Temperature dependence of rate constant

From the relations in 3.5.3 we see that the temperature dependence of the equilibrium constant K can be expressed as 

In 1889, Arrhenius proposed a similar equation for the temperature effect on rate constants, 

Experimental data for a great many reactions over a large range of temperatures show that the Arrhenius equation is usually closely obeyed, showing that Ea is either a constant or a weak function of temperature, and so we can integrate the equation to give 

The activation energy of an overall reaction is made up of the individual contributions of the elementary reactions making up the overall reaction. The magnitude of the activation energy can vary from virtually zero to hundreds of kilojoules per mole and, besides controlling the temperature dependence of the rate constant, provides clues as 

The experimental characterization of a chemical reaction is not completed by deducing the rate law. In addition the way in which the rate constants depend upon temperature, pressure, etc., must be determined. The variation of k with temperature is striking repeated experimentation in the late nineteenth century showed that it could be represented in the form 

Arrhenius was the first to interpret as an activation energy and present, as qualitative justification for this point of view, the rationalization based 

In some cases, either on experimental grounds or to effect comparison with a theoretical model for the rate constant, one wishes to express k(T) in the form 

If m is known on theoretical grounds. A and 0 may be estimated by the least-squares method if In A — m(ln T) is treated as the dependent variable. A direct extension of the least-squares approach is not possible if m is unknown because In T and l/T are not independent quantities. However, by determining Ea for both high- and low-temperature data, m may be estimated. Since (4.42) defines Ea as 

With this and nearby estimates of m it is possible, using the least-squares procedure for known m, to determine both A and 6 as functions of m. The optimal set of parameters may be defined as that which leads to the largest correlation coefficient. An example for which there is definite curvature in a plot of In k vs. l/T is the reaction Cl( P) + CH4 HCl + CH3, illustrated in Fig. 4.7.  

To obtain information about the energetics of a reaction, the temperature dependence of the rate constant is determined. For complex rate laws, this also will involve a study of the concentration dependence of the rate at different temperatures, in order to determine the temperature dependence of the various terms contributing to the rate law. Once the experimental information is available for the specific rate constants, it is usually analyzed in terms of one of the following formalisms. 

Arrhenius seems to have been the first to find empirically that rate constants have a temperature dependence analogous to that of equilibrium constants, given by 

This theory was developed originally for a simple dissociation process in the gas phase and it assumes that the reaction can be described by the following sequence  

The theory proposes that the activated complex or transition state will proceed to products when the A— B bond has a thermal energy k T, so that the rate constant, k, will be proportional to the bond s vibrational frequency, v = k T/h s, with a proportionality constant, K, known as the transmission coefficient (k = Boltzmann s constant, 1.381xl0 erg K h s Planck s constant, 6.626xl0 erg s). It also is assumed that the activated complex is always in equilibrium with the reactant with a normal equilibrium constant, it = [A—B] /[A— B], so that 

If the first-order rate expression, d[B]/dr = k [A—B], is compared to Eq. (1.69), then substitution for K from Eq. (1.70) shows that 

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We are now ready to build a model of how chemical reactions take place at the molecular level. Specifically, our model must account for the temperature dependence of rate constants, as expressed by the Arrhenius equation it should also reveal the significance of the Arrhenius parameters A and Ea. Reactions in the gas phase are conceptually simpler than those in solution, and so we begin with them. 

Solution The analysis could be carried out using mole fractions as the composition variable, but this would restrict applicability to the specific conditions of the experiment. Greater generality is possible by converting to concentration units. The results will then apply to somewhat different pressures. The somewhat recognizes the fact that the reaction mechanism and even the equation of state may change at extreme pressures. The results will not apply at different temperatures since k and kc will be functions of temperature. The temperature dependence of rate constants is considered in Chapter 5. 

Temperature dependence of rate constants, ARRHENIUS EQUATION RLOT Temperature-jump method,... 

Temperature Dependence of Rate Constants a. Arrhenius Expression... 

A vital constituent of any chemical process that is going to show oscillations or other bifurcations is that of feedback . Some intermediate or product of the chemistry must be able to influence the rate of earlier steps. This may be a positive catalytic process , where the feedback species enhances the rate, or an inhibition through which the reaction is poisoned. This effect may be chemical, arising from the mechanistic involvement of species such as radicals, or thermal, arising because chemical heat released is not lost perfectly efficiently and the consequent temperature rise influences some reaction rate constants. The latter is relatively familiar most chemists are aware of the strong temperature dependence of rate constants through, e.g. the Arrhenius law,... 

Figure 23. Temperature dependence of rate constant for excitation transfer in 3He (23S) + 3He, calculated from potentials of Fig. 17 and Table IV. Data are derived from analysis of optical pumping experiments.

Figure 1.1 Examples of temperature dependences of rate constants for the reactions in which the low-temperature rate constant limit has been observed 1, hydrogen transfer in excited singlet state of molecule (6.14) 2, molecular reorientation in methane crystal 3, internal rotation of CH3 group in radical (7.42) 4, inversion of oxyranyl radical (8.18) 5, hydrogen transfer in the excited triplet state of molecule (6.20) 6, isomerization in the excited triplet state of molecule (6.22) 7, tautomerization in the ground state of 7-azoindole dimer (6.15) 8, polymerization of formaldehyde 9, limiting stage of chain (a) hydrobromi-nation, (b) chlorination, and (c) bromination of ethylene 10, isomerization of sterically hindered aryl radical (6.44) 11, abstraction of a hydrogen atom by methyl radical from a methanol matrix in reaction (6.41) 12, radical pair isomerization in dimethylglyoxime crystal (Figure 6.25).

According to various model considerations, one can often obtain more complicated temperature dependences of rate constants than eqn. (43) (see, for example, ref. 3). 

Fig. 7.3. The mutual transformation in system (>Si = 0 + C0<-> >Si + C02). (a) The temperature dependence of rate constant for direct reaction (b) the temperature dependence of rate constant for reverse reaction (c) the calculated structure of the TS for this reaction.

It means that in reality the macroradicals are concentrated in a thin layer near the surface of polymer particles. Fig. 8 shows the temperature dependence of the rate constants of oxidation reactions for the three polymers investigated (curves b, c, d). One can see that in the temperature range investigated this dependence is in agreement with the Arrhenius equation. Let us examine the initial sections of oxidation curves. Analysis of the curves shows that they can be represented as a superposition of two different exponents corresponding to two different rate constants of radical oxidation. The temperature dependence of rate constants determined from the initial sections of oxidation curves of polymethyl-metacrylate is shown in Fig. 8 (curve a). Following fact is of interest ... 

Figure 3. Temperature dependence of rate constant of S -S annihilation.(a) neat benzene and (b) benzene(5.6M) in cyclohexane.

Rate Equations and Stoichiometric Equations 11 Rate Constants and Reaction Orders 13 Temperature Dependence of Rate Constants 13... 

The pressure and temperature dependence of rate constants of combination reactions and their reverse decomposition reactions are complex. Measurements over a wide range of both temperature and pressure may be required to fully characterize the rate parameters. For example, atom... 

Calculate Arrhenius factors and activation energies from measnrements of the temperature dependence of rate constants (Section 18.5, Problems 35-40). 

Table 4.6. I Temperature dependence of rate constants 0/CO2 reaction with H2O in pure water and seawater...

Reaction is complete within 30 min. I st order rate constant 10 min". Activation energy (from temperature dependence of rate constant 6.7 kJ mol". ... 

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