Reaction curvature

Transition state theory can also give us some insight into the non-Arrhenius behaviour of rate coefficients as epitomized by Fig. 2.6 for the OH + ethane reaction. Curvature of the Arrhenius plot can arise from a number of factors. 

Other workers have employed gaussian-type functions for fitting the curvature as a function of the reaction coordinate. In many A -t- BC reactions, curvature is not a simple function of s and the reaction probabilities are very sensitive to it. Attempts to approximate the curvature in such cases led to large quantitative errors however, Carrington eT used their RPH to calculate the lifetime of vinyl-... 

The rate law may change with temperature. Thus for reaction VII-30 the rate was paralinear (i.e., linear after an initial curvature) below about 470°C and parabolic above this temperature [163], presumably because the CuS2 product was now adherent. Non-... 

When the curvature of the reaction vessel is too great for the efficient operation of the bar type magnetic stirrer, a miniature solenoid-operated reciprocating stirrer may be employed (Fig. XII, 2, 19). This stirrer may be easilj constructed from a telephone relay or electric bell. It is advisable to have a control for adjusting the stroke while running. 

To determine the reaction order we plot ln(%p-methoxyphenylacetylene) versus time for a first-order reaction, and (%p-methoxyphenylacetylene) versus time for a second-order reaction (Figure A5.1). Because the straight-line for the first-order plot fits the data nicely, we conclude that the reaction is first-order in p-methoxyphenylacetylene. Note that when plotted using the equation for a second-order reaction, the data show curvature that does not fit the straight-line model. 

Physically the cutting-corner trajectory implies that the particle crosses the barrier suddenly on the time scale of the slow -vibration period. In the literature this approximation is usually called sudden , frozen bath and fast flip approximation, or large curvature case. In the opposite case of small curvature (also called adiabatic and slow flip approximation), coj/coo < sin tp, which is relevant for transfer of fairly heavy masses, the MEP may be taken to a good accuracy to be the reaction path. 

It has been shown that there is a two-dimensional cut of the PES such that the MEP lies completely within it. The coordinates in this cut are 4, and a linear combination of qs-q-j. This cut is presented in fig. 64, along with the MEP. Motion along the reaction path is adiabatic with respect to the fast coordinates q -q and nonadiabatic in the space of the slow coordinates q -qi-Nevertheless, since the MEP has a small curvature, the deviation of the extremal trajectory from it is small. This small curvature approximation has been intensively used earlier [Skodje et al. 1981 Truhlar et al. 1982], in particular for calculating tunneling splittings in (HF)2- The rate constant of reaction (6.45a) found in this way is characterized by the values T<. = 20-25 K, = 10 -10 s , = 1-4 kcal/mol above T, which compare well with the experiment. 

In the bromination of styrene, a po-+ plot is noticeably curved. If the extremes of the curves are taken to represent straight lines, the curve can be resolved into two Hammett relationships with p = —2.8 for electron-attracting substituents and p = —4.4 for electron-releasing substituents. When the corresponding -methylstyrenes are examined, a similarly curved ap plot is obtained. Furthermore, the stereospecificity of the reaction in the case of the -methylstyrenes varies with the aryl substituents. The reaction is a stereoespecific anti addition for strongly electron-attracting substituents but becomes only weakly stereoselective for electron-releasing substituents, e.g., 63% anti, 37% syn, forp-methoxy. Discuss the possible mechanistic basis for the Hammett plot curvature and its relationship to the stereochemical results. 

Figure 3-1 shows calculated plots of Eq. (3-16) for hypothetical systems in which kilk2 has the values 1 and 5. It is evident from the example in which k, = 5 2 that the curvature persists well into the reaction and that unambiguous identification of the terminal linear portion may be difficult. The long extrapolation to find eg is also uncertain. The accuracy of this procedure depends upon the ratios kilk2 and... 

Usually the Arrhenius plot of In k vs. IIT is linear, or at any rate there is usually no sound basis for coneluding that it is not linear. This behavior is consistent with the conclusion that the activation parameters are constants, independent of temperature, over the experimental temperature range. For some reactions, however, definite curvature is detectable in Arrhenius plots. There seem to be three possible reasons for this curvature. 

AC is interpreted as the difference in heat capacities between the transition state and the reactants, and it may be a valuable mechanistic tool. Most reported ACp values are for reactions of neutral reactants to products, as in solvolysis reactions of neutral esters or aliphatic halides. " Because of the slight curvature seen in the Arrhenius plots, as exemplified by Fig. 6-2, the interpretation, and even the existence, of AC is a matter of debate. The subject is rather specialized, so we will not explore it deeply, but will outline methods for the estimation of ACp. 

The second possible cause of nonlinearity is a change in mechanism. Within a reaction series any change in mechanism must be such as to provide a smaller free energy of activation for the reaction (otherwise the mechanism would not change). If a substituent effect can produce a change in mechanism, the result must therefore be curvature that is concave upward. Figure 7-2 is a per plot for the S l solvolyses... 

The curvature may be an artifact of a selection of nucleophiles of mixed structural types chosen to display a wide range in pAo. Buncel et al. ° varied pK by changing the solvent composition over a limited range rather than by changing the structure. They studied the reaction between X-C6H4-CT and p-nitrophenyl acetate in 40-90 mol% dimethylsulfoxide—water mixtures with just three X substituents... 

Curvature in a Br nsted-type plot is sometimes attributed to a change in transition state structure. This is not a change in mechanism rather it is interpreted as a shift in extent of bond cleavage and bond formation within the same mechanistic pattern. Thus, Ba-Saif et al. ° found curvature in the Br nsted-type plot for the identity reactions in acetyl transfer between substituted phenolates this reaction was shown earlier. They concluded that a change in transition state structure occurs in the series. Jencks et al." caution against this type of conclusion solely on the evidence of curvature, because of the other possible causes. 

A qualitative difference in the type of solvation (not simply in the strength of solvation) in a series of nucleophiles may contribute to curvature. Jencks has examined this possibility. " " An example is the reaction of phenoxide, alkoxide, and hydroxide ions with p-nitrophenyl thiolacetate, the Br insted-type plot showing Pnuc = 0.68 for phenoxide ions (the weaker nucleophiles) and Pnu = 0.17 for alkoxide ions. It is suggested that the need for desolvation of the alkoxide ions prior to nucleophilic attack results in their decreased nucleophilicity relative to the phenoxide ions, which do not require this desolvation step. 

This discussion of sources of curvature in Br insted-type plots should suggest caution in the interpretation of observed curvature. There is a related matter, concerning particularly item 5 in this list, namely, the effect of a change in transition state structure. Br nsted-type plots are sometimes linear over quite remarkable ranges, of the order 10 pK units, and this linearity has evoked interest because it seems to be incompatible with Marcus theory, which we reviewed in Section 5.3. The Marcus equation (Eq. 5-69) for the plot of log k against log K of the same reaction series requires curvature, the slope of the plot being the coefficient a. given by Eq. (5-67). A Brjinsted plot, however, is not a Marcus plot, because it correlates rates and equilibria of different reactions. The slope p of a Br nsted plot is defined p = d log kobs/d pK, which we can expand as... 

Discuss possible reasons for the curvature in the Br nsted-type plot of Fig. 7.5 for the nucleophilic reactions of oxygen nucleophiles. 

An implicit assumption of the foregoing treatment is that A// remains independent of temperature over the range investigated. This is very nearly correct in general, and is particularly the case given that studies of reactions in solution are usually conducted over a temperature interval of only some 30-50°. In certain circumstances the temperature profiles show curvature outside the experimental error. Such cases have, or appear to have, temperature-dependent activation enthalpies. Here we explore one of the reasons for that another is given in Section 7.3. 

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