Arrhenius temperature-dependence

The rate constant for elementary reactions is almost always expressed as 

The case of m = Q corresponds to classical Arrhenius theory m = 1/2 is derived from the collision theory of bimolecular gas-phase reactions and m = corresponds to activated complex or transition state theory. None of these theories is sufficiently well developed to predict reaction rates from first principles, and it is practically impossible to choose between them based on experimental measurements. The relatively small variation in rate constant due to the pre-exponential temperature dependence T is overwhelmed by the exponential dependence exp(—Tarf/T). For many reactions, a plot of In(fe) versus will be approximately linear, and the slope of this line can be used to calculate E. Plots of rt(k/T ) versus 7 for the same reactions will also be approximately linear as well, which shows the futility of determining m by this approach. 

Solution The classic way of fitting these data is to plot ln( /7 ) versus T and to extract and Tact from the slope and intercept of the resulting (nearly) straight line. Special graph paper with a logarithmic j-axis and a l/T A-axis was made for this purpose. The currently preferred method is to use nonlinear regression to fit the data. The object is to find values for kQ and Tact that minimize the sum-of-squares  

The model predictions are essentially identical. The minimization procedure automatically adjusts the values for ko and Tact to account for the different values of m. The predictions are imperfect for any value of m, but this is presumably due to experimental scatter. For simplicity and to conform to general practice, we wiU use m = 0 from this point on. 

You may recall the rule-of-thumb that reaction rates double for each 10°C increase in temperature. Doubling when going from 20° C to 30° C means 

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Chemical Reactor Design, Optimization, and Scaleup, Second Edition. By E. B. Nauman Copyright 2008 John Wiley Sons, Inc. 

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The autocatalator model is in many ways closely related to the FONT system, which has a single first-order exothennic reaction step obeying an Arrhenius temperature dependence and for which the role of the autocatalyst is taken by the temperature of the system. An extension of this is tlie Sal nikov model which supports tliennokinetic oscillations in combustion-like systems [48]. This has the fonn ... 

Irreversible reaction with Arrhenius temperature dependence, so that the rate function took the form... 

Calculated and measured conversions agreed when the Arrhenius temperature dependency indicated in Eq. (27-22) was used with the following values for the parameters ... 

We assume the forward and reverse reactions have Arrhenius temperature dependences with Ef < Ef. Setting dbomldT = 0 gives... 

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. 

A good model is consistent with physical phenomena (i.e., 01 has a physically plausible form) and reduces crresidual to experimental error using as few adjustable parameters as possible. There is a philosophical principle known as Occam s razor that is particularly appropriate to statistical data analysis when two theories can explain the data, the simpler theory is preferred. In complex reactions, particularly heterogeneous reactions, several models may fit the data equally well. As seen in Section 5.1 on the various forms of Arrhenius temperature dependence, it is usually impossible to distinguish between mechanisms based on goodness of fit. The choice of the simplest form of Arrhenius behavior (m = 0) is based on Occam s razor. 

Transformation of the independent variables to dimensionless form uses = r/R and jz = z/L. In most reactor design calculations, it is preferable to retain the dimensions on the dependent variable, temperature, to avoid confusion when calculating the Arrhenius temperature dependence and other temperature-dependent properties. The following set of marching-ahead equations are functionally equivalent to Equations (8.25)-(8.27) but are written in dimensionless form for a circular tube with temperature (still dimensioned) as the dependent variable. For the centerline. 

Only numerical solutions are possible when Equation (9.24) is solved simultaneously with Equation (9.14). This is true even for first-order reactions because of the intractable nonlinearity of the Arrhenius temperature dependence. 

Do not infer from the above discussion that all the catalyst in a fixed bed ages at the same rate. This is not usually true. Instead, the time-dependent effectiveness factor will vary from point to point in the reactor. The deactivation rate constant kj) will be a function of temperature. It is usually fit to an Arrhenius temperature dependence. For chemical deactivation by chemisorption or coking, deactivation will normally be much higher at the inlet to the bed. In extreme cases, a sharp deactivation front will travel down the bed. Behind the front, the catalyst is deactivated so that there is little or no conversion. At the front, the conversion rises sharply and becomes nearly complete over a short distance. The catalyst ahead of the front does nothing, but remains active, until the front advances to it. When the front reaches the end of the bed, the entire catalyst charge is regenerated or replaced. 

The kinetic rate coefficients kd, kt are described by Arrhenius temperature dependencies. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

First calculations of the optimum distance between the reactants, R, taking into account the dependence of the probability of proton transfer between the unexcited vibrational energy levels on the transfer distance have been performed in Ref. 42 assuming classical character of the reactant motion. Effects of this type were considered also in Ref. 43 in another model. It was shown that R depends on the temperature and this dependence leads to a distortion of the Arrhenius temperature dependence of the transition probability. 

We may compare these results with a second-order rate law which exhibits Arrhenius temperature dependence ... 

Figure 2 Sketch of typical temperature dependencies of the viscosity r of glass-forming systems. The viscosimetric Tg of a material is defined by the viscosity reaching 1013 Poise. Strong glass formers show an Arrhenius temperature dependence, whereas fragile glass formers follow reasonably well a Vogel-Fulcher (VF) law predicting a diverging viscosity at some temperature T0.

Figure 18 Temperature dependence of the C-H vector (selected, filled symbols) and torsional correlation (open symbols) times for PB from simulation. Also shown is the mean waiting time between transitions for the cis-allyl, trans-allyl, and (3 torsions in PB. The solid lines are VF fits, whereas the dashed lines assume an Arrhenius temperature dependence.

The dispersion of this waiting time distribution, i.e., its second central moment, is a measure that we can use to define a homogenization time scale on which the dispersion is equal to that of a homogeneous (Poisson) system on a time scale given by the torsional autocorrelation time. The homogenization time scale shows a clear non-Arrhenius temperature dependence and is comparable with the time scale for dielectric relaxation at low temperatures.156... 

A. ARRHENIUS TEMPERATURE DEPENDENCE. The effect of temperature on the specific reaction rate f is usually found to be exponential ... 

Example 6.2. The Arrhenius temperature dependence of the specific reaction rate fe is a highly nonlinear function that is linearized as follows ... 

Driscoll [67], Lorimer and Mason [79] and Price [65[ have also obtained inverse Arrhenius temperature dependencies for reactions performed in the presence of ultrasound. For example Driscoll has investigated the polymerisation of styrene and methyl methacrylate in the presence of their respective homopolymers and observed that the lower the reaction temperature the faster was the reaction rate and the higher the final polymer yield (Figs. 5.38 and 5.39). Price on the other hand using a non polymer system has sonicated methyl butyrate (MeOBu) and compared the rates of radical production in the absence and presence of the initiator azobisisobutyronitrile (AIBN) (Tab. 5.18). 

The non-Arrhenius temperature-dependence of the relaxation time. It shows a dramatic increase when the glass transition temperature region is approached. This temperature dependence is usually well described in terms of the so called Vogel-Fulcher temperature dependence [114,115] ... 

It can not be described by means of a single Debye process, but more complicated relaxation functions involving distributions of relaxation times (like the Cole-Cole function [117]) or distributions of energy barriers (like log-normal functions [118]) have to be used for its description. Usually a narrowing of the relaxation function with increasing temperature is observed. The Arrhenius temperature dependence of the associated characteristic time is ... 

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