Volume 41, Issue 4, 15 August 1964

Kinetics of Capillary Condensation in Wedge‐Shaped Pores
View Description Hide DescriptionThe kinetics of capillary condensation into an initially dry pore is treated as an unsteady diffusion problem with a moving boundary at which the vapor concentration satisfies the Kelvin equation. The (justifiable) approximation that the phenomenon is radial leads to important simplifications. An exact similarity solution is found which is valid for the early stages of the condensation process. An approximate method of analysis is developed, using the class of similarity solutions of the two‐dimensional radial diffusion problem with a constant concentration condition at an appropriately moving boundary. The method has some analogies to the Pohlhausen method in fluid mechanics in that the initial and boundary conditions are satisfied exactly, and the diffusionequation is satisfied in an integral sense. The method depends on numerical methods of integration, but approximate analytical solutions are available when the time is sufficiently large.
Numerical solutions are given for the capillary condensation of water. The characteristic equilibration time is highly sensitive to h _{0}, the relative saturation of the atmosphere to which the dry pore is exposed, varying approximately as (1—h _{0})^{—3}. The initial condensation rate is a constant, largely independent of h _{0} when this is close to one. The kinetics of capillary condensation is highly sensitive to temperature, equilibration times for water at 80°C being only 1/233 as great as at 0°C. The major influence of temperature on kinetics is due to the temperature dependence of the saturated vapor density.
A limitation to this approach is that the treatment is essentially macroscopic, although the early stages of condensation may involve relatively few molecules.

Crystal Structure of Potassium Fluoride Tetrahydrate
View Description Hide DescriptionThe crystal structure of KF·4H_{2}O has been determined by x‐ray diffraction using the phase determination method of Hauptman and Karle. It is monoclinic and consists of edge and corner sharing K^{+}(H_{2}O)_{6} and F^{—}(H_{2}O)_{6} octahedra. The similarity in the cation and anion coordination gives rise to pseudo‐orthorhombic symmetry. The K^{+}···O^{—} distances are 2.79±0.02 Å. The F···(H)O distances range from 2.71 to 2.78 Å. The O(H)···O distances which link the octahedra are 2.82 and 2.85 Å.

On the Ionic Hydration Polyhedra in the Crystal Structure of Sodium Hydroxide Tetrahydrate and the Hydrogen Bonding of the Hydroxide Ion
View Description Hide DescriptionThe crystal structure of NaOH·4H_{2}O is compared with that of KF·4H_{2}O, described in the preceding paper, in terms of the hydration polyhedra of the ions. The marked difference in structure corresponds with a difference in the hydrogen‐bond environment of the F^{—} and OH^{—} ions. The environment of the latter is consistent with the formation of five strong acceptor hydrogen bonds and a very weak donor hydrogen bond which completes an octahedral coordination. A similar environment is found in the structures of the mono‐ and heptahydrates.

Franck—Condon Factors for the Ionization of CO, NO, and O_{2}
View Description Hide DescriptionCalculated values of the vibrational overlap integrals for the following transitions are given:
These probabilities are used to construct theoretical ionization‐efficiency curves assuming an energy dependence for the electronic transition integral for both photoionization and ionization by electron impact. Previously published experimental data are discussed in terms of the theoretically calculated Franck—Condon probabilities.

Frictional Coefficient of Polymer Molecules in Solution
View Description Hide DescriptionThe coefficient k_{s} is calculated in the expression for the frictional coefficient f=f _{0}(1+k_{s}c+···), where f _{0} is the frictional coefficient at infinite dilution, and c is the concentration. Results are given for hard spheres and for interpenetrable spheres of uniform segment density. Difficulties caused by the long‐rangedness of hydrodynamic interactions are removed by taking the translational velocity of the spheres relative to the mean solvent velocity; f is computed after transformation to a laboratory frame. All orders of ``reflections'' are correctly summed within the approximation of slow spatial variation of the velocity perturbation arising from a sphere in the vicinity of another. The derived velocity agrees well with that obtained exactly by others for the special case of the line of centers of the two interacting spheres in the direction of the external force. For hard spheres, a value of 7.157 for k_{s} is found when c is the volume fraction of the spheres in the solution. For soft spheres where k_{s} is smaller, the dependence of k_{s} on the segment—segment interaction constant is obtained with the approximation that the binary clusters of overlapping spheres are treated as prolate ellipsoids of varying axial ratio. A substantial drop in k_{s} , but not quite to zero, at the theta temperature is predicted by this model. Also the dependence of k_{s} at the theta point on the polymer molecular weight is briefly discussed.

Assignment of Hyperfine Splittings in Electron Spin Resonance Spectra by Linewidth Analyses
View Description Hide DescriptionA method is developed for the assignment to nuclear positions of the hyperfine splittings in the electron spin resonance spectra of free radicals in solution. The procedure is applicable to the splittings from the nuclei of atoms with a nonvanishing pi‐electron spin density, and it is thus useful for ^{13}C and ^{14}N splittings, but cannot be employed for proton splittings. The technique used depends on linewidth effects. It is shown that one contribution to the linewidths is approximately proportional to the square of the spin density on the atom containing the nucleus of interest (the local spin density), and that, under appropriate conditions, splittings from nuclei at positions with high spin density show greater linewidth effects than those from positions with low spin density. The technique is applied to two of the ^{13}C splittings from nuclei present in natural abundance in the spectrum of the anthracene negative ion radical.

Line Shapes in Electron Spin Resonance Spectra
View Description Hide DescriptionThe theory of the linewidths in the electron spin resonance spectra of free radicals recently developed by Freed and Fraenkel predicts that, in general, composite lines arising from a set of degenerate nuclear spin states should not be Lorentzian in shape, and that the shapes of different lines in the same spectrum should be different. The earlier theory of Kivelson predicted that all the lines should be Lorentzian. To test the differences between the two theories, experimental studies of the line shape in the spectrum of the tetracyanoethylene anion have been made in a solvent consisting of a mixture of absolute ethanol and glycerine. Different lines were found to have different shapes, and the shapes were well represented by a sum of Lorentzian‐shaped components, in agreement with the newer theory. Studies of dimethylsulfoxide solutions of the p‐benzosemiquinone ion were also made. The spectra obtained in this system show only small linewidth variations and, in agreement with the theory, the variations in line shape are negligible. It was also possible to demonstrate that the nitrogen hyperfine splittinga^{N} in the tetracyanoethylene radical is positive.

Configuration‐Interaction Study of the Ground State of the H_{3} ^{+} Molecule
View Description Hide DescriptionA configuration‐interaction calculation on the ground state of H_{3} ^{+} has been done using Slater‐type orbitals of the form 1s, 2s, 2p_{x} , 2p_{y} , 2p_{z} to form the basis set. The calculation was effected by approximating the exact wavefunction for the system by a linear combination of configurations, each of which is a properly antisymmetrized function constructed from the basis orbitals, and by applying the variation theorem to the approximate wavefunction.
The most stable nuclear geometry for H_{3} ^{+} is an equilateral triangle, since a plot of total energy versus central bond angle shows a deep minimum at a bond angle of 60 deg.
The best energy that was obtained for the equilateral triangle was —1.33264 hartree at R=1.6575 bohr, using 12 basis orbitals and 12 symmetry‐adapted configurations. The experimental energy for H_{3} ^{+} is not known, but is estimated to be —1.35 hartree from these calculations.
The first excited singlet state, which is a doubly degenerate ^{1} E′ state, has been examined at many values of R for an equilateral triangle with three basis orbitals, and has been found to give a typical repulsive energy curve.
The fundamental vibrational frequencies for the ground state have been calculated, and it has been found that the symmetric stretch frequency is 3354±60 cm^{—1}, and the doubly degenerate bending frequency is 2790±100 cm^{—1}.

Analytical Self‐Consistent‐Field Functions for Ions Cr^{ n+}
View Description Hide DescriptionAnalytical SCF functions were calculated for ions Cr^{ n+} (n = 1, 2, 3, 4, 5). The cusp condition is identically satisfied. The nodal condition for the radial functions P _{ iλ}(r) is satisfied unusually accurately; an occasional deviation never exceeds 0.00006.

Nuclear Magnetic Resonance Study of Solid Perfluorocyclohexane
View Description Hide DescriptionA nuclear magnetic resonance second‐moment study of perfluorocyclohexane over the temperature range +25° to —130°C gave evidence for a rotational transition at about —95°C. This transition was indicated by a sharp decrease in the second moment from a value of 9.6 to 0.5 G^{2}. Theoretical second moments agreed favorably with the experimental rigid‐lattice value, if the chair form of the molecule was assumed. In addition, spin—lattice relaxation times were measured as a function of temperature. Again rotational motion in the solid was indicated by a sharp minimum in the T _{1} vs T curve at about —93°C.

Pulse Radiolysis Studies. IV. The Solvated Electron in the Aliphatic Alcohols
View Description Hide DescriptionSome of the physical and chemical properties of the solvated electron in the aliphatic alcohols have been investigated by means of the pulse radiolysis technique.
In ethanol the optical absorptionspectrum exhibits a maximum at 700 mμ. The decadic molar extinction coefficient at 5461 Å is determined to be ε_{ e —sol} ^{5461}=(1.1±0.4)×10^{4} M ^{—1} cm^{—1}. From this the oscillator strength of the band from 270 mμ to about 980 mμ is estimated to be f=0.7. The radiation chemical yield of the solvated electron in ethanol is found to be 1.0±0.3 per 100 eV of absorbed energy.
Absolute rate constants for a number of reactions of the solvated electron in ethanol and methanol have been determined. The diphenylide ion, formed by electron attachment to diphenyl in ethanol, shows a fast first‐order decay for which the rate constant has been determined.

Molecular Orbital Description of the Lowest Excited ^{1}Σ_{ g } ^{+} State of H_{2}
View Description Hide DescriptionFurther calculations on the lowest excited ^{1}Σ_{ g } ^{+} state of H_{2} are reported. A lower minimum energy is found with an excited orbital centered at the midpoint of the internuclear axis than with a two‐centered orbital, with coordinates measured from the two nuclei. The electron density distribution of the one‐centered orbital is also more in agreement with expectations.

Exciton Impurity Level and Its Implications to Radiation Chemistry
View Description Hide DescriptionDipole matrix elements corresponding to transitions to exciton states and other associated quantities are calculated for a simple one‐dimensional model of a molecular crystal containing an impurity. Emphasis is placed on transitions to certain exciton levels, characteristic of the presence of an impurity. Absorption of radiation energy by molecular aggregates containing an impurity is discussed in view of the results of the calculation.

Ring Inversion in Cyclohexyl Radical
View Description Hide DescriptionThe ESR spectrum of cyclohexyl radical in solid cyclohexane has been studied, during irradiation, between —85° and 0°C. Continuous electron irradiation is necessary because diffusion and recombination of the radicals occurs in this temperature interval. The broadening of certain of the lines as the temperature is raised has been interpreted in terms of an interconversion between two forms of the radical. The change of linewidth (or alternately, peak height) with temperature corresponds to an activation energy for the interconversion of 4.9±0.5 kcal/mole. The magnitude of the barrier, though smaller than that in cyclohexane itself, suggests that the interconversion is between the two chair forms of the radical. In the calculation of correlation times from the broadening, it was necessary to develop expressions which are applicable to spectra with Gaussian lines. This treatment is verified by the agreement of the data for C_{6}H_{11}· and C_{6}D_{11}·.

Centrifugal Distortion Effects and Structure of Hydrazoic Acid from the Millimeter Wave Rotational Spectra
View Description Hide DescriptionThe rotational spectra of two ^{15}N‐substituted species of hydrazoic acid have been investigated. The values of the rotational constants are: With the aid of the rotational constants for H^{14}N_{3} and D^{14}N_{3} the following structural parameters were obtained:An important part of this study was the investigation of the centrifugal stretching behavior exhibited by HN_{3} and its isotopic species. It has been found that even at low J the introduction of P ^{6} distortion terms for higher K _{—1} transitions is essential to accurately account for the observed rotational frequencies.

Generalized Pippard Equations
View Description Hide DescriptionThe Pippard equations for α, C _{P}, and β^{ T } near a λ‐point transition have been generalized in terms of stress—strain variables. The resulting equations, which interrelate α_{ i }, s_{ij} ^{T} , and C _{P} near a λ point, are tested with data on quartz near its α—β transition. A consequence of the new equations is the prediction that C′=(c _{11}–c _{12})/2 should be constant for cubic crystals near a λ point; this prediction is compared with experimental data on ammonium chloride.

Angular Dependence of Fluorescence from Chlorophyll a in Monolayers
View Description Hide DescriptionAngular distribution of fluorescence from monomolecular films containing chlorophyll a spread on water has been determined. The transition moment of the red fluorescence is found to make an angle ≤20° with the water surface, while the transition moment responsible for the blue absorption band is estimated to make an angle of ≃28° with it. Both angles are measured in the porphyrin plane.

Nuclear Magnetic Resonance Relaxation in Associated Liquids
View Description Hide DescriptionNMR spin—lattice and spin—spin relaxation times have been measured in the associated liquids 1,3‐butanediol, 2‐methyl‐2,4‐pentanediol, glycerol, and 1,2,6‐hexanetriol at resonant frequencies of 10 and 20 Mc/sec and over the temperature range —40° to +40°C using ``spin echo'' techniques. These data were translated into average correlation times using the Kubo—Tomita model of the NMRrelaxation process. By using a log Gaussian type of distribution it was possible in all four liquids to reconcile the average correlation times derived from both relaxation times and at both resonant frequencies. The rigid lattice limiting value of T _{2} and the variation of the distribution width with temperature were also determined.

Diffusion of Helium in Silicon, Germanium, and Diamond
View Description Hide DescriptionThe diffusion coefficients D of helium in silicon,germanium, and diamond were determined from the rate of evolution of helium upon heating boron‐doped crystals after slow neutronirradiation: The 90% confidence level for the activation energies is indicated. Experiments in which helium was introduced into silicon by ion bombardment from a discharge plasma gave values of D concordant with the above. The activation energies were calculated theoretically from the differences in interaction energies of the helium in the interstitial site and at the saddle point between two such sites. An exp‐6 potential was used with parameters estimated from inert‐gas interactions. The theoretical E values were C: 24.8, Si: 6.7, and Ge: 6.4 kcal. The pre‐exponential factors D _{0} as calculated from the vibration frequencies of the interstitial helium were about 10 times higher than the observed.

Electron Spin Resonance Absorption in Recrystallized Free Radicals at Low Fields
View Description Hide DescriptionThe width and shape of the ESR lines of 1,1‐diphenyl‐2‐picryl hydrazyl, picryl‐N‐amino carbazyl, and their recrystallized samples from the various solvents have been determined at different temperatures. It has been found that the ESR lines of recrystallized samples are wider than those of the original samples. The calculated linewidth on the basis of the Kubo and Tomita theory for DPPH agreed very well with the experimental value and hence this theory has been utilized to determine the asymptotic Curie temperature for all the other samples. The temperature variation of linewidth has been utilized to calculate the effective exchange frequency at different temperatures. An empirical formula has also been suggested for this type of variation.