Volume 34, Issue 5, 01 May 1961

Thermodynamic Theory of the Pair Correlation Function
View Description Hide DescriptionThe thermodynamic description of inhomogeneous systems is adapted to a determination of the pair correlation function. The method is most effective for describing the correlation behavior near a critical point. The critical‐point behavior is investigated for a simple model fluid of the van der Waal's type. A novel optical scattering phenomenon is predicted for the interface between two equilibrium phases for temperatures below but very near to the critical temperature. The effect of inclusions on critical point phenomena is noted.

On the Theory of Raman Intensities
View Description Hide DescriptionThe Herzberg‐Teller development for vibronic transition moments is introduced into the Kramers‐Heisenberg dispersion equation. It is shown how ``forbidden'' character (vibrationally induced intensity) in allowed electronic transitions is responsible for the Raman intensities of fundamentals. This suggests a direct link between certain vibronic spectroscopic observations and Raman intensities. The development is carried to the first order in nuclear displacements only—the higher terms give rise to Raman intensities of combinations and overtones. An equation for the polarizability components is obtained which leads directly to the Raman selection rules and in addition provides interesting predictions concerning the Raman and the resonance Raman effect in relation to excited electronic states. Rules are worked out governing the participation of excited electronic states in the Raman effect and these are employed to demonstrate a separability of σ and π contributions to the scattering of totally symmetric modes. The polarizability expression obtained here is compared with that of the semiclassical theory of Shorygin and an interesting resemblance is found. Finally it is shown how, depending on details of the electronic structure, the predicted form of the frequency dependence of scattered intensity ranges between the two expressions that have been commonly used to interpret observations.

Statistical Mechanics of Isotope Effects on the Thermodynamic Properties of Condensed Systems
View Description Hide DescriptionEvidence for the role of molecular structure on the difference in the thermodynamic properties of isotopic molecules in the liquid and solid states is summarized. The properties considered are vapor pressure, heats of vaporization, molal volume, and transition temperatures. It is shown that the molecular structure must be taken into consideration even for small quantum effects. In the approximation of the first quantum correction the difference in thermodynamic properties of isotopic molecules in the condensed state depends upon the atomic masses and an energy parameter associated with each atom in the molecule. The results are extended to higher‐order quantum corrections for a harmonic potential. The rules of the mean are obtained directly. Various frequency distributions for the lattice modes are considered. For the case where the internal frequencies in the condensed phase are similar to the free molecule, the ordered quantum corrections can be used for T>(hv _{max}/12(2)^{½} k)_{mol}; T>(hv _{lattice}/2πk). The role of the gas imperfection and the molal volume of the condensed phase is discussed for equilibria between gaseous and condensed phases. It is shown that the difference in molal volumes of isotopic molecules is a second‐order effect. The difference in molal volumes of isotopic molecules is evaluated by an extension of Gruneisen's equation of state to molecular lattices. The results are in good agreement with available experimental data.

Competitive Reaction Rates of Hydrogen Atoms with HCl and Cl_{2}. Entropy Considerations of the HCl_{2} Transition State
View Description Hide DescriptionThe relative rates of the reactionsandwere determined in the temperature range of 0° to 62°C and found to be given byThe temperature‐independent factor in the above expression is interpreted in terms of the structure and the vibrational frequencies of the HCl_{2} transition state. The doubly degenerate bending frequency of this transition state is found to have a value of about 105 cm^{−1}.

Evidence of Trapped N Atoms in X‐Ray Irradiated NaN_{3}
View Description Hide DescriptionSodium azide irradiated with x rays at liquid‐nitrogen temperature displays a power sensitive ESRspectrum of three equal lines of separation 6.2 oe. This spectrum is interpreted as N^{14} trapped in the crystal lattice.

Nuclear Relaxation Processes of a Nonequivalent Two‐Spin System
View Description Hide DescriptionNuclear magnetic relaxation of a two‐spin system is discussed using Redfield's semiclassical formulation of the relaxation processes. Time dependences of longitudinal magnetizations are given as a function of the ratio J/δν_{0}. An expression for the nuclear induction signal is given, from which the expected shape of multiplet lines and their behaviors upon saturation under various origins of the relaxation are discussed in detail. The discussion leads to the conclusion that the mechanism of nuclear relaxation can be determined by carefully analyzing the spin multiplet lines on the basis of the theory. Finally, it is shown that there is a possibility of finding the absolute sign of the spin‐spin coupling constant in a two‐spin system using the multiple resonance method.

Calculation of Pressure Shifts of Optical Absorption Spectra from Solvent Data
View Description Hide DescriptionFrequency shifts in the absorption spectra of both polar and nonpolar absorbers have been observed at atmospheric pressure in a range of polar and nonpolar solvents and in pentane and ethanol under hydrostaticpressures up to 5000 atm. McRae's equation representing frequency shifts in terms of solventrefractive index and dielectric constant can be modified by the use of the Debye equation to represent frequency shifts in terms of solvent density. If it is assumed that the volume of the cavity occupied by the absorber remains constant with solvent density, frequency shifts resulting from an increase in hydrostaticpressure can be calculated almost to within the limit of experimental error from solvent shift data.

Microwave Spectrum of Isobutylene. Dipole Moment, Internal Barrier, Equilibrium Conformation, and Structure
View Description Hide DescriptionThe microwave spectra of isobutylene and isobutylene‐d‐3 have been studied in the region 17–36 kMc. Observed rotational constants (Mc) for isobutylene are a _{0}=9133.32, b _{0}=8381.75, c _{0}=4615.99; for sym‐isobutylene‐d‐3, a _{0}=9132.59, b _{0}=7788.98, c _{0}=4431.05; for asym‐isobutylene‐d‐3, a _{0}=8819.46, b _{0}=7981.08, c _{0}=4469.59. The most likely structure compatible with these data is r_{c–c} =1.50_{7} A, r_{c=c} =1.34 A, r _{CH}(methyl)=1.08_{5} A, r _{CH}(ethylenic)=1.08_{6} A, <Me—C—Me=115.9°, <HCH(methyl)=108°, <HCH(ethylenic)=117.5°. Measurements of the Stark effect show that the dipole moment μ=0.503±0.009 D. Fine structure leads to a barrier height hindering internal rotation of the methyl groups of 2.21 kcal/mole. The equilibrium conformation is C_{2v } with two methyl hydrogens in the plane of the carbon atoms and the CH bonds pointing away from the symmetry axis. It is concluded that although CC single bonds are shortened when adjacent to a double bond, the double bond length is probably not much affected by the presence of an adjacent CH_{3} group.

Half‐Crystal Energy Constants
View Description Hide DescriptionTables of the summation constants required for the calculation of surface energies of perfect half‐crystals with Lennard‐Jones types of inverse‐power pair potential functions have been calculated using an IBM 709 computer. The tables cover simple cubic, face‐centered cubic, body‐centered cubic, simple hexagonal, and hexagonal close‐packed structures with face plane orientations having Miller indices up to 5 in some cases. The inverse powers included in the tabulation extend from 5 (the lowest power for which these sums converge) to 13. The applications of these results to the calculation of grain boundary energy and spacing and to the estimation of surface distortion in half‐crystals according to methods previously developed by the author are pointed out. The energies of hcp and fcc close‐packed surfaces are compared as an example.

Scattering of High‐Velocity Neutral Particles. XI. Further Study of the He–He Potential
View Description Hide DescriptionResults are presented for two studies of the scattering by room‐temperature helium of beams of helium atoms with controlled energies in the approximate range 150–1500 ev. The interaction potential derived from the two sets of measurements is represented byWhere there are common ranges of the interaction distance r this potential is in reasonable agreement with He–He potentials previously determined from scattering experiments. In its specified range of validity, it is somewhat lower than most of the corresponding values calculated quantum mechanically.

Probability of Initial Ring Closure in the Restricted Random‐Walk Model of a Macromolecule
View Description Hide DescriptionThe probability of initial ring closure in the restricted random‐walk model of a macromolecule is investigated. From a study of the known exact numbers of polygons on the simple quadratic lattice up to 18 sides and on the triangular lattice up to 16 sides, it is concluded that the probability of initial ring closure in two‐dimensions of large ring size k varies inversely as k ^{1.83—θ}, where 0≤θ≤0.05, and this is significantly higher than the dependence on the inverse square of k found by Wall's statistical investigation. It is found that the mean area of initial ring closures in a plane varies as .

Statistical Mechanics of Monatomic Systems in an External Periodic Potential Field. I. Introduction, Virial Expansion, and Classical Second Virial Coefficient
View Description Hide DescriptionIn the first section a general research program is outlined which is concerned with the behavior of an equilibrium macroscopic system (gas, liquid, or solid) of monatomic molecules in an external periodic potential field. The problems of particular interest are: condensationtheory; crystalline state and theory of fusion; ideal degenerate gases; distribution functions; virial expansion for a gas; and the very dilute gas. In the second section the virial expansion is discussed and the classical second virial coefficient for an adsorbed monolayer on a crystal surface is calculated as an illustration. This computation is compared with results obtained from the two conventional cases which are limiting forms of the present treatment: mobile and localized adsorption. The differences are considerable.

Statistical Mechanics of Monatomic Systems in an External Periodic Potential Field. II. Distribution Function Theory for Fluids
View Description Hide DescriptionThe theory of distribution functions for monatomic systems in an external periodic potential field is outlined. Particular attention is given to the equation of state, the second virial coefficient, and the Kirkwood and Born‐Green‐Yvon integral equations for the distribution functions. A brief discussion of a nonperiodic external field is given in an Appendix.

Infrared Spectra of Methanol and Deuterated Methanols in Gas, Liquid, and Solid Phases
View Description Hide DescriptionThe infrared spectra of CH_{3}OH, CH_{3}OD, CD_{3}OH, and CD_{3}OD in the five phases gas, liquid, vitreous solid, α‐crystal, and (except perhaps for CD_{3}OH and CD_{3}OD for which the solid‐solidtransitions have not been studied) β‐crystal have been recorded in the range 4000 to 300 cm^{−1}. The Raman spectrum of liquid CD_{3}OH has been recorded. A complete assignment of the internal modes is given, which differs somewhat from previous assignments for the CH_{3} bending and rocking vibrations. No significant difference in spectrum occurred between the α‐crystal and β‐crystal phases. Under the full symmetry of the β‐phase determined by x‐ray diffraction only one OH out‐of‐plane bending band should occur. Two bands are observed, and it is concluded that the carbon and oxygen atoms in one chain are not coplanar, as is required by the symmetry determined by x‐ray diffraction [K. J. Tauer and W. N. Lipscomb, Acta Cryst. 5, 606 (1952)], but that the chains are puckered and the x‐ray symmetry arises because the puckered chains are irregularly distributed, a structure that had been previously suggested by Tauer and Lipscomb tentatively on the basis of high apparent thermal amplitudes. Bands occur in the crystal spectra near 500 cm^{−1} and 340 cm^{−1} at −180°C. These are interpreted as lattice modes, probably the two infrared‐active modes that involve translations of the molecules.

Difference of Intermolecular Potentials of CH_{3}OH and CH_{3}OD
View Description Hide DescriptionThe intermolecular potentials of CH_{3}OH and CH_{3}OD have been examined using the difference of the heats of vaporization at 0°K obtained from thermal data, and the vibrational frequencies obtained from the preceding paper. They do not differ within the accuracy of the measurements. The intermolecular potential due to the change of the intramolecular zero‐point energy is greater for CH_{3}OH than for CH_{3}OD, and so the other contributions to the intermolecular potential, viz., repulsive, dispersion, electrostatic, etc., energies, are greater by 459±∼260 joule mole^{−1} for the deuterium bond than for the hydrogen bond. This might be due mainly to a higher polarity of CH_{3}OD than of CH_{3}OH.

Condensation of Atomic and Molecular Hydrogen at Low Temperatures
View Description Hide DescriptionWith the use of reflected modulated atomic‐beam techniques, the reflection of hydrogen atoms and molecules at cold surfaces has been examined. It was found that at low temperatures (∼4°K) hydrogen atoms reflect as atoms with a very high probability. The reflection of hydrogen molecules is strongly dependent upon the amount of water vapor condensed on the target previously to or simultaneously with the H_{2}. The fact that condensation of stable gases can be affected by simultaneous condensation of water vapor at temperatures exceeding the gases' normal boiling points leads to an application in the form of an extraordinarily inexpensive way to produce vacuums in the 10^{−5} mm Hg range.

Second‐Order Effects in Low‐Field NMR for Ammonium Ion Solutions
View Description Hide DescriptionLow‐field free‐precession signals from aqueous ammonium nitrate solutions show a complex beat pattern containing the 52.55 cps first‐order spin‐spin splitting observed at high fields and also a 1.35 cps second‐order splitting when the proton precession frequency is 2137 cps. The T _{1} is slightly longer for the ammonium hydrogen than for the water hydrogen; T _{2} is shorter than T _{1} for the ammonium hydrogen and is shorter for the outer lines than for the central line.

Spin‐Lattice Relaxation Time of F^{19} Nuclei in Ag_{2}F
View Description Hide DescriptionThe spin‐lattice relaxation time of F^{19} nuclei in Ag_{2}F crystals has been measured at several temperatures. From the inverse proportionality constant of the relaxation time with respect to temperature in conjunction with the Korringa's theory, it is concluded that the relatively large paramagnetic shift observed is not due to the second‐order paramagnetism but to the hyperfine interaction between conduction electrons and F^{19} nuclei.

Application of a Theory of Nonideal Solutions to Results of Ultrasonic Absorption Measurements
View Description Hide DescriptionIt is shown that results of ultrasonic absorption measurements on solutions can sometimes be explained by application of a theory of nonideal solutions when the corresponding ideal‐solution theory is inadequate. The results of ultrasonic absorption in tertiary butanol solutions in cyclohexane reported recently by R. Musa and M. Eisner are studied using a theory of nonideal solutions. This differs from the corresponding ideal‐solution theory by the appearance of terms proportional to the derivatives of the ratio (k/l) of the relevant rate constants with respect to the concentration of the solute. In ideal solutions, these derivatives are zero. It is assumed that monomers and dimers are the only complexes present and it is concluded that a perturbation of the monomer‐dimer equilibrium is responsible for the measured ``excess absorption.'' The nonideality of the solutions is discussed, and a ``nonideality coefficient'' is introduced. A comparison with the imperfect gas theory of Tabuchi is made.

Mass Spectrometer Investigation of Ionization of N_{2}O by Electron Impact
View Description Hide DescriptionThe N_{2}O molecule has been studied with a nearly monoenergetic electron beam. A result of the positive ion data is a value for D(N_{2}–O) ≤ 1.34±0.2 ev and a value of D(N–NO) ≤ 4.50+0.10 ev. Measurements were made of the kinetic energies of the ions formed. N^{+} and N_{2} ^{+} are found to have zero kinetic energy at threshold. The production of an O^{−} ion is observed with an appearance potential in the range 0 to 0.05 ev.