Rate theories

Rate theory does successfully model the behavior of many receptors. It can also be adapted to accommodate partial agonists and antagonists. Determining kon and fcoff requires more experimental effort than determining KV). 

PROBLEM The value for kon is limited by the rate of diffusion. The diffusion-limited value for kon is 108 M s. Using Equation 5.21, determine the fcoff of a ligand with a Kd of 100 nM. 

SOLUTION Only the right half of Equation 5.21 concerns us. We can rearrange it to directly calculate ko. As always, be careful with the units (100 nM = 10-7 M). fcoff turns out to be 10 s.  

The idea of drug-target residence time places less emphasis on the strength of receptor-ligand interaction and more emphasis on the duration of receptor-ligand interaction. The duration of interaction is defined as the residence time (t) of the ligand, which is equal to the inverse of koff (Equation 5.23). A closely related variable is dissociative half-life (/, 2), also defined by koll (Equation 5.24).31 

SCHEME 5.9 A residence time receptor-ligand binding model 

A few of the many contributors to the classical rate theory of boiling nucleation are Volmer (VI), Becker and Doring (B2), Frenkel (F7), Fisher (F3), and Bernath (B4). All agree that a prime requirement for nucleation to occur in a liquid is that the liquid must be superheated. The bubbles formed are cooler than the liquid therefore nucleation is strictly irreversible. Because of the superheat, a temperature driving force exists between liquid and bubble. However, because surface tension forces are immense for tiny bubbles, a collapsing tendency exists which may counteract the tendency of a bubble to grow by absorbing heat. One problem faced by any theory of nucleation is to explain the formation of a bubble which will not collapse. 

Here Ax signifies an activated cluster of x monomer units (single molecules), A i is a single simple molecule (unactivated), and Ax+i is an 

As the Plate Theory has two serious limitation, viz., first it does not speak of the separating power of a definite length of column, and second it does not suggest means of improving the performance of the column the Rate Theory has been introduced which endeavours to include the vital fact that- the mobile-phase flows continuously, besides the solute molecules are constantly being transported and partitioned in a gas chromatographic column . It is usually expressed by the following expression  

X = Measure of the packing irregularities dp = Particle diameter, y = Tortuosity factor, 

Id = Ratio of the amount of solute in the stationary phase to that in the gas phase, dj = Film thickness (usually in pm), and 

(b) was first advocated by Van Deemter in 1956, and may be rewritten as given below [Eq. (c) wherein all terms except V are constant  

Although HETP is a useful concept, it is an empirical factor. Since plate theory does not explain the mechanism that determines these factors, we must use a more sophisticated approach, the rate theory, to explain chromatographic behavior. Rate theory is based on such parameters as rate of mass transfer between stationary and mobile phases, diffusion rate of solute along the column, carrier gas flowrate, and the hydrodynamics of the mobile phase. 

Glueckauf (14) studied the effect of four factors on the chromatographic process  

Diffusion in the mobile phase normal to the direction of flow. 

The interpretation of the resulting chromatogram will tell how well a separation has been performed. This interpretation can be 

We will look at the three variables that may cause zone spreading, that is, ordinary diffusion, eddy diffusion, and local nonequilibrium. Our approach to this discussion will be from the random walk theory, since the progress of solute molecules through a column may be viewed as a random process. 

1 Multipath term (Wf). Broadening of a chromatographic zone as it passes through the column arises from the variable channels which the 

Furthermore, as diffusion can also take place radially, migration of solute molecules towards the column walls can occur. This could lead to appreciable band broadening on account of, first, the slower flow at the walls of the column, and second, solute molecule interaction with the walls, causing retardation of the zone. However, provided the sample is applied as a narrow concentrated band and the solvent has suitable viscosity, the rate of radial diffusion is not sufficient to become a problem, and under these conditions the column is described as operating in the infinite diameter mode. 

There is also an analogous contribution to the overall plate height due to diffusive spreading in the stationary phase and using similar reasoning as above this may be expressed as 

3 Slow equilibration (Hg). The assumption in the plate theory that the transfer of solute molecules between the mobile and stationary phases 

the system deviates from ideality as there is a finite rate of mass transfer of solute molecules across the chromatographic interface. The contribution to the overall HETP arising from this kinetic control of the sorption-desorption process increases with increasing mobile phase flow-rate. 

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Johnston Fi S 1966 Gas Phase Reaction Rate Theory (Ronaid)... 

The transition from k to on the low-pressure side ean be eonstnieted using iiiidtidimensional unimoleeular rate theory [1, 44], if one knows the barrier height for the reaetion and the vibrational frequeneies of the reaetant and transition state. The transition from to k y ean be deseribed in temis of Kramers theory [45]... 

If Other fall-off broadening factors arising m unimolecular rate theory can be neglected, the overall dependence of the rate coefficient on pressure or, equivalently, solvent density may be represented by the expression [1, 2]... 

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional imimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diflfiision limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Flinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

Hanggi P, Talkner P and Borkovec M 1990 Reaction-rate theory fifty years after Kramers Rev. Mod. Phys. 62 251-341... 

As a result of several complementary theoretical efforts, primarily the path integral centroid perspective [33, 34 and 35], the periodic orbit [36] or instanton [37] approach and the above crossover quantum activated rate theory [38], one possible candidate for a unifying perspective on QTST has emerged [39] from the ideas from [39, 40, 4T and 42]. In this theory, the QTST expression for the forward rate constant is expressed as [39]... 

Berne B J, Borkovec M and Straub J E 1988 Classical and modern methods in reaction rate theory J. Phys. Chem. 92 3711... 

Talkner P and Hanggi P (eds) 1995 New Trends in Kramers Reaction Rate Theory (Dordrecht Kluwer)... 

Haynes G R and Voth G A 1993 The dependence of the potential of mean force on the solvent friction consequences for condensed phase activated rate theories J. Chem. Phys. 99 8005... 

Zhu J J and Cukier R I 1995 An imaginary energy method-based formulation of a quantum rate theory J. Chem. Phys. 102 4123... 

The theory of isolated resonances is well understood and is discussed below. Mies and Krauss [79, ] and Rice [ ] were pioneers m treating unimolecular rate theory in temis of the decomposition of isolated Feshbach resonances. 

Rosenstock H M, Wallenstein M B, Wahrhaftig A L and Frying H 1952 Absolute rate theory for isolated systems and the mass spectra of polyatomic molecules Proc. Natl Acad. Sci. USA 38 667-78... 

Hase W L 1983 Variational unimolecular rate theory Acc. Chem. Res. 16 258-64... 

Wardlaw D M and Marcus R A 1984 RRKM reaction rate theory for transition states of any looseness Chem. Rhys. Lett. 110 230-4... 

Mies F H 1969 Resonant scattering theory of association reactions and unimolecular decomposition. Comparison of the collision theory and the absolute rate theory J. Cham. Phys. 51 798-807... 

Hase W L, Cho S-W, Lu D-H and Swamy K N 1989 The role of state specificity in unimolecular rate theory Chem. Phys. 139 1-13... 

Finally, exchange is a kinetic process and governed by absolute rate theory. Therefore, study of the rate as a fiinction of temperature can provide thennodynamic data on the transition state, according to equation (B2.4.1)). This equation, in which Ids Boltzmaim s constant and h is Planck s constant, relates tlie observed rate to the Gibbs free energy of activation, AG. ... 

Upson J B, Beiderhase T W, Molina L T and Molina M J 1999 Production of HCI in the OH + CIO reaction laboratory measurements and statistical rate theory calculations J. Phys. Chem. A 103 6540-51... 

Fleming GR 1986 Chemical Applications of Ultrafast Spectroscopy (Oxford Oxford University Press) Jolmston Ft S 1966 Gas Phase Reaction Rate Theory (Ronald)... 

Hanggi, R, Talkner, P., Borkovec, M. Reaction-rate theory fifty years after Kramers. Revs Modern Phys. 62 (1990) 251-341. 

Mathematical developments of reaction rate theory are given in... 

The Arrhenius rate theory, an empirical derivation, holds for the sterilization process ... 

Electrode kinetics lend themselves to treatment usiag the absolute reaction rate theory or the transition state theory (36,37). In these treatments, the path followed by the reaction proceeds by a route involving an activated complex where the element determining the reaction rate, ie, the rate limiting step, is the dissociation of the activated complex. The general electrode reaction may be described as ... 

The original microscopic rate theory is the transition state theory (TST) [10-12]. This theory is based on two fundamental assumptions about the system dynamics. (1) There is a transition state dividing surface that separates the short-time intrastate dynamics from the long-time interstate dynamics. (2) Once the reactant gains sufficient energy in its reaction coordinate and crosses the transition state the system will lose energy and become deactivated product. That is, the reaction dynamics is activated crossing of the barrier, and every activated state will successfully react to fonn product. 

This result comes from the idea of a variational rate theory for a diffusive dynamics. If the dynamics of the reactive system is overdamped and the effective friction is spatially isotropic, the time required to pass from the reactant to the product state is expected to be proportional to the integral over the path of the inverse Boltzmann probability. 

An important feamre of this method is the connection to variational rate theory The best reaction pathway is one that minimizes the reaction time or maximizes the reaction... 

BJ Berne. In P Hanggi, G Eleming, eds. Theoretical and Numerical Methods m Rate Theory. London World Scientific, 1993. 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

There are two fundamental chromatography theories that deal with solute retention and solute dispersion and these are the Plate Theory and the Rate Theory, respectively. It is essential to be familiar with both these theories in order to understand the chromatographic process, the function of the column, and column design. The first effective theory to be developed was the plate theory, which revealed those factors that controlled chromatographic retention and allowed the... 

The original Rate Theory which describes dispersion in packed beds evolved over a number of years, probably starting with the work of Lapidus and Amundson [6] in 1952, extended by that of Glueckauf [7] and Tunitski [8] in 1954. The final form of the equation that described dispersion in packed beds as a function of the linear... 

In a chromatographic separation, the individual components of a mixture are moved apart in the column due to their different affinities for the stationary phase and, as their dispersion is contained by appropriate system design, the individual solutes can be eluted discretely and resolution is achieved. Chromatography theory has been developed over the last half century, but the two critical theories, the Plate Theory and the Rate Theory, were both well established by 1960. There have been many contributors to chromatography theory over the intervening years but, with the... 

Rate Theory to describe the mechanism of band dispersion and which will be the major topic of Part 2 of this book. 

Dispersion in Columns and Mobile Phase Conduits, the Dynamics of Chromatography, the Rate Theory and Experimental Support of the... 

The theory that results from the investigation of the dynamics of solute distribution between the two phases of a chromatographic system and which allows the different dispersion processes to be qualitatively and quantitatively specified has been designated the Rate Theory. However, historically, the Rate Theory was never developed as such, but evolved over more than a decade from the work of a number of physical chemists and chemical engineers, such as those mentioned in chapter 1. 

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