Volume 31, Issue 3, 01 September 1959

Statistical Mechanical Theory of Transport Processes. XII. Dense Rigid Sphere Fluids
View Description Hide DescriptionThe theory of transport in a dense fluid of rigid spheres is developed from classical statistical mechanics by the use of phase space transformation functions. A modified Maxwell‐Boltzmann integro‐differential equation for the distribution function in μ space is derived, and the difference between this equation and the Enskog equation is discussed. To obtain a formulation of the stress tensor and heat flux solely in terms of binary collisions, it is necessary to allow the time of coarse graining to be very short. The implications of this are discussed with relation to the general principles of the statistical mechanics of transport. The viscosity and thermal conductivity of the dense rigid sphere fluid are calculated. The viscosity is the same as that first computed by Enskog, but the thermal conductivity differs from his calculations.

Statistical Mechanical Theory of Transport Processes. XIII. Kinetic Theory of Dense Rigid Sphere Fluids
View Description Hide DescriptionThis paper discusses the details of the calculation of the coefficients of shear viscosity and thermal conductivity for a moderately dense fluid of rigid spheres based on a modified Boltzmann equation previously derived. The physical significance of short time smoothing and its effect on the computed transport co‐efficients is briefly considered.

Elastic Scattering of Slow Ions in Gases
View Description Hide DescriptionDifferential cross sections for the elasticscattering of slow ions by neutral atoms or molecules have been calculated by numerical integration of the classical equations of motion. The model used for the ion‐neutral potential energy function consists of three terms: an r ^{—4} attraction, an r ^{—6} attraction, and an r ^{—12} repulsion. The attractive terms are theoretically sound, but the repulsive term is empirical. Quantum effects have been ignored, but an approximate formula indicates at what point the classical calculations begin to fail in any particular case. The results should be of most use in the very low energy region where experimentation is difficult.

Molecular and Crystal Structure of B_{8}Cl_{8}. I. Preliminary X‐Ray Diffraction Study
View Description Hide DescriptionUnit cell dimensions and space groups of two modifications of B_{8}Cl_{8} are described. The density measurements indicate that the formula hitherto represented as (BCl_{0.9})_{ x } is in best agreement with B_{8}Cl_{8}. Complete three‐dimensional x‐ray diffraction data of one of the modifications have been taken. The unit cell of this form is orthorhombic, of symmetry P2_{1}2_{1}2_{1} and four B_{8}Cl_{8} molecules in a unit cell of dimensions a=13.64, b=7.85, and c=12.91 A.

Molecular and Crystal Structure of B_{8}Cl_{8}. II. Solution of the Three‐Dimensional Structure
View Description Hide DescriptionWith only the information that B and Cl are present in the approximate ratio of 10 to 9, the molecular structure of B_{8}Cl_{8}, a new boron chloride, has been determined from single crystals by x‐ray diffraction methods. The chlorine positions, solved by a point‐by‐point, three‐dimensional Patterson superposition method, enabled the boron atoms to be located from electron density maps. The boron atoms form a dodecahedron with triangular faces and D _{2d } symmetry. One chlorine atom is attached to each boron atom at an average distance of 1.70 A. The shortest boron‐boron distance is 1.78 A. The over‐all agreement factor is .

Molecular Structure of B_{10}H_{12}(CH_{3}CN)_{2}
View Description Hide DescriptionAn x‐ray diffraction study of single crystals of B_{10}H_{12} (CH_{3}CN)_{2} has yielded the positions of all atoms, including hydrogen atoms. The structure is regarded as a substitution derivative of the B_{10}H_{14} ^{—2}structure having C _{2v } symmetry, two bridge hydrogens and two BH_{2} groups. A B–N single bond replaces a B–H of each BH_{2} group and the acetonitrile residue is linear with normal bond distances, except for C–CH_{3}=1.45 A in agreement with C–CH_{3}=1.46 in acetonitrile itself.
The unit cell is monoclinic, of symmetry I2/c, and has parameters a=7.81, b=11.31, c=14.18, and β=96°52′. Final values of R=0.15 and r=0.12 were obtained, as described in the text.

Nonresonant Microwave Absorption and Electric Dipole Moment of NO in the Gaseous State
View Description Hide DescriptionThe microwave absorption in NO has been measured by a resonant cavity method at frequencies near 9 and 23 kMc/sec and at pressures in the range 3 to 27 atmos. The major part of the absorption is due to direct Λ‐type doublet transitions, although at the higher pressures, a significant contribution also comes from the low frequency wings of the pure rotational spectrum. Since the Λ‐doublet separation is much smaller than the applied frequency for all significant transitions, the resulting spectrum has the nonresonant or Debye shape. From the intensity of this spectrum, the electric dipole moment was found to be 0.148±0.002 debye. The effective collision cross section for random spacial reorientation of the molecules is approximately twice the kinetic collision cross section.

Kinetics of Coordinate Bond Formation. III. Method of Temperature Patterns and the Reaction NH_{3}+BF_{3}
View Description Hide DescriptionThe method for determining rate constants from the temperature pattern in dilute spherical diffusionflames is critically examined, and several possible sources of error are corrected for. The method is tested by a re‐examination of the reaction NH_{3}+BF_{3} over a wide range of experimental parameters. On the whole, previous results are confirmed, but the high‐pressure limiting rate is found to be 1.2×10^{11} cm^{3}/mole‐sec, somewhat less than half of the former value, and some new quantities can be measured. Ammonia is found to be about 3 times as effective as nitrogen in energy‐exchange collisions with the excited product of the initial addition reaction. In the absence of N_{2}, the rate constant is ½ the high‐pressure limit when the NH_{3} concentration is about 8×10^{—8} mole/cm^{3}. The heat of the gas‐phase addition reaction is 27.5±2 kcal/mole and the equilibrium constant for the reverse reaction is ≪2×10^{—10} mole/cm^{3}. The activation energy, if any, is less than 3 kcal/mole.

Microwave Spectrum and Structure of Perrhenyl Fluoride
View Description Hide DescriptionRotational absorption lines of the symmetric‐top molecule ReO_{3}F have been observed in the K‐band regions of 21 400 and 28 500 Mc, with a Stark modulation spectrometer designed specially to accommodate this and other chemically reactive molecules The spectrum exhibits a hyperfine structure arising from interaction of the Re nuclear electric quadrupole moment with the surrounding charge. The hfs pattern establishes the Re nuclear spin as 5/2 and yields a quadrupole coupling constant eqQ (Re^{187}) = (—48.4±1.3) Mc for the ground vibration state. Additional groups of lines associated with molecular states of vibrational excitation were also present. Of the six normal vibration frequencies, three were sufficiently low to contribute an observable fraction of molecules to their excited states: one non‐degenerate, ν_{3} (a _{1}), and two doubly degenerate, ν_{5} (e) and ν_{6} (e). The corresponding rotation‐vibration and l type doubling constants were measured to be α_{3} = (+12.30±0.06) Mc, α_{5} = (—10.91±0.06) Mc, ql _{5} = (16.31±0.25) Mc, α_{6} = (+2.52±0.06) Mc, ql _{6} = (5.00±0.24) Mc. Strong dependence of quadrupole coupling on molecular vibration was observed, the values of eqQ (Re^{187}) being —27, —35, and —58 Mc respectively in the singly‐excited states ν_{3}, ν_{5}, and ν_{6}. Comparative measurements of line intensities relative to those of the ground vibration state yield the following approximate values for the three lowest normal vibration frequencies: ν_{3} = (325±40) cm^{—1}, ν_{5} = (410±25) cm^{—1}, ν_{6} = (345±40) cm^{—1}. Measurements of the Stark effect in ReO_{3}F give as the dipole moment of this molecule a value μ = (0.85±0.05) debye.
Precise determination of molecular structure was effected from frequency measurements of lines belonging to the slightly asymmetric top ReO_{2} ^{16}O^{18}F. The results give Re–O = (1.692±0.003) A, Re–F = (1.859±0.008) A, F–Re–O = 109° 31′±16′. Comparison is made between these and the known values of two other structurally similar molecules, ReO_{3}Cl and MnO_{3}F.

Quadrupole Coupling and Bond Structure in ReO_{3}F, ReO_{3}Cl, and MnO_{3}F
View Description Hide DescriptionA theoretical explanation is offered to account for the magnitude and sign of the quadrupole coupling constants in ReO_{3}F, ReO_{3}Cl, and MnO_{3}F. Subject to the requirement that the valence electron bonding orbitals of Re and Mn have a tetrahedral configuration, a combination of s, p, and d atomic orbitals is chosen which tends to minimize the energies of bond formation. Results of the analysis reveal that, in order to account for the observed quadrupole couplings, the effect of atom core polarization must be considered. It is shown in particular that the contribution to the electric field gradient at the Re nucleus from the polarized electron core is more than 25 times that of the external polarizing ions themselves.
Comparison of the quadrupole coupling in ReO_{3}F with that of atomic Re shows that the amount of unbalanced p electron along the molecular symmetry axis is nearly zero and thus provides a reasonable explanation for the pronounced dependence of eqQ(Re) on the state of molecular vibration.

Thermodynamic and Spectroscopic Study of Pyrrolidine. I. Thermodynamic Properties in the Solid, Liquid, and Vapor States
View Description Hide DescriptionThe low‐temperature heat capacity, heat of combustion, heat of vaporization, and vapor pressure of pyrrolidine have been measured. Information obtained from these measurements is as follows: the heat capacity of the solid from 14 to 215°K; the liquidheat capacity from 215 to 312°K; a heat of transition of 127 cal mole^{—1} at 207.14°K; a heat of fusion of 2053 cal mole^{—1} at the melting point, 215.31°K; the entropy of the liquid at 298.15°K, 48.78 cal mole^{—1} deg^{—1}; the standard heat of formation from the elements at 298.15°K, —9.81 kcal mole^{—1}; the heat of vaporization at 298.15°K, 8990 cal mole^{—1}; and the vapor pressure, log_{10} P(mm) = 6.88816 — 1157.40/(t+202.332). From a comparison of the calorimetric entropy of the ideal gas with statistically calculated values based upon newly obtained spectroscopic data, pyrrolidine vapor appears to have the indefinitely puckered cyclopentane type structure. Thermodynamic properties of the solid,liquid, and ideal gas have been tabulated at selected temperatures.

Thermodynamic and Spectroscopic Study of Pyrrolidine. II. Vibrational Spectra and Configuration
View Description Hide DescriptionThe Raman spectrum of pyrrolidine liquid at several temperatures and the infrared spectra of pyrrolidine in the vapor, liquid, and solid states, in the 320–3500 cm^{—1} region and over the temperature range +150°C to —120°C have been recorded. Similar but less extensive studies were made on pyrrolidine N–D. Evidence for the presence of several configurational isomers in the liquid and solid states was obtained. A vibrational assignment, complete except for one degree of freedom, has been made and used to calculate the entropy of the ideal vapor. Comparison with the measured entropy has shown that pyrrolidine in the vapor is a free or an almost free pseudorotator.

Oscillator Strengths in the Far Ultraviolet. I. Nitric Oxide
View Description Hide DescriptionAbsolute integrated absorption intensities have been measured for the nitric oxide beta, gamma, delta, and epsilon bands in the range 1700 to 2300 A. From these data the electronic absorption oscillator strengths of the beta and gamma band systems have been calculated to be 0.00151±7% and 0.00240±5%, respectively. The experimental method involved the photometric detection of pressure broadened nitric oxide spectra. Also described is the construction of a logarithmic load resistance which allows the direct tracing of spectra in absorbance units.

Oscillator Strengths in the Far Ultraviolet. II. Oxygen Schumann‐Runge Bands
View Description Hide DescriptionAbsolute integrated absorption intensities have been measured for the oxygen Schumann‐Runge band system. From these data the total band system absorption oscillator strength was found to be between 0.00027 and 0.00031 while from the band strengths the electronic absorption oscillator strength was calculated to be 0.163±10%. The experimental method involved the photometric detection of pressure‐broadened oxygen spectra from 1750 to 2000 A.

Forbidden Band Systems in Nitrogen. II. The System in Absorption
View Description Hide DescriptionFour bands of the system in nitrogen have been observed in absorption at high resolution in a path of 3.4 meteratmospheres, at 1444 A (1–0), 1414 A (2–0), 1358 A (4–0), and 1331 A (5–0). They consist of Q branches only (ΔJ=0). The observation of these bands, highly forbidden by the usual selection rules, is explainable by perturbations with the nearest ^{1}II_{ u } states lying between about 98 000 cm^{—1} and 109 000 cm^{—1} above the ground state.
The rotational analysis of each of the bands resulted in rotational constants and vibrational intervals for a′^{1}Σ_{ u } ^{—} which agree very well with those obtained from the emission system . The electronic energy (T _{00}) of state a′^{1}Σ_{ u } ^{—} is now precisely located at 67738.18 cm^{—1} (8.398 ev); this value combined with other data gives also the following electronic energies (T _{00}): w ^{1}Δ_{ u }, 71697.68 cm^{—1} (8.889 ev); x ^{1}Σ_{ g } ^{—}, 113210.98 cm^{—1} (14.036 ev); and y ^{1}II_{ g }, 114165.18 cm^{—1} (14.154 ev). T _{00} of a′^{1}Σ_{ u } ^{—} is close to that of a ^{1}II_{ g }: 68951.21 cm^{—1} (8.548 ev).
The new energy for y ^{1}II_{ g } is 0.339 ev lower than the minimum value deemed possible by Lofthus and Mulliken on the basis of an observed predissociation at v=0, J=10 of state y. That predissociation is now reinterpreted as most probably an accidental one resulting in the formation of ^{2}P+^{4}S atoms instead of ^{2}D+^{2}D atoms as formerly assumed.
The intensity distribution in the system is found to be approximately proportional to J(J+1)(2J+1) exp—[B _{0}″J(J+1)hc/kT], where B _{0}″=1.9898 cm^{—1} and T=300°K, in agreement with theory.
From appearance pressures of the five forbidden systems of nitrogen now known in absorption, it is possible to estimate the transition probabilities. From these, together with a recent molecular beam determination of the mean lifetime of the state a ^{1}II_{ g }, it is possible to estimate the lifetimes of the metastable states: A ^{3}Σ_{ u } ^{+}, 2.6×10^{—2} sec, a′^{1}Σ_{ u } ^{—}, 4.0×10^{—2} sec. By comparison of the transition probability of the atmospheric oxygen bands with that of the Vegard‐Kaplan bands of nitrogen , together with the known lifetime of b ^{1}Σ_{ g } ^{+}, the value 6.5×10^{—3} sec is found for the lifetime of A ^{3}Σ_{ u } ^{+}, in fair agreement with the value given above.

Angular Dissymmetry of the Critical Opalescence in Liquid Mixtures
View Description Hide DescriptionVisible lightscattered by a one‐component homogeneous liquid in the vicinity of its critical point or by a homogeneous mixture of two liquids in the vicinity of their critical mixing temperature shows angular dissymmetry which becomes the more pronounced the nearer the temperature comes to the critical temperature. It is shown how this dissymmetry can be used as a measure for the range of molecular forces in the case of ordinary molecules. In the case of polymer molecules, observation of the dissymmetry provides a method for measuring the size of polymer coils in a size range too small to be measured by the usual dissymmetry method applied to diluted solutions.

Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid
View Description Hide DescriptionBy finding the saddle point in the expression derived in Paper I (see reference 8) for the free energy of a nonuniform system, we have derived the properties of a critical nucleus in a two‐component metastable fluid.
At very low supersaturations, we find that the properties of the nucleus approach those predicted by the classical theory that assumes the nucleus to be homogeneous with an interfacial energy that does not vary with curvature. However, with increasing supersaturation, the following changes occur in the properties of the critical nucleus. (a) The work required for its formation becomes progressively less than that given by the classical theory, and approaches continuously to zero at the spinodal. (b) The interface with the exterior phase becomes more diffuse until eventually no part of the nucleus is even approximately homogeneous. (c) The composition at the center of the nucleus approaches that of the exterior phase. (d) The radius and excess concentration in the nucleus at first decrease, then pass through a minimum and become infinite again at the spinodal.
These properties are deduced without resort to any specific solution model. In addition, they are evaluated for a regular solution to permit a quantitative comparison with the predictions of previous treatments.

Absorption Spectrum of NiCl_{2} in Molten LiCl/KCl
View Description Hide DescriptionThe absorption maxima in the spectra of NiCl_{2} in molten LiCl/KCl are discussed from the point of view of the ligand fieldtheory. It is concluded that the nickel ion is surrounded by four chloride ions in a nearly tetrahedral arrangement. Variations in the intensity of one of the bands gives a sensitive indication of the degree of distortion of this structure.

Rapid Procedure for Rigorous Analysis of Electron Diffraction Data
View Description Hide DescriptionA refined procedure for the analysis of electron diffraction patterns of gas molecules has been developed and programed for rapid computation by the IBM 650 digital computer. In addition to increasing greatly the speed and accuracy of data processing, it reduces to routine practicability the most rigorous methods of structural analysis presently available. Improved methods for handling standard electron diffraction computations and several new schemes facilitating molecular structure determinations are given including an objective determination of parameters by the method of steepest ascents. A simplified means of estimating standard errors is developed.

Absorption Spectrum of Manganous Fluoride
View Description Hide DescriptionThe absorptionspectrum of single crystal MnF_{2} has been measured in the range 2700–6000 A. In the room temperature axial spectrum peaks are observed which are identified with transitions from the ^{6} A _{1g }(^{6} S) ground level to various quartet levels. The wave numbers in units of 10^{3} cm^{—1} corresponding to transitions to the various levels are: ^{4} T _{1g }(^{4} G), 19.44; ^{4} T _{2g }(^{4} G), 23.5; ^{4} A _{1g }(^{4} G), 25.19; ^{4} E_{g} (^{4} G), 25.50; ^{4} T _{2g }(^{4} D), 28.12 and 28.37; ^{4} E_{g} (^{4} D), 30.23; ^{4} T _{1g }(^{4} P), 33.06. The spectrum is interpreted in terms of a perturbation of the free ion energy levels by an octahedral ligand field including the diminution of the magnitude of the d _{γ} orbitals on the Mn^{++} ion by a factor (1 — ε)^{½} in calculating the coulomb and exchange integrals. ε is the ``covalency parameter'' of Koide and Pryce. Energy matrices in the strong field representation and the free‐ion representation are calculated. The spectra are fitted with ε = 0.064 and the octahedral ligand field parameter Dq = 0.78×10^{3} cm^{—1}. From measurements of the π and σ spectra with polarized light it is concluded that the transitions are electric dipole ones. The spectra were also photographed with a grating spectrograph at temperatures down to 20°K. At 20°K sharp lines are seen corresponding to the transitions to the ^{4} A _{1g }(^{4} G), ^{4} T _{2g }(^{4} D), and ^{4} E_{g} (^{4} D) levels. A shift of about 0.06×10^{3} cm^{—1} in the transition to the ^{4} A _{1g } level between 77° and 20°K is in agreement with that expected from the effect on the energy levels of the antiferromagnetic ordering energy.