Axial Equilibrium Research Articles

Potential Energy Surface of the Ring Puckering Motion in 1-Chloro-1-silacyclobutane

The equatorial–axial conformational equilibrium of 1-chloro-1-silacyclobutane has been investigated by microwave spectroscopy andab initiocalculations. The potential energy function of the ring puckering motion has been determined: four vibrational states are localized in the equatorial well and two vibrational states in the axial well, all the remaining higher energy states lying above the interconversion barrier. This study has shown that the equatorial conformer is more stable than the axial one by 185(40) cm−1. This value is in good agreement with that provided byab initiocalculations using the 6-31G* basis set.

General axial shapes of sodium clusters

AbstractThe shell correction method is applied to calculate the shapes and binding energies of Na clusters. The axial equilibrium shapes are calculated by minimizing five deformation parameters simultaneously. Strong deviations from spheroidal shapes including reflection asymmetric ones are found. The influence of cluster deformation on the separation energies and ionization potentials and on the splitting of the photo ‐ resonance are compared with the available experimental data. The N‐dependence of the cluster shape and its relation to the N‐ and Z‐dependences of the nuclear shape is discussed.

Vibrational study of the structure and polarons in piperazine hexahydrate

AbstractThe Raman spectrum of piperazine hexahydrate was observed over the whole wavenumber region at various temperatures (300—29 K). The splitting between the N—H stretch and the polaron was found to be 21 cm−1 at 44 K. The bands due to the axial configuration of piperazine were above 200 K in both the H and D systems. The enthalpy difference in the equatorial—axial equilibrium was calculated to be 8.1 and 7.6 kJ mol−1 in the H and D systems, respectively. The phase transition was found at ca. 170 K, and is ascribed to the order—disorder type associated with the axial configuration. The polaron was discussed group theoretically and reinvestigated on the basis of the coupling of the N—X stretch with optical lattice modes. It is noted that the polaron is little affected by the phase transition at 170 K or disorder in hydrogen positions.

Determination of equivalent thermal response parameters for evaluating the structural response of beams subjected to transient thermal environments

A methodology is presented which provides the basis for evaluating the structural response of building components, which are subjected to a complex temperature distribution, normally of the curvilinear form resulting from transient heat conduction through the thickness. An equivalent linear temperature distribution is equated by means of axial and moment equilibrium considerations. The concept is applied initially to homogeneous materials where the use of Lagrange polynomial interpolation permits explicit solutions for the equivalent thermal parameters. This formulation also leads to the optimization of sampling points for experimental measurement by the establishment of a Gauss point rule. Secondly a spreadsheet formulation is developed for the numerical solutions of more complex problems, which has utility for analysis of the response of multilayer composite constructions or scenarios where material properties are temperature dependent.

Nonlinear analysis of orthotropic beams of solid cross-sections

The paper presents a nonlinear formulation and a finite-differences based numerical solution for the structural behavior of generic orthotropic beams of solid cross-sections. The analysis includes a detailed description of the three-dimensional out-of-plane warping and the solution is achieved by successive iterations while preserving all geometrical nonlinearities. The axial body forces are introduced consistently through the axial differential equilibrium equation. The formulation yields some basic closed-form analytic solutions for homogeneous beams, and the numerical results provide an insight into the coupling effects mechanisms in generic laminated composite beams, including the resulting interlaminar stresses.

Plasma heating and dynamics in the Coaxial Slow Source

Simple semi-analytical models are presented to calculate the temporal evolution of the plasma temperature and length or thickness in the Coaxial Slow Source (Nucl. Fusion 27 (1987) 1478) for both tearing formation and programmed formation. It is assumed that energy is delivered to the plasma via Ohmic heating and compressional work and is lost through impurity line radiation. The plasma is considered to be always fully ionized and in pressure balance; the external magnetic field is taken to be a known function of time, and particle losses are neglected. In tearing formation, a long and thin plasma sheet is initially formed. This can be studied using a 1-D model; it is shown that the higher the external field and the smaller the line integrated density, the faster the temperature increases. In programmed formation, an axial equilibrium is quickly established and a 2-D model is required. It is shown that when the external magnetic field exceeds a critical value, which depends on the temperature, the impurity fraction and the effective plasma thickness, the radiated power overcomes Ohmic heating and a radiative collapse occurs. Various time histories of the external magnetic field are analysed in order to determine the conditions that result in the fastest increase in plasma temperature

Coordination Chemistry of F430: Axial ligation equilibrium between square-planar and bis-aquo species in aqueous solution

Abstract X-ray absorption spectroscopic characterization of axial ligand coordination to factor F430, the nickel-tetrapyrrole cofactor of the S-methyl-coenzyme M (CH3SCoM) methyl reductase enzyme from methanogenic bacteria, is presented. The nickel of isolated F430 is hexacoordinate at 10 K in aqueous solution (as is the enzyme-bound cofactor), whereas the epimerized and ring-oxidized derivatives of F430 have four-coordinate nickel. Reduction of the ring-oxidized derivative, F560, with dithionite yields F430 in its native configuration, with axial ligands indistinguishable from those present when the cofactor is obtained directly from the holoenzyme. Thus, we conclude that the axial ligands to F430 in aqueous solution are water molecules. Analysis of the nickel extended x-ray absorption fine structure is consistent with this conclusion. Resonance Raman spectra obtained at room temperature contain features characteristic of both 4- and 6-coordinate forms of the cofactor. We have found that the resonance Raman, optical, and x-ray absorption spectra of aqueous solutions of F430 are temperature-dependent due to a ligand-binding equilibrium involving the square-planar and 6-coordinate bis-aquo forms of the cofactor. At low temperatures (less than 250 K) the 6-coordinate form predominates, whereas higher temperature solutions contain both 4- and 6-coordinate species in a dynamic equilibrium. Similar behavior is observed in other weakly coordinating solvents such as methanol and ethanol. The 4-coordinate form is predominant in solvents with strong electron-withdrawing substituents such as 2,2,2-trifluoroethanol and 2-mercaptoethanol. The relevance of this facile ligand exchange to the active site structure and enzymatic mechanism of the parent enzyme is discussed.

Thermolysis of the CoC bond in adenosylcobalamin (coenzyme B 12)—IV. Products, kinetics and CoC bond dissociation energy studies in ethylene glycol

Following an introduction outlining the key questions surrounding the adenosylcobalamin (AdoB 12) bond homolysis problem, the full details of product, kinetic and CoC5′ bond dissociation energy studies of the thermolysis of AdoB 12 in ethylene glycol are presented. The anaerobic thermolysis of AdoB 12 in the absence of the nitroxide radical trap TEMPO proceeds with four isosbestic points to yield 100±2% Co(II)B 12 and two nucleoside products, 8,5′-anhydroadenosine and 5′-deoxyadenosine. In the presence of >10 −2 M TEMPO, the TEMPO-trapped Ado . product (T-Ado) and Co(II)B 12 account quantitatively for the starting AdoB 12. From HPLC studies of the concentration of the nucleoside products vs [TEMPO], absolute rate constants at 110°C for Ado . cyclization and its H . abstraction from glycol solvent are obtained for the first time, k c (110°C) ⋍ 5 × 10 5 s −1 and k a (110°C) ⋍ 7 × 10 3 M −1 s −1. Kinetic studies in the presence of added, authentic Co(II)B 12 show an inverse, linear dependence of 1/ k obs vs [Co(II)B 12], thereby providing quantitative evidence for the long sought demonstration of the AdoB 12 ⇌ Ado .+Co(II)B 12 equilibrium outside of the enzyme-cofactor complex. The Co(II)B 12 dependence data also provide the previously unavailable rate constant for recombination of Co(II)B 12 and Ado . in ethylene glycol of k t ⋍ 3 × 10 8 M −1 s −1. Next, the assumption used previously of slow homolysis by the base-off form, k h,off ⪡ k h,on , an assumption necessary to simplify the AdoB 12 thermolysis kinetics, is tested directly by the synthesis and thermolysis of adenosylcobinamide +OH − and found to be valid. The base-on homolysis rate constants ( k h,on ) were measured from 110 to 80°C. When combined with independently measured temperature-dependence parameters for the off ⇌gon axial base equilibrium, Δ H = −7.6±0.2 kcal mol −1 and Δ S = −20.2±0.7 e.u., values for the base-on homolysis activation parameters of ethylene glycol are obtained, Δ H h,on ‡ = 34.5±0.8 kcal mol −1, and Δ S ‡ = 14±1 e.u. Use of the Δ H h,on ‡ data to provide an estimate of the AdoB 12CoC5′ BDE in ethylene glycol is discussed, as are a number of additional points and conclusions that result from the present work. Key considerations that led to the nitroxide radical trapping method, and the desirable features of this method, are then discussed, with an eye towards aiding future kinetic studies and BDE determinations of CoR and other MR compounds.

RC Rectangular Slabs with Edge Restraints

Reinforced concrete slabs with edges restrained against lateral displacement are capable of carrying ultimate loads which are considerably higher than the loads predicted by using conventional yield‐line theory. An analytical relation for the load‐deflection behavior is derived in the paper by using a plastic‐flow theory. The compatibility condition introduced in the paper for the deformation of the centerline of the slab incorporates boundary displacements as well as the in‐plane deformations of the slab. Combining the compatibility condition, the flow rule associated with the load‐moment yield criterion, and the axial equilibrium condition, a nonlinear differential equation is obtained for the membrane force in the slab. The load on the slab is then calculated for a given deflection by balancing the external and internal energy dissipation. The predicted load‐deflection curves are compared with those obtained from two test series and a good agreement is obtained up to the peak load. The load‐deflection curve beyond the peak load is found to be predicted better by the incorporation of an analysis which approximates strain softening of the concrete.

Conformational enthalpy and volume changes in bromocyclohoexane determined by Raman spectroscopy

Raman spectra of bromocyclohexane have been recorded as a function of temperature between 22 and 135°C and as a function of pressure from 1 bar to 7.0 kbar. The integrated intensities of the bands at 657 and 687 cm −1 have been used to monitor the relative populations of the axialand equatorial forms respectively. From these data, enthalpy and volume differences of 1.1 ± 0.3 kJ mol −1 and −2.2 ± 0.3 cm 3 mol −1 respectively were determined for the equatorial—axial equilibrium.

Conformational enthalpy and volume changes in chlorocyclohexane determined by Raman spectroscopy

AbstractThe Raman spectrum of chlorocyclohexane has been recorded as a function of temperature from −40 to +110 °C and the conformational enthalpy for the equatorial to axial equilibrium was determined as 1.10±0.13 kJ mol−1 for ΔH. Changes in the Raman spectrum as a function of pressure up to 5.6 kbar have been interpreted in terms of a shift in the conformational equilibrium to favour the axial form at high pressure. The associated conformational volume change has been calculated as 2.33±0.27 cm3 mol−1.

Modes of perturbation of a diffuse linear pinch: the Bessel function model

The low frequency wave and instability spectrum of a cylindrical, uniform density, zero beta plasma in a Bessel function magnetic field is analysed. The analysis has application to wave heating of reversed field pinches and energy propagation in, and instabilities of, twisted solar coronal loops. A surface Alfven wave which can be Alfven resonance damped is identified, and its connection to the unstable external kink mode is discussed. A distinctive extra mode is identified as a poloidal fast Alfven wave with wave fields concentrated at the position of zero axial equilibrium magnetic field and with a dispersion relation dependent dominantly on the poloidal wavenumber and only weakly on the axial wavenumber. Finally, the spectrum of discrete Alfven waves, or global Alfven eigenmodes, is also identified.

Microwave spectrum of axial cyclohexyl chloride and conformational equilibrium in cyclohexyl chloride

The high-resolution microwave spectrum of the axial conformer of cyclohexyl chloride has been assigned for both the Cl(35) and Cl(37) isotopic species, in natural abundance. Moreover, the high-resolution microwave spectrum of the equatorial conformer, previously assigned in the ground state [ Chem. Phys. Lett. 21, 592–594 (1973)], has been further investigated. Several low-energy vibrational satellites have been studied for the Cl(35) isotopic species of both the axial and the equatorial isomers. The ground state centrifugal distortion constants for both the isotopic species of the two rotamers, and the quadrupole coupling constants of Cl(35) and Cl(37) axial isomer and of Cl(37) equatorial isomer (not reported previously) have been determined. From line relative intensity measurements the energy difference between the two conformers has been calculated to be 0.51 ± 0.15 kcal/mol, the equatorial isomer being more stable. In the same way, the vibrational energies of the investigated vibrational satellites have been computed. A structural model for axial cyclohexyl chloride is proposed on the basis of the available experimental data.

Instability of optimal equilibria in the minimum mass design of uniform shallow arches

Necessary and sufficient conditions for the minimum mass design of arbitrarily loaded uniform shallow arches are derived. The problem is posed as an optimal control problem with mass as the criterion, initial curvature and axial load as design variables, and with the differential equations of axial and transverse equilibrium of the arch as side conditions. Thus, an optimal equilibrium is associated with each optimal design, and the stability of these equilibria becomes an integral part of the problem solution. As an example, the design process is carried out for the sinusoidally loaded hinged-hinged arch with a fixed span. It turns out that, depending on the given load amplitude, the optimal equilibrium can be unstable, stable after snap-through, and nonunique with one equilibrium unstable and the other stable after snap-through, at the design load of the arch. In addition, a necessary condition for a local minimum is the same as the usual critical point condition in stability analysis, thus assuring the instability of the arch at the optimum. A brief survey of earlier work on the optimal design of arches and curved beams is also included.

Low resolution microwave spectroscopy and the conformational analysis of 4,4-difluoropiperidine

The low resolution and high resolution microwave spectra of 4,4-difluoropiperidine and the N deutero analog have been recorded. Based on a low resolution microwave adaptation of Kraitchman's equations, the species giving rise to the single observed low resolution band series is seen to be the N-H equatorial conformer. This result is verified by obtaining the principal axis coordinates of the amine hydrogen atom via an assignment of the high resolution microwave spectra of the normal and N-D compounds. Consideration of the least intensity of the N-H axial conformer's spectrum that would be necessary for it to be observed, leads to the result: ΔG ⩾ O.89 kcal/mole for the equatorial ⇌ axial equilibrium.

Plasma equilibrium and stability in the Scyllac toroidal sector experiments

Experiments are reported on helical plasma equilibrium and stability in the Scyllac toroidal θ-pinch sectors (120°) which have major radii of 2.375 and 4.0 m with coil arc lengths of 5.0 and 8.4 m, respectively. In these experiments the outward toroidal drift force was compensated by a combination of ℓ = 1 helical and ℓ = 0 bumpy fields which are generated by shaping the inner surface of the compression coil or by driven ℓ = 1 windings. Time-resolved measurements were made of the gross plasma-column motion, the plasma radius, the magnetic flux excluded by the plasma, the external magnetic field, the plasma density, the electron and ion temperatures, and the plasma β at axial locations of minimum and maximum plasma radius. These data are used to study the approach to the theoretically predicted toroidal equilibrium (including axial pressure equilibrium). The plasma column remained in stable equilibrium for 7 – 10 μs in the 8-m sector compared with 4 – 7 μs in the 5-m experiment, at which times the onset of a terminating m = 1, k ≈ 0 sideways motion occurred. The results show that the plasma achieved axial pressure equilibrium (nkT = const) in 4 – 6 μs, while maintaining equilibrium in the toroidal plane for 10 μs or longer. The measurements of the plasma radius, β and magnetic field in the various experiments have confirmed in detail the stable toroidal equilibrium observed in the streak photographs during the first 4-10 μs of the discharge. The observed toroidal equilibria of the high-β, θ-pinch plasma are in quantitative agreement with MHD sharp-boundary theory and confirm the theoretical scaling of the equilibrium field between the 5-m and the 8-m sector experiments.

Dynamic behavior of cylindrical membranes

An often encountered difficulty in non-linear membrane theories, is that of discussing static or dynamic solutions to the non-linear membrane equations when the equilibrium tensions are small. In this paper a theory of cylindrical membranes is formulated, which is essentially the dynamic equivalent of the Bromberg-Stoker theory. It is shown that a cylindrical membrane may develop compressions in the course of its vibration, unless the axial equilibrium tension exceeds a definite positive value. In addition, a short discussion is given of the dynamic behavior of cylindrical membranes.

Gradient Elution in Liquid Chromatography Under Conditions of Axial Equilibrium

Gradient Elution in Liquid Chromatography Under Conditions of Axial Equilibrium

AXISYMMETRIC RESISTIVE INSTABILITIES IN THE HARD-CORE PINCH

The hard-core pinch configuration with azimuthal equilibrium magnetic field only is considered. Provided finite conductivity is included, unstable axisymmetric modes are found. These instabilities are apparently due to nonlocalized plasma diffusion across the magnetic flux lines. On the laboratory scale, the growth rates do not appear to be significant, when the material of the conducting shell is mercury. For a hydrogen plasma shell, however, it appears that the growth rates may be of practical importance. It is found that the ratio of the current in the shell to the hard-core current has a significant effect on the "spectrum" of instabilities. This effect in part accounts for the observed relative stability of the hard-core system when there is no axial equilibrium magnetic field.

Études Conformationnelles IX. Epimérisation des 4‐tertiobutylcyclohexanols stéréoisomères par le potassium dans du benzène. Analyse par la loi d'action des masses

AbstractThe epimerisation of the cis and trans isomers of 4‐tertiobutylcyclohexanol is studied quantitatively in the case of potassium in benzene as an agent. A general theory is formulated for the epimerisation of conformationally homogeneous cyclohexanols by monovalent agents. It has been shown that, in such conditions there is a linear relationship between the equilibrium titres of the trans derivative and the titre of the cyclohexanolate present. By extrapolation one may obtain experimental values of the conformational equatorial axial equilibrium of both the free alcohol and the corresponding alcoholates.