Compensation laws

Comparisons of reactivity at different temperatures may be misleading if the Compensation Law or isokinetic relationship applies. i 152c... 

In a series of reactions for which an acceUrative decrease in the activation energy is accompanied by a decelerative decrease in the entropy of activation (Compensation Law ), or the two increase together, there wiU be an isokinetic temperature (between 0-200° C for three-fourths of the 79 reactions tabulated by Leffler ). The rate vs. temperature curves for all the reactions in the series pass through this single point. Comparisons are affected since the isokinetic temperature is a point of inversion of relative reactivity in the series. 

It is also a point of change in control of the reaction rate by the energy of activation below it to control by the entropy of activation above it. The effect of changes in structure, solvent, etc., will depend on the relation of the experimental temperature to the isokinetic temperature. A practical consequence of knowing the isokinetic temperature is the possibility of cleaning up a reaction by adjusting the experimental temperature. Reactions are cleaner at lower temperatures (as often observed) if the decrease in the experimental temperature makes it farther from the isokinetic temperature. The isokinetic relationship or Compensation Law does not seem to apply widely to the data herein, and, in any case, comparisons are realistic if made far enough from the isokinetic temperature. 

The problem of relationship between the activation parameters-the so called isokinetic relationship or compensation law—is of fundamental importance in structural chemistry, organic or inorganic. However, there are few topics in which so many misunderstandings and controversies have arisen as in connection with this problem. A critical review thus seems appropriate at present, in order to help in clarifying ideas and to draw attention to this treatment of kinetic or equilibrium data. The subject has already been reviewed (1-6), but sufficient attention has not been given to the statistical treatment which represents the heaviest problems. In this review, the statistical problems are given the first place. Theoretical corollaries are also dealt with, but no attempt was made to collect all examples from the literature. It is hoped that most of the important... 

Figure 3-34 The compensation law for diffusion of some species in (a) water where In 20.12 + 0.404F, and (b) a silicate melt where In 4 -19.29+ 0.0453F.

The compensation law is a very rough empirical correlation between the activation energy and the pre-exponential factor of diffusion. Winchell (1969) showed that the logarithm of the pre-exponential factor (A) is roughly linear to the activation energy ( ) ... 

Therefore, the compensation law is equivalent to the statement that in a In D versus 1/T plot, all lines for various species intersect at one common point. 

In addition to applications to diffusion in the same phase, the compensation law has also been applied to the diffusion of a given species in many phases (Bejina and Jaoul, 1997). This is equivalent to the assumption that at some critical temperature, the diffusion coefficients of the species in all phases would be the same. The relation again is expected to be very approximate. 

Not many practical uses have been found for the compensation law because it is not accurate enough. One potential use of the compensation law is that if one knows the diffusivity at one single temperature, then both the pre-exponential factor A and the activation energy E may be estimated. That is, the temperature dependence of the diffusivity may be inferred. In practice, however, because the compensation "law" itself is not accurate, the uncertainty of the approach is very large (intolerable in geologic applications). Hence, the approach is not recommended. 

Examine the applicability of the compensation law using the following examples (data can be found in the Appendix 4, plus your own search of data from literature). 

The suggestion has been made recently that the origin of the relationship lies in solute-solvent interactions (Laidler, 1959). This cannot be the whole story, however, since the compensation law appears also to hold for gas phase reactions and equilibria. Furthermore it is by no means universal reaction series are known in which either AH or AS is constant, or in which AH and AS vary independently. Riietschi (1958) has recently noted that the basic cause of compensation appears to be the invariance of the shape of the potential energy surfaces for a series of similar reactants and shows that this can lead to the proper relation between the frequency and the dissociation energy of similar bonds. The suggestion that solute-solvent interactions contribute to this compensation in solution processes can be accommodated by supposing that changes in structure alter the frequencies related to restricted rotation, perhaps of solvent molecules (Laidler, 1959 Willi, 1961). 

The existence of an enthalpy-entropy relationship has some important mechanistic implications. As the subject has been reviewed (Leffler, 1955) it will suffice here to make only a few brief comments. One important consequence of the compensation law is that linear free-energy relationships appear to apply to reactions with variable entropy only when the entropy is a linear function of the enthalpy (Jaffe, 1953 Taft, 1956c). 

OSHAhas estimated a safe maximum noise level of 85 dB. The time-weighted average (TWA) is an exposure for an 8-h to a noise level not exceeding 90 dB. If this level exceeds 85 dB, OSHA requires the employer to institute a hearing conservation program (HCP). Therefore, if a company wants to avoid loss claims under worker compensation laws, it must not only meet the prescribed legal standards, but also attempt to reduce noise to the lowest possible level (< 80 dB). 

Jaoul O. and Sautter V. (1999) A new approach to geospeedometry based on the compensation law . Phys. Earth Planet. Inter. 110, 95-114. 

A widely used interpretation of the compensation law is based on a two-site model proposed by Hoffman et al. (22) to describe crystalline relaxations in n-paraffins. Molecular movements are assumed to involve an entire short-chain molecule, the length of the molecules corresponding to the thickness of crystallites. Under these assumptions, the relaxation time is expressed by an Eyring equation (Equation 1E9), with... 

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