Rate constant microscopic

It is a remarkable fact that the microscopic rate constant of transition state theory depends only on the equilibrium properties of the system. No knowledge of the system dynamics is required to compute the transition state theory estimate of the reaction rate constant... 

The recipe (5.58) is even more sensitive to the high-frequency dependence of kjj than similar criterion (5.53), which was used before averaging over kinetic energy of collisions E. It is a much better test for validity of microscopic rate constant calculation than the line width s j-dependence, which was checked in Fig. 5.6. Comparison of experimental and theoretical data on ZR for the Ar-N2 system presented in [191] is shown in Fig. 5.7. The maximum value Zr = 22 corresponding to point 3 at 300 K is determined from the rate constants obtained in [220],... 

The microscopic rate constant is derived from the quantum mechanical transition probability by considering the system to be initially present in one of the vibronic levels on the initial potential surface. The initial level is coupled by spin-orbit interaction to the manifold of vibronic levels belonging to the final potential surface. The microscopic rate constant is then obtained, following the Fermi-Golden rule, as ... 

The microscopic rate constants for association and dissociation at a site within an electric field (for block by charged drugs) are exponential functions of the membrane voltage ... 

Note that the ratios of observable to microscopic rate constants (in square brackets) determine the fractional degree to which the microscopic rate constant determines the rate with protiated substrate (i.e., [(kcat/ M)H/ oiH] = 1 when ioiH = ( cat/ M)H and moiH is fully rate-limiting [(kcat/ M)H/ ioiH] =0 when ioiH ( cat/ M)H and moiH has no effect on the rate). These quantities are weighting factors for the individual isotope effects ... 

Consider a multistep reaction with a single isotope-sensitive step. Let the observable isotopic rate constants be denoted kyi, and fex and the microscopic rate constants be denoted Not isotope sensitive) and fesH. sd. and st (isotope Sensitive). Let sh/ sd = 5 and = 10-2, so that sd/ st=1-9... 

The above theory allows one to estimate the off-rate constant for biopolymer disassembly, but it does not take into account the possibility of having distinctly different rate constants at each end of the polymer. Thus, one obtains the sum of the off-rate constants rather than the exact constants for each end. Another limitation is that one does not know how many growing points there are at each microtubule end, and from the discussion of the multiplicity of helical starts (Amos et al., 1976), the observed off-constant should represent the product of n (the number of growing points) and the actual microscopic rate constant for each of these growing points. As will be seen below, axoneme-promoted assembly of microtubules (Bergen and Borisy, 1980) may help to obviate the former limitation, but the latter requires further characterization of the true growing points. 

The rate constants so far discussed have been microscopic rate constants for specific electronic and vibrational levels. Such rate constants are not always directly available from experiment. Experiments usually measure a rate constant statistically averaged over different energy levels, each level being weighted by its population. An experimentally observed rate constant for intersystem crossing from the appropriate average of 1 2u vibrational levels to all triplet levels we denote by kIso. 

Although the common features 1, 2, and 3 above were for certain idealized prepared states, it seems reasonable to expect that the actual prepared state appropriate to a given experiment may be similar enough that these same common features will be exhibited for the microscopic rate constants. These common features of the microscopic rate constant may in turn provide qualitative predictions for the averaged rate constant Arise- These qualitative predictions are expected to be characteristic of intersystem crossing due to a zero-order crossing. 

It was pointed out earlier that the reaction rate for low substrate concentrations is given by v = (kcJKM)[ E (l[S] (equation 3.3) that is, kcJKM is an apparent second-order rate constant. It is not a true microscopic rate constant except in the extreme case in which the rate-determining step in the reaction is the encounter of enzyme and substrate. 

Steady state kinetic measurements on an enzyme usually give only two pieces of kinetic data, the KM value, which may or may not be the dissociation constant of the enzyme-substrate complex, and the kcM value, which may be a microscopic rate constant but may also be a combination of the rate constants for several steps. The kineticist does have a few tricks that may be used on occasion to detect intermediates and even measure individual rate constants, but these are not general and depend on mechanistic interpretations. (Some examples of these methods will be discussed in Chapter 7.) In order to measure the rate constants of the individual steps on the reaction pathway and detect transient intermediates, it is necessary to measure the rate of approach to the steady state. It is during the time period in which the steady state is set up that the individual rate constants may be observed. 

Just as in enzyme kinetics (see chapter 7), Km here is an algebraic function of the microscopic rate constants for binding, dissociation, and translocation of the substrate in either direction. 

The interpretation of kinetic data begins with a hypothetical sequence of ele mentary reaction steps, each characterized by two microscopic rate constants, one for the forward and one for the reverse reaction. From this proposed mechanism a rate equation is derived, predicting the dependence of the observed reaction rate on concentrations and on microscopic rate constants, and its form is tested against the observations. If the form of the rate equation meets the test of experiment, it may be possible to derive from the data numerical values for the microscopic rate constants of the proposed elementary reaction steps. While inconsistency is clear grounds for modifying or rejecting a mechanistic hypothesis, agreement does not prove the proposed mechanism correct. 

Macroscopic and microscopic rate constants Except in the simplest mechanisms, the observed rate constant for the reaction as a whole will not correspond to any one of the microscopic rate constants k characterizing the individual steps. The term observed rate constant, kobs, is used for the overall rate constant for the complete reaction. 

Comparison between the transition state expression (2.61) and the Arrhenius equation (2.50) may be made if both are applied to the microscopic rate constant for a single reaction step.42 The correspondence is as follows 43... 

Define the terms microscopic rate constant and observed rate constant. 

Branching may also occur from an intermediate and Scheme 9.8b shows the simplest scheme where P and Q are formed irreversibly from a Bodenstein intermediate (I). Ki-netically, this mechanism is not distinguishable from that where branching occurs at the reactant. The product ratio [P]/[Q] is the ratio of the rate constants of the forward processes at the branch point. The reaction again shows first-order behaviour with respect to the reactant R, with the overall experimental pseudo-first-order rate constant dependent on all the microscopic rate constants. 

In this connection kinetic models can also be separated into microscopic and macroscopic models. The relations between these models are established through statistical physics equations. Microscopic models utilize the concepts of reaction cross-sections (differential and complete) and microscopic rate constants. An accurate calculation of reaction cross-sections is a problem of statistical mechanics. Macroscopic models utilize macroscopic rates. 

There are two reaction pathways for addition of neutral and anionic nucleophiles to p-1 (Scheme 39) direct nucleophile addition with rate constant kNu (M 1 s 1) and specific acid-catalyzed nucleophile addition kHNu (M 2 s 1), through the protonated intermediate p-H-l+ with an acidity constant and microscopic rate constants of ks and k- u (Scheme 39) for addition of solvent or nucleophilic anion to form product. Eqs. (2a) and (2b) show the... 

Multiply and divide by the microscopic rate constants for the protiated substrate ... 

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