Volume 59, Issue 8, 15 October 1973

Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I Theory
View Description Hide DescriptionA method has been developed to investigate the sensitivity of the solutions of large sets of coupled nonlinear rate equations to uncertainties in the rate coefficients. This method is based on varying all the rate coefficients simultaneously through the introduction of a parameter in such a way that the output concentrations become periodic functions of this parameter at any given time t. The concentrations of the chemical species are then Fourier analyzed at time t. We show via an application of Weyl's ergodic theorem that a subset of the Fourier coefficients is related to 〈∂c_{i} /∂k_{l} 〉, the rate of change of the concentration of species i with respect to the rate constant for reactionl averaged over the uncertainties of all the other rate coefficients. Thus a large Fourier coefficient corresponds to a large sensitivity, and a small Fourier coefficient corresponds to a small sensitivity. The amount of numerical integration required to calculate these Fourier coefficients is considerably less than that required in tests of sensitivity where one varies one rate coefficient at a time, while holding all others fixed. The Fourier method developed in this paper is not limited to chemical rate equations, but can be applied to the study of the sensitivity of any large system of coupled, nonlinear differential equations with respect to the uncertainties in the modeling parameters.

Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. II Applications
View Description Hide DescriptionThe Fourier amplitude method developed in Paper I as a diagnostic tool for determining the sensitivity of the results of complex calculations to the parameters which enter these calculations has been applied to two chemical reaction systems involving sets of coupled, nonlinear rate equations. These were: (a) a five reaction set describing the high temperature (6000 °K) dissociation of air and (b) a nine reaction set describing the constant temperature (2000 °K) combustion of H_{2} and O_{2}. We have evaluated the Fourier amplitudes for all the species at a number of different times for both reaction systems. The analysis of these results verifies the claims made in Paper I. The relative magnitudes of the Fourier amplitudes showed a several order of magnitude distribution which permitted a clear distinction of the relative sensitivity of the species concentration to uncertainties in the rate coefficients. The conclusions based on the Fourier amplitude method for these two reaction systems are in excellent agreement with sensitivity predictions which could be made on the basis of previous kinetic studies of these systems.

Molecular collisions. XX. First order generalized phase shift approximation for symmetric top molecules
View Description Hide DescriptionThe development of a generalized phase shift approach to the theory of inelastic scattering of molecules of an arbitrary type is begun and carried out in detail for nonvibrating symmetric top molecules. The latter results are compared with the sudden approximation and the semiclassical limit of the restricted distorted wave approximation.

Hydrogen bond studies. 77. Electron density distribution in α‐glycine: X‐N difference Fourier synthesis vs ab initio calculations
View Description Hide DescriptionAccurate x‐ray and neutron diffraction data have been combined to study the electron density distribution in the simplest amino acid, α‐glycine. Ab initio electron density calculations have also been made using an MO‐LCAO SCF approach employing a Gaussian basis set of ``double‐ζ'' type. Highly satisfactory qualitative agreement is found.

Critical phenomena in fluids. The Percus‐Yevick radial distribution functions
View Description Hide DescriptionThe Percus‐Yevick theory is studied in the neighborhood of the critical point for the Lennard‐Jones 6:12 potential. It is shown that near the critical point the PY radial distribution functionsg(r) display a functional form different from that at other temperatures and densities. The asymptotic behavior and the behavior of the second peak of g(r) are particularly striking.

Magnetic moment of ^{181}Ta
View Description Hide DescriptionThe nuclear magnetic moment of ^{181}Ta is measured by NMR of a tantalum solution. For tantalum dissolved in HF/HNO_{3} broad lines arising from ^{181}Ta nuclei in complexes are observed at room temperature. A study of linewidth vs temperature shows motional narrowing of the quadrupole broadened line with increasing temperature. Applying the quadrupolar correction, we obtain a nuclear gyromagnetic ratio for ^{181}Ta of 2π(0.5098 ± 0.0002) × 10^{3} sec^{−1} G^{−1} and an uncorrected nuclear moment of 2.342 ± 0.001 nm. The chemical shift correction gives 2π(0.5097 ± 0.0002) × 10^{3} sec^{−1} G^{−1} for the nuclear gyromagnetic ratio. Correcting the nuclear moment for core diamagnetism, we get 2.361 ± 0.010 nm.

Absolute measurements of total cross sections for electron scattering by Na_{2} and K_{2} (0.5–50 eV)
View Description Hide DescriptionAbsolute measurements have been made of total cross sections for the scattering of electrons by sodium and potassium diatomic molecules, over the electron energy range 0.5–50 eV, at temperatures of 405 and 335 °C, respectively, using the molecular beam recoil technique. We find the molecular cross sections to lie about 60% and 33% above the respective sodium and potassium atomic cross sections above 2 eV. The measuredsodium and potassium molecular total cross sections range from 305 and 430 Å^{2}, respectively, at 0.5 eV and 85 and 97 Å^{2}, respectively, at 50 eV. The over‐all uncertainty in the sodium dimer results is at most 16%, and in the potassium dimer results is at most 20%.

EPR spectra of Cu^{2+} in single crystals of chabazite
View Description Hide DescriptionAn EPR study of Cu^{2+} ions in chabazite single crystals has revealed that the copper is presented in three magnetically inequivalent sites. Copper ions at two of these sites differ only in symmetry axes which are 7° apart. This is believed to be a result of twinning or strain in the crystals. These ions are characterized by g _{∥} = 2.324, g _{⊥} = 2.068, A _{∥} = 0.0159 cm^{−1}, A _{⊥} = 0.0018 cm^{−1} and Q = 0.0009 cm^{−1}. They are thought to be in the site 2 position, which is located near the center of the six‐membered rings. The symmetry axis of the ions essentially coincides with the symmetry axis of the crystal. Copper ions in the third type of site are less abundant and are in a more isotropic environment which suggests that they are located at the site 1 position in the center of the distorted hexagonal prisms.

Ionic solution theory for nonideal solvents
View Description Hide DescriptionA statistical‐mechanical study is made of the deviations from the primitive model (PM) of an ionic solution. In the PM, the solvent is treated as an ideal continuum that manifests itself only through the appearance of a dielectric constant ε ≠ 1. This paper begins with the introduction of a simple model that complements the PM, the NCDM (no charge‐dipole model) in which the granularity of the solvent is treated exactly. A second more realistic model, the EISIM (exact ion‐solvent interaction model), is then defined and discussed, as is the limit in which the PM approximation becomes an exact result. Like the PM the EISIM can be used as an ``ideal‐solvent'' approximation in which the solvent‐solvent interaction is idealized so that it manifests itself only through the appearance of ε. However, in the EISIM the solute‐solvent interaction is treated exactly. A cluster expansion for , the n ‐point potential of mean force, is derived and a thermodynamic perturbation theory for the problem is developed; an alternative treatment is also noted in which the mean‐spherical approximation can be used to bypass the McMillan‐Mayer theory and the attendant necessity for knowing , n > 2. The connection between the expressions herein derived and the phenomenologically developed Gurney‐Frank picture is pointed out; one term in the EISIM is shown to be identifiable with the solute‐solvent contribution to the Gurney cosphere‐intersection potential. Part of that term can be further isolated and identified as the exclusion‐volume contribution to the cosphere‐intersection potential.

Simple bond‐charge model for bending force constants of linear symmetric triatomic molecules
View Description Hide DescriptionA bond‐charge model is used to derive a simple universal relation, K _{θθ} = 0.026K _{11}, between the bond stretching force constant K _{11} and bond bending force constant K _{θθ} for linear symmetric triatomic molecules. The relation is shown to be reasonably accurate for 17 molecules.

Solvent effects on the coupling processes of polystyryl radicals
View Description Hide DescriptionThe molecular weight distributions of polystyrene prepared in various solvents at 60 °C showed broadening with increasing solventviscosity, whereas the effects of solvent power were not conspicuous. Simulations of polymerization reactions with several models showed that the coupling probability of polystyryl radicals is not independent of chain length at 60 °C and that the chain length dependence becomes apparent with increasing solventviscosity. These results were discussed in terms of the dynamical behavior of polymer radicals in solution and it was tentatively concluded that the chain length dependence of coupling probability of polystyryl radicals originates mainly from the hydrodynamic interactions between polymer segments.

Molecular rotational Zeeman effect in HOF, a comparison with H_{2}O, F_{2}O, and other fluorine containing molecules; and dipole moments of HOF and DOF
View Description Hide DescriptionThe linear and quadratic rotational Zeeman effect in HOF has been observed with a high magnetic field. The magnetic susceptibilityanisotropies are 2χ_{ aa } −χ_{ bb } −χ_{ cc } = (−19.6±0.6) × 10^{−6} erg/G ^{2}·mole and 2χ_{ bb } −χ_{ aa } −χ_{ cc } = 12.8±1.2 (the a axis is nearly along the OF bond and the c axis is perpendicular to the molecular plane); the molecular g values are g_{aa} = 0.642±0.001, g_{bb} = −0.119±0.001, and g_{cc} = −0.061±0.001 and the molecular quadrupole moments are Q_{aa} = (0.2±0.4)×10^{−26} esu · cm^{2}, Q_{bb} = 1.9±0.8, and Q_{cc} = −2.1±1.1. These results are compared with H_{2}O and F_{2}O. The effect of in‐plane fluorine substitution on the out‐of‐plane minus the average in‐plane magnetic susceptibilityanisotropy, Δχ, is discussed for a number of molecules and additive values are obtained for each successive fluorine substitution for planar molecules. These contributions to Δχ are first fluorine = +4.5±1.2, second F = +2.5±1.1, and third F = +0.6±0.8. These trends are explained in terms of changes in hybridization of the atom to which the fluorine is bonded. These values are used with results on mono‐ and difluorobenzene to estimate Δχ for benzene and trifluorobenzene. The electric dipole moments were obtained by the Stark effect and are HOF: μ_{ a } = 0.37±0.10 and μ_{ b } = 2.20±0.10. The total electric dipole moment of HOF is μ = 2.23±0.10 D. The electronic structure of HOF is discussed in light of these results.

ESR spectrum and structure of the n ‐propyl radical
View Description Hide DescriptionWe have observed the electron spin resonance(ESR) spectrum of the n ‐propyl radical in argon at 4 °K, following its production by the photolysis at 2537 Å of n ‐propyl iodide in the argon matrix. The spectrum is much better resolved than previously observed spectra of this radical, and thus could be analyzed in detail to obtain the following complete set of magnetic constants for the radical: g _{1} = 2.00290, g _{2} = 2.00302, g _{3} = 2.00249; A _{1} ^{(α)} = −28.7. A _{2} ^{(α)} = −17.8, A _{3} ^{(α)} = −21.0 Oe; and A _{1} ^{(β)} = 37.7, A _{2} ^{(β)} = 33.3, A _{3} ^{(β)} = 33.9 Oe. Axis 1 is the C_{β}C_{α} bond, and the plane determined by axes 1 and 2 approximately bisects the β‐CH_{2} bond angle. Comparison of the observed hyperfine structure (hfs) constants with calculated values shows that the radical is rapidly reorienting between approximately equivalent conformations in which the α‐CH_{2} plane makes angles of approximately ± 30 ° with the 1–2 plane of the principal axes of the magnetic tensors. However, a small triplet splitting (5 Oe) of the center line of the spectrum indicates that the β protons, and thus the aforementioned conformations of the radical, are not exactly equivalent. The outer lines of this triplet can be accounted for if certain radicals are reorienting either between a pair of conformations in which the α‐CH_{2} plane makes angles of 35 ° and − 15 ° with the 1–2 plane (state I) or between the pair of conformations with angles of 15 ° and − 35° (state II), which pairs of conformations are equivalent except for an interchange of the β‐proton hfs constants. The center line of the triplet can be explained if either some but not all of the radicals occupy matrix sites which permit rapid reorientation between states I and II, or if there is tunnelling between states I and II at a frequency of the same order of magnitude as the difference between the β‐proton hfs constants in a single state. As a sidelight to this study, we observed the ESR spectrum of the ethyl radical. The resulting hfs constants for ethyl are in good agreement with those obtained by McDowell et al., and, in addition, we obtained the following components of the g factortensor:g _{∥} = 2.00268, and g _{⊥} = 2.00273, where the parallel axis is the CC bond.

Absolute rate constant for the O + NO chemiluminescence in the near infrared
View Description Hide DescriptionInfrared chemiluminescence from the process O + NO (+ M) → NO_{2} + h ν (+ M) has been studied between 1.3 and 4.1 μm. The wavelength dependence of the continuum between 1.3 and 3.3 μm is in fair agreement with previous studies and the measured radiative rate constant at 1.51 μm, , of (2.4 ± 0.8) × 10^{−17} cm^{3} sec^{−1} · μm^{−1} establishes the NO–O glow in this spectral range as a secondary emission standard. Comparison with previous studies of the visible region of the glow implies that the over‐all radiative rate constant I ^{o} lies in the range (9.4–11.2) × 10^{−17} cm^{3} sec^{−1}. In the region 3.3–4.1 μm, the previously observed broad band, peaking at 3.7 μm, shows a complex kinetic dependence on [O] and [M].

Radiative lifetimes and pressure dependence of the relaxation rates of some vibronic levels in N_{2} ^{+}, N_{2}, CO^{+}, and CO
View Description Hide DescriptionRadiative lifetimes of the B^{2}Σ_{ u } ^{+}(ν′ = 0) level in N_{2} ^{+}, the C ^{3}π_{ u } (ν′ = 0, 1, 2) levels in N_{2}, the B ^{1}Σ^{+}(ν′ = 0) and C ^{1}Σ^{+}(ν′ = 0) levels in CO, and the B ^{2}Σ^{+}(ν′ = 0, 1) levels in CO^{+} have been measured to be 60.4 ± 0.4, 40.4 ± 0.5, 40.6 ± 0.5, 38.5 ± 0.6, 23 ± 3, 2.2 ± 0.8, 53.6 ± 0.7, and 64.1 ± 0.8 nsec, respectively. The measurements were made by studying the time distribution of photons emitted in the decay of these levels after collisional excitation by a pulsed‐proton beam. The apparent lifetime of the levels was studied as a function of target pressure over a range 1–200 mtorr. Collisional quenching and resonance trapping were observed in varying degrees. The radiative lifetimes are derived from the extrapolated zero pressure relaxation rates, and ν′ ‐ progression transition probabilities are determined by using our measured B‐A and C‐A electronic branching ratios in CO, and the CO^{+} band strength measurements of Judge and Lee.

Observation of strong nonlinearities in the high field Zeeman spectrum of NO at 1876 cm^{−1}
View Description Hide DescriptionStrong nonlinearities have been observed in the Zeeman effect in NO and quantitatively explained by taking account of off‐diagonal Zeeman interaction terms in an accurate Hamiltonian diagonalization calculation. Zeeman spectra were obtained by tuning the Q(3/2)_{3/2} and Q(5/2)_{3/2} transitions of the fundamental band (1876 cm^{−1}) through the P(15)_{9,8} laser line of CO.

Exact solution of the mean spherical model for simple polar mixtures
View Description Hide DescriptionWertheim's solution of the mean spherical model (MSM) for pure fluids composed of hard spheres with embedded dipoles [J. Chem. Phys. 55, 4291 (1971)] is extended to multicomponent polar fluid mixtures. The components are restricted to have equal hard sphere radii but may have different dipole moments. The anisotropic part of the pair correlation functions for an m‐component fluid characterized by hard sphere diameter d, temperature parameter β, dipole moments μ_{1}, μ_{2}, ···,μ_{ m }, and densities ρ_{1}, ρ_{2}, ···,ρ_{ m } are shown to be expressable in terms of the corresponding functions for an effective pure MSM polar fluid with the same hard sphere radius and the same temperature parameter but with an effective dipole moment and an effective density . The excess thermodynamic properties of the mixture (relative to a pure hard sphere Percus‐Yevick fluid) are shown to be those of the effective pure fluid. The dielectric constant of the mixture is also that of the effective pure fluid. The case of polar‐nonpolar mixtures is considered by allowing one or more of the dipole moments to vanish. It is found that the potential of mean force and spatial distribution of the nonpolar molecules is independent of the magnitude of the dipoles while the anisotropiccorrelations between the polar molecules are independent of the presence of the nonpolar species.

Electron spin resonance of transition metal ions in calcium tartrate tetrahydrate host lattice I. Mn^{2+}, Cu^{2+}, and Ni^{2+} as dopants
View Description Hide DescriptionThe electron spin resonance of Mn^{2+}, Cu^{2+}, and x‐irradiated Ni^{2+}dopedcalcium tartrate tetrahydrate single crystals was studied. This and earlier results were discussed together in the light of the known crystal structure of the host lattice. The nickel center was interpreted as the d ^{7}Ni^{3+} ion. A possible doping mechanism for transition metal ions in calcium tartrate tetrahydrate was suggested.

Electron spin resonance of transition metal ions in calcium tartrate tetrahydrate. II. VO^{2+} as dopant
View Description Hide DescriptionAs a continuation of a previous work, the ESR of VO^{2+}dopedcalcium tartrate tetrahydrate was examined. Three vanadium centers were observed of which detailed analysis of two were carried out. It was concluded that, like Mn^{2+}, Cu^{2+}, and Ni^{2+}, VO^{2+} entered the lattice in interstitial sites.

Investigations of a nonrandom numerical method for multidimensional integration
View Description Hide DescriptionA numerical integration technique based upon the use of nonrandom number sequences is examined with test integrations of a simple, analytical function. A comparison of the nonrandom technique with the familiar Monte Carlo method shows that the error of the new method decreases faster as more points are used in the calculation. Moreover, the new method needs fewer points to calculate an integral to an accuracy of 10% than does the Monte Carlo method.