Energy, of activation

The rate constants, of most reactions increase with increasing temperature. This function is described quantitatively by the Arrhenius equation (Arrhenius, 1889), 

Fundamentals of Electrochemistry, Second Edition, By V. S. Bagotsky Copyright 2006 John Wiley Sons, Inc. 

FIGURE 14.1 Potential energy-distance curves for reactants and products in a chemical reaction. 

The total energy effect of the reaction is given by = - AH = Hy Hy. When it is assumed that the forward and reverse reactions pass through the same transition state, then, evidently. 

According to the theory of rate processes (Eyring et al., 1941), reaction rate constants are determined by the expression 

In previous sections, apparent rate coefficients of electrode reactions have been described as a function of the electrode—electrolyte potential difference. As in other chemical processes, their dependence on temperature can be expressed by the Arrhenius equation 

The determination of absolute or ideal energies of activation is handicapped by the fact that the variation of the cell temperature introduces undertainties either at the reference electrode—solution interface, when both working and reference electrodes are at the same temperature, or from the thermal liquid junction potential in a non-isothermal measurement [5,36]. 

Since electrode processes involve, in most cases, reactions in solution at constant pressure with no significant change of volume, energies of activation coincide with the so-called heat or enthalpy of activation which are found in the electrochemical literature. 

From eqns. (68) and (69) and ignoring double layer effects and surface geometric factors arising from adsorbed species 

Since the electrode potential, E, includes an undeterminable potential difference due to the reference electrode [eqn. (6)], an apparent heat of activation can be obtained at constant overpotential, regardless of the reference electrode, by replacing E with E ° + rj in eqn. (101) [36]. 

To determine the energy of activation of a reaction, it is necessary to measure the rate constant of a particular reaction at different temperatures. A plot of nkr versus /T yields a straight line with slope —/S.E /R (Fig. 1.12). Alternatively, integration of Eq. (1.58) as a definite integral with appropriate boundary conditions. 

This equation can be used to obtain the energy of activation, or predict the value of the rate constant at T2 from knowledge of the value of the rate constant at T, and of AE.  

A parameter closely related to the energy of activation is the Z value, the temperature dependence of the decimal reduction time, or D value. The Z value is the temperature inerease required for a one-logio reduction (90% decrease) in the D value, expressed as 

The Z value can be determined from a plot of logjo D versus temperature (Fig. 1.13). Alternatively, if D values are known only at two temperatures, the Z value can be determined using the equation 

It can easily be shown that the Z value is inversely related to the eneigy ofacavation 2smRT,n 

Using equation 16.102 it is possible to obtain an approximate value of the energy of activation of the reaction between the three atoms. London considered the special case where the configuration of the three atoms was linear, when the energy of activation is a minimum. 

Initially atom t is a considerable distance from atoms a and b and there is no interaction, if weak van der Waal s forces are ignored, between a and b on the one hand and c on the other. Hence we may state 

These terms can in fact be neglected when Rae or Rf exceeds 3 or 4 A. In these circumstances the expression 16.102 becomes 

In order to simplify the mathematical treatment, London assumes that the distance between a and b in the molecule a—b remains unchanged on the approach of the atom c, and only the interaction with c is taken into consideration. Thus the value of Aab remains constant and the variation of the energy depends only on Ab i-e. on the distance between b and c. The value of increases with decrease of the distance between b and c until this distance is identical with the a—b distance. The energy of the system is then a maximum, which can be determined by taking the differential coefficient of E with respect to Abe and equating to zero, viz 

On solution of equation 16.105 we find that the energy has a maximum value when Abe = Aabl2. Substituting this expression in equation 16.104, we find that at the transitional state the energy is given by 

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For the steady-state case, Z should also give the forward rate of formation or flux of critical nuclei, except that the positive free energy of their formation amounts to a free energy of activation. If one correspondingly modifies the rate Z by the term an approximate value for I results ... 

Within this framework, by considering the physical situation of the electrode double layer, the free energy of activation of an electron transfer reaction can be identified with the reorganization energy of the solvation sheath around the ion. This idea will be carried through in detail for the simple case of the strongly solvated... 

From stochastic molecnlar dynamics calcnlations on the same system, in the viscosity regime covered by the experiment, it appears that intra- and intennolecnlar energy flow occur on comparable time scales, which leads to the conclnsion that cyclohexane isomerization in liquid CS2 is an activated process [99]. Classical molecnlar dynamics calcnlations [104] also reprodnce the observed non-monotonic viscosity dependence of ic. Furthennore, they also yield a solvent contribntion to the free energy of activation for tlie isomerization reaction which in liquid CS, increases by abont 0.4 kJ moC when the solvent density is increased from 1.3 to 1.5 g cm T Tims the molecnlar dynamics calcnlations support the conclnsion that the high-pressure limit of this unimolecular reaction is not attained in liquid solntion at ambient pressure. It has to be remembered, though, that the analysis of the measnred isomerization rates depends critically on the estimated valne of... 

Finally, exchange is a kinetic process and governed by absolute rate theory. Therefore, study of the rate as a fiinction of temperature can provide thennodynamic data on the transition state, according to equation (B2.4.1)). This equation, in which Ids Boltzmaim s constant and h is Planck s constant, relates tlie observed rate to the Gibbs free energy of activation, AG. ... 

As an example, experimental kinetic data on the hydrolysis of amides under basic conditions as well as under acid catalysis were correlated with quantitative data on charge distribution and the resonance effect [13]. Thus, the values on the free energy of activation, AG , for the acid catalyzed hydrolysis of amides could be modeled quite well by Eq. (5)... 

The jump frequency is related in an exponential fashion to the free energy of activation between the ground state and the saddle point ... 

Energy of activation (Section 3 2) Minimum energy that a re acting system must possess above its most stable state in or der to undergo a chemical or structural change... 

If the fraction of sites occupied is 0, and the fraction of bare sites is 0q (so that 00 + 1 = 0 then the rate of condensation on unit area of surface is OikOo where p is the pressure and k is a constant given by the kinetic theory of gases (k = jL/(MRT) ) a, is the condensation coefficient, i.e. the fraction of incident molecules which actually condense on a surface. The evaporation of an adsorbed molecule from the surface is essentially an activated process in which the energy of activation may be equated to the isosteric heat of adsorption 4,. The rate of evaporation from unit area of surface is therefore equal to... 

Rates of nitration determined over a range of temperatures in two-phase dispersions have been used to calculate energies of activation from 59—75 kj/mol (14—18 kcal/mol). Such energies of activation must be considered as only apparent, since the tme kinetic rate constants, NO2 concentrations, and interfacial area all change as temperature is increased. 

Order of thermal stabiUty as determined by differential thermal analysis is sebacic (330°C) > a2elaic = pimelic (320°C) > suberic = adipic = glutaric (290°C) > succinic (255°C) > oxahc (200°C) > malonic (185°C) (19). This order is somewhat different than that in Table 2, and is the result of differences in test conditions. The energy of activation for decarboxylation has been estimated to be 251 kj/mol (60 kcal/mol) for higher members of the series and 126 kJ/mol (30 kcal/mol) for malonic acid (1). 

The kinetics of the ethylene hydration reaction have been investigated for a tungstic oxide—siHca gel catalyst, and the energy of activation for the reaction deterrnined to be 125 kJ/mol (- 30 kcal/mol) (106,120). The kinetics over a phosphoric acid-siHca gel catalyst have been examined (121). By making some simplifying assumptions to Taft s mechanism, a rate equation was derived ... 

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