Rate constant energy dependence

The attenuation of ultrasound (acoustic spectroscopy) or high frequency electrical current (dielectric spectroscopy) as it passes through a suspension is different for weU-dispersed individual particles than for floes of those particles because the floes adsorb energy by breakup and reformation as pressure or electrical waves josde them. The degree of attenuation varies with frequency in a manner related to floe breakup and reformation rate constants, which depend on the strength of the interparticle attraction, size, and density (inertia) of the particles, and viscosity of the Hquid. 

This model permits one to immediately relate the bath frequency spectrum to the rate-constant temperature dependence. For the classical bath (PhoOc < 1) the Franck-Condon factor is proportional to exp( —with the reorganization energy equal to... 

Such reactions are often exothermic and the role of the third body is to carry away some of the energy released and thus stabilize the product molecule. In the absence of a collision with a third body, the highly vibrationally excited product molecule would usually decompose to its reactant molecules in the timescale of one vibrational period. Almost any molecule can act as a third body, although the rate constant may depend on the nature of the third body. In the Earth s atmosphere the most important third-body molecules are N2 and O2. 

A thermodynamically unstable structure can exist when its conversion to some other structure proceeds at a negligible rate. In this case we call the structure metastable, inert or kinetically stable. Since the rate constant k depends on the activation energy Ea and the temperature according to the Arrhenius equation,... 

The forward and backward activation free energies and the corresponding rate constants thus depend on an extrinsic factor, the standard free energy of the reaction, AG° = E — E°, and an intrinsic factor, the standard activation free energy, that reflects the solvent and internal reorganization energy, Aq and A [equation (1.31)]. 

Eor an electrochemical reaction the rate of reaction v and the rate constant k depend on potential E specifically, the potential difference across electrode-solution interphase Acf) through the electrochemical activation energy AGf. Thus, the central problem here is to find the function... 

The constant k is used as the rate constant and depends only on temperature and activation energy. If the temperature of a reaction medium is altered, the k value also changes. 

This quadratic dependence of the activation energy on the reaction free energy leads to the prediction of an inverted region in which the reaction rate constant (which depends on AG ) falls when the overall reaction free energy becomes more favourable. This is readily seen from the simple picture shown in Figure 4.14. When the intersection point of the wells leads to A G = 0 the reaction becomes free of an activation barrier, but as the products well sinks deeper the point of intersection rises again. 

If external mass transport limitations strongly dominate, the rate becomes equal to the mass transfer rate, (eq 15). Hence, a first-order dependency is observed and, since the mass transfer coefficient is fairly independent of the temperature the apparent activation energy is negligible. However, due to the existing correlations, the observed rate constant is dependent on the flow rate and particle size. 

G is the ratio of the dimensionless number SD in the disturbed region to its value in the normally-operating part of the bed. SD contains the activation energy, heat of reaction, inlet temperature and bed height, all of which have fixed constant values in all regions of the bed. It also contains the possibly variable quantities C, k and F. C is the average heat capacity of the fluid, and depends on the local phase ratio. kg is the specific rate constant, and depends on the local catalyst density and the phase holdup. F is the local average linear velocity, which can vary from point to point for a variety of reasons. 

Like equilibrium constants, rate constants also depend on environmental factors such as pressure and, especially, temperature. An increase in temperature usually gives rise to an increase in the chemical reaction rate, because molecules are moving faster and colliding more frequently with greater energy. If rate constants are known for two different temperatures, the rate constant for any other temperature can be calculated using the Arrhenius rate law,... 

The conclusion that most amine quenching occurs at the pre-equilibrium limit and therefore rate constants will depend, in part, on the equilibrium constant for exciplex formation, which in turn depends on the amine electron-donating ability, raises the possibility of an alternative mechanism. Here formation of the exciplex would simply facilitate the electronic to vibrational energy transfer discussed for solvent quenching where kA (cf. Eq. (21)) replaces fcisc in Eq. (34). This would be much more effective than solvent quenching where interaction simply involves encounter complexation. That such a mechanism does not operate is demonstrated in Figure 6 which shows a plot [82] of the first-order constant for decay of 02( Ag) luminescence in benzene as a function of DABCO and DABCO-2HI2 [83], There is clearly no isotope effect and the mechanism of Eq. (31) appears very firmly established. A similar conclusion has been drawn from recent work [84] which shows that, as expected, hydrazines also quench 02(1Ag) via the same mechanism. The hydrazine 4 is a particularly efficient quencher with kq values in benzene and acetonitrile about twice those of strychnine. 

AG. The similar rate maxima occur because the continuum of final proton states provides many opportunities for the quantum transition. This is in contrast to etpt, where both initial and final electron/proton states are bound, and the rate constant values depend quite sensitively on the number and energies of the final bound states. 

Values of the rate constants and activation energies as far as they have been obtained are given in Table II. For these reactions as for the halide reactions the specific rate constants are dependent on ionic strength. 

The forward and reverse rate constants,k/,kr. quantify the fraction of collisions that result in successful reactions. We expect these rate constants to depend on temperature because the frequency and energy of collisions increases with the mean velocity of the molecules which increases with temperature. A kinetic view of the equilibrium condition is obtained by setting the net reaction rate to zero. 

Finally, we return to the point that unless V-V energy exchange is eliminated, experimental measurements are likely to provide a rate constant which describes the net result of removal of molecules from more than one excited level. If the state-specified rate constants increase rapidly with v then the observed rate constant may depend on the extent of the initial excitation. This problem is most likely to arise with reactive systems, since the rate constants for removal of molecules from successively higher levels can increase rapidly for such systems. 

Equations (22) point out an interesting fact the coefficients and aj2 are weakly dependent on the temperature, as the activation energies E5 and E have values which are very close to one another. These coefficients will therefore be calculated only once at a temperature in the middle of the range being studied. On the contrary, it is clear that the rate constant kjg depends strongly on the temperature, as is the case for k3 andk. ... 

A space-charge region is also formed, and the bands bent, when a potential is apphed to the electrode. As above, the band edges remain pinned at the electrode/solution interface, which arises because the potential drop between the bulk semiconductor and the solution is essentially entirely across the space-charge region rather than at the semiconductor interface. As a consequence, the intrinsic electron transfer rate constant is independent of applied potential. Nevertheless the current (and hence the effective rate constant) does depend on the apphed potential because the concentration of electrons (the majority carriers) at the electrode surface relative to its bulk concentration has a Boltzmann dependence on the energy difference between the band edge and the interior of the electrode. (The Fermi Dirac distribution reduces to a Boltzmann distribution when E > Fp-)... 

The energy barrier is related to the activation energy, inherent to each reaction. If the activation energy is higher than the energy barrier, the reaction will occur. Thus, Arrhenius defined a reaction rate constant which depends mainly on the temperature and is a function of activation energy E. The Arrhenius equation shows that the reaction rate constant varies exponentially with the temperature, according to Equation 3.30. 

Because of the activation energy of about 125 kJ/mol which is necessary for such reactions, these times depend considerably on the temperature. The rate constant also depends somewhat on the solvent, as with BPO (Table 20-2), but not so much as the rate constants of ionic reactions. Therefore the mean decomposition constant of initiator mixtures can be compiled additively from the individual constants. 

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