Rate constants, calculating

One of the traditional ways of examining data from a kinetic analysis is that of preparing a table of the results and looking for consistency. In this method, the data correlated by a particular rate law are used to calculate 

TABLE 3.1 Calculated Rate Constants for the Hydrolysis of Ethyl Acetate Using the Experimental Data Shown in Table 2.1. 

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Observed and Calculated Rate Constants for Second-Order Gas-Phase Reactions... 

In principle it should be possible to predict quantitatively the reactivity of such species containing nucleophilic homolytic leaving groups towards diazonium ions, by using a dual parameter equation. One parameter serves as a measure of the donor property of the particle the other parameter is the redox potential. However, the complex nature of kinetics of homolytic dediazoniations is likely to be a great obstacle in attempts to calculate rate constants referring only to the radical-generation step. 

All the data (linear correlation of k2 and k3 in Scheme 11-1, and the values for Na,Np-rearrangement for complexed and free diazonium ions) indicate that the dediazoniation of the complexed diazonium ions proceeds only through the CT complex. The calculated rate constants for dediazoniation of complexed diazonium ions ( 3 in Scheme 11-1) are, however, not identical with k2 in Scheme 11-2, as the complexed species are present partly as CT complexes and partly as IC complexes. We see, at least at present, no possibility of determining the ratio [ArNJ. .. Crown]CT/ [ArNJ. .. Crown]IC which would be necessary for calculating k2 and for checking whether k2 is really zero. Nevertheless, k2 is likely to be much smaller than k2. 

The method of least squares provides the most powerful and useful procedure for fitting data. Among other applications in kinetics, least squares is used to calculate rate constants from concentration-time data and to calculate other rate constants from the set of -concentration values, such as those depicted in Fig. 2-8. If the function is linear in the parameters, the application is called linear least-squares regression. The more general but more complicated method is nonlinear least-squares regression. These are examples of linear and nonlinear equations ... 

This relationship can be used to calculate rate constant k at any time f of the isothermal run. This is illustrated by the following example. 

A quantum-mechanical description of spin-state equilibria has been proposed on the basis of a radiationless nonadiabatic multiphonon process [117]. Calculated rate constants of, e.g., k 10 s for iron(II) and iron(III) are in reasonable agreement with the observed values between 10 and 10 s . Here again the quantity of largest influence is the metal-ligand bond length change AR and the consequent variation of stretching vibrations. 

Calculating rate constants for the formation and breakdown of tetrahedral intermediates is possible provided the corresponding equilibrium constant is known. The mechanism must be known or postulated these mechanisms often involve proton transfer steps. 

Theoretical calculations published concurrently appeared to support these suggestions. Inclusion of tunneling corrections for the calculated rate constants was found to lower the apparent and A values, consistent with the experimentally... 

Statistical rate theories have been used to calculate rate constants for gas-phase Sn2 reactions.1,7 For a SN2 reaction like Cl" + CH3Clb, which has a central barrier higher than the reactant asymptotic limit (see Figure 1), transition state theory (TST) assumes that the crossing of the central barrier is rate-limiting. Thus, the TST expression for the SN2 rate constant is simply,... 

Errors in Calculated Rate Constants Caused by Analytical Errors... 

It has been found that both the anhydrous Form III and dihydrate phases of carbamazepine exhibit fluorescence in the solid state [78]. The fluorescence intensity associated with the dihydrate phase was determined to be significantly more intense than that associated with the anhydrate phase, and this difference was exploited to develop a method for study of the kinetics of the aqueous solution-mediated phase transformation between these forms. Studies were conducted at temperatures over the range of 18 40 °C, and it was found that the phase transformation was adequately characterized by first-order reaction kinetics. The temperature dependence in the calculated rate constants was used to calculate activation energy of 11.2 kCal/ mol (47.4 cal/g) for the anhydrate-to-dihydrate phase conversion. 

Recently Bogdal [48] observed, using kinetic studies, greater MW rate enhancements when the Knoevenagel reaction of salicylaldehyde with ethyl malonate (vide supra, Scheme 4.15) was performed in toluene than when ethanol was used as the solvent. The calculated rate constants in toluene solution were more than three times higher under MW irradiation than under conventional conditions, whereas the rate constants of the reaction in ethanol were the same, within experimental error, under both heating methods. 

Calculated Rate Constants, Activation Energies, Equilibrium Constants, and Related Experimental Data, for Addition of CO and H2 to 3Fe(CO)4 at 300 K... 

We note at this point that the nonadiabatic-transition state method used here (6,19,77) is not expected to be able to give quantitative agreement with experimental rate constants. There are too many factors that are treated approximately (or not at all) in this theory for such performance to be possible. One of the key difficulties is that calculated rate constants are very sensitive to the accuracy of the potential energy surface at room temperature, an error of lkcalmol-1 on the relative energy of the MECP relative to reactants will equate, roughly speaking, to an error by a factor of five on the calculated rate constant. Even though we... 

The TGA system was a Perkin-Elmer TGS-2 thermobalance with System 4 controller. Sample mass was 2 to 4 mgs with a N2 flow of 30 cc/min. Samples were initially held at 110°C for 10 minutes to remove moisture and residual air, then heated at a rate of 150°C/min to the desired temperature set by the controller. TGA data from the initial four minutes once the target pyrolysis temperature was reached was not used to calculate rate constants in order to avoid temperature lag complications. Reaction temperature remained steady and was within 2°C of the desired temperature. The actual observed pyrolysis temperature was used to calculate activation parameters. The dimensionless "weight/mass" Me was calculated using Equation 1. Instead of calculating Mr by extrapolation of the isothermal plot to infinity, Mr was determined by heating each sample/additive to 550°C under N2. This method was used because cellulose TGA rates have been shown to follow Arrhenius plots (4,8,10-12,15,16,19,23,26,31). Thus, Mr at infinity should be the same regardless of the isothermal pyrolysis temperature. A few duplicate runs were made to insure that the results were reproducible and not affected by sample size and/or mass. The Me values were calculated at 4-minute intervals to give 14 data points per run. These values were then used to... 

Earlier (see Chapter 4) a great discrepancy was noticed between the calculated rate constants for the reactions... 

Rate constants and turnover calculations are sensitive to the initial data. For example, sensitivity analysis showed that mixing the NHC into a larger amount of soil impacts the calculated rate constants. For example, if only the SOC in the 0-15 cm zone was considered (SOC = 26,750 kg C ha-1) for data from Larson et al. (1972), then kSU( was 0.14 g (g SOC year)-1. However, if the 0-30 cm soil zone was considered (SOC = 53,500 kg C ha-1), then NHC was 0.28 g C (g SOC year)-1. In these calculations, increasing the soil depth did not impact ksoc... 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

The preferred general method is the vacuum reaction calorimeter because of its wide range and flexibility, and because the enthalpy of the reactions is a good indicator of whether a polymerisation has gone to completion in any case, tests for residual monomer by glc must not be omitted. The complete reaction curve, however acquired, can reveal not only the internal order of a reaction, and whether it changes with conversion, and it is a far firmer base for calculating rate-constants than the initial rate or a maximum rate. 

The new treatment of kinetic data described here is generalised in this paper and used to calculate rate constants. 

Cho and McGrath [19] calculated rate-constants ( pzn in their notation) for the polymerisations of nBVE and ethyl vinyl ether (EVE) in hexane and CC14 at various temperatures for initiation by HI + I2. The polymerisations had z = 0 and were of first order with respect to [HI]C and [I2]0. so that... 

Find the order of reaction. Calculate rate constant and the rate of decomposition of A, when [A] = 0.45 mol dm 3. 

Vibrational frequencies for various normal modes must be estimated and active as well as inactive energies should be decided. Numerical methods may be used to calculate rate constant k at various concentrations obtained by RRKM theory. The rate constant has been found to be same as given by conventional transition state theory, i.e. 

Table 7 Calculated rate constants of outer-sphere ET model reactions of OsCl6 /A and OsCl6 l A type for substrates of three representative standard potentials, 1.5, 1.9 and 2.2 V. ... 

Table 3 shows the atmospheric lifetime for eleven PAH with respect to gas-phase reaction with OH and NO3 radicals, O3 and N2O5. This was calculated from the estimated and calculated rate constants. It is evident that most of the nitroarenes formed under ambient atmospheric conditions were produced by reaction of PAH with OH. The PAH reaction with NO3 radical was also considered as an important step because it resulted in the formation of nitroarenes from the N2O5 reaction with gas-phase PAH. 

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