Volume 38, Issue 8, 15 April 1963

Linear Velocity‐Gradient Term in Time‐Dependent Pair Distribution Function
View Description Hide DescriptionA Chapman—Enskog‐type perturbationsolution, in gradients of energy, velocity, and density, is proposed for the hierarchy of integro‐differential equations obeyed by the time‐dependent Ursell—Mayer functions. The hierarchy is then terminated by an ansatz relating the three‐particle Ursell functions to those of lower order, which yields a closed system for the perturbations to the singlet and pair distribution functions (it is assumed the equilibrium functions are known). Attention is focused on the equation for the two‐particle functions, and on the term therein proportional to the divergence of fluid velocity, from which the bulk viscosity may be calculated. An expansion for this term is assumed in powers of scalar products of particle separations and velocities, in which the coefficients in turn are expanded in Sonine polynomials in the velocities. The functions which multiply these polynomials satisfy a system of integral equations, for which a first approximation is written down and the solution discussed. Finally, it is shown that substitution of the solution for the pair distribution function into the hierarchy equation for the singlet distribution yields an infinite number of conditions which may be used to evaluate a corresponding number of parameters in the ansatz employed to close the hierarchy.

Dependence of Surface Tension on Temperature
View Description Hide DescriptionIt is shown that the equation of Reiss et al. for the surface tension of liquids composed of hard‐core molecules does not usually provide the proper dependence of surface tension on temperature for typical non‐ionic liquids. Because of this, an empirical computation procedure is adopted, whereby the hard‐core diameter is given a temperature dependence similar to that of the collision cross section for gas viscosity. When Lennard‐Jones 6–12 or Sutherland intermolecular‐potential data are then used to calculate a hard‐core diameter, the agreement between computed and measured dependence of surface tension on temperature is usually very good, even for liquids composed of such polar, aspherical molecules as water, ethanol, and ammonia. As an illustration of the potential value of the empirical use of gas‐viscosity data for improving computations of the dependence on temperature of physical properties and thermodynamic values for liquids, the Lennard‐Jones 6–12 and Sutherland data are applied to two rigid‐sphere relationships for liquid compressibility.

Gaseous Diffusion in Porous Media. III. Thermal Transpiration
View Description Hide DescriptionA previously proposed model for the diffusion of gases in porous media at uniform and nonuniform pressures has been extended to allow for temperature gradients. The porous medium is visualized as a collection of ``dust'' particles constrained to remain stationary in space. As before, by formally treating the ``dust'' particles as giant molecules, it is possible to cover the entire range of intermediate mechanisms from the Knudsen to the normal region by varying the mole fraction of the real gas. For binary systems (a single gas and ``dust''), the phenomenon of thermal transpiration is accounted for by the model in a consistent way for all aspects which are diffusive in character but, as before, needs modification at high pressures by the addition of an expression to account for the viscous backflow. With this modification, an equation is obtained which describes thermal transpiration over the entire pressure range. This equation discloses two new relationships not previously noticed: one between the maximum in the thermal transpiration curve and the Knudsen minimum in the permeability curve, and one between the height of the thermal transpiration maximum and the translational heat conductivity of the gas.
It was found that it was possible to extend the model to include capillaries as well as porous media. This extension disclosed that the relationships applicable to porous media would also describe the analogous behavior that occurs in capillaries with definite geometry. The capillary results reproduce previous semi‐empirical and empirical equations of Weber and Liang for thermal transpiration, and the permeability equation of Knudsen. The present equations, however, have fewer adjustable parameters than do the previous equations. A simple connection, not previously mentioned, between Weber's and Liang's equations was also noticed: Liang's equation is really a special form of Weber's equation written in a differential‐approximation form.
A remarkable feature of the results, as applied to capillary systems, is that one can calculate rotational relaxation times in gases from the height of the thermal transpiration maxima, which suggests much simpler experimental techniques than those previously employed for the measurement of this phenomenon.

Calculation of Mean Atomic Positions in Vibrating Polyatomic Molecules
View Description Hide DescriptionA new method is described for calculating effects of anharmonic vibrations on bond lengths and angles in polyatomic molecules. It is shown that the quantum mechanical analog of Newton's second law of motion leads to simple equilibrium conditions from which it is possible to calculate mean displacements of atoms from positions of minimum potential energy. The method resembles the standard first‐order perturbation method insofar as it relates mean displacements to mean‐square displacements which can be computed from the zeroth‐order harmonic problem. It gives a much more direct calculation, however, circumventing the laborious nonlinear transformations between internal and normal coordinates encountered in the perturbation method. Advantages and limitations are discussed.
Applications to several problems are outlined briefly, including isotope effects and effects of temperature on mean molecular structures. The approach, extended to crystal lattices, is shown to account for Grueneisen's relation which correlates thermal expansion and heat capacity. Numerical results of calculations are presented for H_{2}O, D_{2}O, NH_{3}, and ND_{3}, comparing the present method with other methods. Contrary to prevailing opinion, the mean apical angle of NH_{3} appears to be smaller than that of ND_{3}.

Slater—Condon Parameters from Spectral Data
View Description Hide DescriptionSlater—Condon parameters have been fitted by least‐squares methods, as far as spectroscopic data are available, for all the neutral atoms of the first three periods of the periodic system, and for their isoelectronic mono‐positive ions, and for the isoelectronic di‐ and tri‐positive ions of the first two periods. Separate evaluations were performed for each configuration, and for pooled data from all configurations. It is concluded that the approximations involved in the Slater—Condon method are at least as serious as the further approximations introduced by pooling of data from several configurations.

Inelastic Atomic Scattering. II
View Description Hide DescriptionCross sections for the transitions in an alkali atom in collision with a rare‐gas atom have been calculated for thermal impact energies by the use of a modified ``distorted‐wave'' approximation. These calculations are part of a continuing comparison study on the effectiveness of several different approximations in predicting intermultiplet transitions. Numerical results agree qualitatively with experiment but this agreement is fortuitous and is not the immediate aim of the calculation. The critical features of such calculations are briefly discussed.

Second Virial Coefficients of Mixed Hydrocarbon Vapors. Principle of Congruence
View Description Hide DescriptionBrønsted's principle of congruence, according to which the excess thermodynamic functions of a liquid mixture of n‐alkanes depend only on the average chain length in the mixture, is shown to apply also to the second virial coefficients of gaseous hydrocarbon mixtures.

Absolute Intensity Measurement for the NO Fundamental at 5.3 Microns
View Description Hide DescriptionThe intensity of the fundamental vibration—rotation band of NO has been measured using the Wilson—Wells—Penner—Weber self‐broadening technique. The reported value is 111±7 cm^{—1}/cm atm at 273°K, considerably higher than some previously published values, and in marginal agreement with other less precise previous determinations.

Free Energy of Nonequilibrium Polarization Systems. II. Homogeneous and Electrode Systems
View Description Hide DescriptionThe polar contribution to the free energy is calculated for a system containing ``particles'' imbedded in a continuum having nonequilibrium dielectric polarization. Each particle may either be a single molecule or a whole collection of molecules (such as an electrode) at specified nuclear configuration and may possess a general permanent and induced charge distribution. Thereby, the results of Part I of this series are extended to a variety of systems. It is also shown that the polar contribution to the free energy can be written as the sum of free energies of equilibrium polarization systems, thereby permitting the immediate application of literature expressions for the latter to calculating the former. This expression is also derived for systems possessing partial dielectric unsaturation under a certain typical condition. Applications are made to the theory of electron‐transfer reactions in solution and at electrodes and to the theory of the shift of electronic spectra of polar solutes by polar solvents.

Field‐Emission Observations of Carbon on Tantalum
View Description Hide DescriptionThe carbon—tantalum system, observed with field emission, shows the temperature effects on the solubility and precipitation of the carbide phase. Carbon, once deposited on tantalum, cannot be removed by high‐temperature treatment, as in the case of tungsten. With low‐carbon contamination the 334 planes appear as dark areas, just as for carbon on tungsten. At temperatures in the region of 950°K, platelets, presumably Ta_{2}C, form and migrate to the [111] zones of the emitter single‐crystal tip. The energy of activation of this surface migration process is 54 kcal.

Field‐Emission Study of Carbon Monoxide on Tantalum
View Description Hide DescriptionAdsorbed carbon monoxide shows three states of binding on tantalum. The weakest of these is desorbed at temperatures above 125°K with no observable surface migration. The bonding with the surface is of the van der Waals type. This is in contrast with the second state, which is desorbed above 650°K. The third state is dissociated before desorption to give the oxygen‐on‐tantalum pattern. Surface migration involving the second adsorbed state occurs with an activation energy of 38 kcal. The work function of a carbon monoxide‐covered tantalumsurface, obtained by spreading the carbon monoxide on a shadowed tip at 40°K, is 0.8 eV greater than that of the corresponding clean tantalum. Fowler—Nordheim plots of clean and shadowed tips show that at least under these conditions, the infinite‐field extrapolation of logi/V ^{2} is proportional to the log of the emitting areas.

Evaluation of Vapor‐Pressure Data for Mercury, Lithium, Sodium, and Potassium
View Description Hide DescriptionVapor‐pressure data for the elements mercury,lithium,sodium, and potassium were correlated using an equation of state,PV = RT+BP, where B was evaluated empirically with the relationship B = b/T^{m}. Values of B thus obtained are compared with values of the second virial coefficient calculated from a Lennard‐Jones 6–9 potential curve. Calculated values for vapor pressures and fugacity‐to‐pressure ratios are presented for these elements as a function of temperature.

On Ionic Catalysis of Hydrogen‐Atom Recombination
View Description Hide DescriptionThe feasibility of using hydrogen molecular ions, produced in a discharge, for the homogeneous catalysis of H‐atom recombination in a second‐order process has been investigated. The catalytic cycle involves H_{3} ^{+} and H_{4} ^{+} ions and the efficiency of the H‐atom recombination depends critically upon the stability of the H_{4} ^{+} ion complex against unimolecular decomposition. Rate constants for the various elementary reactions have been calculated, and the limiting values of the rate constants for the second‐order recombination reaction and for the catalytic efficiency have been determined. It is shown that the detrimental loss of ions through electron—ion recombination can be minimized through sequential pulsed ionization. The results obtained indicate that hydrogen‐atom recombination may be catalyzed effectively by molecular ions owing to the large reaction cross section of ion—atom reactions. The possibility of obtaining high catalytic efficiency of H‐atom recombination with pulsed molecular ions of the type (X_{ n }H_{ m })^{+} would seem to warrant some extensive experimental studies.

Theoretical Rate of Decomposition of Cyclopropane
View Description Hide DescriptionSlater's original application of his theory was for the isomerization of cyclopropane to propylene. This calculation was based on inadequate spectroscopic data, and hence a recalculation of the Slater parameters was undertaken using more recent and complete spectroscopic data. Since n = 14 is the maximum number allowed by the theory for cyclopropane it was also decided to test the effect of removal of symmetry on this value. It is concluded that a value of n = 21 for the isomerization of cyclopropane‐d _{2} is not demanded by n = 14 for light cyclopropane, but is dependent upon the normal coordinate system; hence removing a major objection to the theory. The value of n is further shown to depend in a sensitive way on the choice of a reaction coordinate. The appropriate values of amplitude factors and frequency factors are given for the various critical modes.

Isomerization of Nitrous Acid: An Infrared Photochemical Reaction
View Description Hide DescriptionNitrous and deuteronitrous acid suspended in nitrogen matrices at 20°K isomerize when illuminated with infrared light. Kinetic studies indicate that both the cis to trans and the trans to cisisomerizations occur. Filter studies show that a narrow range of frequencies is responsible, 3650–3200 cm^{—1} for HONO and 4100–3500 cm^{—1} for DONO. An isomerization mechanism is proposed that involves a highly efficient intramolecular transfer of energy between vibrational modes. The height of the potential barrier to isomerization is estimated to be 9.7±0.7 kcal/mole in the matrix and 8.7±1 kcal/mole for gaseous HONO. Comparison to methyl nitrite suggests that the bondings in these two molecules are quite similar and that the reported structure of methyl nitrite is incorrect.

Relation between the Chemical Composition and the Electronic Properties of the Charge‐Transfer Complex, p‐Chloranil: p‐Phenylenediamine
View Description Hide DescriptionThe charge‐transfer complex p‐chloranil:p‐phenylenediamine has been prepared by three methods: (a) precipitation from benzene, (b) precipitation from methylene chloride, (c) direct reaction between the dry solid constituents. A 1:1 complex was found in all cases, with entrapped solvent present in the cases of samples precipitated from solution. It has been shown that the concentration of unpaired electron spins and the conductivity are characteristic of the complex and do not depend upon the method of preparation. The thermal activation energy for conductivity has been found to be 0.43 eV, and for spin concentration it has been found to be 0.13 eV. Theoretical difficulties involved in the consideration of activation energies are discussed. A mechanism is considered in the light of the thermal activation energies and the Seebeck coefficient, which was found to be +1.4 mV/°C.

Analysis of ESR Spectra Observed in Irradiated Aromatic Systems
View Description Hide DescriptionSeveral structures have been proposed to explain the large hyperfine splittings observed in aromatic systems which have been exposed to ionizing radiation or hydrogen‐atom bombardment. The observed extra hyperfine structure was used to obtain electron densities on the various carbon atoms of the cyclohexadienal radical, and McLachlan's treatment was applied to calculate the splitting of the two hydrogen atoms on the single carbon atom of the C_{6}H_{7} radical. The calculated values are in satisfactory agreement with the 45‐ to 50‐G splitting between lines of the observed triplet. The other proposed structures are shown to be unlikely to produce such large splittings, or not applicable to purely aromatic systems.

Paramagnetic Resonance Spectrum of Fe^{3+} in Calcite
View Description Hide DescriptionThe paramagnetic spectrum of an impurity in natural crystals of calcite has been examined and identified as that of Fe^{3+}. The value of g is assumed to be two. Approximate values for  D  and  a—F  are 2.9 and 0.44 Gc, respectively. The value of the cubic field splitting parameter a was not determined.

NMR Studies of Hydrated Sodium Tetraborate Minerals. I. Boron—Oxygen Polyion in Borax and Tincalconite
View Description Hide DescriptionThe quadrupolar splittings of the B^{11}nuclear magnetic resonance(NMR) signals in single crystals of borax, Na_{2}B_{4}O_{7}·10H_{2}O, and tincalconite, Na_{2}B_{4}O_{7}·5H_{2}O, have been investigated. The electric quadrupole coupling constants, eqQ/h, and asymmetry parameters η of the electric field gradient tensors at the two unique B^{11} sites in the two crystals were determined to beThe orientations of the principal axes of the electric field gradient tensors for the above sites and their symmetry‐related sites have also been determined.
The picture of the boron—oxygen polyion in borax as given by NMR is in complete agreement with the x‐ray results as to numbers and coordinations of boron atoms, symmetry of the polyion and orientation of the planes of the triangularly coordinated boron atoms. The NMR results also indicate that the same boron—oxygen polyion occurs in tincalconite as in borax.

Fluorescence and Pre‐Ionization in Nitrogen Excited by Vacuum Ultraviolet Radiation
View Description Hide DescriptionFluorescence in nitrogen following absorption of vacuum ultraviolet radiation has been found to begin at a wavelength of 661.3 Å and to extend to at least 580 Å, the shortest wavelength available in this experiment. The energy of the longest wavelength at which the fluorescence appears is almost exactly equal to the energy required to excite the N_{2} molecule from the ground state,X ^{1}Σ_{ g } ^{+}, to the B ^{2}Σ_{ u } ^{+} state of the N_{2} ^{+} ion. It is thought that the first negative bands of N_{2} ^{+}, B ^{2}Σ_{ u } ^{+}→X ^{2}Σ_{ g } ^{+}, are responsible for the observed fluorescence. The fluorescence is excited by dispersed radiation from a stabilized helium gas discharge which produces the Hopfield continuum in the region 580–1100 Å. Fluorescence is also detected in carbon monoxide and in oxygen, but it is too weak for accurate measurements of the appearance wavelength. Experiments with a quartz cell window, in the presence of the electric field from the end window of the photomultiplier, enabled the pre‐ionized bands of nitrogen to be observed.