Time dependent rate coefficient

Since the study of diffusion-limited reactions in solution seeks to discover more about the nature of the reaction path, the nature of the encounter pair, the energetics of the reaction and possibly the rate of reaction of the encounter pair, ftact, it is to be recommended that experimentalists actively seek to measure the diffusion coefficients of the reactants (or similar species), as well as any other parameters which may have an important bearing on the rate coefficient. By so doing, some of the uncertainty in estimating encounter distance may be removed and inconsistencies between diffusion coefficients measured independently and those obtained from an analyses of rate coefficient time dependence may provide valuable insight into the nature of the diffusion process at short distances. 

Specific interface in gas/liquid systems Mass-transfer coefficient Time-dependent dispersion coefficient Knudscn number Reaction rate constant... 

If a system experienced a complicated thermal history, the rate coefficient would depend on time and the solution to the rate equation would be more complicated. For the special case of reaction kinetics described by one single rate coefficient, the concentration evolution with time can be solved relatively easily. 

These equations are linear differential equations with time-varying coefficients since the hazard rates are time-dependent and may be presented in matrix form as... 

Reaction Rate The time dependent evolution of the concentrations of reactant ca or product c taking the stoichiometric coefficients into account, is a measure for the reaction rate or its velocity, cf. Equation (3). The convention requires a negative sign for reactants, indicating consumption, and a positive sign for the products, indicating formation ... 

As it has appeared in recent years that many hmdamental aspects of elementary chemical reactions in solution can be understood on the basis of the dependence of reaction rate coefficients on solvent density [2, 3, 4 and 5], increasing attention is paid to reaction kinetics in the gas-to-liquid transition range and supercritical fluids under varying pressure. In this way, the essential differences between the regime of binary collisions in the low-pressure gas phase and tliat of a dense enviromnent with typical many-body interactions become apparent. An extremely useful approach in this respect is the investigation of rate coefficients, reaction yields and concentration-time profiles of some typical model reactions over as wide a pressure range as possible, which pemiits the continuous and well controlled variation of the physical properties of the solvent. Among these the most important are density, polarity and viscosity in a contimiiim description or collision frequency. 

There is one important caveat to consider before one starts to interpret activation volumes in temis of changes of structure and solvation during the reaction the pressure dependence of the rate coefficient may also be caused by transport or dynamic effects, as solvent viscosity, diffiision coefficients and relaxation times may also change with pressure [2]. Examples will be given in subsequent sections. 

Thachuk M and Schatz G C 1992 Time dependent methods for calculating thermal rate coefficients using flux correlation functions J. Chem. Phys. 97 7297-313... 

Figure B2.5.7 shows the absorption traces of the methyl radical absorption as a fiinction of tune. At the time resolution considered, the appearance of CFt is practically instantaneous. Subsequently, CFl disappears by recombination (equation B2.5.28). At temperatures below 1500 K, the equilibrium concentration of CFt is negligible compared witli (left-hand trace) the recombination is complete. At temperatures above 1500 K (right-hand trace) the equilibrium concentration of CFt is appreciable, and thus the teclmique allows the detennination of botli the equilibrium constant and the recombination rate [54, M]. This experiment resolved a famous controversy on the temperature dependence of the recombination rate of methyl radicals. Wliile standard RRKM theories [, ] predicted an increase of the high-pressure recombination rate coefficient /r (7) by a factor of 10-30 between 300 K and 1400 K, the statistical-adiabatic-chaunel model predicts a...

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

The rate-acidity profile for pyrimidin-2-one indicated reaction on the free base but since the derived second-order rate coefficient is 104 times greater than that for 2-pyridone, and the acidity dependence in the H0 region was also greater, the slope of log kt versus —H0 plot being 0.45, cf. 0.15 for 2-pyridone reaction was, therefore, postulated as occurring via a covalent hydrate, hydration taking place at the 4 position. Methyl substitution increased the rate as expected and N-methyl substitution produced a larger effect than 4,6-dimethyl substitution and this may be due to alteration of the amount of covalent hydration at equilibrium. The data... 

These relations between the various coefficients are valid provided that the transfer rate is linearly related to the driving force and that the equilibrium relationship is a straight line. They are therefore applicable for the two-film theory, and for any instant of time for the penetration and film-penetration theories. In general, application to time-averaged coefficients obtained from the penetration and film-penetration theories is not permissible because the condition at the interface will be time-dependent unless all of the resistance lies in one of the phases. 

A dynamic ordinary differential equation was written for the number concentration of particles in the reactor. In the development of EPM, we have assumed that the size dependence of the coagulation rate coefficients can be ignored above a certain maximum size, which should be chosen sufficiently large so as not to affect the final result. If the particle size distribution is desired, the particle number balance would have to be a partial differential equation in volume and time as shown by other investigators ( ). 

A few results have been reported on the oxidation of cyclohexanol by acidic permanganate In the absence of added fluoride ions the reaction is first-order in both alcohol and oxidant , the apparent first-order rate coefficient (for excess alcohol) at 25 °C following an acidity dependence k = 3.5-1-16.0 [H30 ]sec fcg/A , depends on acidity (3.2 in dilute acid, 2.4 in 1 M acid) and D2o/ H20 is f-74. Addition of fluoride permitted observation of the reaction for longer periods (before precipitation) and under these conditions methanol is attacked at about the same rates as di-isopropyl ether, although dioxan is oxidised over twenty times more slowly. The lack of specificity and the isotope effect indicates that a hydride-ion abstraction mechanism operates under these conditions. (The reactivity of di-isopropyl ether towards two-equivalent oxidants is illustrated by its reaction with Hg(II).) Similar results were obtained with buffered permanganate. 

The relative rate coefficient, k Jk -j, can be determined by assuming a steady-state with respect to arsenic(IV). According to the data given in Table 11 the value of k lk j depends on the ionic strength and the pH. Dependence on the pH can be explained by the fact that arsenic(IV) reacts with FeOH ions, present at low acid concentrations, 90 times faster than with Fe ions. 

Study the effect of varying rate coefficients k], k2, k3, in the range (0.001 to 0.1) on the time-dependent concentrations of A, B, C and D. Note that Cb and Cc pass through maximum values. 

Fig. 3.3.7 Time dependence of the axial dispersion coefficients D for water flow determined by NMR horizontal lines indicate the asymptotic values obtained from classical tracer measurements. (a) Water flow in packings of 2 mm glass beads at different flow rates and (b) water flow in catalyst.

Equation (75) shows that (u(t) is an exponentially decaying function for long times with a decay constant /p. For very massive B particles M N mN with M/mN = q = const, the decay rate should vary as 1 /N since p = mNq/ (q + 1). The time-dependent friction coefficient (u(t) for a B particle interacting with the mesoscopic solvent molecules through repulsive LJ potentials... 

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