Temperature dependence of the rate

Rates of reaction usually go up when the temperature increases, although in catalysis this is only partly true as we will see later in this chapter. As a (crude ) rule of thumb, the rate of reaction doubles for every 10 K increase in temperature. 

For an elementary reaction the temperature dependence of the rate constant is given by the Arrhenius equation 

Arrhenius proposed his equation in 1889 on empirical grounds, justifying it with the hydrolysis of sucrose to fructose and glucose. Note that the temperature dependence is in the exponential term and that the preexponential factor is a constant. Reaction rate theories (see Chapter 3) show that the Arrhenius equation is to a very good approximation correct however, the assumption of a prefactor that does not depend on temperature cannot strictly be maintained as transition state theory shows that it may be proportional to 7. Nevertheless, this dependence is usually much weaker than the exponential term and is therefore often neglected. 

We can determine the activation energy from a series of measurements by plotting the logarithm of the rate constant against the reciprocal temperature, as rearrangement of Eq. (45) sho vs  

The slope of such a plot yields the activation energy and the intercept the prefactor. This procedure is represented by the follo ving elegant mathematical expression, which we will often use in the course of this book  

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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... 

The rate-conttolling step to chlorate is the bimoleculat formation of chlorite, which reacts rapidly with hypochlorite. The temperature dependence of the rate constants is expressed by the equations = 2.1 x 10 g-io3.8/i T 3 2 x 10 g-87.o/i T L/(mol-s) (144). The uncataly2ed decomposition to... 

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].

The temperature dependence of the rate constant of rapid hydrogen transfer, measured by Mordzinski and Kuhnle [1986], is given in fig. 1. A similar dependence has been found by Grellmann et al. [1989] for one-proton transfer in half of the above molecule (6.16), which does not include the two rightmost rings. 

The temperature dependence of A predicted by Eq. (5-11) makes a very weak contribution to the temperature dependence of the rate constant, which is dominated by the exponential term. It is, therefore, not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted dependence of A is observed experimentally. Uncertainties in estimates of A tend to be quite large because this parameter is, in effect, determined by a long extrapolation of the Arrhenius plot to 1/T = 0. 

The numerical values of AG and A5 depend upon the choice of standard states in solution kinetics the molar concentration scale is usually used. Notice (Eq. 5-43) that in transition state theory the temperature dependence of the rate constant is accounted for principally by the temperature dependence of an equilibrium constant. 

Arrhenius acid Species that, upon addition to water, increases [H+], 86 Arrhenius base Species that, upon addition to water, increases [OH-], 86 Arrhenius equation Equation that expresses the temperature dependence of the rate constant In k2/ki = a(l/Ti — 1 IT2)IR, 302-305... 

The kinetic expression was derived by Akers and White (10) who assumed that the rate-controlling factor in methane formation was the reaction between the adsorbed reactants to form adsorbed products. However, the observed temperature-dependence of the rate was small, which indicates a low activation energy, and diffusion was probably rate-controlling for the catalyst used. 

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). 

The increased fluctuations of protein structures at elevated temperatures is reflected in the thermal expansion of proteins [433], that may contribute to the marked temperature dependence of the rate of Ca transport. 

Parameter estimation. Integral reactor behavior was used for the interpretation of the experimental data, using N2O conversion levels up to 70%. The temperature dependency of the rate parameters was expressed in the Arrhenius form. The apparent rate parameters have been estimated by nonlinear least-squares methods, minimizing the sum of squares of the residual N2O conversion. Transport limitations could be neglected. 

The temperature dependence of the rate constants follows the Arrhenius equation ... 

Proteins may be covalently attached to the latex particle by a reaction of the chloromethyl group with a-amino groups of lysine residues. We studied this process (17) using bovine serum albumin as a model protein - the reaction is of considerable interest because latex-bound antigens or antibodies may be used for highly sensitive immunoassays. The temperature dependence of the rate of protein attachment to the latex particle was unusually small - this rate increased only by 27% when the temperature was raised from 25°C to 35°C. This suggests that non-covalent protein adsorption on the polymer is rate determining. On the other hand. the rate of chloride release increases in this temperature interval by a factor of 17 and while the protein is bound to the latex particle by only 2 bonds at 25°C, 22 bonds are formed at 35°C. 

The temperature dependency of the rate constants on Miike coal was determined between 350 C and 450 C. The result is shown in Table 4. 

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... 

The concentration of A in the entrance stream is 1.5 lb moles/ft3. The following table provides some information about the temperature dependence of the rate constants and fc j. 

In order to be able to evaluate the integrals in equations 10.2.1 and 10.2.3, one must know not only the temperature dependence of the rate terms but also the relationship between the fraction conversion and the temperature of the system. 

Apparent activation energy, kcal/mole, determined from the temperature dependence of the rate n at ethane and hydrogen partial pressures of 0.030 and 0.20 atm, respectively. 

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. 

Witte, F., Urbanik, E., Zetzsch, C. (1986) Temperature dependence of the rate constants for the addition of OH to benzene and to some mono substituted aromatics (aniline, bromobenzene, and nitrobenzene) and the unimolecular decay of the adducts. Kinetics into a quasi-equilibrium. J. Phys. Chem. 90, 3251-3259. 

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. 

The photolyses of diazirines 9a and 9b were similarly studied in Ar matrices at 10-34.5 K 59 Eq. 10. Benzylchlorocarbene (10a) and its ct,a-d2 analogue (10b) were observed by UV or IR (10b) spectroscopy, and their decay to styrenes 11 and 12 could be monitored. Tunneling in these 1,2-H(D) shifts was indicated by (a) much higher rates of carbene decay at 10 K than could be anticipated from extrapolation of the 298 K LFP kinetic data, (b) a kinetic isotope effect (KIE) for the 1,2-H(D) shifts estimated at 2000, and (c) little temperature dependence of the rate at low temperature.59 Accepting that QMT is important in the very low temperature H shifts of carbenes 10 and 18, the obvious question becomes is QMT important at higher or even ambient temperatures ... 

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