was measured for both pure methane and samples doped with about 0.8 per cent of oxygen as a spin-species conversion catalyst. With the conversion catalyst it was possible to obtain equilibrium with the spin system within about 30 min and to measure heat capacities down to 0.4 K. For pure methane the spin-species conversion becomes very slow below 10 K; however, one can measure non-equilibrium heat capacities on the basis of no change in spin-species composition. Entropy values are calculated from the heat capacities on the basis of both fixed and equilibrium spin-species composition and compared with the statistical calculation for gaseous methane. The results agree within possible experimental error with the expected zero-point entropies for the two spin situations.

The energy states are calculated for an oxygen molecule in a crystal field of cubic symmetry that would be appropriate for a substitutional site in solid methane. The resulting heat capacity is also calculated and discussed in relation to data for oxygen-doped methane. The catalytic effect of oxygen on the spin-species conversion of methane is treated by consideration of the dipole-dipole interaction of the electron spin of oxygen with the proton spin of methane. The matrix elements are calculated for the low-energy states of methane and of oxygen. It is found that oxygen should be an effective catalyst for spin-species conversion above about 0.3°K but that its effectiveness may decrease rapidly below that temperature.

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A system of equations for the thermodynamic properties of electrolytes is developed on the basis of theoretical insights from improved analysis of the Debye-Hückel model as well as recently published numerical calculations for more realistic models. The most important result is the recognition of an ionic strength dependence of the effect of short-range forces in binary interactions. By modifying the usual second virial coefficients to include this feature, one obtains a system of equations which are only slightly more complex than those of Guggenheim but yield agreement within experimental error to concentrations of several molal instead of 0.1 M. If one compares instead with the recently proposed equations of Scatchard, Rush, and Johnson, the present equations are very much simpler for mixed electrolytes (and somewhat simpler for single electrolytes) yet appear to yield comparable agreement with experimental results for both single electrolytes and mixtures.

The system of equations developed in the first paper of this series is successfully applied to the available free energy data at room temperature for 227 pure aqueous electrolytes with one or both ions univalent. The experimental data are represented substantially within experimental error from dilute solutions up to an ionic strength varying from case to case but typically 6 M. Where the data extend to high concentration, three parameters are evaluated for each solute, but one of these has negligible effect and is omitted if there are data only for the dilute range. This yields a very compact set of tables from which these important and useful properties can be reproduced. These parameters will also be of importance in treating mixed electrolytes. A simplified graphical presentation is given for activity coefficients of 1–1 electrolytes. In most cases the new equations are fitted to the osmotic coefficient data as recommended by Robinson and Stokes but for hydroxides, zinc halides, hydrogen halides, and a few other cases we have based our evaluation on the original data from several sources. The implications of our parameters are also discussed in terms of solvent structure and interionic forces.

The peculiar behavior of 2–2 and higher valence type electrolytes is discussed in terms of various theories some of which assume, while others do not, an equilibrium between separated ions and ion pairs as distinct chemical species. It is recognized that in some cases a distinct species of inner-shell ion pairs is indicated by spectroscopic or ultrasonic data. Nevertheless, there are many advantages in representing, if possible, the properties of these electrolytes by appropriate virial coefficients and without chemical association equilibria. It is shown that this is possible and is conveniently accomplished by the addition of one term to the equations of Parts I and II of this series. The coefficients of these equations are given for nine solutes. It is also noted that these equations have been successfully applied to mixed electrolytes involving one component of the 2–2-type.

An equation has been developed with the guidance of recent statistical theories of electrolytes which is designed for convenient and accurate representation and prediction of the thermodynamic properties of aqueous electrolytes including mixtures with any number of components. The three previous papers have given the theoretical background and the evaluation of parameters for pure electrolytes of various charge types. The equation is here applied to a wide variety of mixed aqueous electrolytes at room temperature and at ionic strengths up to 6 M in many cases and occasionally even higher. The first objective is the prediction of properties of mixed electrolytes using only the parameters for pure electrolytes; on this basis standard deviations in ln γ or φ for 69 sets of mixtures are less than 0.01 in 36 cases and above 0.05 in only seven cases all involving Cs+ or OH⁻. A second objective is the determination of parameters giving the differences in short-range interaction of ions of the same sign where these differ significantly from zero. As expected, these difference terms, while always small, are relatively most important for singly charged ions (and especially for OH⁻ and Cs⁺) and less important for ions of higher charge. The equations, including difference terms where known from binary mixtures with a common ion, were finally tested on 17 sets of mixtures involving four or more ions without any further adjustment of parameters. The standard deviation is less than 0.01 in all cases and is 0.003 or less in 11 cases. Thus these equations appear to yield accurate predictions of properties of mixed aqueous electrolytes.

The contribution of higher-order electrostatic terms (beyond the Debye-Hückel approximation) to the thermodynamic properties of mixed and pure electrolytes is investigated. It is found that these effects are important for cases of unsymmetrical mixing, especially when one ion has a charge of three units or more. The appropriate correction can be made by a purely electrostatic function since the mutual repulsion of ions of the same sign keeps them far enough apart that short-range forces have little effect. This function is evaluated, and several convenient approximations are also given. Application is made to systems mixing ions of the type 1–2 and 1–3. Higher-order limiting laws exist for symmetrical mixtures and for pure, unsymmetrical solutes, but these effects were not found to be significant in relationship to existing activity or osmotic-coefficient data.

Equations previously developed and widely applied to the thermodynamic properties of strong electrolytes are extended to solutions involving a dissociation equilibrium. Excellent agreement is obtained with the data for pure phosphoric acid to 6 M and for phosphate buffer solutions. The parameters of the strong electrolyte components of the buffer solutions are taken from other work, and the remaining parameters for H⁺, H₂PO⁻₄, and H₃PO₄ are evaluated, including a pK of 2.146. The present method avoids ambiguities which formerly arose in treating weak acids with as small pK as this.

Although the thermodynamic properties of sulfuric acid above 0.1 M and near 25°C are well established numerically, they have not been represented accurately by equations which are based upon the ionic species present, H⁺, HSO₄⁻, and SO₄²⁻. We have developed and fitted such equations over the range from 0 to 6 M in a system compatible with those for fully dissociated, strong electrolytes. The enthalpy is treated as well as the activity and osmotic coefficients. These equations also establish the solute standard state and the relationship between the properties of sulfuric acid in that state with those for the pure acid. Among the results obtained (for 25°C) are the dissociation constant 0.0105 and the heat of dissociation –5.61 kcal mol⁻¹ for HSO₄⁻ and the entropy of SO₄²⁻, 4.2 ± 0 .2, and of HSO₄⁻, 32.1 ± 0.3 cal K−1 mol⁻¹. Also for the reaction H₂SO₄(1) = 2H⁺(aq) + SO₄²⁻(aq), ΔH° = − 22844, ΔG° = − 12871 cal mol⁻¹.

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Mineral solubilities in binary and ternary electrolyte mixtures in the system Na-K-Mg-Cl-SO₄-OH-H₂O are calculated to high temperatures using available thermodynamic data for solids and for aqueous electrolyte solutions . Activity and osmotic coefficients are derived from the ion-interaction model of PITZER (1973, 1979) and co-workers, the parameters of which are evaluated from experimentally determined solution properties or from solubility data in binary and ternary mixtures. Excellent to good agreement with experimental solubilities for binary and ternary mixtures indicate that the model can be successfully used to predict mineral-solution equilibria to high temperatures. Although there are currently no theoretical forms for the temperature dependencies of the various model parameters, the solubility data in ternary mixtures can be adequately represented by constant values of the mixing term θ_ij and values of ψ_ijk which are either constant or have a simple temperature dependence. Since no additional parameters are needed to describe the thermodynamic properties of more complex electrolyte mixtures, the calculations can be extended to equilibrium studies relevant to natural systems . Examples of predicted solubilities are given for the quaternary system NaCl-KCl-MgCl₂-H2O.

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The free energies of hydration of the alkali and halide ions are found to agree reasonably well with the simple expression of Born (−ΔF=(1−1/D)Ne²/2r_s.) for solution of charged spheres in a dielectric medium, provided the crystal radii are suitably modified so as to correspond to the radii of the cavities in the dielectric medium. The results show that the dielectric constant of water remains large even in the intense field next to the ion. The entropies of hydration are also found to be consistent with these radii. Because of the simplicity of this calculation, the resulting free energies of solution of individual ions are considered to be a priori the most probable and are used to calculate a value of −0.50 volt for the absolute potential of the calomel half-cell.

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The intermolecular potential energy between two inert gas molecules is considerably altered when these molecules are next to a solid surface as in physical adsorption. The change in the interaction is evidenced by the additional long-range repulsion that is often observed between the molecules of a monolayer and also by the additional attractions that must play a role in multilayer formation.

In this article, the two-molecule-surface potential is derived from quantum mechanical third-order perturbation theory. It is shown that this potential consists of two parts just as the energy giving the van der Waals attraction of a single molecule to a surface does. The first part exists only when the surface has a net electrostatic field and this is equivalent to the classical polarization effect. The second part arises from the fluctuations of the surface fields and is of the same origin as the dispersion forces. The third-order energy, i.e., the new intermolecular interaction caused by the surface, is directly related to the zero-coverage heat of adsorption and except for this experimental quantity, the results do not require specific assumptions about the surface. Thus, the theory is applicable to either metal or insulator surfaces. When both the two-molecule-surface and the one-molecule-surface interactions are available experimentally (for example, from the application of virial coefficients treatment in physical adsorption) the electrostatic field of the surface can be estimated.

The fluctuation or dispersion part of the third-order energy is shown to yield a repulsion between two molecules in a monolayer that amounts to 20–40% of the gas phase Lennard-Jones potential minimum ∈₀. The same energy yields an additional attraction of about 10–20% of ∈₀ when the two molecules are on top of one another as in multilayer formation. The theory is applicable also when more than two molecules at a time need be considered on the surface.

A solution of an alkali metal in a fused halide of the same metal is discussed in terms of the mixing of the two negative species, halide ions and F-center-like electrons in cavities, within the irregular lattice of positive ions. The positive energy of mixing is related to the excess energy of a hypothetical metal comprising an ionic lattice of F-centers and positive ions over that of the true metal. The theory predicts substantial positive excess entropies of mixing. The solubility data of Bredig and collaborators are shown to be consistent with this model and certain parameters are evaluated for each system.

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The conclusion of classical Debye-Hückel theory that a phase separation may occur in highly unsymmetrical plasmas or electrolytes is shown to be false and to arise from a serious error in the treatment of the interaction of pairs of the most highly charged ions. After an approximate correction for this error, no phase separation is predicted. Specific application to iron in the solar plasmas is discussed.

Solutions with composition extending continuously from molecular liquids such as water to fused salts are relatively unusual but of considerable interest. Conductance and thermodynamic properties are considered for several examples. New equations for the activities of the respective components represent the data more accurately than previous treatments and delineate the similarities and differences between such systems and nonelectrolyte solutions.