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Anisotropic interparticle correlations in the simple cubic lattice of single-domain ferroparticles (SCLF) are studied using both theory and computer simulation. The theory is based on the Helmholtz free energy expansion like classical virial series up to the second virial coefficient. The analytical formula for the Helmholtz free energy is incorporated in a logarithmic form to minimize the effects of series truncation. The new theoretical approach, including discrete summation over lattice nodes coordinates, is compared critically against the classical virial expansion of the Helmholtz free energy for the dipolar hard sphere fluid; the main differences between the Helmholtz free energy of SCLF and dipolar hard sphere fluid are discussed. The theoretical results for the Helmholtz free energy, the magnetization, and the initial magnetic susceptibility of the SCLF are compared against Molecular Dynamic simulation data. In all cases, theoretical predictions using logarithmic form of the Helmholtz free energy are seen to be superior, but they only have an applicability range of the effective dipolar coupling constant $\lambda_e < 1.5$. For highest values of $\lambda_e$, the structural transition of the magnetic dipoles in SCLF is observed in Molecular Dynamic simulation. It has been shown that for $\lambda_e \gtrsim 2$, an antiferromagnetic order appears in the system., Comment: 13 pages, 8 figures, published in Physica A

This paper is concerned with the Cauchy problem for the one-dimensional compressible Euler equations with critical time-dependent overdamping. In this critical case, the damped Euler system is proved to admit a unique global solution that converges time-asymptotically to the corresponding nonlinear diffusion wave. Particularly, the optimal convergence rate in logarithmic form is first derived by the technical time-weighted method.

Studies on allometric models are essential for quantifying the potential of a stand, especially when the objective is to know the wood stock through estimates. This study aimed to select height, volume and artificial form factor equations for a teak stand with data from a sample with 81 trees at 96 months in different diameter classes. The correlation analysis was performed between the variables total height, diameter with and without bark at 1.30 m soil height (dbhcc and dbhsc), total volume with and without bark (VTcc and VTsc) and artificial form factor with and without bark (ffcc and ffsc), using the Pearson coefficient. Six models were evaluated for each category: height, volume and artificial form factor, which were selected by the following criteria: analysis of variance, adjusted coefficient of determination, standard error of estimate, Akaike information criterion and graphical analysis of residues. The selected equations were validated by applying the L & O test. The variables dbhcc, dbhsc, HT, VTcc and VTsc presented significant correlations, ranging from moderate to very strong, the others presented weak correlation. The model that presented the best relation of total height to diameter was the model of Azevedo and contributors. For the total volume with and without bark, Schumacher and Hall model in its logarithmic form was the one that presented the best results. The adjusted models for artificial form factor did not present satisfactory results and, therefore, the use of average form factor is recommended. The mean form factor values by diameter class ranged from 0.4498 to 0.7139 and 0.4480 to 0.7075 for diameter with and without bark, respectively.

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 8) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

The problem of guaranteed cost control for the networked control systems (NCSs) with time-varying delays, time-varying sampling intervals and signals quantization was investigated, wherein the physical plant was continuous-time one, and the control input was discrete-time one. By using an input delay approach and a sector bound method, the network induced delays, quantization parameter and sampling intervals were presented in one framework in the case of the state and the control input by quantized in a logarithmic form. A novel Lyapunov function with discontinuity, which took full advantages of the NCS characteristic information, was exploited. In addition, it was shown that Lyapunov function decreased at the jump instants. Furthermore, the Leibniz-Newton formula and free-weighting matrix methods were used to obtain the guaranteed cost controller design conditions which were dependent on the NCS characteristic information. A numerical example was used to illustrate the effectiveness of the proposed methods.

The rheology of solutions of randomly methylated (1.8) β-cyclodextrin (β-CD) in 1-ethyl-3-methyl imidazolium acetate ([EMIM][Ac]) was studied in detail by rotational and oscillatory shear measurements. It was found that a gel structure was formed even at a relatively low β-CD concentration, which induced intriguing results in oscillation experiments. The solutions showed abrupt changes of the structurally dependent rheological moduli and reproducible transitions from gel to sol state at specific shear stress. For all β-CD solutions, a non-Newtonian flow behavior including shear thinning was obtained. The influence of temperature and β-CD concentration on the flow behavior was studied for all solutions, and the flow activation energies were calculated from the logarithmic form of the Arrhenius equation for nonassociating electrolytes.

We propose some nonlinear Schrodinger equations by adding some higher order terms to the Lagrangian density of Schrodinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrodinger equation is weak.

In Laser Beam (L.B.) and Electron Beam (E.B.) drilling, the energy distribution significantly affects both the penetrating efficiency and working performance, both of which are usually estimated by numerical skill or experimental measure. Through the application of a stimulation model, an unstable solution with the finite difference method will result near the solidliquid interface unless much finer grid sizes are set up. To improve on the above defect, nonuniform grids are therefore utilized; this will complicate the built-up of the program and also easily causes the simulated energy distribution to be divergent in the iteration process. In this study, an estimated small Peclet number and observed narrow-deep cavity made the convective and radial diffusion terms small enough to be neglected in the governing equation. From these assumptions, the model was then used to investigate the drilling efficiency where two-phase flow convection could be simplified further into one dimension and thus the analytical solution becomes possible by transferring the penetration velocity into the logarithmic form. When compared with the experiment made by Allmen [1] , the present model shows good agreement in higher energy density and relative errors are no more than 10%.

A critical examination is made of the mathematical principles that are used for the kinetic interpretation of nonisothermal thermoanalytical rate measurement by a representative approximate method, the widely applied Coats–Redfern (CR) equation. It is concluded that the dominant feature in the identification of that isothermal kinetic equation, which most satisfactorily expresses the rate characteristics, is the (effective) exponent in the set of equations comparatively considered. Consequently, the form of the CR equation possesses only limited ability to distinguish between ‘fits’ to alternative kinetic models. This is entirely consistent with literature conclusions expressing the view that data from a single nonisothermal experiment is insufficient to identify the three kinetic parameters: kinetic model, g ( α )= kt , and both Arrhenius terms, A and E . An usual first step in kinetic analysis by the CR (and other related) equations is to incorporate the kinetic model ( g ( α )= kt ) into the expression used to calculate A and E . Because the rate equation (effective) exponent is a dominant and, without supporting observations, is an unknown parameter, this introduces the well-known ambiguity that alternative kinetic parameters are obtained by the uncritical use of this method. Accordingly, the following replacement calculation sequence is recommended as being more trustworthy. At least two, preferably several, nonisothermal experiments are undertaken, each at a different (usually constant) rate of temperature increase. For a comprehensive range of constant increments of reaction, Δ α i , the different rates (d α /d t ) at different reaction temperatures, T i , are determined from the several experiments and the activation energy, E i , for each successive reaction interval can be calculated. The constancy, or otherwise, of E i with α i throughout the reaction is thus established. From these data, the rate constant for each α i value can be extrapolated to a selected representative temperature T R , perhaps at the mid-point of a reaction ( α =0.5). From these data a pseudoisothermal α – t ( T R constant) curve can be constructed, suitable for analysis by the usual methods. Various advantages from this approach are perceived. The unfortunate role of the (effective) kinetic model exponent in combination with the logarithmic form of the CR equation in analyzing the data, which inhibits recognition of the kinetic model, is avoided. The laborious calculations will not be a problem for modern high-speed computers, unlike the situation existing when the CR equation was first introduced in 1964. Suitable programs may be used to maximize the accuracy of the method, through the use of several nonisothermal experiments and small Δ α increments in the analysis. Kinetic comparisons can be extended to a wider range of kinetic models than the limited selection that often restrict the possibilities for nonisothermal rate data interpretation by the most widely used approximate methods.

We develop the Newton-Kantorovich method to solve the system of2×2nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.

We show the critical Sobolev inequalities in the Besov spaces with the logarithmic form such as Brezis-Gallouet-Wainger and Beale-Kato-Majda. As an application of those inequalities, the regularity problem under the critical condition to the Navier-Stokes equations, the Euler equations in ${mathbb R}^n$ and the gradient flow to the harmonic map to the sphere are discussed. Namely the Serrin-Ohyama type regularity criteria are improved in the terms of the Besov spaces.

The turbulent transport of three coal off-gas mixture fractions is coupled to a prescribed joint β -probability density function ( β -PDF) mixing model. This physical transport and subgrid joint β -PDF mixing model is used to explore the incorporation of coal off-gas compositional disparities between the devolatilization and the char oxidation regime in detailed pulverized-coal combustion simulations. A simulation study of the University of Utah pulverized-coal research furnace is presented to evaluate the sensitivity of different mixing model assumptions. These simulation studies indicate that using a variable composition to characterize the process of coal combustion does not appreciably change the predicted gas-phase temperature field. Moreover, neglecting fluctuations in the char off-gas stream was found to change gas-phase temperature predictions by approximately 15%. State space variable sensitivity to the assumed shape of the PDF (clipped Gaussian vs. joint β ) is presented. Simulation results indicate differences in temperature profiles of as much as 20% depending on the chosen shape of the PDF. Integration accuracy issues for the joint β -PDF are presented and are found to be acceptable. A robust β -PDF function evaluation procedure is presented that accommodates arbitrarily high β -PDF distribution factors. This robust algorithm simply transforms the joint β -PDF function evaluation into a logarithmic form. The assumption that a joint PDF, as rigorously required within a prescribed subgrid mixing model, can be written as the product of N − 1 statistically independent probability density functions is quantified and shown to be less accurate.

The ability of a large-eddy simulation to represent the large-scale motions in the interior of a turbulent flow is well established. However, concerns remain for the behaviour close to rigid surfaces where, with the exception of low-Reynolds-number flows, the large-eddy description must be matched to some description of the flow in which all except the larger-scale ‘inactive’ motions are averaged. The performance of large-eddy simulations in this near-surface region is investigated and it is pointed out that in previous simulations the mean velocity profile in the matching region has not had a logarithmic form. A number of new simulations are conducted with the Smagorinsky (1963) subgrid model. These also show departures from the logarithmic profile and suggest that it may not be possible to eliminate the error by adjustments of the subgrid lengthscale. An obvious defect of the Smagorinsky model is its failure to represent stochastic subgrid stress variations. It is shown that inclusion of these variations leads to a marked improvement in the near-wall flow simulation. The constant of proportionality between the magnitude of the fluctuations in stress and the Smagorinsky stresses has been empirically determined to give an accurate logarithmic flow profile. This value provides an energy backscatter rate slightly larger than the dissipation rate and equal to idealized theoretical predictions (Chasnov 1991).

The logarithmic form of complex Tricomi function Ψ(a,c;x), holding for c - a positive integer, is used in case a =c/2-jk - complex, k - real, -∞ 01 mode is explored, too.

We propose a new method to obtain approximate solutions for the Schr\"{o}dinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows in many cases to find analytical solutions. It offers a convenient way to study the qualitative features of the energy spectrum of bound states in any potential. In particular, we illustrate our method by solving the case of central potentials with power-law form and with logarithmic form. For these types of potentials, we propose very accurate analytical energy formulae which improve a lot the corresponding formulae that can be found in literature., Comment: 2 figures

The paper considers non-linear self-consistent electric potential equation (Sec. I), due to a cloud made of a single species of electric charges, satisfying a Boltzmann distribution law (Sec. II). Exact solutions are obtained in a simple logarithmic form, in three cases: (Sec. III) spherical radial symmetry; (Sec. IV) plane parallel symmetry; (Sec. V) a special case of azimuthal-cylindrical symmetry. All these solutions, and their transformations (Sec. VI), involve the Debye-Huckel radius; the latter was originally defined from a solution of the linearized self-consistent potential equation. Using an exact solution of the self-consistent potential equation, the distance at which the potential vanishes differs from the Debye-Huckel radius by a factor of 2. The preceding (Secs. II–VI) simple logarithmic exact solutions of the self-consistent potential equations involve no arbitrary constants, and thus are special or singular integrals not the general integral. The general solution of the self-consistent pot...

Numerical results for the first gap of the Lyapunov spectrum of the self-dual random-bond Ising model on strips are analysed. It is shown that finite-width corrections can be fitted very well by an inverse logarithmic form, predicted to hold when the Hamiltonian contains a marginal operator., Comment: LaTeX code with Institute of Physics macros for 7 pages, plus 2 Postscript figures; to appear in Journal of Physics A (Letter to the Editor)

In Reply. —The ghost of the "Great Trans-atlantic Acid Base Debate" 1 roams again. Imagine my delight in being found in JAMA 2 sandwiched between founders of acid-base physical chemistry: "In the days of Arrhenius, Severinghaus, and Henderson-Hasselbalch..." in Neiberger's letter. Henderson and Hasselbalch, the first transatlantic acid-base nomenclatura, never met to consummate their hyphen. 3 At Harvard, Lawrence J. Henderson applied the mass action law to the CO 2 HCO 3 relationship (1907), while 10 years later, the Dane Karl Albert Hasselbalch wrote it in the logarithmic form (1917). Svante Arrhenius demonstrated ionization in a nearly disapproved thesis at Uppsala, Sweden, in 1884, which won him the Nobel Prize in 1903. 4 Good company, both in age and distinction! Why not switch from pH to H + , as Neiberger suggests? The H + activity , which is what is measured by a pH electrode, approximates the H + ion concentration, but in no

Adsorption on polarized Hg electrode of formate ions from aqueous solutions of pure HCOONa using differential capacity measurements was studied. Anion specific adsorption was indicated by the cathodic shift of potential of zero charge with increase in bulk-electrolyte activity. Variation of charge due to surface excess of sodium ions (Γ + Z+ F) and of charge due to specifically adsorbed formate ions (q1), with electrode charge (qM) indicated super-equivalent adsorption of formate ions at all positive charges and for all concentrations studied. Esin—Markov coefficient was found to be unreliable criterion of the occurence of specific adsorption. Logarithmic form of constant charge adsorption isotherms were found to be followed. The plots of φM − 2 vs q1 at constant qM were not linear and resembled those for F−, BF4− and CH3COO− ions; which is attributed to relatively weak specific adsorption of anions.

We describe an experimental study of the modified Beer-Lambert law or applied to the resonance transition of atomic lead at λ = 283.3 nm (Pb(7s(3Po1) ← 6p2(3P0))). γ is determined experimentally by two methods. The first involves the standard empirical procedure for time-resolved resonance line absorption following, in this case, the pulsed irradiation of lead tetraethyl in the presence of a buffer gas, yielding γ = 0.41 ± 0.04. The second employs a direct calibration of I0/Itr against the absolute concentration of atomic lead vapour in the temperature range T = 600–783 K, in equilibrium with solid lead, using the technique of phase sensitive detection. This yields γ = 0.54 ± 0.01. γ is also generated by calculation from the slope of the logarithmic form of the curve of growth using the three-layer model. Whilst such calculations can generate γ values considerably less than unity at high optical densities, they do not yield such values using realistic parameters at the optical densities at which the calibration using absolute atomic concentrations was carried out. It is therefore concluded that the disagreement between the measured and calculated values arises from limitations in the three-layer model. The time-resolved resonance absorption signals obtained in the empirical measurements of γ permit one to estimate a rate constant k for the reaction. of 2 × 10−10 cm3 molecule−1 s−1 (300 K)

In the relativistic Hamiltonian dynamics of particle world lines, a dynamically determined scalar evolution varible is employed to avoid the ''no-interaction theorem''. Apart from the usual ten generators which realize the Poincare algebra, there is a new generator x which is related to dynamical evolution of a particle system in an inertial frame. For one single free particle, the generator H was found to be -m ln(p/sub o//m). We show that this logarithmic form of H actually holds in general for N innteracting particles: H/sub N/ = -M ln(P/sub o//M), where M = ..sigma../sup parallel//sub a/ = 1m/sub a/, P/sub o/ = ..sigma../sup parallel//sub a/ = 1p/sub a/o, N> or =2, provided that there is a common potential and that one uses a similar evolution variable which is determined dynamically.

The electronic polarisability function is calculated for a simple model of a noninteracting 2D electron gas on an infinitely long strip of width w. Periodic boundary conditions are applied both along and across the strip. An explicit formula for the dynamic polarisability function is given at temperature T which takes account of collisional broadening gamma . Detailed calculations of the static polarisability for T=0 and finite gamma are made as functions of wavenumber q and width w. As w is increased so that more and more sub-bands have minima below the Fermi level h(cross)2kF/2m*, the singular behaviour at q=2kF changes from the broadened logarithmic form characteristic of a 1D electron gas through an intermediate region exhibiting similar peaks associated with each occupied sub-band to reach a 2D limit in which the peaks coalesce to give the broadened cusp singularity which is characteristic of a 2D electron gas.

Arrhenius parameters are frequently evaluated incorrectly by applying the least squares method to the logarithmic form of the Arrhenius equation without simultaneously transforming the statistical weights as required for the change of variable from k to ln k. This has been mentioned briefly in an earlier paper. In the present communication the correct procedures are discussed and illustrated by several examples of evaluations. In particular, Arrhenius parameters calculated by the Taylor series expansion of the exponential form of the Arrhenius equation are compared with those calculated from the logarithmic form, using an exact and two approximate transformations of the statistical weights. The comparisons indicate thatthe preferred procedure for obtaining Arrhenius parameters is either the Taylor series method or the logarithmic method with proper transformation of the experimentally determined statistical weights of the rate constants ki. The common approximation of assuming equal statistical weights of ln ki when the logarithmic form of the Arrhenius expression is used is shown not to be always appropriate, and reasons forthis are given.

The values of force constant, compressibility, i.r. absorption frequency, Debye temperature, lattice energy, Gruneisen parameter, Anderson-Gruneisen parameter, and Moelwyn-Hughes parameter are reported for the cubic NaCN. KCN and RbCN crystals using a logarithmic form of potential and a recent method of calculation. Using the computed lattice energies we estimate ΔH°f .(CN– )(g) = 44.6 kJ mol–1 ,ΔH°solv (CN– )(g) = –337.4 kJ.mol–1 ,the proton affinity of the gaseous CN- ion to be–1 445.7 kJ mol–1 ,electron affinity of the cyanide radical to be390.5 kJ.mol–1 .Using the calculated values of thermodynamic data for CN– ion, energetics for several other cyanide crystals are also estimated. The computed values of various parameters compare well with the available literature data.

The majority of reported values of activation energy and pre-exponential factor have been obtained by fitting experimental results to the logarithmic form of the Arrhenius equation, In k = In A — E a a RT where E a and A are the apparent activation energy and pre-exponential factor and both are, to a first approximation, considered to be independent of temperature. The statistical technique of line-fitting by least-squares is frequently used to obtain Ea and In A. Theoretical treatments of rate constant give equations in which the activation energy and pre-exponential factor may vary with temperature so that the graph of In k vs . 1 T is actually a curve and not a straight line. The apparent and theoretical activation energies will, therefore, differ. In this paper the differences between Ea derived by the least-squares method and the theoretical activation energy have been calculated for a wide range of experimental conditions and for two types of theoretical equation. In addition, conditions for detection of curvature in the graph of In k vs l T have been determined. Because in practice the curvature is slight and not usually detected, the most interesting conclusion drawn from these calculations is that differences between apparent and theoretical values as great as 5 kcal/mole can exist while the experimental results still appear to fit the Arrhenius equation. Further, if apparent activation energies from two different reactions are compared, as a rough approximation, only differences greater than 3–4 kcal/mole are likely to be significant.

Multiple-beam white light fringes of equal chromatic order are used for the study of the growth from solution of sodium chlorate. They are sharper and much more informative than monochromatic Fizeau fringes, and have proved to be very sensitive for the measurement of variation of concentration. The experimental results are found to be in agreement with the assumption of the existence of the quasi-stationary state theory introduced by Rieck and further developed recently by Frank. According to this theory, the radial gradient of concentration is analogous to a static potential field multiplied by a Gaussian cut off factor. The law relating the concentration to the radius is found to be of logarithmic form near the crystal and exponential at larger distances.

Many catalysts are likely to have surfaces which are energetically heterogeneous. The overall rate of reactions on such surfaces will, therefore, be the sum of rates of concurrent reactions each proceeding with different activation energies. Apparent activation energies are, however, commonly derived by treating the overall rate as if it were governed by a single activation energy. The majority of reported values of activation energy have been obtained by fitting experimental results to the logarithmic form of the Arrhenius equation and measuring the slope. The method of linefitting used is often the statistical technique of least-squares or some visual equivalent. In this paper, equations are derived which relate the apparent activation energy derived by the least-squares technique to any postulated type of heterogeneity and the experimental error in the rate determination. In this derivation the dependence upon temperature and surface concentration of the activation enthalpy and entropy is included in the calculations. The graph of log (rate constant) vs. 1 T for catalysts with heterogeneous surfaces may not be linear. The degree of curvature introduced by any postulated heterogeneity has been calculated and may be compared with experimental error in order to decide whether it would be detectable.

A kinetic study of the synthesis of nitrides using carbothermic reduction of UO2 and PuO2 in flowing N2 was conducted. Reactio n rate constants as functions of the reaction temperature and N2 flow rate were determined. The measured rate constants varied with temperature according to the Arrhenius equation at constant N2 flow rates. In logarithmic form, the measured rate constants also varied linearly with N2 flow rate at constant temperature, up to a flow rate of 120 cm3/min-g of reactant. The rate constant for a particular reaction temperature and N2 flow rate can be calculated from the logarithmic form of the Arrhenius equation by adding a flow- rate term, F. The expression ln k = (-86,000/ RT) + 18.1 + 0.021F fits the kinetic data from 1430 to 1625 deg C. Close agreement was obtained between calculated and experimenta rate constants determined for 50-g batch production runs. The reaction rates measured are considered rapid enough for an economical nitride process.

Negative magnetoresistance observed in many heavily doped semiconductors is considered as due to a change in the population of states for which the product Δ(E) of the mobility and the density of states varies rapidly with energy. A logarithmic form of Δ(E) is found to give an excellent fit to experimental data which can thereby be described as a simple function of H/(T + θ). The form of Δ(E) suggests that the Fermi level lies in a conduction-band tail with a width of 2kθ, typically 0.2 meV for Ge and CdS. Possible repopulation mechanisms are considered.

In this paper, we recover the characteristic polynomial of an arrangement of hyperplanes by computing the rational equivalence class of the variety defined by the logarithmic ideal of the arrangement. The logarithmic ideal was introduced in [arXiv:0907.0896v2] in a study of the critical points of the master function. The above result is used to understand the asymptotic behavior the Hilbert series of the logarithmic ideal. As an application, we prove the Solomon-Terao formula under the tame hypothesis by identifying each side of the formula with a certain specialization of the Hilbert series of the logarithmic ideal., Comment: 17 pages

A kinetic study of the synthesis of nitrides using carbothermic reduction of UO2 and PuO2 in flowing N2 was conducted. Reactio n rate constants as functions of the reaction temperature and N2 flow rate were determined. The measured rate constants varied with temperature according to the Arrhenius equation at constant N2 flow rates. In logarithmic form, the measured rate constants also varied linearly with N2 flow rate at constant temperature, up to a flow rate of 120 cm3/min-g of reactant. The rate constant for a particular reaction temperature and N2 flow rate can be calculated from the logarithmic form of the Arrhenius equation by adding a flow- rate term, F. The expression ln k = (-86,000/ RT) + 18.1 + 0.021F fits the kinetic data from 1430 to 1625 deg C. Close agreement was obtained between calculated and experimenta rate constants determined for 50-g batch production runs. The reaction rates measured are considered rapid enough for an economical nitride process.

IN chemical kinetics the effect of temperature on reaction rates is often evaluated in terms of the activation parameters E and A of the Arrhenius equation1,2 where k is an appropriate rate coefficient at the absolute temperature T, A is a constant related to the entropy of activation, E is the activation energy and R is the gas constant. In practice the numerical value of E is derived from the slope m of the graph log k versus l/T, and E = 2.303 Rm when common logarithms are used; log A is usually obtained by substituting the corresponding values of k, E and T in the logarithmic form of the equation.

A new model to describe interionic binding in diatomic crystals is proposed. With the inclusion of an overlap repulsion of logarithmic form along with charge-charge and van der Waals interaction terms, the model is found to yield improved values for the cohesive energy and other parameters of alkali halide crystals. Unlike previous forms for the interionic binding in an LiF crystal, in which the potential is negative at distances much smaller than the distance of closest approach of Li + and F- ions, the present model tends to infinity in the positive energy region before such distances are approached

An empirical extension of the logarithmic form of the classical Arrhenius equation is written as a simple three-parameter equation which fits precise experimental data on the variation of rate-constant with temperature better than any three-parameter extension of the Arrhenius equation yet proposed.The proposed equation has been compared with the three and four parameter equations of Hyne and Robertson for a series of sets of rate-constant against temperature data on solvolysis systems. The equation proposed here is simpler to apply, and provides at least as good a statistical fit to the data as does the three-parameter equation of Hyne and Robertson.