Reaction rate constants Arrhenius forms

Section 5.1 shows how nonlinear regression analysis is used to model the temperature dependence of reaction rate constants. The functional form of the reaction rate was assumed e.g., St = kab for an irreversible, second-order reaction. The rate constant k was measured at several temperatures and was fit to an Arrhenius form, k = ko exp —Tact/T). This section expands the use of nonlinear regression to fit the compositional and temperature dependence of reaction rates. The general reaction is... 

Solving for rates of production of chemical species requires as input an elementary reaction mechanism, rate constants for each elementary reaction (usually in Arrhenius form), and information about the thermochemistry (Aff/, 5, and Cp as a function of temperature) for each chemical species in the mechanism. 

The reaction rate constant for each elementary reaction in the mechanism must be specified, usually in Arrhenius form. Experimental rate constants are available for many of the elementary reactions, and clearly these are the most desirable. However, often such experimental rate constants will be lacking for the majority of the reactions. Standard techniques have been developed for estimating these rate constants.A fundamental input for these estimation techniques is information on the thermochemistry and geometry of reactant, product, and transition-state species. Such thermochemical information is often obtainable from electronic structure calculations, such as those discussed above. 

Precision of Activation Energy Measurements. The activation energy of a reaction can be determined from a knowledge of the reaction rate constants at two different temperatures. The Arrhenius relation may be written in the following form. 

Comparison of this equation with the Arrhenius form of the reaction rate constant reveals a slight difference in the temperature dependences of the rate constant, and this fact must be explained if one is to have faith in the consistency of the collision theory. Taking the derivative of the natural logarithm of the rate constant in equation 4.3.7 with respect to temperature, one finds that... 

Substituting the Arrhenius form of the reaction rate constants,... 

Here Cp is a heat capacity, k the thermal conductivity, and Q the reaction exothermicity. The term A exp [ — E/RT is simply the Arrhenius form of the reaction rate constant. 

Reaction rate constants usually are highly temperature dependent. A modified three-parameter Arrhenius form,... 

The question now is how these energy profiles relate to reaction rates. Remember that the reaction rate constant r of an elementary process is described in the Arrhenius form as (5, 106)... 

Comparison of Reaction Rate Constants. The calculated values for ki and k2 are listed in Table I. Figure 4 contains a plot of log (k2) vs. 1/T for the reaction between bisphenol A-phenoxide salt and 4,4 -dichlo-rodiphenylsulfone. This yielded an activation energy of 20.3 kcal../mole with a standard deviation of 0.9 kcal./mole. The other activation energies in Table I were determined by using the values for k at just two temperatures and the following form of the Arrhenius equation ... 

Calculate the reaction rate constant at the reactor operating temperature. Since the temperature in the reactor is not 25°C (where the value for the reaction rate constant is known), the rate constant must be estimated at the reactor temperature. The Arrhenius form of the rate constant may be used to obtain this estimate ... 

The temperature dependence of the forward reaction rate constant can be described by Arrhenius form... 

Mathematically, the combustion process has been modelled for the most general three-dimensional case. It is described by a sum of differential equations accounting for the heat and mass transfer in the reacting system under the assumption of energy and mass conservation laws At present, it is impossible to obtain an analytical solution for the three-dimensional form. Therefore, all the available condensed system combustion theories are based on simplified models with one-dimensional or, at best, two-dimensional heat and mass transfer schemes. In these models, the kinetics of the chemical processes taking place in the phases or at the interface is described by an Arrhenius equation (exponential relationship between the reaction rate constant and temperature), and a corresponding reaction order with respect to reactant concentrations. 

Fig. IV.3. Graphical methods of representing the temperature dependence of specific reaction rate constants, (a) For the Arrhenius equation in the form In A = In A — E/RT,...

Thermal effects constitute a significant portion of the study devoted to catalysis. This is true of electrochemical reactions as well. In general the reaction rate constants, diffusion coefficients, and conductivities all exhibit Arrhenius-type dependence on temperature, and as a rule of the thumb, for every 10°C rise in temperature, most reaction rates are doubled. Hence, temperature effects must be incorporated into the parameter values. Fourier s law governs the distribution of temperature. For the example with the cylindrical catalyst pellet described in the previous section, the equation corresponding to the energy balance can be written in the dimensionless form as follows ... 

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