Showing papers on "Rate equation published in 1976"
TL;DR: In this paper, a fundamental rate equation for the dissolution of calcite in a pure 0.7 M KC1 solution has been determined, where the reaction rate is pH independent above pH = 7.5.
194 citations
TL;DR: Partially purified glucose isomerase from a Streptomyces species was immobilized on porous glass particles and studied for various characteristics concerning its use as an industrial catalyst as mentioned in this paper, and the activities were investigated in relation to the reaction parameters and the enzyme deactivation was studied systematically under various reaction conditions.
Abstract: Partially purified glucose isomerase from a Streptomyces species was immobilized on porous glass particles and studied for various characteristics concerning its use as an industrial catalyst. The activities were investigated in relation to the reaction parameters and the enzyme deactivation was studied systematically under various reaction conditions. The half-life of the immobilized enzyme was found to exceed 200 days at 50°C. The rate equation of the reversible glucose ⇄ fructose reaction was derived and the kinetic constants were determined. The rate equation was found to be in good agreement with experimental data for both forward and reverse reactions. The degree of diffusional effects was experimentally measured and theoretically analyzed.
77 citations
TL;DR: In this paper, an experimental investigation of the shock initiation of liquid nitromethane is described, which supports the thermal ignition model of one-dimensional shock initiation, and the interferometer initiation time data are analyzed to obtain kinetic parameters in a first order overall rate law.
70 citations
TL;DR: In this paper, the authors presented detailed numerical calculations of surface-catalyzed chemical reaction rates in the Brownian-motion rate theory of Kramers, for a wide range of values of the friction coefficient, the principal parameter of the theory, are obtained for the first time.
Abstract: This paper presents detailed numerical calculations of surface-catalyzed chemical reaction rates in the Brownian-motion rate theory of Kramers. Accurate values of the rate for a wide range of values of the friction coefficient, the principal parameter of the theory, are obtained for the first time. A one-parameter empirical formula is obtained which gives the rate in quite general circumstances. Application of the results to calculation of catalytic rates at metal surfaces is discussed.
66 citations
TL;DR: In this article, computer molecular dynamics studies are carried out on a binary mixture of hard disks in which one species can convert to the other by photoinduced transition or by collision-induced reactions.
Abstract: Computer molecular dynamics studies are carried out on a binary mixture of hard disks in which one species can convert to the other by photoinduced transition or by collision‐induced reactions. Phenomenological rate laws are postulated for a simple model system which is then shown to possess multiple steady‐state concentrations under suitable conditions. The computer results for a 450‐disk system show that such solutions do exist, and that in general the postulated rate equations adequately describe the system behavior. Concentration fluctuations are examined and found to agree with the predictions of a master equation. Fluctuations are found to be enhanced near a point of marginal stability.
57 citations
TL;DR: In this paper, precise absorption measurements on vibration-rotation transitions pumping far-infrared (FIR) laser transitions in HCOOH, CH 3 OH, CH3 I and CH 3 F were carried out using a frequency-stabilized and intensity-controlled CO 2 laser.
Abstract: Precise absorption measurements on vibration-rotation transitions pumping far-infrared (FIR) laser transitions in HCOOH, CH 3 OH, CH 3 I and CH 3 F were carried out using a frequency-stabilized and intensity-controlled CO 2 laser. Linear absorption coefficients and saturation intensities as a function of pressure are obtained. With the help of a rate equation model it is concluded that within the pressure range relevant for CW FIR lasers the rate of deexcitation of pumped molecules is limited by rotational relaxation. For CH 3 F, limitation by diffusion of excited molecules to the cell wall is observed.
55 citations
TL;DR: In this paper, it was shown that the rate of an endothermic decomposition reaction may also depend upon rates of diffusion in the reactant phase, rate of transfer of the solid reaction product at the reaction interface, and/or the thermodynamic stability of the product.
Abstract: When the solid product of an endothermic decomposition reaction is porous, the rate-limiting chemical step is usually assumed to be a surface step of the gaseous product or of a precursor of that product It is shown here that the rate of such a reaction may also depend upon rates of diffusion in the reactant phase, the rate of transfer of the solid reaction product at the reactant-product interface, and/or the thermodynamic stability of the solid product Rate equations are derived for the six possible limiting cases when either a single step or a coupled pair of steps of a decomposition reaction significantly influence its rate Data for calcite (CaCO/sub 3/) decomposition are shown to be most simply explained as reflecting formation of a known metastable modification of calcium oxide with near equilibrium conditions maintained for each reaction step except desorption of carbon dioxide If this explanation is correct the free energy of formation of the metastable oxide from the stable oxide should be found to be about +7500 - 5T cal/mole (auth)
49 citations
TL;DR: In this paper, rate equations for an optically-pumped, far infrared laser were developed and solved for the case of a high intensity, pulsed pump beam, conditions for saturated absorption and for saturated stimulated emission were established.
42 citations
TL;DR: In this article, the rate equations governing the temporal evolution of photon densities and level populations in pulsed F+H2→HF+H chemical lasers are solved simultaneously for all relevant vibrationalrotational levels and vibrational-rotational P•branch transitions.
Abstract: The rate equations governing the temporal evolution of photon densities and level populations in pulsed F+H2→HF+H chemical lasers are solved for different initial conditions. The rate equations are solved simultaneously for all relevant vibrational–rotational levels and vibrational–rotational P‐branch transitions. Rotational equilibrium is not assumed. Approximate expressions for the detailed state‐to‐state rate constants corresponding to the various energy transfer processes (V–V, V–R,T, R–R,T) coupling the vib–rotational levels are formulated on the basis of experimental data, approximate theories, and qualitative considerations. The main findings are as follows: At low pressures, R–T transfer cannot compete with the stimulated emission, and the laser output largely reflects the nonequilibrium energy distribution in the pumping reaction. The various transitions reach threshold and decay almost independently and simultaneous lasing on several lines takes place. When a buffer gas is added in excess to the...
38 citations
TL;DR: In this article, a theoretical prediction based on a rate equation model involving only one parameter, the effective phonon frequency of the host was presented, within the assumptions of small ion-lattice coupling and small oscillator strength variation.
36 citations
TL;DR: Steady state solutions of master equations with one variable are constructed in this article, based on a transformation of the original equation for the probability into one for a slowly varying function, which is of general applicability and particularly useful in obtaining solutions in the case where detailed balance does not hold.
Abstract: Steady state solutions of master equations with one variable are constructed. The method of solution is based on a transformation of the original equation for the probability into one for a slowly varying function. The method is of general applicability and is particularly useful in obtaining solutions in the case where detailed balance does not hold. Examples of such systems in chemical reaction models and the two photon laser are discussed.
TL;DR: In this paper, a tunable flash lamp pumped dye laser was used for excitation of the sodium contained in a vapour cell, and three ways of describing the interaction of the beam with the atomic ensemble were discussed and appropriate rate equations for population was derived.
TL;DR: In this article, the authors suggest a method of obtaining initial rates that lead to a simple and accurate method of determining rate constants, particularly valuable when side reactions complicate a mechanism and thus limit the use of an integrated rate equation.
Abstract: The authors suggest a method of obtaining initial rates that lead to a simple and accurate method of determining rate constants. The method is particularly valuable when side reactions complicate a mechanism and thus limit the use of an integrated rate equation.
TL;DR: In this paper, a comparison between quantum and classical mechanical reaction probabilities and rate constants for collinear H + Cl2 and D+ Cl2 systems is made, and different quasi-classical methods are compared and the QCRF method which fulfils the principle of detailed balance is suggested.
TL;DR: In this article, the reaction rate for F+H2→HF+H was determined to be 1.00±0.08×1013 cm3mole−1⋅sec−1 at 298°K.
Abstract: The reactions between F and H2, H and F2, and F and CF3H were studied by using a fast flow reactor with electron spin resonance detection. The reaction rate constant for F+H2→HF+H was determined to be 1.00±0.08×1013 cm3 mole−1⋅sec−1 at 298 °K. The rate of the reaction H+F2→HF+F was determined in a system where known amounts of F and F2 were injected into a stream of H2 in He. Using finite difference techniques to solve the rate equations, the rate constant was found to be 0.26±0.06×1013 cm3 mole−1⋅sec−1. These reaction rates are in good agreement with other published values. The rate constant determined for the reaction F+CF3H→HF+CF3 was 0.90±0.08×1011 cm3 mole−1⋅sec−1. Techniques are presented which permit ESR to be used in studies of complex reaction mechanisms and in nonuniform flow velocities.
TL;DR: In this article, the thermal decomposition of nitrogen trifluoride has been investigated behind incident shock waves in dilute (1%) NF3/Ar mixtures over the temperature range 1150-1530 K and at 0.80-1.81 atm total pressures.
Abstract: The thermal decomposition of nitrogen trifluoride has been reinvestigated behind incident shock waves in dilute (1%) NF3/Ar mixtures over the temperature range 1150–1530 K and at 0.80–1.81 atm total pressures. The reaction kinetics were followed by monitoring NF2 radical concentrations in absorption at 260 nm. In several experiments, the infrared emission at 5.17 μ was also monitored. Under the experimental conditions used, the reversible dissociation NF3+M?NF2+F+M is known to be first order in both NF3 and Ar concentrations, and the low pressure limit second‐order rate constant was found to be k1(cm3 mol−1 s−1) =1016.61±0.13 ×exp[−(48.0±0.8 kcal mol−1)/RT] using an approximate iterative procedure. A numerical solution of the differential rate equation yielded comparable results. The temperature dependence of the absorption coefficient for NF2 and the equilibrium constant were also determined.
TL;DR: In this article, a general equation for instant current in an electrode reaction coupled with a preceding first-order chemical reaction was derived with the condition (k 1 + k 2 )≫1, k 1 and k 2 being the rate constants of the chemical reaction.
TL;DR: In this article, a time dependent, rate equation model is proposed to describe the main features of pump absorption and gain in pulsed CH 3 F lasers, which is relevant to the design of more efficient pulsed FIR sources.
TL;DR: In this paper, the effect of 10.8 μ radiation from a cw N2O laser on the gas surface N 2O-Cu reaction has been measured and the rate equations for energy transfer, optical excitation, and chemical reactions have been examined and, by adjusting the probability of the latter processes for reaction of ground state and vibrationally excited N 2 O, reasonable agreement with the experimental results is achieved.
Abstract: The effect of 10.8 μ radiation from a cw N2O laser on the gas–surface N2O–Cu reaction has been measured. The rate equations for energy transfer, optical excitation, and chemical reactions have been examined and, by adjusting the probability of the latter processes for reaction of ground state and vibrationally excited N2O, reasonable agreement with the experimental results is achieved. The probabilities of reaction were found to be 2×10−7 for the N2O(001) mode, and 3×10−11 for all remaining N2O modes. Although experimental results are not sufficiently accurate to specify these values precisely, they do demonstrate the type of information which can be gained from a detailed analysis of laser chemistry results.
TL;DR: In this article, the authors investigated the kinetics of iodide-ion oxidation by H2SO5 and H2PO5 and found an inverse linear dependence of the logarithms of the rate constants on the basiclty of the leaving group.
Abstract: The kinetics of iodide-ion oxidation by H2SO5 and H2PO5 have been investigated. The rate of the first reaction is given by equation (i) where k= 1.73 × 103 dm3 mol–1 s–1 in aqueous solution at 25 °C and I= 0.2 mol dm–3. Rate =k[HSO5–][I–](i) The rate equation for rhe second reaction is given by the three-term expression (ii) with k1= 1.57 × 103. k2= 57.5, and k3= 0.418 dm3 mol–1 s–1 under the same conditions. Rate =(k1[H3PO5]+k2[H2PO5–]+k3[HPO52–])[I–](ii) The effect of electrolyte addition has been investigated. An inverse linear dependence of the logarithms of the rate constants on the basiclty of the leaving group has been found. The mechanism is consistent with nucleophilic attack by I– on the external peroxidic oxygen leading to breaking of the oxygen–oxygen bond.
TL;DR: In this paper, the authors examined the gas kinetic and discharge properties of waveguide CO 2 laser and calculated the dependence of the population inversion and laser small-signal gain on gas pressure, gas mixture, pumping rate (discharge current), tube bore diameter, and wall temperature.
Abstract: Using a simple rate equation approach we examine the gas kinetic and discharge properties of waveguide CO 2 lasers. We calculate the dependence of the population inversion and laser small-signal gain on gas pressure, gas mixture, pumping rate (discharge current), tube bore diameter, and wall temperature. The results indicate, for example, that at a pressure of 50 torr and a tube-bore diameter of 0.125 cm, the gain is optimized with a gas mixture in the ratio CO 2 :N 2 :He of 1:0.75: 1.5. At higher pressures the gain is optimized by using more helium-rich mixtures. We also calculate the dependence of laser tunability on the gas kinetic properties and cavity losses. We find that for low-loss cavities the laser tunability may substantially exceed the molecular full width at half-maximum. Furthermore, the more helium-rich gas mixtures give greater tunability when cavity losses are small, and less tunability when cavity losses are large. The roles of the various gases in the waveguide CO 2 laser are the same as those in conventional devices. By contrast with conventional lasers, however, the waveguide laser transition is homogeneously broadened. Thus the dependence of gain on gas pressure and other kinetic properties differs substantially from that predicted by scaling results from conventional low-pressure lasers.
TL;DR: In this article, a kinetic study of the reaction of phenol with formaldehyde has been carried out at temperatures of 65° ± 0.05°C, 70° ± 1.20, and 80° ± 2.00°C using hydrochloric acid as catalyst.
Abstract: A kinetic study of the reaction of phenol with formaldehyde has been carried out at temperatures of 65° ± 0.05°C, 70° ± 0.05°C, 75° ± 0.05°C, and 80° ± 0.05°C using hydrochloric acid as catalyst. The pH maintained was 1.14, 1.32, 2.20, and 3.00. The reaction follows a second-order rate law. The rate is found to increase with decrease in pH. The overall rate constants are resolved into step rate constants. The values of the Arrhenius parameters and the entropy of activation for the overall reaction as well as for the step reactions have been calculated. A mechanism conforming to the energies and entropies of activation of the reaction has also been suggested.
TL;DR: In this article, the authors extend the results of Szepe and establish necessary and sufficient conditions for a constant-conversion optimal policy in a continuous stirred-tank reactor, and a proof of the same for a plug-flow reactor with distributed control of temperature for a general irreversible reaction with separable kinetics.
TL;DR: In this article, it was shown that the rate law also applies to the reaction of ozone with ethylene and presumably with all lower alkenes, which accounts for the simplifed rate law found in the presence of excess oxygen.
Abstract: The rate law − d[O3]/ dt = k1[A][O3] + k3[A][O3]2/ (k4 + k5[O2]) has been found to obtain for the reaction of ozone with allene and with 1,2-butadiene. We now find that this rate law also applies to the reaction of ozone with ethylene and presumably with all lower alkenes. This generalizes the inhibiting effect of oxygen and accounts for the simplifed rate law found in the presence of excess oxygen. Oxygen itself is a product of the ozone–ethylene reaction, and we find that as [O3]0 increases, the (O2 formed)/(O3 used) ratio approaches 1.5. Values of k1, k3/k5 for ethylene are compared with those for allene, 1,3-butadiene, and propene. A generalized mechanism is postulated for the reaction of ozone with alkenes involving a chain sequence that produces oxygen and which accounts for the observed rate law. A specific mechanism is postulated for the reaction of O3 with ethylene, and the thermochemistry of the chain sequence is examined in detail.
TL;DR: In this article, a nonlinear dispersion relation for beam plasma oscillations is obtained by expanding the usual dispersion relations about both the most unstable root and the perturbed beam electron orbits.
Abstract: A nonlinear dispersion relation for beam plasma oscillations is obtained by expanding the usual dispersion relation about both the most unstable root and the perturbed beam electron orbits. This procedure leads to a nonlinear rate equation similar to the van der Pol equation.
TL;DR: Using the time dependent rate equations, the dynamical behavior of a high power pulsed N(2) laser radiating at 337.1 nm is derived and the output peak power and pulse width as a function of length and initial inversion density are shown.
Abstract: Using the time dependent rate equations, the dynamical behavior of a high power pulsed N2 laser radiating at 337.1 nm is derived. The threshold conditions are given. The output peak power and pulse width as a function of length and initial inversion density are shown. The theoretical results derived are compared with experimental data and are found to be in good agreement.
TL;DR: In this article, the excitation and subsequent decay of the E 3 Σg+ state of N2 was studied in a delayed coincidence experiment with a pulsed electron beam by monitoring the time and pressure dependence of the second positive C 3 Πu→B Πg(0,0) radiation at 3371 A.
Abstract: The excitation and subsequent decay of the E 3Σg+ state of N2 has been studied in a delayed coincidence experiment with a pulsed electron beam by monitoring the time and pressure dependence of the second positive C 3Πu→B 3Πg(0,0) radiation at 3371 A. The rate equations for the populations of the E and C states have been analyzed for the conditions of the experiment and the relevant equations obtained. The analysis shows the 3371 A radiation detected, at times long compared to the C state lifetime, to be associated with the initial E state population. Analysis of the time and pressure dependence of this long lived radiation yields a value of 3.8×103 sec−1 mtorr−1 for the collisional deactivation rate coefficient for the E state and a value of 1.9×103 sec−1 mtorr−1 for the energy transfer rate coefficient between the E and C states.
01 Jan 1976
TL;DR: Dimensional analysis provides one of the simplest and most versatile techniques for detecting algebraic mistakes and checking results available in enzyme kinetics.
Abstract: This chapter discusses the principles which explain the chemical kinetics occurring within the molecules. A chemical reaction can be classified either according to its molecularity or according to its order. The molecularity is defined by the number of molecules that are altered in the reaction. The order is a description of the kinetics of the reaction. For a simple reaction that consists of a single step, or for each step in a complex reaction, the order is usually the same as the molecularity. For a complex reaction, it is often not meaningful to define an order, as the rate often cannot be expressed as a product of concentration terms. The binding of a substrate molecule to an enzyme molecule is a typical example of a second-order bimolecular reaction in enzyme kinetics. The conversion of an enzyme substrate complex into products or into another intermediate is an example of a first-order unimolecular reaction. Constants of integration must always be included and calculated when kinetic equations are integrated as they are rarely found to be zero. Some reactions are observed to be zero order, that is, the rate is found to be constant, independent of the concentration of reactant. The simplest means of determining the order of a reaction is to measure the rate at different concentrations of the reactants. Dimensional analysis provides one of the simplest and most versatile techniques for detecting algebraic mistakes and checking results available in enzyme kinetics. The reaction velocities are profoundly influenced by temperature.
TL;DR: In this paper, the authors studied the kinetics of the reaction between nitrite and iodide in perchloric acid medium at 25 ± 0.1°C with an ionic strength of 0.2, and showed that in the presence of citric acid the rate increases, while the bromide ion exerts an inhibitory effect.
TL;DR: Saturated absorption of CO2-laser radiation in HCOOH gas was measured as a function of pressure and a simple rate equation model was used to calculate the saturation photon densities as mentioned in this paper.
Abstract: Saturated absorption of CO2-laser radiation in HCOOH gas was measured as a function of pressure. A simple rate equation model is used to calculate the saturation photon densities. It is concluded from these data that vibrational relaxation is collision predominant, and that decreased absorption, observed at high pressure and high radiation intensities, cannot be interpreted as diffusion bounded vibrational relaxation.