Activation enthalpy

The pressure- and shear-stress-dependent activation enthalpy AH(P, z) for dislocation motion provides the necessary connection between the t = 0 kink-pair 

We begin in the t = 0 limit, where the change in enthalpy to form either a kinked or unkinked screw dislocation at constant pressure P can be written as 

Dislocations and Plasticity in bcc Transition Metals at High Pressure 

To obtain the second line of Eq. (11), we have re-expanded the derivative term in the first hne about and noted that at constant pressure 

Using Eq. (11) in Eq. (10), one finds that the pressure terms cancel and one is left with the resnlt, correct to first order in AO, 

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It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic dreoty of gases, and their interrelationship tlu ough k and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests tlrat this should be a dependence on 7 /, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to for the case of molecular inter-diffusion. The Anhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, tlren an activation enthalpy of a few kilojoules is observed. It will thus be found that when tire kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation entlralpy will be a few kilojoules only (less than 50 kJ). 

Typical values of the energy to form vacancies are for silver, lOSkJmol and for aluminium, 65.5kJmol These values should be compared with the values for the activation enthalpy for diffusion which are given in Table 6.2. It can also be seen from the Table 6.2 that die activation enthalpy for selfdiffusion which is related to the energy to break metal-metal bonds and form a vacant site is related semi-quantitatively to the energy of sublimation of the metal, in which process all of the metal atom bonds are broken. 

The rearrangement of the simplest possible case, 1,5-hexadiene, has been studied using deuterium labeling. The activation enthalpy is 33.5kcal/mol, and the entropy of activation is — 13.8eu. The substantially negative entropy reflects the formation of the cyclic transition state. 

Table 11.2. Effect of Phenyl Substituents on Activation Enthalpy and Entropy of Cope...

Another interesting idea that can be drawn from Table 6 is with regard to the relative rate of substitution of the last reactive ring position in dimethylol derivatives. It is clear that there is no favorable change in the activation enthalpy... 

The behavior of strained,/Zuorimiret/ methylenecyelopropanes depends upon the position and level of fluorination [34], l-(Difluoromethylene)cyclopropane is much like tetrafluoroethylene in its preference for [2+2] cycloaddition (equation 37), but Its 2,2-difluoro isomer favors [4+2] cycloadditions (equation 38). Perfluoromethylenecyclopropane is an exceptionally reactive dienophile but does not undergo [2+2] cycloadditions, possibly because of stenc reasons [34, 45] Cycloadditions involving most possible combinations of simple fluoroalkenes and alkenes or alkynes have been tried [85], but kinetic activation enthalpies (A/f j for only the dimerizations of tetrafluoroethylene (22 6-23 5 kcal/mol), chlorotri-fluoroethylene (23 6 kcal/mol), and perfluoropropene (31.6 kcal/mol) and the cycloaddition between chlorotnfluoroethylene and perfluoropropene (25.5 kcal/mol) have been determined accurately [97, 98] Some cycloadditions involving more functionalized alkenes are listed in Table 5 [99. 100, 101, 102, 103]... 

Equation (5-43) has the practical advantage over Eq. (5-40) that the partition functions in (5-40) are difficult or impossible to evaluate, whereas the presence of the equilibrium constant in (5-43) permits us to introduce the well-developed ideas of thermodynamics into the kinetic problem. We define the quantities AG, A//, and A5 as, respectively, the standard free energy of activation, enthalpy of activation, and entropy of activation from thermodynamics we now can write... 

It should be noted that the experimental activation enthalpy for the Diels-Alder reaction is 33 kcal/mol (estimated from the reverse reaction and the experimental reaction energy i.e. the MP2/6-31G(d) value is 14kcal/mol too low. Similarly, the calculated reaction energy of —47 kcal/mol is in rather poor agreement with the... 

The activation enthalpies and entropies are in principle dependent on temperature (eq. 12.22)), but only weakly so. For a limited temperature range they may be treated as constants. Obtaining these quantities experimentally is possible by measuring the reaction rate as a function of temperature, and plotting ln(k/T) against T" (eq. 12.24). 

The acid hydrolysis of diaziridines has been investigated kinetic-ally. The reaction is first order and shows a relatively high temperature coefficient. Thus one finds a relatively high activation enthalpy (23-28 kcal) and a positive activation entropy (2-6 eu). The influence of substitution on nitrogen is small. The velocity of the diaziridine hydrolysis depends only in the weakly acid region on the acid concentration. Between pH 7 and 3 the fc-values rise by nearly 10 . For the... 

Further studies by Garcia, Mayoral et al. [10b] also included DFT calculations for the BF3-catalyzed reaction of acrolein with butadiene and it was found that the B3LYP transition state also gave the [4+2] cycloadduct, as happens for the MP2 calculations. The calculated activation energy for lowest transition-state energy was between 7.3 and 11.2 kcal mol depending on the basis set used. These values compare well with the activation enthalpies experimentally determined for the reaction of butadiene with methyl acrylate catalyzed by AIGI3 [4 a, 10]. 

For small enough temperature steps (< lOK) during small step annealing the vacancy concentration practically remains constant and corresponds to the instantaneous aimealing temperature. This allows for an easy analysis of SRO-kinetics yielding SRO-relaxation times and SRO-activation enthalpies, which by usual interpretation correspond to H +Hf. 

M. Migschitz and W Pfeiler, Vacancy activation enthalpies of Au-Fe alloys determined from SRO-induced resistivity changes, A/a/. Sci. Eng. A206 I55 (1996)... 

Though solid electrolytes for multivalent ions offer the advantage of a larger charge transfer, their conductivities are much lower than those of monovalent ions at ambient temperature because of a higher activation enthalpy for the ionic motion... 

Table 1. Conductivities, activation enthalpies, and other aspects of fast lithium-ion solid conductors...

Another way of looking at high ionic conductivities of solid electrolytes is to consider the activation enthalpy as illustrated in Fig. 8. Generally, the activation enthalpy is strongly correlated with the room-temperature ionic conductivity the higher the room-temperature ionic conductivity, the lower the activation enthalpy. The straight lines in the Arrhenius... 

An example of the determination of activation enthalpies is shown in Figs. 11 and 12. A valuable indication for associating the correct minimum with the ionic conductivity is the migration effect of the minimum with the temperature (Fig. 11) and the linear dependence in the cr(T versus 1/T plot (Fig. 12). However, the linearity may be disturbed by phase transitions, crystallization processes, chemical reactions with the electrodes, or the influence of the electronic leads. 

Table 3 shows that the activation enthalpies determined by various authors can be very different. These differences cannot be correlated to discrepancies in reaction orders since, even when these are the same, activation energies can vary. Since the theoretical difference between activation enthalpy and activation energy is low (2RT = 3kJ mol"1) with regard to the differences found in experimental determinations, the values discussed below are either enthalpies or energies of activation (For more detailed information see Table 3). 

Due to the differences in the values relative to any one system, conclusions cannot easily be drawn from the activation parameters listed in Table 3. However, an analysis of the results relative to 1,2-ethanediol, 2,2-dimethyl-l,3-propanediol, 1,5-pentanediol, 1,10-decanediol and diethylene glycol shows that a slight difference can be observed between aromatic and aliphatic acids the activations enthalpies and entropies are in the ranges 70, 100 kJ mol"1 and -SO, -130 J K"1 mol-1 for aromatic acids, and in the ranges 50, 70 kJ mol"1 and -200, -100 J K"1 mol-1 for the aliphatic acids. 

Several authors studied the influence of substituents on activation parameters. Bad-dar et al.315 who studied the polyesterification of y-arylitaconic anhydrides and adds with 1,2-ethanediol found that in the non-catalyzed reaction a p-methoxy substituent decreases both the activation enthalpy and the entropy whereas an increase is observed with a p-chloro substituent. On the other hand, Huang et al., who studied the esterification of 2,2-dimethyl-l,3-propanediol with benzoic, butanedioic, hexanedioic, decanedioic and o-phthalic add found the same values since the activation enthalpy is 64 kJ mol-1 for the first reaction and 61 kJ mol-1 for the others. 

Both for reaction in and IV the order with respect to catalyst is 0.5. The activation enthalpies are 96.6 3.4 and 97.6 3.4 kJ mol-1 respectively when Ti(OBu)4 is used as the catalyst. This is not too far from the activation enthalpies200 for the Sn(II)-cata-lyzed esterification of B with isophthalic acid (85.1 4.9) and with 2-hydroxyethyl hydrogen isophthalate (85.8 4.2). It is also close to the Ti(OBu)4-catalyzed esterification of benzoic acid with B (85.8 2.5)49. This is probably due to the formation of analogous intermediate complexes and similar catalytic mechanisms. On the other hand, the activation entropies of reactions III and IV are less negative than those of the reaction of benzoic or isophthalic acid with B. This probably corresponds to a stronger desolvation when the intermediary complex is formed and could be due to the presence of the sodium sulfonate group. 

Habid and Malek49 who studied the activity of metal derivatives in the catalyzed esterification of aromatic carboxylic acids with aliphatic glycols found a reaction order of 0.5 relative to the catalyst for Ti(OBu)4, tin(II) oxalate and lead(II) oxide. As we have already mentioned in connection with other examples, it appears that the activation enthalpies of the esterifications carried out in the presence of Ti, Sn and Pb derivatives are very close to those reported by Hartman et al.207,208 for the acid-catalyzed esterification of benzoic and substituted benzoic acids with cyclohexanol. These enthalpies also approach those reported by Matsuzaki and Mitani268 for the esterification of benzoic acids with 1,2-ethanediol in the absence of a catalyst. On the other hand, when activation entropies are considered, a difference exists between the esterification of benzoic acid with 1,2-ethanediol catalyzed by Ti, Sn and Pb derivatives and the non-catalyzed reaction268. Thus, activation enthalpies are nearly the same for metal ion-catalyzed and non-catalyzed reactions whereas the activation entropy of the metal ion-catalyzed reaction is much lower than that of the non-catalyzed reaction. 

It was concluded that the variations in rate are due to variations in activation enthalpy rather than entropy, and since the rates of substitution rates at the para positions of toluene and /-butylbenzene varied by only 4 % for a change in reactivity of 6,430, it was concluded that the Baker-Nathan reactivity order does not arise from a solvent effect (c/. Table 57). 

Some quantities associated with the rates and mechanism of a reaction are determined. They include the reaction rate under given conditions, the rate constant, and the activation enthalpy. Others are deduced reasonably directly from experimental data, such as the transition state composition and the nature of the rate-controlling step. Still others are inferred, on grounds whose soundness depends on the circumstances. Here we find certain features of the transition state, such as its polarity, its stereochemical arrangement of atoms, and the extent to which bond breaking and bond making have progressed. 

The activation parameters bring out several features. Note that the activation enthalpy and activation energy for kn, which represents a very rapid reaction, are quite small. Of course, a fast reaction can have a higher activation energy, if the value of AS is more positive, so as to compensate. The activation entropy associated with k is particularly large and negative, as is most often the case for a second-order reaction that occurs by a bimolecular step. In such cases, AS reflects the loss of entropy from the union of the two reaction partners into a single transition state. 

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