Showing papers on "Rate equation published in 1971"
TL;DR: In this paper, the authors show that the kinetics of mass transfer among aluminosilicates and aqueous solutions can be described by a simple parabolic rate law, which is consistent with diffusional mass transfer and applicable to incongruent as well as congruent reactions.
213 citations
01 Jan 1971
TL;DR: In this paper, high temperature third and fourth stage grain growth has been found to conform to the theoretical rate equation, where the activation energy for third stage growth is considerably larger and is comparable to that of volume self-diffusion.
Abstract: High temperature third and fourth stage grain growth has been found to conform to the theoretical rate equation\(^2 - {\text{ }}\overline D _0^2 {\text{ = }}K {\text{(}}t{\text{ - }}t_{\text{0}} {\text{)}}\). The activation energy for fourth stage grain growth has been measured and found to be similar to that of grain boundary self-diffusion. The activation energy for third stage grain growth is considerably larger and is comparable to that of volume self-diffusion.
31 citations
01 Jan 1971
TL;DR: In this paper, measurements and calculations of E-I characteristics of cascade arcs burning in noble gases under atmospheric pressure are reported, assuming Saha equilibrium [complete local thermal equilibrium] is not in agreement with the detailed solution of balance equations.
Abstract: Measurements and calculations of temperatures, densities, and field-strength-current characteristics of cascade arcs burning in noble gases under atmospheric pressure are reported. The evaluation of measured arc data assuming Saha equilibrium [complete local thermal equilibrium (LTE)] is not in agreement with the detailed solution of the balance equations. The temperatures of electrons and heavy particles and the density of electrons and neutrals have to be determined from the set of rate equations, from the equation of state, connection with the electron energy balance and the equation of state, the energy balance of the electron gas, and of the total plasma. Solutions of these equations are compared with results following from measured line intensities only solving the rate equations in connection with the electron energy balance and the equation of state. For helium, both methods give results which agree within a few percent. The deviations from Saha equilibrium are caused by diffusion and the overpopulation of ground state atoms. The excited atoms, however, are nearly in equilibrium with free electrons in the range of electron densities reached in our experiment (partial LTE). Measurements of E-I characteristics agree with calculated data, if diffusion is taken into account. A simple criterion for the limit between diffusion-dominated plasma and a plasma in thermal equilibrium is derived.
30 citations
TL;DR: A rate expression for kf, the third order forward rate constant in ion-molecule clustering reactions, by means of an energy transfer mechanism was derived in this paper.
Abstract: A rate expression has been derived for kf, the third order forward rate constant in ion-molecule clustering reactions, by means of an energy transfer mechanism. The rate expression has been tested against all the available data concerning the temperature dependence of such reactions and the predictions as to the effect of cluster size and nature of third body upon the magnitude of kf have been considered. The rate equation appears to provide a useful method for the estimation of kf values to within an order of magnitude.
26 citations
TL;DR: In this paper, a model for CO2 is used that assumes that the energy levels of each vibrational mode are evenly spaced and that each mode can be assigned a vibrational temperature.
Abstract: Macroscopic equations are formulated for the nonequilibrium interaction of vibrational and radiative rate processes in CO2. A model for CO2 is used that assumes that the energy levels of each vibrational mode are evenly spaced and that each mode can be assigned a vibrational temperature. The bending and symmetric‐stretching modes are, however, taken to be in mutual equilibrium so that they have the same vibrational temperature. Collisional rate equations for the V—T and V—V processes are derived on a phenomenological basis. The resulting phenomenological coefficients are interpreted, with the help of basic knowledge of the microscopic physics, in terms of characteristic relaxation times and parameters measuring the relative amounts of energy exchanged by the various modes during V—V transitions. Radiative transfer equations are developed for the three strongest infrared bands, located in the spectrum at 15, 4.3, and 2.7 μ, by appropriately summing a previously derived microscopic transfer equation. It is found that the absorption coefficient is a function of the kinetic and the vibrational temperatures in a manner that depends on the band in question. The source function, which is also different for each band, depends strongly on the temperature of the vibrational modes involved in the transition and weakly on the kinetic temperature. The rate and transfer equations are then incorporated with the conservation equations of gas dynamics for application to linearized nonequilibrium flows of carbon dioxide gas. As a particularly simple example, a solution for the spectrophone is presented, including expressions for the phase lag in the pressure signal that results from radiative excitation via each of the bands individually. One of these expressions can be used in conjunction with experimental results from other sources to draw conclusions about predominant V—V transitions in CO2. The results of doing this using the 4.3‐μ band have been reported previously. The utility of a spectrophone can also be extended by exploiting the possibility of selectively exciting CO2 through the different bands and making use of all the phase‐lag expressions. All equations were derived without restriction to room temperature, allowing for possible use of spectrophones at higher temperatures. The approach used to derive the macroscopic rate equations can equally well be applied to other polyatomic molecules.
20 citations
TL;DR: In this paper, the stoichiometric number of the rate-determining step is explicitly taken into consideration and all equations are derived starting with a general rate equation rather than the equation used by Vetter, which is applicable only to charge-transfer-controlled processes.
18 citations
TL;DR: In this paper, specific specific rate constants of bimolecular reactions have been derived from the viewpoint of the absolute reaction rate theory, and corresponding reaction cross sections have been obtained.
Abstract: Detailed specific rate constants of bimolecular reactions have been derived from the viewpoint of the absolute reaction rate theory, and the corresponding reaction cross sections have been obtained. The energy dependence of the reaction cross sections of the absolute reaction rate theory in the high energy range has been discussed, and it has been shown that the cross sections of the absolute reaction rate theory converge as the energy approaches infinity, if the effect of the exclusion of the disallowed states is properly taken into account.
17 citations
TL;DR: In this article, the thermal decomposition of polytetrafluoroethylene (TFE, Teflon), high and low density polyethylene, Delrin Acetal (DA), AVCO Phenolic Fiberglass (APFG), and carbon phenolic (CP), were studied by a thermogravimetric technique which utilized a constant heating rate.
Abstract: The thermal decomposition of polytetrafluoroethylene (TFE, Teflon), high and low density polyethylene (HDP and LDP), Delrin Acetal (DA), AVCO Phenolic Fiberglass (APFG), and carbon phenolic (CP), were studied by a thermogravimetric technique which utilized a constant heating rate. Loss in sample weight was recorded as a function of time or temperature from room temperature to approximately 700°. Reaction orders were established from logarithmic rate versus temperature plots. Arrhenius frequency factors and overall activation energies were determined from computerized integrations of the appropriate rate equations in which the results were treated on the basis of first-order reaction mechanisms for specific temperature regions. Zero-order mechanisms were estimated by the usual graphical methods.
17 citations
TL;DR: In this article, the authors used pressure measurements on the end wall of a shock tube to infer the continuously varying rate of vibrational excitation behind incident shock waves in undiluted carbon monoxide.
Abstract: Measurements of pressure on the end wall of a shock tube have been used to infer the continuously varying rate of vibrational excitation behind incident shock waves in undiluted carbon monoxide. Results are presented in the form of separate relaxation-time plots for each shock wave studied. The data cover the temperature range 2400-6000°K and are in excellent agreement with the previous results of Matthews 1 and Hooker and Millikan.2 The relaxation-time curves from all the experimental runs tend to fall on a single straight line on a conventional Landau-Teller plot, with the exception of data obtained near equilibrium. These results support use of the Landau-Teller rate equation to describe vibrational excitation in CO for situations involving large departures from equilibrium, but point out the possible need to modify the rate equation for situations in which the vibrational energy mode is highly excited.
16 citations
TL;DR: In this article, a simple model consisting of three simultaneous rate equations is used to explain the conversion of infrared excitation to green emission by LiY0.799 Yb0.2 Ho0.001 F4.
15 citations
TL;DR: In this paper, the rate of sulfur dioxide oxidation over activated carbon was measured in gaseous stream of sars dioxide-oxygen-nitrogen mixtures and it was found that catalyst fouling during the reaction was caused by sulfur trioxide formed on carbon.
Abstract: The rate of sulfur dioxide oxidation over activated carbon was measured in gaseous stream of sulfur dioxide-oxygen-nitrogen mixtures. It was found that catalyst fouling during the reaction was caused by sulfur trioxide formed on carbon. According to the Langmuir-Hinshelwood model, the over-all rate equation was derived on the basis of the mechanism where the rate-determining step is a surface reaction. The rate data were well correlated with the rate equation.
TL;DR: In this article, the rate equations used in non-isothermal kinetic systems have aroused some controversy and two groups have objected to the use of the partial differential equation illustration for the rate equation.
Abstract: OUR recent communication1 on the rate equations used in non-isothermal kinetic systems has aroused some controversy and two groups2,3 have objected to the use of the partial differential equation illustration
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TL;DR: In this article, a simple one-dimensional mathematical model is developed which is used to calculate the burning rate of a spray of droplets with an initial Rosin-Rammler size distribution.
TL;DR: The reaction cross section of a bimolecular reaction has been calculated from the corresponding rate constant by the inverse Laplace transform and the computation of the energy density function associated with the definition of the weighted-average cross section has been discussed.
Abstract: The reaction cross section of a bimolecular reaction has been calculated from the corresponding rate constant by the inverse Laplace transform. In carrying out the inverse Laplace transform, the method of steepest-descent has been used. As applications, the reaction cross sections for the rate constant of k(T) = k′ Tne-E0/kT and for the rate constant of the absolute reaction rate theory have been calculated. The computation of the energy density function associated with the definition of the weighted-average cross section has also been discussed.
TL;DR: In this article, Hill has argued that no change is necessary because in the expression ======€€€£€£££ £€£ ££ £ £ £££ −£ £ −£ − £ £ − ££ t is effectively zero.
Abstract: THERE has been some controversy recently concerning the chemical rate equation to be used in nonisothermal systems. MacCallum and Tanner1 have proposed to modify the rate equation that is used for isothermal systems. In an answer to their paper, Hill2 has argued, quoting Kissinger3, that no change is necessary because in the expression
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(∂x/∂T)t is effectively zero.
TL;DR: In this article, the second-order rate constants are little influenced by the solvent, while the first-order rates vary with its nucleophilicity and steric requirements, and the results are discussed in terms of a suggested reaction mechanism.
TL;DR: The reaction of solid CuCl of average particle radius 8.75 µm with Cl2 gas at pressures of a few cmHg has been followed manometrically in the temperature range 75 to 130°C as discussed by the authors.
Abstract: The reaction of solid CuCl of average particle radius 8.75 µm with Cl2 gas at pressures of a few cmHg has been followed manometrically in the temperature range 75 to 130°C. Up to about CuCl1.2, the reaction proceeds at constant rate and shows zero or slightly negative apparent activation energy. In this region, the product continuously recrystallizes exposing fresh CuCl surface. At higher extent of reaction, the product layer builds up on each crystallite and diffusion becomes rate-controlling. From CuCl1.25 to CuCl1.62, the rate law is the appropriate modification of the parabolic law (a modified Ginstling-Brounshtein equation integrated over the particle size distribution), with tarnishing constant 2.18 × 10–12 cm2 s–1 at 130°C and activation energy 11.3 kcal mol–1. These values are compatible with migration of a defect at concentrations of the order of 0.1 % of cation concentrations in CuCl2. Propane affects the reaction rate in a manner depending on solid phase composition according to a two-peaked curve.
TL;DR: In this paper, a microwave-pulse temperature jump technique was used to study the proton transfer reaction between 2,4-dinitrophenol and tri-n-butylamine in a number of solvents.
Abstract: A microwave-pulse temperature-jump technique has been used to study the proton-transfer reaction between 2,4-dinitrophenol and tri-n-butylamine in a number of solvents. The effect of solvent viscosity of the rate constant at room temperature indicates that diffusion processes influence the reaction. From a study of the reaction in chlorobenzene over the temperature range +38 to –14°, the apparent activation energy for the forward reaction is –3.5 kJ mol–1. Possible reasons for the inadequacy of the Stokes-Einstein equation in the present systems are discussed.
TL;DR: The programme CORNEK is essentially a least-squares refinement programme applied to non-linear systems that uses directly the differential forms of the first derivatives of mass-balance and rate equations, thus avoiding the time-consuming derivations of near-equilibrium rate equations.
TL;DR: In this paper, the rate constant and initial conditions for first-order reactions from experimental data are evaluated using a stoicheiometric equation of the type of Guggenheim equation.
Abstract: The rates of chemical reactions are frequently followed by measuring the change of a physical property, such as pressure changes in a gaseous reaction, with time. In many cases a value of the physical property at zero time is necessary to calculate the rate constant. Often it is not practical to measure this quantity directly. Several methods are available to evaluate the rate constant and initial conditions for firstorder reactions from experimental data. The methods of Guggenheim,l H a r t l e ~ , ~ and Swinbourne3 depend on a reaction obeying the first-order rate law for a considerable part of the reaction. Swinbourne4 also described a method for determining the initial and final pressures and the rate constant of a gas reaction of order n. Roseveare5 has described a method which can be applied to a second-order reaction of a single reactant or equal concentrations of two reactants. All of these methods require orthogonal data. These methods are not satisfactory when the reaction is complicated by the decon~position of product as a consecutive reaction, or where a product inhibits or accelerates the rate of reaction. For such cases it is necessary to determine the initial physical property and rate constant from data over the early part of the reaction. In gaseous elimination reactions, such as the pyrolyses of alliyl halides and the hydrogen halide-catalysed pyrolyses of alcohols, the reaction can be represented by a stoicheiometric equation of the type
TL;DR: The concentration profile of a reactant species of an association-dissociation reaction occurring in an inhomogeneous diffusion barrier and the reaction rate profiles under stationary state conditions are obtained as solutions satisfying certain non-linear differential equations.
TL;DR: In this article, the rate equation r for the vapor phase reduction of nitrobenzene to aniline on copper-magnesia catalyst in the temperature range from 175°C to 220°C was obtained as follows;r=kPH/1+KNPN+KAPA+KWPW where k; rate constantPH, PN, PA and PW; partial pressure of hydrogen, Nitrobenzinzene, anilines, and waterKN, KA and KW; adsorption equilibrium constant of n-bensene,
Abstract: The kinetics of the catalytic vapor phase reduction of nitrobenzene to aniline over copper-magnesia catalyst in the temperature range from 175°C to 220°C was studied using a differential reactor. The rate of reaction was measured at atmospheric pressure with a high mole ratio of hydrogen to nitrobenzene. From the analysis of the experimental data, it was concluded that the rate-controlling step was the adsorption process of hydrogen on the copper-magnesia catalyst surface. According to the Langmuir-Hinshelwood mechanism, the final rate equation r for the vapor phase reduction of nitrobenzene on copper-magnesia catalyst in the temperature range from 175°C to 220°C was obtained as follows;r=kPH/1+KNPN+KAPA+KWPWwhere k; rate constantPH, PN, PA and PW; partial pressure of hydrogen, nitrobenzene, aniline and waterKN, KA and KW; adsorption equilibrium constant of nitrobenzene, aniline and waterThis equation will be useful if it can be used to predict the rate over a wide range of reactions and hence, be suitable for reactor design.
TL;DR: In this article, an exact generalized kinetic equation for the pseudo-unimolecular reaction X + A ⇌ B + X is presented, where the limits of long time and low X-particle density are taken, this equation reduces to the usual form of a chemical rate equation.
TL;DR: In this paper, a new approach to the thermodynamic analysis of the chemical transport reaction of the PCl3-Ga-H2 system is made, which suggests that the Ga-P stoichiometry condition does not hold in their gas phase components.
Abstract: A new approach to the thermodynamic analysis of the chemical transport reaction of the PCl3–Ga–H2 system is made. In this approach, the chemical reactions in reaction zone and deposition zone are analyzed individually. The result suggests that the Ga–P stoichiometry condition does not hold in their gas phase components and that the amount of gallium in gas phase in deposition zone is about 2.3 times as large as that of phosphorus. Growth rate of GaP was obtained from the calculated result as a function of temperature for various PCl3 to H2 feed ratios. A maximum growth rate occurs at around 730° and this temperature rises with increasing of PCl3 to H2 feed ratio. GaAs growth rate in the AsCl3–Ga–H2 reaction system is also obtained.
TL;DR: In this article, the authors investigated the kinetics and mechanism of the reaction of fluorine with CCl3F in a static system over the temperature range 491-598 K. The reaction was at least partly heterogeneous and independent of surface area to volume ratio, independent of diluent and uninhibited by either oxygen or chlorine monofluoride.
Abstract: The kinetics and mechanism of the reaction of fluorine with CCl3F have been investigated in a static system over the temperature range 491–598 K. The reaction was at least partly heterogeneous, independent of surface area to volume ratio, independent of diluent and uninhibited by either oxygen or chlorine monofluoride. The rate equation had the form : –d[CCl3F]/dt=k[F2]½[CCl3F]. The rate constants were reproducible using different Pyrex reaction vessels and fitted the Arrhenius equation : log k(l.½ mol–½ s–1)=(6.77 ± 0.10)–(19 250 ± 240)/2.303 RT. A full mechanism is presented which, by using steady-state assumptions, agrees with the experimentally derived rate equation.
TL;DR: In this paper, the physical interpretation of the phenomenological rate constants of chemical kinetics is examined for perfect gas systems, and the traditional view of the rate constants for elementary reactions as...
Abstract: The physical interpretation of the phenomenological rate constants of chemical kinetics is examined. For perfect gas systems, the traditional view of the rate constants for elementary reactions as ...
TL;DR: In this article, the hydrogen bromide-catalysed decomposition of t-butylamine into isobutene and ammonia has been studied in the temperature range 395-460°C.
Abstract: The hydrogen bromide-catalysed decomposition of t-butylamine into isobutene and ammonia has been studied in the temperature range 395–460°C. Individual runs follow the first-order rate law. The reaction is homogeneous, of the first order in each reagent, and molecular. Possible transition states for the catalysed reaction are examined. The variation of rate constant with temperature is described by the equation k2= 1012·21 exp (–29,329/RT) cm3 mol–1 s–1
TL;DR: In this paper, the effect of large electric fields on the ionization and recombination rate in a cesium plasma has been investigated and the effects of departures from the local thermodynamic equilibrium (LTE) conditions on the rate of ionization are discussed.
TL;DR: In this paper, the electron and population densities in nonequilibrium plasmas are computed by using the radiative transport equation, which is achived by using a set of rate equations dependent on geometrical dimensions.
Abstract: The electron and population densities in nonequilibrium plasmas are computed. For these computations rate equations dependent on geometrical dimensions were used. Such rate equations result if the unknown radiation intensity is eliminated. This is achived by using the radiative transport equation. Our density computations are valid for a quasi neutral plasma. The total pressure or the density of the heavy particles and the distribution of the electron energy have to be assumed. For homogeneous plasmas the inelastic collision processes and the radiation processes are considered. For inhomogeneous plasmas the diffusion processes of the excited atoms are considered in addition to the collision and radiation processes. Using this method of calculation the electron and population densities of a cesium plasma are computed. We assumed maxwellian distributions and a total pressure of 1 Torr.
TL;DR: In this paper, the rate equation corresponding to the title equations has been integrated by a numerical method, and it is shown that the form of the solution thus obtained depends only on the value of the equilibrium constant for the second step, K, and is independent of the value for the first step, k.
Abstract: The rate equation corresponding to the title equations has been integrated by a numerical method. It is shown that the form of the solution thus obtained depends only on the value of the equilibrium constant for the second step, K, and is independent of the value of the rate constant for the first step, k. In favourable cases it is possible by this numerical method to estimate the value of K as well as to determine the value of the rate constant from results of a single kinetic run. Various approximate solutions of the rate equation are discussed, and it is shown that accurate rate constants can be obtained by using the simple expressions corresponding to K= 1 and K=∞. The former approximation is useful only for small values of K, but the latter approximation is of use for any value of K greater than about 4.