Temperature Dependence of Reaction Rate Constant

The rates of chemical and biochemical reactions usually increase with temperature. The dependence of the reaction rate on temperature can usually be represented by the following Arrhenius-type equation over a wide temperature range  

The slope of the straight line through the data points (-EJR) is -6740 K. Thus, the activation energy is given as  

Rates of reactions with larger values of activation energy are more sensitive to temperature changes. If the activation energy of a reaction is approximately 50 000 kJ kmol , the reaction rate will be doubled with a 10 C increase in reaction temperature at room temperature. Strictly, the Arrhenius equation is valid only for elementary reactions. Apparent activation energies can be obtained for non-elementary reactions. 

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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... 

Experiment Relations between decompositian rate and temperature Dependences of reaction rate constants on temperature were evaluated. Experiments... 

Since data are almost invariably taken under isothermal conditions to eliminate the temperature dependence of reaction rate constants, one is primarily concerned with determining the concentration dependence of the rate expression [0(Ct)] and the rate constant at the temperature in question. We will now consider two differential methods that can be used in data analysis. 

We find a clue in the observed temperature dependence of reaction rate constants. The rates of many reactions increase extremely rapidly as temperature increases typically a 10°C rise in temperature may double the rate. In 1889 Svante Arrhenius suggested that rate constants vary exponentially with inverse temperature. 

Ever since the rates of chemical reactions were first systematically measured, chemical kineticists have sought to measure, characterise, and understand the effect of temperature on these rates. One of the earliest and most celebrated efforts in this direction was the proposal by Svante Arrhenius [1] that the temperature-dependence of reaction rate constants can be expressed in terms of what we now know as the Arrhenius equation ... 

Arrhenius expression for temperature dependence of reaction rate constants... 

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. 

Seeley J V, Morris R A, Viggiano A A, Wang FI and Flase W L 1997 Temperature dependencies of the rate constants and branching ratios for the reactions of Cr(Fl20)g 3 with CFIjBr and thermal dissociation rates for CI (CFl3Br) J. Am. Chem. Soc. 119 577-84... 

Although the Arrhenius equation does not predict rate constants without parameters obtained from another source, it does predict the temperature dependence of reaction rates. The Arrhenius parameters are often obtained from experimental kinetics results since these are an easy way to compare reaction kinetics. The Arrhenius equation is also often used to describe chemical kinetics in computational fluid dynamics programs for the purposes of designing chemical manufacturing equipment, such as flow reactors. Many computational predictions are based on computing the Arrhenius parameters. 

Fig. 1. Examples of temperature dependence of the rate constant for the reactions in which the low-temperature rate-constant limit has been observed 1. hydrogen transfer in the excited singlet state of the molecule represented by (6.16) 2. molecular reorientation in methane crystal 3. internal rotation of CHj group in radical (6.25) 4. inversion of radical (6.40) 5. hydrogen transfer in halved molecule (6.16) 6. isomerization of molecule (6.17) in excited triplet state 7. tautomerization in the ground state of 7-azoindole dimer (6.1) 8. polymerization of formaldehyde in reaction (6.44) 9. limiting stage (6.45) of (a) chain hydrobromination, (b) chlorination and (c) bromination of ethylene 10. isomerization of radical (6.18) 11. abstraction of H atom by methyl radical from methanol matrix [reaction (6.19)] 12. radical pair isomerization in dimethylglyoxime crystals [Toriyama et al. 1977].

An illustration of this method can be found in fitting the temperature dependence of the rate constant for the following dimerization reaction ... 

The temperature dependence of a rate is often described by the temperature dependence of the rate constant, k. This dependence is often represented by the Arrhenius equation, /c = Aexp(- a/i T). For some reactions, the temperature relationship is instead written fc = AT" exp(- a/RT). The A term is the frequency factor for the reaction, which reflects the number of effective collisions producing a reaction. a is known as the activation energy for the reaction, and is a measure of the amount of energy input required to start a reaction (see also Benson, 1960 Moore and Pearson, 1981). 

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... 

Huie, R.E., Herron, J.T. (1975) Temperature dependence of the rate constants for reaction of ozone with some olefins. Int l. J. Chem. Kinet. SI, 165. 

On the other hand, the low temperature dependance of the rate constants with activation energies around 5 kcal/mole indicates a diffusion limited reaction rate which could refer to diffusion of oxygene into the fibers of the board, i.e. into the fiberwalls. The corresponding negative activation energy for the groundwood based hardboard and the effect of fire retardants there upon are difficult to understand. 

Then, they depend also on the viscosity of the system. Specific diffusion control is characteristic of fast reactions like fluorescence quenching. In polymer formation, specific diffusion control is responsible for the acceleration of chain polymerization due to the retardation of the termination by recombination of two macroradicals (Trommsdorff effect). Step reactions are usually too slow to exhibit a dependence on translational diffusion also, the temperature dependence of their rate constants is of the Arrhenius type. 

Any energy change (from the ground state to the transition state in a chemical reaction) that accounts (a) for the presence of a reaction barrier and (b) for the temperature dependence of a rate constant. This quantity is symbolized by and may also be referred to as the Arrhenius energy of activation and activation energy. 

Methyl acetate probably originates from the reaction of methanol with the intermediate cobalt-acyl complex. The reaction leading to the formation of acetaldehyde is not well understood. In Equation 8, is shown as the reducing agent however, metal carbonyl hydrides are known to react with metal acyl complexes (20-22). For example, Marko et al. has recently reported on the reaction of ri-butyryl- and isobutyrylcobalt tetracarbonyl complexes with HCo(CO) and ( ). They found that at 25 °C rate constants for the reactions with HCo(CO) are about 30 times larger than those with however, they observed that under hydroformylation conditions, reaction with H is the predominant pathway because of the greater concentration of H and the stronger temperature dependence of its rate constant. The same considerations apply in the case of reductive carbonylation. Additionally, we have found that CH C(0)Co(C0) L (L r PBu, ... 

To a first approximation over the relatively small temperature range encountered in the troposphere, A is found to be independent of temperature for many reactions, so that a plot of In k versus T l gives a straight line of slope —E.d/R and intercept equal to In A. However, the Arrhenius expression for the temperature dependence of the rate constant is empirically based. As the temperature range over which experiments could be carried out was extended, nonlinear Arrhenius plots of In k against T 1 were observed for... 

Even more unusual behavior is obseived for the temperature dependence of the rate constant. Figure 6.11 shows these data in Arrhenius form for the reactions of toluene and 1,2,3-trimethylbenzene. At the higher temperatures, the Arrhenius plot is linear with a normal activation energy (i.e., the rate constant increases with increasing temperature). However, as the temperature is lowered, there is a sharp discontinuity in the plot and at lower temperatures the temperature dependence is reversed i.e., the rate constants decrease with increasing temperature. 

Huie, R. E., and C. L. Clifton, Temperature Dependence of the Rate Constants for Reactions of the Sulfate Radical, S04, with Anions, J. Phys. Chem.., 94, 8561-8567 (1990). 

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