Transition-state theory entropy

MICHAELIS-MENTEN KINETICS PREEXPONENTIAL FACTOR ARRHENIUS EQUATION COLLISION THEORY TRANSITION-STATE THEORY ENTROPY OF ACTIVATION PRENYL-PROTEIN-SPECIFIC ENDOPEP-TIDASE... 

IV. Entropy of Activation and Structure From the inception of transition state theory, entropies of activation have been discussed from the twin aspects of molecular structure and reaction mechanism. Even though there is considerable overlap between these two aspects we shall utilize a formal separation, reserving much of the discussion of mechanism for the next section. In this section our primary concern shall be the effect that structural change in a non-reacting part of a molecule has upon the entropy and enthalpy of activation for that molecule. The nature of interactions (polar, steric, and resonance) between the substituent group and the reaction center clearly relates to the problem of reaction mechanism, the solution of which involves, in the final analysis, a detailed description of the disposition of the atoms in the transition state and the interactions among them. 

The experimental side of the subject explores the effects of certain variables on the rate constant, especially temperature and pressure. Their variations provide values of the activation parameters. They are the previously mentioned energy of activation, entropy of activation, and so forth. The chemical interpretations that can be realized from the values of the activation parameters will be explored in general terms, but readers must consult the original literature for information about those chemical systems that particularly interest them. On the theoretical side, there is the tremendously powerful transition state theory (TST). We shall consider its origins and some of its implications. 

The activation parameters from transition state theory are thermodynamic functions of state. To emphasize that, they are sometimes designated A H (or AH%) and A. 3 4 These values are the standard changes in enthalpy or entropy accompanying the transformation of one mole of the reactants, each at a concentration of 1 M, to one mole of the transition state, also at 1 M. A reference state of 1 mole per liter pertains because the rate constants are expressed with concentrations on the molar scale. Were some other unit of concentration used, say the millimolar scale, values of AS would be different for other than a first-order rate constant. 

Hence, according to the transition state theory, adsorption becomes more likely if the molecule in the mobile physisorbed precursor state retains its freedom to rotate and vibrate as it did in the gas phase. Of course, this situation corresponds to minimal entropy loss in the adsorption process. In general, the transition from the gas phase into confinement in two dimensions will always be associated with a loss in entropy and the sticking coefficient is normally smaller than unity. 

Nonlinear Hamiltonian system, geometric transition state theory, 200-201 Nonlinear thermodynamics coefficients linear limit, 36 entropy production rate, 39 parity, 28-29... 

Holroyd (1977) finds that generally the attachment reactions are very fast (fej - 1012-1013 M 1s 1), are relatively insensitive to temperature, and increase with electron mobility. The detachment reactions are sensitive to temperature and the nature of the liquid. Fitted to the Arrhenius equation, these reactions show very large preexponential factors, which allow the endothermic detachment reactions to occur despite high activation energy. Interpreted in terms of the transition state theory and taking the collision frequency as 1013 s 1- these preexponential factors give activation entropies 100 to 200 J/(mole.K), depending on the solute and the solvent. 

Three possibilities were considered to account for the curved Arrhenius plots and unusual KIEs (a) the 1,2-H shift might feature a variational transition state due to the low activation energy (4.9 kcal/mol60) and quite negative activation entropy (b) MeCCl could react by two or more competing pathways, each with a different activation energy (e.g., 1,2-H shift and azine formation by reaction with the diazirine precursor) (c) QMT could occur.60 The first possibility was discounted because calculations by Storer and Houk indicated that the 1,2-H shift was adequately described by conventional transition state theory.63 Option (b) was excluded because the Arrhenius curvature persisted after correction of the 1,2-H shift rate constants for the formation of minor side products (azine).60... 

It is clear by comparing the transition state theory with the collision model that the corresponding entropy of activation can be calculated from the value... 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

The standard enthalpy difference between reactant(s) of a reaction and the activated complex in the transition state at the same temperature and pressure. It is symbolized by AH and is equal to (E - RT), where E is the energy of activation, R is the molar gas constant, and T is the absolute temperature (provided that all non-first-order rate constants are expressed in temperature-independent concentration units, such as molarity, and are measured at a fixed temperature and pressure). Formally, this quantity is the enthalpy of activation at constant pressure. See Transition-State Theory (Thermodynamics) Transition-State Theory Gibbs Free Energy of Activation Entropy of Activation Volume of Activation... 

See Transition-State Theory Transition-State Theory (Thermodynamics) Transition-State Theory in Solutions Entropy of Activation Volume of Activation... 

Pi)Ay /Rr. Thus, In 2 = (3000 - l)Ay /(82.05 X 298) = 5.7 cm. This exercise indicates that reaction rate is relatively insensitive to pressure changes if Ay is small. See Transition-State Theory Expressed in Thermodynamic Terms Gibbs Free Energy of Activation Enthalpy of Activation Entropy of Activation lUPAC (1979) Pure Appl. Chem. 51, 1725. 

TRANSITION-STATE THEORY GIBBS FREE ENERGY OF ACTIVATION ENTROPY OF ACTIVATION VOLUME OF ACTIVATION ENTROPY... 

As seen in Tables 22—25, the Arrhenius preexponential factors Aa for the initiation step are very low, 10 in 7, 10 in 20, 10 " in 41 and 1in 44. These are very low values for bimolecular reactions for which values of about 10 ° are observed and also predicted by the Transition State Theory Thus step (a) belongs to a class of slow reactions , some of which might have ionic transition states . The activation entropies AS obtained from the Transition State Theory rate constant expression... 

The preexponential factor involves the entropy change in going from reactants to the transition state the more highly ordered and tightly bound is the transition state, the more negative A S° will be and the lower the preexponential factor will be. Transition state theory thus automatically takes into account the effect of steric factors on rate constants, in contrast to collision theory. 

Transition state theory itself is a beautiful union of wave mechanics in one respect and thermodynamics in the other. However, it cannot be applied fully to the systems that we have been discussing. In these circumstances the theory is used in an approximate way, it may even be used as a language to present results. Very often the entropy of activation that is reported for a reaction is not much more than a formal translation of data in a mechanical way, and it is dangerous to read all transition state implications into the result. 

I am suggesting that often the applicability of Barkley-Butler type plots, that is, the linear relationship between the entropy and enthalpy of activation in a series may come about because of there being a distribution of reaction paths. Small variations in the importance of low activation energy, low probability paths could then account for the data in Dr. Taube s table. By contrast, transition state theory in its approximate application, invariably leads to diagrams of energy vs. reaction path which, in spite of all protest, one reaction path, whatever it is, one transition state, and one energy. 

R. A. Marcus My interests in variational microcanonical transition state theory with J conservation goes back to a J. Chem. Phys. 1965 paper [1], and perhaps I could make a few comments. First, using a variational treatment we showed with Steve Klippenstein a few years ago that the transition-state switching mentioned by Prof. Lorquet poses no major problem The calculations sometimes reveal two, instead of one, bottlenecks (transition states, position of minimum entropy along the reaction coordinate) [2], and then one can use a method described by Miller and partly anticipated by Wigner and Hirschfelder to calculate the net dux. 

The high effective concentration of intramolecular groups is one of the most important reasons for the efficiency of enzyme catalysis. This can be explained theoretically by using transition state theory and examining the entropy term in the rate equation (2.7). It will be seen that effective concentrations may be calculated by substituting certain entropy contributions into the exp (AS /R) term of equation 2.7. 

The transition state theory indicates that the rate of a reaction is not a matter of energy alone, but also requires a favorable configuration by a change of entropy. In addition, the rate of a reaction can be speeded up through the following methods. These methods are the guiding principles in the search for the most efficient AOPs ... 

Since the concept of an activation entropy arises out of transition state theory, it is useful to review briefly the principles of the theory. For a more complete treatment the reader is referred to standard texts (Glasstone et al., 1941a Laidler, 1950). 

---