Volume 40, Issue 7, 01 April 1964

Role of Thermodynamic Activity in Rate Processes
View Description Hide DescriptionThis paper contains an investigation of the relation between rates of diffusion and evaporation and thermodynamic activity. The method of proceeding involves the use of the one‐dimensional Ising model for which an exact statistical treatment can be given. The actual system treated may be thought of as an edge dislocation with adsorbed impurity atoms. As a by product, some interesting results concerning the formation of cluster and pinning points along a dislocation are derived. However, the main purpose of this work remains the investigation of the connection between rate and activity.
For the one‐dimensional model (for any degree of coupling between molecules) it can be shown exactly that the rate of evaporation is proportional to the thermodynamic activity with constant coefficient and that the rate of diffusion is similarly proportional to the gradient of the thermodynamic activity with constant coefficient. These results, which have been suggested before, therefore receive theoretical justification in terms of this simple model.
Certain not‐too‐restrictive conditions must be applied, however. The most important of these is the requirement that every subsystem of the system be at equilibrium with respect to all variations save the one which corresponds to the rate process.

Thermal Activation Relations
View Description Hide DescriptionA material is considered which has one or more thermally activated linear response functions that obey Boltzmann statistics to good approximation. These response functions may, for example, be mechanical or dielectric relaxation times, diffusion coefficient, viscosity, etc. Neither the entropy change, ΔS, nor the enthalpy change, ΔH, in the activation process need be temperature‐independent. An expression for ΔG, the Gibbs free‐energy change in the thermally activated process, is considered which applies to a system with a nonzero or zero glass transition temperature, T_{g}, and allows the product c _{1} ^{ g } c _{2} ^{ g } of the WLF polymerequation to be interpreted and its variation with material explained by an activation energy model. Explicit expressions for the resulting temperature‐dependent ΔS and ΔH are given which involve the material parameter E; this is the enthalpy and activation energy of the process only when T_{g} =0 and thus an Arrhenius equation applies. The results pertain to mechanical and dielectric dispersion experiments on amorphous polymers and other glass‐forming materials. In addition, they apply to such processes as diffusion by a vacancy mechanism, to viscous flow, and, when T_{g} =0, to intrinsic semiconduction and many other processes. It is found that the Lawson—Keyes relation, ΔS/ΔH≃4α, should not be applied when T_{g} ≠0 and may be very inaccurate even in the case where T_{g} =0. An improved relation between ΔS and ΔH which holds for T_{g} =0 is presented; it furnishes a possible explanation for the negative value of ΔS sometimes found experimentally. An expression for ΔG correct to first order in pressure and temperature is given which applies to all situations where an Arrhenius equation is found. On identifying the semiconductor energy gap for thermal activation as a Gibbs free energy difference, the results are illustrated by analyzing pressure and temperature data pertaining to intrinsic semiconduction in Si and Ge. Experimental dielectric dispersion results for isoamyl bromide (T_{g} ≠0) are also analyzed and compared with results of previous work. Finally, temperature‐dependent aspects of the transient and frequency response of a distributed, linear activated system with T_{g} ≠0 are examined when either the pre‐exponential factor or the activation parameter E is distributed, or when they are simultaneously distributed and are linearly related.

Effect of Surface Coverage and Temperature on the Sticking Coefficient
View Description Hide DescriptionA theoretical equation is derived that quantitatively describes the peculiar shape of the curves relating the sticking coefficients for the adsorption of nitrogen and carbon monoxide on tungsten as a function of the amount adsorbed and the temperature. The model used to describe chemisorption into a β state with a high binding energy involves a precursor state of low binding energy as assumed by others. However, emphasis is placed on the importance of nonsteady state of the precursor for low coverage and very high coverage. It is found that if the precursor state is actually the α states (for N_{2} and CO) discovered by Ehrlich, the experimental curves can be predicted within experimental error.

Deviations from the Virial Relationship in Many‐Center Variational Functions
View Description Hide DescriptionAn equation is derived which relates kinetic and potential energies that have been calculated from unscaled many‐center variational functions. The expression finds use in accurately determining diatomic equilibrium nuclear separations and force constants from many‐center variational wavefunctions that are not properly scaled.

Linewidth Studies in Electron Spin Resonance Spectra : The Para and Ortho Dinitrobenzene Anions
View Description Hide DescriptionLarge variations have been found among the linewidths of the different hyperfine lines in the low‐temperature electron spin resonance spectra of the p‐ and o‐dinitrobenzene anions generated electrolytically in N,N‐dimethylformamide solutions. The magnitude of the variations in the para compound is so great that the spectrum is superficially uninterpretable. Detailed analysis of the spectrum of this radical shows, however, that most of the linewidth differences can be accounted for by modulation through molecular tumbling of the intramolecular anisotropic dipolar and g‐tensor interactions. There may also be a small contribution from modulation of the isotropic protonhyperfine splittings, the mechanism that accounts for the alteranting linewidth phenomenon in a number of radicals, but an alternation of the widths is not observed in the p‐dinitrobenzene anion spectrum because the contribution from this interaction is small. The sign of the isotropic nitrogen hyperfine splittinga ^{N} has been determined by a new method involving only pure dipolar interactions with the nitrogen nuclei and protons and not depending on any assumptions about the magnitudes of the components of the gtensor. A number of features of the relaxation matrix determining the linewidths are discussed, and the problems encountered when the components of a degenerate line have different widths are analyzed. The spectra obtained in dimethylformamide solutions are much better resolved than those previously reported in acetonitrile solutions. Changes of hyperfine splittings with both solvent and temperature are observed.

Retarding Potential Measurements of Electrons Photoemitted by N_{2}, CO, and O_{2}
View Description Hide DescriptionRetarding potential measurements have been made on electrons emitting during the photoionization of N_{2}, O_{2}, and CO. More than half of the ions produced by radiation in the 700 to 500 Å interval were observed to be in states of electronic excitation. Vibrational excitation of the ground states of CO^{+} and O_{2} ^{+} occurred in the wavelength intervals between the first and second ionization potentials. Part of the excitation of vibrational states was attributed to autoionization phenomena.

Spin—Lattice Relaxation in Some TCNQ Ion—Radical Salts
View Description Hide DescriptionSpin—lattice relaxation as a function of temperature has been studied for the salts (φ_{3}AsCH_{3})^{+}(TCNQ)_{2} ^{—}, (φ_{3}PCH_{3})^{+}(TCNQ)_{2} ^{—}, and (Cs^{+})_{2}(TCNQ)_{3} ^{=}. A plot of spin—lattice relaxation time (T _{1}) as a function of the inverse of temperature yields a V‐shaped curve for the above salts. Correlation times are calculated from the spin—lattice relaxation data and are shown to vary as the inverse of the exchange frequency. Such measurements furnish an independent means to calculate exchange frequencies. This represents the first case in which a minimum in T _{1} due to random modulation of the dipolar interaction has been observed in ESR, although such phenomena are common in NMR.

Growth of Durene from the Melt
View Description Hide DescriptionMicroscopic observation of the solidification and melting of durene at small undercoolings demonstrated that the habit face grows by layer spreading. It was also found that certain other unidentified faces grow by layer spreading. The critical supercooling required for an appreciable rate of two‐dimensional nucleation on the habit face is consistent with Cahn's analysis of the layer‐spreading mechanism for diffuse interfaces. Melting appeared to occur by the reverse of the freezing mechanisms.

Chemically Pumped Molecular Lasers
View Description Hide DescriptionThe feasibility of efficient conversion of chemical energy to radiation in chemically pumped lasers is discussed. Suitable specific systems are suggested and analyzed.

Effect of Pressure in the Radiolysis and Photolysis of Methane
View Description Hide DescriptionThe photolysis and radiolysis of equimolar CH_{4}–CD_{4} mixtures were investigated as a function of pressure. The fact that, in the presence of NO, the ethane fraction consists entirely of C_{2}D_{6}, C_{2}D_{4}H_{2}, C_{2}H_{4}D_{2}, and C_{2}H_{6} in comparable amounts indicates that CH_{2} and CD_{2} are produced. The relative yield of these ethanes which are formed by insertion of methylene into methane increases with pressure in both the photolysis and radiolysis. In the radiolysis, the G value reaches a value of 0.35±0.1 at pressures above 15 atm. Information about the effect of pressure on the production of the ethyl ion was obtained by investigating the radiolysis of CH_{4}–C_{4}D_{10} and CD_{4}—C_{3}H_{8} mixtures from 1.5 cm to 130 atm. The data indicate that there is a gradual decrease of the ethyl ion yield with increase in pressure while the parent ion yield increases with increase in pressure to a pressure of at least 15 atm.

Statistical Mechanics of Defect‐Containing Solids. I. General Formalism
View Description Hide DescriptionA formalism is set up for treating the contributions of point defects to the thermodynamic functions of crystalline solids by means of a cluster expansion of the partition function. Similar expansions for the defectdistribution functions and potentials of average force are derived. The assumptions involved and the necessary differences between the formalism and the McMillan—Mayer theory of solutions are discussed. Problems arising in the explicit evaluation of the coefficients of the virial expansions are discussed and possible applications of the formalism suggested.

Statistical Mechanics of Defect‐Containing Solids. II. Ionic Crystals
View Description Hide DescriptionThe cluster expansions for the partition function and defectdistribution functions derived previously are studied in detail for the case of ionic crystals with the object of calculating activity coefficients, defect concentrations, and defectdistribution functions at low defect concentrations. A diagram classification procedure analogous to that in the Mayer theory of ionic solutions is used to obtain nondivergent expansions for defect activity coefficients and distribution functions. The discreteness of the lattice requires some modification of the diagram summation techniques employed in the solutions theory. The theory of association of defects of the sort considered by Lidiard and Teltow is formulated more precisely in terms of the defectdistribution functions. The formal multicomponent expressions are studied in more detail for the case of Schottky defects and impurity ions in a sodium chloride lattice. The results parallel those of the Mayer ionic‐solution theory, the principal difference being that the Debye—Hückel potential of average force appearing in the final Mayer expressions is everywhere replaced by Ae ^{2} exp(—κRξ)/RD, where A and ξ are structure and concentration dependent and go to unity in the continuum limit.
As an example for the case of activity coefficients, calculations of the contributions from cycle diagrams and terms of next lowest order in concentration have been made for divalent impurity ions and cation vacancies in sodium chloride. The pair correlation function for oppositely charged defects and the degree of association have been calculated for the doped crystal. The theory reduces to that of Lidiard in the limit of zero concentration but differs at finite concentrations. However, calculation of the contribution of ``triangle diagrams'' to the activity coefficients indicates that below 500°C the expansions do not converge rapidly enough to be of value at concentrations of experimental interest because of the low dielectric constant. It was found that in the range of temperature and composition for which the theory converged, the parameters A and ξ differed little from unity.

Resolvent Operator Formulation of Stationary State Perturbation Theory
View Description Hide DescriptionBy starting with an exact operator equation and using different methods of expanding the resolvent operator, the Schrödinger, Wigner—Brilloin, similarity transformation, gauge transformation, and first‐order perturbationiteration method, perturbation expansions are generated in a rigorous and straightforward manner. It is also shown how additional perturbation and perturbationiteration methods can be generated.

Optical Absorption Spectrum of NiF_{2}
View Description Hide DescriptionThe optical absorptionspectrum of NiF_{2} is reported for the paramagnetic and antiferromagnetic states. A prominent satellite appears in the ^{3} A _{2}→^{1} E transition below the Néel temperature. This new line is explained in terms of the interaction of the excited Ni^{2+} ion with its six antiparallel Ni^{2+} neighbors.

Experimental Determination of the Thermal Conductivity of Molten Lithium from 320° to 830°C
View Description Hide DescriptionA comparative, axial‐heat‐flow apparatus with compensating guard heating was developed to determine the thermal conductivity of 99.8+ wt% molten lithium. Contrary to the data for two other alkali metals(sodium and potassium), the conductivity of lithium was found to increase linearly from 0.445 W/cm·°C at 320°C to 0.606 at 830°C. The total uncertainty in the results was conservatively estimated to vary from ±8% to ±15% from the lower to the higher temperatures. Unpublished data from 282° to 521°C by the Naval Research Laboratory average 4% below the present results. The values predicted by two correlations—the equation of Ewing et al. and the Wiedemann—Franz relationship—are uniformly 10% and 6% higher, respectively, than the present data.

Analysis of A_{2}B_{2} NMR Spectra. II. Nonsymmetrical 1,2‐Disubstituted Ethanes
View Description Hide DescriptionThe features of A_{2}B_{2}protonmagnetic resonance spectra of nonsymmetrical 1,2‐disubstituted ethanes are discussed in detail. It s shown that under certain conditions the magnitudes and signs of all the coupling constants may be obtained for this type of A_{2}B_{2} spectra. Typical calculated spectra along with several experimental spectra are given to illustrate these conditions. With one exception, experimental data are given only for molecules for which all four of the coupling constants are obtained.
It is found that the geminal coupling constants are of opposite sign from the rotationally averaged values for the vicinal couplings. The values of the geminal couplings are approximately those found by other workers in the corresponding monosubstituted methanes. Small vicinal substituent effects on the geminal coupling constants are reported for several of the substituent groups. Assignment of the chemical shift of the particular methylene group is based on the comparison of the A_{2}B_{2}chemical shifts with the chemical shifts in the related ethyl compounds.

Description of the Distribution of Electrons in the Methane Molecule
View Description Hide DescriptionA formal scheme is presented for describing the electron distribution in the methane molecule. Purely theoretical electron density contour maps for the ground state of the molecule are displayed, and various aspects of the electronic distribution are discussed. The electronic wavefunction employed is an analytical form composed of atomic orbitals of s through g type centered on the carbon nucleus.
The electronic charge density ρ is expressed as a series of normalized (to 4π) tetrahedral harmonics T_{l} (θ,φ), which are certain symmetrical linear combinations of spherical harmonics of the given l value. Namely,where the A_{l} (r) depend only on r, the radial distance from the carbon nucleus. The functions A_{l} (r) are tabulated, and contour maps of ρ are developed. The molecular octupole moment is predicted to be about 1.7×10^{—24} e·cm^{3}.
The distribution of electrons in a single carbon—hydrogen bond is considered from several points of view, with the purpose, among others, of elucidating the question of the sign and magnitude of the ``CH bond moment.''

Calculation of Magnetic Susceptibilities of Diatomic Molecules. IV. Application to LiH Molecule
View Description Hide DescriptionA previously proposed method for the calculation of magnetic susceptibilities of diatomic molecules has been applied to the LiH molecule, the result is χ=—9.39×10^{—6} cgs units per mole. The experimental value is not known. From a measurement of the rotational magnetic moment it is possible to obtain an experimental value for the paramagnetic contribution to the susceptibility for a specific choice of gauge, this value is 7.14×10^{—6} cgs units which should be compared with our theoretical result 5.48×10^{—6} for the same quantity.

Calculation of Intensity Distribution in the Vibrational Structure of Electronic Transitions: The B ^{3}Π_{0+ u }—X ^{1}Σ_{0+ g } Resonance Series of Molecular Iodine
View Description Hide DescriptionFranck—Condon overlap integrals have been calculated which predict within experimental error the intensity distribution of the sixty measured lines in the visible fluorescencespectrum of molecular iodine, B ^{3}Π_{0+ u }(v′ = 15, 16, or 26)→X ^{1}Σ_{0+ g }(v″ = 0 to 69). Rydberg—Klein—Rees potentials were used for both electronic states, and exact vibrational eigenfunctions were obtained by direct numerical solution of the radial Schrödinger equation, including vibration—rotation interaction. The electronic transition moment was assumed to be independent of internuclear distance. Overlap integrals derived in the same way for Morse potentials fail to give even qualitative agreement with experiment for lines with v″≳10. Because of the rapid oscillation of the vibrational wavefunctions for high v′ and v″, a shift in the potential of only 0.002 Å is found to alter appreciably the calculated intensity distribution; thus the agreement obtained provides a very severe test of the RKR potentials and the Franck—Condon principle. The radiative lifetime of the B state has also been calculated from the absolute intensity of a single line and the integrated intensity of the band system, and the results compare favorably with direct lifetime measurements.

Simple Basis Set for Molecular Wavefunctions Containing First‐ and Second‐Row Atoms
View Description Hide DescriptionThe self‐consistent field functions for the ground state of the first‐ and second‐row atoms (from He to Ar) are computed with a basis set in which two Slater‐type orbitals (STO's) are chosen for each atomic orbital. The reported STO's have carefully optimized orbital exponents. The total energy is not far from the accurate Hartree—Fock energy given by Clementi, Roothaan, and Yoshimine for the first‐row atoms and unpublished data for the second‐row atoms. The obtained basis sets have sufficient flexibility to be a most useful starting set for molecular computations, as noted by Richardson. With the addition of 3d and 4f functions, the reported atomic basis sets provide a molecular basis set which duplicate quantitatively most of the chemical information derivable by the more extended basis set needed to obtain accurate Hartree—Fock molecular functions.