Isokinetic temperature calculation

Thus, y is the slope of the plot of A// against AG at the harmonic mean temperature, and from y the isokinetic temperature P is calculated. Tomlinson has shown many examples of this type of analysis. 

By repeating the calculation for various values of x, one can obtain y and Sx as functions of x and find the minimum of the latter by successive approximations. The value of x at this minimum (xo) gives the estimate of the isokinetic temperature Xo The corresponding values yo and So are obtained from eqs. (52) and (53) So has... 

When E = 0, log A is equal to B, and this corresponds to the specific rate constant for reaction at the isokinetic temperature kti. It is also easily shown that e — (Rfi) . The variation of ft with e over the temperature range of interest, in the reactions considered in the present review, is given in Fig. 1, which includes dashed lines for values of = + 0.05c, 0.10c, and 0.20e, corresponding to the influence of various levels of inaccuracy of data upon the calculated precision with which )3 may be determined. For example, when ode = 0.05 and /I = 700 K the deviation is + 30 K and this reduces to 15 K. when fi = 300 K other values are very readily found. 

Isokinetic temperatures (Tis), logarithms of apparent rate constants at this temperature (B) and parameters of Gamma-distribution function (a and b) of some pyrogenic oxides surface on chemisorption activation energy of organosilicon compounds calculated from the kinetic isotherms obtained in [4,134] and activation energies on free surface from [79]... 

The best way to correct determination of chemisorption activation parameters on heterogeneous oxides surface includes the calculation of its both distribution functions on the activation energy and on logarithms of the pre-exponential factor and their central moments from the kinetic isotherms measured at different temperatures. Obtained from such calculations the isokinetic temperature and logarithm of rate constant at this temperature reflect the gaseous organic compound reactivity toward active sites of the heterogeneous oxides surface. 

Equation 12.3 can be used for calculations of isokinetic temperatures corresponding to different decomposition modes. Table 12.1 presents as an example of the calculated results for the decompositions of CaCOs, Mg(OH)2, and Li2S04 H2O under the condition = 1 Torr. The experimentally obtained... 

There exists a linear dependence between the enthalpy and entropy values (Fig. 1). The slope of the straight line, which was considered as the isokinetic temperature, amounts to 3K. It was calculated by the least squares method. Similar values of the isokinetic temperature were obtained by means of the method of intersection of the Arrhenius lines. 

Second-order rate constants for the reaction of an amine with substituted benzyl bromides in methanol solution at several temperatures are shown in Table 6.18. Calculate AH and AS for the reaction of each of the benzyl bromides. Do the data exhibit an isokinetic relationship If so, what is the isokinetic temperature ... 

Temperature dependence of the fluorescence quantum yields and fluorescence lifetimes of frans-4,4 -di-fert-butylstilbene in n-hexane and n-tetradecane allowed to define the index of refraction dependence of the radiative rate constants, kf= (3.9 — 1.8) X 10 s, and fluorescence lifetime [78]. This relationship was used to calculate torsional relaxation rate constants ktp> for traws-4,4 -dimethyl- and frans-4,4 -di-ferf-butylstilbene in the n-alkane solvent series. It was found that activation parameters for ktp, based on Eyring s transition state theory, adhered to the medium-enhanced thermodynamic barrier model relationship, AHtp = AHt + aEr, and to the isokinetic relationship. The isokinetic relationship between the activation parameters for the parent frans-stilbene led to an isokinetic temperature of P = 600K and brings it into agreement with the isokinetic temperature for activation parameters based on estimated microviscosities, qp, experienced by stilbene in its torsional motion. The authors concluded that only microviscosities raflier than shear viscosities, q, can be employed in the expression ktp = ktSq — b, when a = b. These data clearly indicated the important role of the media dynamics in the stilbene cis-trans photoisomerization. 

Copolyester Synthesis. Copolycondensation has attracted considerable interest as a method of producing polymers with novel properties." In a series of studies the kinetics of the sequence distribution formation during the preparation of fully aromatic copolyesters were examined using p.m.r. spectroscopy and lanthanide-shift reagents." The calculated values of degrees of randomness confirmed the theory that the copolyesters obtained at the isokinetic temperature (200 °C) had a random distribution of comonomer units. Above and below this temperature there was a tendency for block copolymers to form. 

The existence of compensation behaviour can be accounted for as follows. All samples of calcite undergo dissociation within approximately the same temperature interval, many kinetic studies include the range 950 tolOOO K. The presence of COj (product) may decrease reactivity and a delay in heat flow into the reactant will decrease the reaction temperature. Thus, imder varied conditions, the reaction occurs close to a constant temperature. This is one of the conditions of isokinetic behaviour (groups of related reactions showing some variations of T within the set will nonetheless exhibit a well-defined compensation plot [61]). As already pointed out, values of A and E calculated for this reaction, studied under different conditions, show wide variation. This can be ascribed to temperature-dependent changes in the effective concentrations of reaction precursors, or in product removal [28] at the interface, and/or heat flow. The existence of the (close to) constant T, for the set of reactions, for which the Arrhenius parameters include wide variations, requires (by inversion of the argument presented above) that the magnitudes of A and E are related by equation (4.6). 

However, even Cremer (d), who reviewed this subject more than 10 years ago, expressed the suspicion that some cases of observed compensation might be caused by experimental error. Later NoUer and Schwab (7) were able to show how the diffusion of reactants into the pores of the solid catalysts may result in a frequency factor-activation energy relationship. Recently, Exner (5) demonstrated that published isokinetic relations are often fictitious because of the peculiar character of error distribution in the calculation of the frequency factor and activation energy from rate constants measured at different temperatures. 

For example, the oxidation of l-amino-3-arylguanidines by Ce(IV) was studied by Schelenz et al. (1982) in perchloric acid solutions for a homologous series of 17 substituted compounds at four different temperatures. The predominant products of this eight-equivalent oxidation are nitrogen and 3,6-diarylamino-l,2,3,5-terrazine compounds. For such well-defined systems, the resulting isokinetic and Hammet correlation relations calculated from the kinetic data present significant mechanistic information. 

In order to obtain a steady state from Eqs. 38 dissipative heat must be removed from the system. This is achieved by the last (thermostatting) terms of the last two equations in Eqs. 38. In this respect it is essential to observe that accurate values for Uj and A are needed. Any deviations from the assumed streaming and angular velocity profiles (biased profiles) will exert unphysical forces and torques which in turn will affect the shear-induced translational and rotational ordering in the system [209,211,212]. The values for the multipliers and depend on the particular choice of the thermostat. A common choice, also adopted in the work of McWhirter and Patey, is a Gaussian isokinetic thermostat [209] which insures that the kinetic and rotational energies (calculated from the thermal velocities p" and thermal angular velocities ot) - A ) and therefore the temperature are conserved. Other possible choices are the Hoover-Nose or Nose-Hoover-chain thermostats [213-216]. 

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