Joseph E. Mayer

The Statistical Thermodynamics of Multicomponent Systems

Distribution functions, Fn(z, {n}), for multicomponent systems are defined proportional to the probability density that n molecules in an infinite isothermal system of fugacity set z will occupy the configurational coordinates symbolized by {n}. All thermodynamic functions may be obtained as certain sums of integrals of these distribution functions. These sums are always convergent, but impractically slow in convergence for numerical use without further transformation. In particular, the grand‐partition function, exp [VP(z)/kT], may be expanded in a power series in the fugacities z with coefficients given by integrals of the distribution functions Fn(0, {n}) at the fugacity set 0. As has been previously demonstrated for one component systems, this is shown to be a special case of a more general relation permitting the calculation of the distribution functions (and therefore the thermodynamic functions) for one fugacity set from those at another set. The function —kT ln Fn(z, {n}) is the potential of avera...

Zur Gittertheorie der Ionenkristalle

ZusammenfassungDie Gittertheorie der Ionenkristalle wird durch drei Änderungen des Energieansatzes verschärft: Das Abstoßungspotential wird nicht als Potenz des Gitterabstandes, sondern als Exponentialfunktion angenommen. Dabei wird das Gesetz der Additivität der Ionenradien berücksichtigt. Endlich werden die van der Waalsschen Kohäsionskräfte mit in Rechnung gesetzt. Es wird gezeigt, daß hierdurch die verschiedene Stabilität der Gittertypen NaCl und CsCl verständlich gemacht werden kann.

The Theory of Ionic Solutions

The virial development for the osmotic pressure of a solution may be used, if the potentials of average force of the solute molecules at infinite dilution are known, to compute the deviations from perfect solution behavior. The expression for the logarithm of the activity coefficient can thus be obtained as a sum of coefficients multiplied by powers of the concentration. For an ionic solution, with 1/R2 forces, the series is only conditionally convergent. By summing certain additive terms occurring in the coefficients over all powers of concentration convergence can be attained.The integrations necessary to obtain terms correct up to and including c32 are performed. The results are given in terms of certain functions which can readily be computed.

The Theory of the Racemization of Optically Active Derivatives of Diphenyl

T. L. Hill has recently shown that it is possible to compute the magnitude of steric strain in certain organic molecules. The present paper summarizes the results of a similar but independent investigation directed towards a computation of the rate of racemization of optically active (i.e., sterically hindered) derivatives of diphenyl. The energy of the planar form of a sterically hindered diphenyl can be approximated by the equation: E= ∑ i12aiqi2+Aiexp (−d1/ρ1)+A2exp (−d2/ρ2), where the qi are the normal coordinates of the unstrained molecule in question and the exponential terms are approximations (over the limited range of interest) to the steric repulsions of the non‐bonded groups which repel each other. There are two such exponential terms, for in most sterically hindered diphenyls the repulsion is caused by two pairs of ortho substituents. The equation for E0, the activation energy for racemization, has been found for the symmetrical case by minimizing E; the constants ai, A, and ρ, and the dimensi...

The Lattice Energies of the Silver and Thallium Halides

The lattice energies and lattice constants of the silver and thallium halides are calculated assuming ionic crystals with a van der Waals potential. The latter term, which is large, accounts for the low solubilities (high lattice energies) of the salts. The calculations appear to be quantitatively satisfactory for the thallium halides, TlCl, TlBr and TlI, and for the three silver halides AgF, AgCl and AgBr. Definite evidence is found for assuming the existence of a homopolar potential in AgI of about 10 percent of the total lattice energy. One reason for believing this, is the stability of the zincblend instead of the rocksalt lattice, which latter should be stable were the compound purely ionic. There is presumably some homopolar binding in AgBr but it cannot be large. The thallium salts probably are entirely ionic. The theoretically calculated and experimental (chemical) lattice energies are, respectively, in K cal., AgF, 219, 217.7; AgCl, 203, 205.7; AgBr, 197, 201.8; AgI, 190, 199.2; TlCl, 167, 170.1;...

Statistical Mechanics of Imperfect Gases

The application of statistical mechanical equations to the calculation of thermodynamic properties of imperfect gases has been hindered by the occurrence of highly multiple integrals in these equations. By observing that some of these multiple integrals are related to the iterated kernels of an integral equation involving the potential energy function of a pair of molecules, a technique is developed which expresses these integrals in terms of the characteristic values of the integral equation. This technique is applied to the calculation of third virial coefficients and the molecular distribution function (which is essentially the probability of finding two specified molecules in two small volume elements a distance r from each other) at various temperatures for imperfect gases with Lennard‐Jones potential energy functions.

The Mutual Repulsive Potential of Closed Shells

A method is developed for representing, in terms of a small number of integrals, the mutual potential of two ions or atoms having a rare gas electron configuration. The accuracy is the usual first approximation of the Heitler‐London method. The integrals are evaluated for two like atoms, and numerical calculations made for two neon atoms. The total (repulsive) potential at R=1.8×10‐8 cm is 344×10‐14 erg; at R=2.3×10‐8 cm, 35×10‐14 erg; and at R=3.2×10‐8 cm, 0.4×10‐14 erg.

Vapor Pressures, Heats of Vaporization, and Entropies of Some Alkali Halides

Vapor pressures of crystalline KCl, KBr, KI, and NaCl have been measured in the pressure range of 10−1 to 10−7 mm by a surface ionization method. Heats and entropies of vaporization are calculated from the data. The heats of vaporization at 0°K and entropies at 298°K are also calculated. From the data, saturated KCl vapor is shown to be less than 2 percent associated at 800°K and the heat of dissociation of (KCl)2 is shown to be less than 47 kcal.

Contribution to Statistical Mechanics

The method of the grand partition function may be used to calculate distribution functions Fn(z, {n}) proportional to the probability that n molecules in a system of fugacity z, and fixed temperature, occupy the position of their coordinates symbolized by {n}. The method makes use of the distribution functions Fn(0, {n}) at zero fugacity. The distribution functions may be written Fn(z, {n})=exp[−Wn(z, {n})/kT], in which Wn(z, {n}) is the potential of average force of n molecules at the fugacity z, which becomes equal to the ordinary potential energy at zero fugacity. The equations may be generalized to permit the calculation of the distribution functions at any fugacity assuming a knowledge of them at any other fugacity. Using methods previously employed for imperfect gases, the pressure, and also the density in molecules per unit volume, may be developed in a power series of difference of fugacity around any arbitrary fugacity. The coefficients of these developments are calculable at all fugacities by th...

Integral Equations between Distribution Functions of Molecules

Integral equations are derived that relate variations in the potentials of average force between molecules of a system at two different densities or activities. These permit the calculation of the change in thermodynamic properties, or of the change in the distribution of molecules in space, in a liquid or crystalline phase, if either the temperature is varied, or if the the mutual potentials between the molecules is assumed to change. The equations are in a somewhat complex, but still distinguishable, matrix form. A matrix operates on the variations in potential occurring at one activity and transforms them into those occurring at a second activity. The matrix elements are combinations of the distribution functions at the second activity, to which the transformation is made, multiplied by powers of the difference of the two activities. The matrix approaches the unit matrix in value as this activity difference approaches zero. The product of the two matrices, one which transforms from activity α to activi...