Arrhenius-form rate constant

Equation (25-2) is frequently used for a kinetic modehng of a burner using mole fractions in the range of 0.15 and 0.001 for oxygen and HC, respectively. The rate constant is generally of the following Arrhenius form ... 

III. The rate constant k is a function of the absolute temperature in the form that Arrhenius (1889) found ... 

We can make two different uses of the activation parameters AH and A5 (or, equivalently, E and A). One of these uses is a very practical one, namely, the use of the Arrhenius equation as a guide for interpolation or extrapolation of rate constants. For this purpose, rate data are sometimes stored in the form of the Arrhenius equation. For example, the data of Table 6-1 may be represented (see Table 6-2) as... 

Integr ation may lead to a relation for rate constant with temperature dependency in the form of Arrhenius law ... 

From the Arrhenius form of Eq. (70) it is intuitively expected that the rate constant for chain scission kc should increase exponentially with the temperature as with any thermal activation process. It is practically impossible to change the experimental temperature without affecting at the same time the medium viscosity. The measured scission rate is necessarily the result of these two combined effects to single out the role of temperature, kc must be corrected for the variation in solvent viscosity according to some known relationship, established either empirically or theoretically. 

Reaction rates almost always increase with temperature the rare ones that do not have a negative activation energy will be dealt with later. The expression of the temperature dependence is always given for the rate constant, rather than the rate. For now, only elementary reactions will be considered, with composite reactions and other more complicated situations deferred to Section 7.5. Two forms are commonly used to express the rate constant as a function of temperature. The first is the familiar Arrhenius equation,... 

If a data set containing k T) pairs is fitted to this equation, the values of these two parameters are obtained. They are A, the pre-exponential factor (less desirably called the frequency factor), and Ea, the Arrhenius activation energy or sometimes simply the activation energy. Both A and Ea are usually assumed to be temperature-independent in most instances, this approximation proves to be a very good one, at least over a modest temperature range. The second equation used to express the temperature dependence of a rate constant results from transition state theory (TST). Its form is... 

As in collision theory, the rate of the reaction depends on the rate at which reactants can climb to the top of the barrier and form the activated complex. The resulting expression for the rate constant is very similar to the one given in Eq. 15, and so this more general theory also accounts for the form of the Arrhenius equation and the observed dependence of the reaction rate on temperature. 

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... 

More complicated rate expressions are possible. For example, the denominator may be squared or square roots can be inserted here and there based on theoretical considerations. The denominator may include a term k/[I] to account for compounds that are nominally inert and do not appear in Equation (7.1) but that occupy active sites on the catalyst and thus retard the rate. The forward and reverse rate constants will be functions of temperature and are usually modeled using an Arrhenius form. The more complex kinetic models have enough adjustable parameters to fit a stampede of elephants. Careful analysis is needed to avoid being crushed underfoot. 

The temperature dependence of enz5anatic reactions is modeled with an Arrhenius form for the main rate constant k . The practical range of operating temperatures is usually small, but the activation energies can be quite large. Temperature dependence of the inhibition constants can usually be ignored. 

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. 

The variation of a rate constant with temperature is described by the Arrhenius equation. According to its logarithmic form (Equation ), a plot of in k vs. 1 / Z, with temperature expressed in keIvins, shou id be a straight line. 

The presence of diffusion limitations has a strong effect on the apparent activation energy one measures. We can express both the rate constant, k, and the diffusion constant, Defr, in the Arrhenius form ... 

This section focuses on the problem of determining the temperature dependence of the reaction rate expression (i.e., the activation energy of the reaction. Virtually all rate constants may be written in the Arrhenius form ... 

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,... 

If reaction 1 is to be enhanced and reaction 2 depressed, the ratio of the rate constants must be made as large as possible. This ratio may be written in the Arrhenius form as... 

In summary, to apply the Marcus theory of electron transfer, it is necessary to see if the temperature dependence of the electron transfer rate constant can be described by a function of the Arrhenius form. When this is valid, one can then determine the activation energy AEa only under this condition can we use AEa to determine if the parabolic dependence on AG/ is valid and if the reaction coordinate is defined. 

Influenced by the form of the van t Hoff equation, Arrhenius (1889) proposed a similar expression for the rate constant kAin equation 3.1-2, to represent the dependence of (-rA) on T through the second factor on the right in equation 3.1-1 ... 

As introduced in sections 3.1.3 and 4.2.3, the Arrhenius equation is the normal means of representing the effect of T on rate of reaction, through the dependence of the rate constant k on T. This equation contains two parameters, A and EA, which are usually stipulated to be independent of T. Values of A and EA can be established from a minimum of two measurements of A at two temperatures. However, more than two results are required to establish the validity of the equation, and the values of A and EA are then obtained by parameter estimation from several results. The linear form of equation 3.1-7 may be used for this purpose, either graphically or (better) by linear regression. Alternatively, the exponential form of equation 3.1-8 may be used in conjunction with nonlinear regression (Section 3.5). Seme values are given in Table 4.2. 

The rate constants kf and kr are then calculated at Topt from the Arrhenius equation, and the maximum rate is obtained from equation 5.3-15 in the form... 

The equilibrium constant Kf for reaction i is computed from thermodynamic considerations using Gibbs free energies. The Arrhenius form for the rate constants is written in terms of the pre-exponential factor A,, the temperature exponent /( , the activation energy Ei, and the universal gas constant R in the same units as the activation energy. 

A change in the reaction temperature affects the rate constant k. As the temperature increases, the value of the rate constant increases and the reaction is faster. The Swedish scientist, Arrhenius, derived a relationship that related the rate constant and temperature. The Arrhenius equation has the form k = Ae-E /RT. In this equation, k is the rate constant and A is a term called the frequency factor that accounts for molecular orientation. The symbol e is the natural logarithm base and R is universal gas constant. Finally, T is the Kelvin temperature and Ea is the activation energy, the minimum amount of energy needed to initiate or start a chemical reaction. 

The rate constants of the electron transfers vary with the electrode potential. In particular, in their Arrhenius form, they are expressed by ... 

The idea that an activated complex or transition state controls the progress of a chemical reaction between the reactant state and the product state goes back to the study of the inversion of sucrose by S. Arrhenius, who found that the temperature dependence of the rate of reaction could be expressed as k = A exp (—AE /RT), a form now referred to as the Arrhenius equation. In the Arrhenius equation k is the forward rate constant, AE is an energy parameter, and A is a constant specific to the particular reaction under study. Arrhenius postulated thermal equilibrium between inert and active molecules and reasoned that only active molecules (i.e. those of energy Eo + AE ) could react. For the full development of the theory which is only sketched here, the reader is referred to the classic work by Glasstone, Laidler and Eyring cited at the end of this chapter. It was Eyring who carried out many of the... 

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