Exact Arrhenius temperature dependence

The temperature dependence of the rate constant for the step A - B leads to the term /(0) in the dimensionless mass- and heat-balance eqns (4.24) and (4.25). The exact representation of an Arrhenius rate law is f(9) — exp[0/(l + y0)], where y is a dimensionless measure of the activation energy RTa/E. As mentioned before, y will typically be a small quantity, perhaps about 0.02. Provided the dimensionless temperature rise 9 remains of order unity (9 10, say) then the term y9 may be neglected in the denominator of the exponent as a first simplification. 

In this section, therefore, we briefly investigate the stationary-state and Hopf bifurcation patterns that are found with the exact Arrhenius temperature dependence. 

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We have already discussed the expressions resulting from a full Hopf bifurcation analysis of the thermokinetic model with the exponential approximation (y = 0). We may do the same for the exact. Arrhenius temperature dependence (y 0). Although the algebra is somewhat more onerous, we still arrive at analytical, expressions for the stability of the emerging or vanishing limit cycle and the rate of growth of the amplitude and period at... 

Some problems in functional optimization can be solved analytically. A topic known as the calculus of variations is included in most courses in advanced calculus. It provides ground rules for optimizing integral functionals. The ground rules are necessary conditions analogous to the derivative conditions (i.e., df jdx = 0) used in the optimization of ordinary functions. In principle, they allow an exact solution but the solution may only be implicit or not in a useful form. For problems involving Arrhenius temperature dependence, a numerical solution will be needed sooner or later. 

The multiplicity analysis described above can be extended to cope with the exact form of the Arrhenius temperature dependence (non-zero y). The stationary-state condition has a slightly more cumbersome form ... 

The Arrhenius temperature dependence can be represented exactly in this way with... 

The temperature dependence of the reaction rate constant closely (but not exactly) obeys the Arrhenius equation. Both theories, however, predict non-Arrhenius behavior. The deviation from Arrhenius behavior can usually be ignored over a small temperature range. However, non-Arrhenius behavior is common (Steinfeld et al., 1989, p. 321). As a consequence, rate constants are often fitted to the more general expression k = BTnexp( —E/RT), where B, n, and E are empirical constants. 

This quantity describes how much the temperature dependence of structural relaxation deviates from an Arrhenius behavior, which is the most basic concept embodied by the term fragility. Within AG theory, this quantity exactly equals... 

In contrast to the formally analogous van t Hoff equation [10] for the temperature dependence of equilibrium constants, the Arrhenius equation 1.3 is empirical and not exact The pre-exponential factor A is not entirely independent of temperature. Slight deviations from straight-line behavior must therefore be expected. In terms of collision theory, the exponential factor stems from Boltzmann s law and reflects the fact that a collision will only be successful if the energy of the molecules exceeds a critical value. In addition, however, the frequency of collisions, reflected by the pre-exponential factor A, increases in proportion to the square root of temperature (at least in gases). This relatively small contribution to the temperature dependence is not correctly accounted for in eqns 2.2 and 2.3. [For more detail, see general references at end of chapter.]... 

The classical RRK theory thus predicted that the high pressure limiting rate coefficient is exactly of the Arrhenius form. In its applications the values of A and Ep are generally deduced from the experimental temperature dependence of the high pressure limit. 

Note that if the kinetics of the process under consideration is dominated by a single mechanism, then the rate obtained in eqn (7.61) has a temperature dependence of exactly the Arrhenius form mentioned at the beginning of this section. An additional comment to be made about our assessment of the transition rate is that it presupposed a knowledge of the pathway connecting the various wells. 

In this section, the discussion will begin with the simplest case that can realistically be considered—a zero-order irreversible chemical reaction. In this example, the reaction rate is a function only of temperature until all reactant is consumed and the reaction stops. The exact fimction governing the temperature dependence of the reaction rate is not defined in this initial analysis, but it can be, it is assumed approximated to be linear over the small temperature interval of the modulation. The more general case where the chemical reaction can be considered to be a function of time (and therefore conversion) and temperature is then treated. Finally, the Arrhenius equation is dealt with, as this is the most relevant case to the subject of this book. 

This equation is called the Arrhenius expression and is the fundamental equation representing the temperature dependence of reaction rate constants. Comparing the Arrhenius expression Eq. (2.39), with rate constant Eq. (2.33) by the collision theory and (2.38) by the transition state theory, the temperature dependence of the exponential factor is exactly the same as derived by these theories, and Ea of the Arrhenius expression corresponds to the activation energy Ea of the transition state theory. A plot of the logarithm of a reaction rate constant, In k against MRT, is called an Arrhenius plot, and the experimental value of activation energy can be obtained from the slope of the Arrhenius plot. This linear relationship is known to hold experimentally for numerous reactions, and the activation energy for each reaction has been obtained. 

Figure 5.8 demonstrates the temperature dependence of exact and JSA rate constants for the D+H2(t = l,j) reaction. The temperature range is 200 - 1000/if, plotted in inverse Kelvin. The exact rate constants were obtained from Eq. (5.4) and the cross sections in Fig. 5.7, and the JSA rate constants were obtained from Eqs. (5.41)-(5.43) and the J = 0 reaction probabilities in Fig. 5.4. Very good agreement is obtained for all j values. In all cases, the JSA predicts the correct Arrhenius activation energy (i.e. negative of the slope of log k(T) vs. 1/ksT) and is qualitatively correct in predicting the Arrhenius prefactor (i.e. y-intercept). The overall agreement is truly excellent for = 1 and 2. However, the JSA systematically underestimates the rate constant by roughly 40% for j = 0 and 3.

Here (Le) = Dcpc/K is the Lewis number, the ratio of molecular to thermal diffusivities, and c is the dimensionless flame velocity. The variable 0 is a measure of the temperature rise due to self-heating. A suitable form might be 0 = (T - To)/(T6 — To). The function f 6) represents the temperature dependence of the reaction rate constant. The exact Arrhenius form has f 6) = exp 50/(l-FeB0), where e = RTqIE mdB = EqaafcpoRlQ = (Tf, - To)EIRTq is the maximum (adiabatic) temperature rise expressed in the same form. Typically, B will have a large value and generally e [Pg.507]

In fig. 26 the Arrhenius plot ln[k(r)/coo] versus TojT = Pl2n is shown for V /(Oo = 3, co = 0.1, C = 0.0357. The disconnected points are the data from Hontscha et al. [1990]. The solid line was obtained with the two-dimensional instanton method. One sees that the agreement between the instanton result and the exact quantal calculations is perfect. The low-temperature limit found with the use of the periodic-orbit theory expression for kio (dashed line) also excellently agrees with the exact result. Figure 27 presents the dependence ln(/Cc/( o) on the coupling strength defined as C fQ. The dashed line corresponds to the exact result from Hontscha et al. [1990], and the disconnected points are obtained with the instanton method. For most practical purposes the instanton results may be considered exact. 

The dependence of the reaction rate on temperature can be approximated by the simple Arrhenius law. The further analysis is based on the assmnption that the measurable onset in a DTA represents exactly the moment vdien the equilibrium state described above is left. Such a situation must also be observable under plant conditions. If the equation of critical balance for the measuring system is divided by the corresponding equation for the plant system, the following equation is obtained ... 

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