Forward activation energies

Figure 4.77 The optimized structure of the transition state II for the ethylene-insertion reaction II III (4.106), with forward activation energy A > = 6.90 kcalmol-1 relative to the metal-ethylene complex II.

Fig. 7-7. Potential energy curves for an elementary step of reaction in equilibrium (solid curve) and in nonequilibrium (dashed curve) 4glq = activation energy in equilibrium 4gj s forward activation energy in nonequilibrium p>. , -electrochemical potential of activated partide in equilibrium p = symmetry factor Zi = charge number of reacting partide.

In the case of electrode reactions, the activation energy depends on the electrode potential. We now consider an elementary step in which a charged particle (charge number, zi) transfers across the compact double layer on the electrode interface as shown in Fig. 7-7. In the reaction equilibrium, where the electrochemical potentials of reacting particles are equilibrated between the initial state and the final state (Pk o = Pf( i)), the forward activation energy equals the backward activation energy (P , - Pi = P, i- Pr) P , is the electrochemical potential of the reacting particle at the activated state in equilibrium. 

The result of Glukhovtsev et al. [43], who applied G2 theory to reaction (9), is also somewhat high, but there is particularly close accord with the results of Hassan-zadeh and Irikura [10]. Begovi et al. [44] used density functional theory (DFT) to obtain values that are too negative by 14 kj mor. The value of Berry et al. [11], which is significantly below the present value, was derived from the difference between the measured forward activation energy Ea for... 

The energy required to attain the transition state is the energy difference between the activated complex, A B C, and the minima (zero-point energies) of the reactants for the forward reaction direction or of the products for the reverse direction where the Gibbs energies of these are described by their respective standard chemical potentials (n°). The standard forward activation energy barrier, AG-... 

Why is the forward activation energy for the following reaction so large ... 

A catalyst causes a lower total activation energy by providing a different mechanism for the reaction. The total of the activation energies for both steps of the catalyzed pathway [Fai(fwd) + a2(fwd)] is Isss than the forward activation energy of the uncatalyzed pathway. 

Figure 5, Steady states of the cooperative isomerization model, showing sub-critical (t) = 1), critical (t) — 4), and supercritical (t) = 5 curves of mole fraction x as a function of the forward activation energy t. The deterministic transitions for r/ = 5 are indicated by arrows the dashed line denotes an equal areas construction which determines the unique equilibrium transition (22).

Thermodynamic consistency of individual elementary reactions implies that the backward activation energy AE, is related to the forward activation energy AE,jr and the standard enthalpy change of the reaction AH in the following way ... 

As tire reaction leading to tire complex involves electron transfer it is clear that tire activation energy AG" for complex fonnation can be lowered or raised by an applied potential (A). Of course, botlr tire forward (oxidation) and well as tire reverse (reduction) reaction are influenced by A4>. If one expresses tire reaction rate as a current flow (/ ), tire above equation C2.8.11 can be expressed in tenns of tire Butler-Volmer equation (for a more detailed... 

The dehydration reactions have somewhat higher activation energies than the addition step and are not usually observed under strictly controlled kinetic conditions. Detailed kinetic studies have provided rate and equilibrium constants for the individual steps in some cases. The results for the acetone-benzaldehyde system in the presence of hydroxide ion are given below. Note that is sufficiently large to drive the first equilibrium forward. 

These results predict that the hydroxycarbene to formaldehyde reaction will proceed significantly more easily than the fonvard reaction. However, for this problem, electron correlation is needed for good quantitative values. For example, the MP4/6-31G(d,p) level predicts a value of 86.6 kcal mol" for the activation energy of the forward reaction. 

This is the general expression for film growth under an electric field. The same basic relationship can be derived if the forward and reverse rate constants, k, are regarded as different, and the forward and reverse activation energies, AG are correspondingly different these parameters are equilibrium parameters, and are both incorporated into the constant A. The parameters A and B are constants for a particular oxide A has units of current density (Am" ) and B has units of reciprocal electric field (mV ). Equation 1.114 has two limiting approximations. 

The relationship between activation energies for the forward and reverse reactions can be expressed mathematically. The activation energy is denoted by the symbol A// (read delta-//-cross ) and the heat of the reaction by AH. Hence we may write ... 

Figure 8-8 shows the analogous situation for a chemical reaction. The solid curve shows the activation energy barrier which must be surmounted for reaction to take place. When a catalyst is added, a new reaction path is provided with a different activation energy barrier, as suggested by the dashed curve. This new reaction path corresponds to a new reaction mechanism that permits the reaction to occur via a different activated complex. Hence, more particles can get over the new, lower energy barrier and the rate of the reaction is increased. Note that the activation energy for the reverse reaction is lowered exactly the same amount as for the forward reaction. This accounts for the experimental fact that a catalyst for a reaction has an equal effect on the reverse reaction that is, both reactions are speeded up by the same factor. If a catalyst doubles the rate in one direction, it also doubles the rate in the reverse direction. 

Catalysts increase the rate of reactions. It is found experimentally that addition of a catalyst to a system at equilibrium does not alter the equilibrium state. Hence it must be true that any catalyst has the same effect on the rates of the forward and reverse reactions. You will recall that the effect of a catalyst on reaction rates can be discussed in terms of lowering the activation energy. This lowering is effective in increasing the rate in both directions, forward and reverse. Thus, a catalyst produces no net change in the equilibrium concentrations even though the system may reach equilibrium much more rapidly than it did without the catalyst. 

FIGURE 13.27 (a) The activation energy for an endothermic reaction is larger in the forward direction than in the reverse and so the rate of the forward reaction is more sensitive to temperature and the equilibrium shifts toward products as the temperature is raised, (b) The opposite is true for an exothermic reaction, in which case the reverse reaction is more sensitive to temperature and the equilibrium shifts toward reactants as the temperature is raised. 

A catalyst speeds up a reaction by providing an alternative pathway—a different reaction mechanism—between reactants and products. This new pathway has a lower activation energy than the original pathway (Fig. 13.34). At the same temperature, a greater fraction of reactant molecules can cross the lower barrier of the catalyzed path and turn into products than when no catalyst is present. Although the reaction takes place more quickly, a catalyst has no effect on the equilibrium composition. Both forward and reverse reactions are accelerated on the catalyzed path, leaving the equilibrium constant unchanged. 

L-mol 1 -min 1 and the rate constant for the reverse reaction is 392 L-mol 1 -min. The activation energy for the forward reaction is 39.7 kj-mol 1 and that of the reverse reaction is 25.4 kj-mol" (a) What is the equilibrium constant for the reaction (b) Is the reaction exothermic or endothermic (c) What will be the effect of raising the temperature on the rate constants and the equilibrium constant ... 

C15-0083. If a reaction has an activation energy of zero, how is A for the forward reaction related to for the reverse reaction Draw an activation energy diagram illustrating your answer. 

C15-0084. Consider the exothermic reaction AC -b B AB -b C. (a) Draw an activation energy diagram for this reaction, (b) Label the energies of reactants and products, (c) Show A reaction by a double-headed arrow, (d) Show a for the forward reaction by a single-headed arrow, (e) Label and draw a molecular picture of the activated complex. 

In principle this is derived from an Arrhenius plot of In r+ versus 1/T but such a plot may deviate from a straight line. Hence, the apparent activation energy may only be valid for a limited temperature range. As for the orders of reaction, one should be very careful when interpreting the activation energy since it depends on the experimental conditions. Below is an example where the forward rate depends both on an activated process and equilibrium steps, representing a situation that occurs frequently in catalytic reactions. 

A pre-exponential factor and activation energy for each rate constant must be established. All forward rate constants involving alkyne adsorption (ki, k2, and ks) are assumed to have equal pre-exponential factors specified by the collision limit (assuming a sticking coefficient of one). All adsorption steps are assumed to be non-activated. Both desorption constants (k.i and k ) are assumed to have preexponential factors equal to 10 3 sec, as expected from transition-state theory [28]. Both desorption activation energies (26.1 kcal/mol for methyl acetylene and 25.3 kcal/mol for trimethylbenzene) were derived from TPD results [1]. 

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