Volume 35, Issue 3, 01 September 1961

Wave Functions and Intensity Calculations for a Rotating Anharmonic Oscillator
View Description Hide DescriptionWave functions which are good through second‐order perturbation theory are given for a harmonic oscillator perturbed by a potential of the form V = ℏw (ax+bx ^{2}+cx ^{3}). These results are then applied to the case of a rotating anharmonic oscillator. Expressions are also given for overlap integrals 〈n̄  m〉 between different unperturbed harmonic oscillatorwave functions. A brief comparison of this treatment with similar treatments is given and a discussion of rotation‐vibration interaction in non ^{1}Σ states is included. The application to intensity calculations in electronic and infrared spectra is discussed. An approximate expression for the dipole moment matrix element 〈1J′  u  0J〉 is given.

Infrared Spectra of Sodium and Potassium Cyanide
View Description Hide DescriptionThe infrared absorption spectra of sodium cyanide and potassium cyanide vapors have been measured in the region 3500 to 200 cm^{‐1}. Two absorptions assigned as fundamental frequencies of the monomeric species were found in each system. The absorption frequencies and their present assignments are: NaCN ν_{2}=238, ν_{3}=2176 cm^{‐1}; KCN ν_{2}=207, ν_{3}=2158 cm^{‐1}. A third absorption at 2076 cm^{‐1} in sodium cyanide vapor is assigned to the dimer of NaCN.
The bond‐stretching force constant of the CN group in XCN molecules is sensibly independent of the electronegativity of X. The angle‐bending force constant varies directly with the electronegativity of X.

Structure of the Band Spectrum of CuCl Molecule. I. Additional Knowledge in the Coarse Structure
View Description Hide DescriptionA number of new bands in the A, B, C, D, and Esystems of CuCl, recorded in the spectrum of the molecule excited in an electrodeless discharge tube by a microwaveoscillator, are reported. Because of the employment of high dispersion and the use of a single isotopic species of CuCl (Cu^{63}Cl^{35}), it has been possible to investigate the phenomenon of formation of head of heads in the Δv=−4 sequence of the Esystem. This phenomenon has also been detected in the Δv=−3 sequence of the Asystem.
The weak Fsystem has been photographed for the first time under suitable dispersion so that a satisfactory vibrational analysis of the same could be made. The R heads of this system of the Cu^{63}Cl^{35} molecule are represented by the formulaIt has been shown that the lower state of the Fsystem is the same as the common lower state of all the other known band systems, which is also the ground state of the molecule. Data on the A and Fsystems of the Cu^{65}Cl^{35} molecule also are given.

Vapor Pressure of Antimony by the Torsion‐Effusion Method
View Description Hide DescriptionThe pressures and mean molecular weights of the vapors over solid antimony (420–550°C) and solid zinc (250–335°C) have been determined by simultaneous measurements by the torsion‐momentum and Knudsen weight loss methods. The torsion measurement was calibrated with mercury at 30° using a mercuryvapor pressure of 2.96×10^{‐3} mm Hg determined by concurrent weight loss experiments. Zinc vapor is confirmed to be monatomic. The zincpressure data are represented by logP _{mm}=(8.741±0.218) — (6630±125) /T giving ΔH _{298}° (vap)=30.73±0.57 and 31.18±0.12 kcal/mole by the second law and third law of thermodynamics, respectively, in excellent agreement with previously reported results. The vapor over solid antimony is all Sb_{4} molecules within experimental error. Least‐squares analysis of the antimony data yields logP _{mm}=(10.571±0.090) — (10 300±68) /T,pressures 10%‐40% lower than those previously reported. The heat of vaporization of Sb_{4}(g) at 298.16°K is 49.83±0.31 kcal/mole by the second law and 49.45±0.09 by the third law of thermodynamics. This thermodynamic consistency and the agreement of the zinc results with accepted measurements indicate that the antimonypressures presented should be preferable to those previously reported.

Self‐Nucleation in the Sulfuric Acid‐Water System
View Description Hide DescriptionHoward Reiss' ``Vector model'' was adapted to calculate the self‐nucleation rate for the sulfuric acid‐water system and similar systems.
A numerical calculation was carried out for the typical case of 50% relative humidity at 25°C. It was concluded that rapid self‐nucleation would take place at sulfuric acid partial pressures in the range 10^{‐8}‐10^{‐10} mm Hg. The considerable uncertainty is principally due to lack of data on the partial pressure of sulfuric acid above its aqueous solutions and on the dominant sulfuric acid‐bearing species in humid atmospheres.

Electrical Conductivity of Synthetic Crystalline Zeolites
View Description Hide DescriptionThe electrical conductivity of anhydrous Linde zeolite types A, X, and Y in various cationic forms has been measured between 20 and 400°C. The conductivity is ohmic, essentially noncapacitive, and appears to be strictly ionic with unusually low activation energy. From the effect of zeolite channel size, type of cation and cation density on ΔH, ΔG, and ΔS for the conduction process, some conclusions as to the zeolite internal structure and surface energy have been obtained. The heterogeneity of the internal surface energy found by the measurement of adsorption heats is discussed in the light of these conclusions.

Transition Probabilities in Multilevel Systems: Calculation from Impulsive and Steady‐State Experiments
View Description Hide DescriptionThe study of collision‐induced transitions of molecules in a system of many energy levels is discussed in an analysis of time‐dependent and steady‐state experiments. These results are used in a discussion, with numerical examples, of the difficulties in calculating the transition probabilities from observed population distributions.
An intimate conceptual and mathematical relationship is established between two types of experiments which are possible at present. One is an impulse, or time‐dependent experiment such as excitation with a fast flash lamp and snapshot observation after a known delay time. The other is a steady‐state experiment in which molecules are fed into the system (a gas or a surface) at a single energy level, for example, and undergo transitions among the available levels in competition with a first‐order removal process such as spontaneous radiation or desorption from a surface.
The ideas summarized above are used to develop procedures for calculating transition probabilities from observed population distributions. Numerical examples and qualitative considerations indicate that when transitions involving more than single quantum jumps are involved, the experimental data must be known to much higher accuracy than that desired in the transition probabilities. The difficulties increase rapidly as the number of levels in the system increases.

Atomic Wave Functions for the Ground States of Na^{+}, Ne, and F^{‐}
View Description Hide DescriptionCalculations of wave functions for the ′Sground states of Na^{5}, Ne, and F^{‐} which essentially correlate the outer shell are presented here. The ``root'' function in each case is an analytic Hartree‐Fock function. The improved 13‐configuration function yields an improvement in energy of about 50% of the estimated correlation energy.

Law of Corresponding States for Fused Salts
View Description Hide DescriptionReduced equations of state are developed for strictly ionic salts through an analysis of the configuration integral. In this way, appropriate laws of corresponding states are developed for vapor pressures,surface tensions, and melting points. Agreement between theory and experiment is satisfactory for a large class of salts.

Least‐Squares Local‐Energy Method for Molecular Energy Calculations Using Gauss Quadrature Points
View Description Hide DescriptionThe least‐squares local‐energy method previously described and tested in calculations on H_{2} ^{+} has been further tested on H_{2} ^{+}, He, and H_{2}. The theory of Gauss quadrature is used to provide a better basis of points over which to average the local‐energy. The electronic energy is easily calculated to six significant figures for H_{2} ^{+} and to three or four significant figures for He and H_{2}.

Momentum‐Correlation Function in a Rayleigh Gas
View Description Hide DescriptionThe time‐correlation function in an equilibrium ensemble of the momentum of a heavy particle imbedded in a gas of noninteracting light particles is calculated. The technique used is a density expansion of the correlation function. No statistical hypothesis beyond that of canonical ensemble at the initial time is used.

Elastic and Inelastic Scattering of Low‐Velocity H^{+} and H_{2} ^{+} in Hydrogen
View Description Hide DescriptionCross sections have been measured in the energy range 4 to 400 ev for elasticscattering and charge exchange of H^{+} and H_{2} ^{+} ions in hydrogen. The results are compared with similar measurements for deuterium ions in deuterium. Charge exchange was observed over the entire energy range for H_{2} ^{+}, but was detected only above 50 ev for H^{+}. Constants for empirical potential functions have been evaluated and are tabulated.

Electron Coupling of Proton Spins in Heterocyclic Molecules. N‐benzylthieno[3,2‐b]pyrrole
View Description Hide DescriptionThe high‐resolution protonmagnetic resonancespectrum of the four protons attached to the heterocyclic rings of N‐benzylthieno[3,2‐b]pyrrole (I) has been examined in detail at 16.2 Mc/sec. The spectrum is of the type ABCD perturbed by two nuclei X _{2} which couple with A, and it involves three long‐range proton‐proton couplings, 0.5, 1.3, and 0.3 cps over five, six, and six bonds, respectively, which can be accounted for only by pi‐electron interactions. The analysis of the spectrum indicates that all of the coupling constants have the same sign, which is presumed to be positive on theoretical grounds. This analysis illustrates the great value of obtaining unambiguous magnitudes of the coupling constants and chemical shifts from high‐frequency spectra in order to determine the relative signs of the coupling constants from low‐frequency spectra. The coupling constants found in each of the rings in I are compared with those of the monocyclic compounds, thiophene and pyrrole. Although angular effects may govern the coupling via sigma electrons in HCCH groups, the pi‐electron contribution appears to be sensitive to electronegativity differences. In addition, heteroatoms and ionic structures are important in determining the long‐range coupling via the pi electrons.

Photoionization of Ce^{3+} in Glass
View Description Hide DescriptionThe photoionization of Ce^{3+} is investigated by determining the optical absorption changes and the electron spin resonance changes that occur when cerium‐containing silicate glasses are illuminated with ultraviolet light. The results are: (1) The room temperature quantum yield of photoelectrons from Ce^{3+} is 0.1. (2) The optical absorption and the electron spin resonance bands caused by trapped photoelectrons are found. (3) A Ce^{3+} ion from which a fourth electron has been removed by photoionization is different from a Ce^{4+} ion.

General Theory of Monolayer Physical Adsorption on Solids
View Description Hide DescriptionBy suitable modifications of the general equations given previously [W. A. Steele and M. Ross, J. Chem. Phys. 33, 464 (1960)], a theoretical monolayeradsorption isotherm is given which is written as a power series. The terms in the series involve integrals over the interaction energies of the adsorbed atoms with each other and with the solid. This treatment is based on a virial expansion of the density of the film in powers of the activity of the film, but differs from previous two‐dimensional virial expansions in that it explicitly includes the effect of periodic variation of the solid‐gas atom potential energy as the atom moves over the surface of the solid. It is shown that this general equation reduces to a sitewise model for large parallel variations in the solid‐gas atom energy, and to a perfectly mobile film for small variations. However, when the potential energy for a gas atom over a simple solid is calculated as a function of the relative sizes of the gas and solid atoms, it appears that many adsorption systems may obey neither of the two limiting models. Some possible methods of dealing with adsorption systems which are neither completely localized or completely mobile are outlined and it is shown that some information about the nature of the solid‐gas atom potential function may be obtained from an analysis of the experimental data for suitable systems.

Monolayer Adsorption of Helium on Argon
View Description Hide DescriptionExperimental isotherms and heats of adsorption of helium on an argon surface are reported for temperatures from 10° to 20°K. The potential energy of interaction of a helium atom with an argon surface is calculated by summing the He–Ar pair interaction over the solid lattice. The experimental data are analyzed in terms of a quantum mechanical modification of the theoretical treatment presented previously [W. A. Steele and M. Ross, J. Chem. Phys. 33, 464 (1960) (I); 35, 850 (1961) (II)]. It is shown that the adsorption properties calculated from the a priori potential function are in quantitative agreement with the experiments, in the region where a quantitative comparison can be made, and that the remainder of the data is in agreement with semiquantitative theoretical estimates. A detailed description of the probable adsorption process for the He–Ar system is deduced from the comparison between experiment and theory. It is concluded that the first half of the helium atoms comprising a monolayer on this surface are highly localized at equivalent positions over the argon lattice unit cells, and are nearly all in their quantum mechanical ground state at temperatures up to 20°K. The second half of the atoms in the monolayer is adsorbed over a different position relative to the solid lattice, and is more weakly bound than the first. Although the quantitative comparison between the theory and the experiment is somewhat limited because the quantum version of the theory is not as complete as the classical, it is concluded that the agreement between experiment and theory indicates that this approach to the problem of physical adsorption may prove useful in the analysis of data taken on other suitable systems.

Multilayer Adsorption of Helium on Argon
View Description Hide DescriptionMultilayer isotherms and heats are reported for helium adsorbed on an argon surface at 4.2°K. The data show that, on the uniform surface used in this work, the layer capacities and energies of the first two or three layers of adsorbed helium can be estimated directly from the heats without recourse to the isotherm data. The adsorption energies are shown to be in good agreement with theoretical estimates. Two alternate approaches to the theoretical calculation of layer capacity are presented: one of the methods involves a computation of the perturbation of the density of the bulk liquid due to the presence of the surface; the other calculation is based on a computation of the density dependence of the lateral interaction energy in a layer plus the surface interaction energy of the layer. The density of a completed layer can be estimated by equating the total energies per atom in successive layers. It is shown that the results of these calculations are in good agreement with each other, and with the experiments. The high coverage isotherm data are analyzed in terms of a modified form of the Frenkel‐Halsey‐Hill isotherm equation, and it is shown that the parameters required to fit the experiments to this equation are consistent with other estimates. Finally, the data are fitted to the B.E.T. equation, and to several of the modified B.E.T. isotherms which have been suggested as more realistic representations of helium adsorption data. It is concluded that no B.E.T.‐type model is completely successful, but that the isotherm equation proposed by Steele gives the most accurate results.

Electrical Conductivity and Thermodynamic Equilibrium in Nickel Oxide
View Description Hide DescriptionThe electrical conductivity of nickel oxide has been studied at temperatures between 600° and 1350°C and partial pressures of oxygen between 1 atm and 10^{−4} atm. The slope of log conductivity vs reciprocal temperature plots increases when the temperature is high enough for the sample to come into thermodynamic equilibrium with the atmosphere, and the temperature at which the change in slope occurs is observed to depend on the rate of heating or cooling of the crystal. The heat of formation of nickelvacancies may be determined from the magnitude of the change in slope by a simple calculation. Nickel oxide is a p‐type semiconductor [M. Verwey, M. Haaijman, H. Romeijn, and M. van Oosterhout, Philips Research Repts. 5, 173 (1950)] in which the conductivity is proportional to the concentration of Ni^{3+} ions in the lattice. When thermodynamic equilibrium is established with the atmosphere the conductivity is proportional to the ⅙th power of the oxygen partial pressure.
The analysis of the conduction mechanism and the conduction data may be self‐consistently correlated with nickeldiffusion,nickelvacancydiffusion, and weight changes resulting from equilibration with different oxygen partial pressures. The following constants may be determined from these correlations: heat of formation of nickelvacancies,activation energy for nickelvacancy migration, activation energy for electron hole migration, and concentrations of nickelvacancies and Ni^{3+} ions as a function of temperature and oxygen pressure. Combined results give the following for the concentration of nickelvacancies: 0.11 (P _{O2})^{⅙} exp(−17 800/RT) vacancies per ion pair.

General Theory of Polymer Solutions
View Description Hide DescriptionA general theory of the thermodynamic properties of polymer solutions is given. The theory is intended to be valid for every concentration range. The formalism begins with the assumption of some functional relation between the local free energy density, for an assigned polymer configuration, and the local concentration of segments belonging to various polymer chains. That is, the thermodynamic properties of the segments are presumed known, and the problem is taken to be the determination of changes in the thermodynamic properties which are induced by a joining of the segments into a polymer chain. The local segment concentration is assumed to be close enough to the bulk or average concentration that a Taylor series expansion of the local free energy can be made and cut off at the term containing the square of the fluctuation in local concentration. The resulting free energy of the whole solution can then be cast into a form which includes as one part the free energy of a random distribution of segments, and as a second part the free energy of an imperfect gas with a pairwise additive intermolecular potential, albeit a concentration‐dependent potential. At low concentrations the results are identical to previous treatments based on imperfect gas theory. At high concentrations a previously introduced calculation of the radial distribution function is employed to show that the thermodynamic properties are accurately described as those of a random distribution of segments. The Flory‐Huggins formula is an example of a free‐energy function which can be assumed to hold locally at any concentration, but is then found to hold macroscopically only at high concentrations.

Scattering of High‐Velocity Neutral Particles. XII. He–CH_{4}; He–CF_{4}. CH_{4}–CH_{4} and CF_{4}–CF_{4} Interactions
View Description Hide DescriptionCollision cross sections have been measured for helium atoms with energies between 500 and 2100 ev, scattered in room‐temperature CH_{4} and CF_{4}. The results have been analyzed to obtain the average potential between a helium atom and a CH_{4} molecule,for r between 1.92 and 2.37 A, and the average potential between a helium atom and a CF_{4} molecule,for r between 2.43 and 2.74 A. This He–CH_{4} potential is consistent with one valid at larger separation distances which has been obtained by combining potentials derived from high‐temperature viscosity and second virial coefficients of helium, and from high‐temperature viscosity coefficients of methane.
A procedure which assumes the centers of force to reside in the peripheral H or F atoms has been used to analyze these average atom‐molecule interactions in terms of the effective He–H and He–F interatomic potentials. These interatomic potentials have been combined with the potential previously determined for He–He to obtain effective H–H and F–F potentials for atoms in different CH_{4} and CF_{4} molecules. These effective interatomic potentials were then summed and averaged over all molecular orientations to yield the average potential between two CH_{4} molecules,for r between 2.47 and 3.06 A, and the average potential between two CF_{4} molecules,for r between 3.43 and 3.77 A. This CH_{4}–CH_{4} potential is consistent with one valid at larger separation distances which has been derived from measurements on gaseous viscosity at high temperatures.