Rate constant, defined

From Equation (2.4) we see that the overall activation energy for the enzyme-catalyzed reaction is related to the second-order rate constant defined by the ratio... 

The simplest model to assume is based on the possibility that the reactivities of the o- and p-methylol groups are independent of the structure of the molecule to which they are attached. According to this model, three rate constants koo kop and kpp are sufficient to characterize the kinetic behavior of the reaction at a given temperature. (The subscripts associated with these rate constants define... 

The kinetics for all guests were fit to the sum of two exponentials. The recovered observed rate constants differed by a factor of 4 for the kinetics of these guests with ct-DNA and by a factor of 10 for the kinetics of the guests with the polydeoxynuc-leotides. For this reason, the kinetics were analyzed by determining an apparent observed rate constant defined by the fractional amplitudes (A,) and the individual rate constants ... 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

Since the enzymatic reactions are generally rapid, it may be assumed that the steady-state approximation applies. Note, however, that although true is most systems, this is not always the case, as exemplified in Section 5.2.5. Each half-reaction is characterized by three rate constants, defined in Scheme 5.1. They may alternatively be characterized by the following... 

This technique has been taken to new heights through the use of 2-D nuclear magnetic resonance " to assign many, if not all, exchangeable protons in terms of the residues to which they are attached. As a result, one can assess individual rate constants defining the kinetics of exchange. 

Setting the right-hand side of Eq. (E) to 0 and using the composite rate constant defined by Eq. (F), one can show (see Problem 11) that... 

Also, in most cases, double-resonance experiments provide the forward, kf, and reverse, k, rate constants defined in Eq. (2). 

Equation 18.21 is occasionally used to estimate barrier heights assuming that k is 1.0 and v = kBT/h (Chapter 2, section A, and the rate constant defined as in equation 18.10), but its main use is the calculation of changes in AG D—i.e., AAG d—-for which the terms vk cancel out. For example, if a mutant folds with a rate constant k f, compared with kf for wild type, then the change in activation energy on mutation is given by... 

The kinetics of the condensation of the Cr(H20)63+ ion and its corresponding deprotonated species have been studied in the pH region 3.5-5.0 [25°C, I = 1.0 M(NaC104)] (201). The study of this reaction is complicated by the formation of higher oligomers. Chromatographic analysis of the products as a function of time established the dinuclear species to be the main product for the first 5% of reaction, and the initial-rate kinetics of condensation were studied by a pH-stat technique. The observed pH dependence of the rate was interpreted in terms of the second-order rate constants defined by Eq. (44), and values for... 

At the standard potential, kfh = kb h = ks h, for all reactants in the standard state, where ks h is the standard heterogeneous rate constant, defined according to the Arrhenius equation as... 

The shape and location of the current-potential curves are strongly dependent on the kinetics of the electrode reaction through the dimensionless rate constant, defined as... 

Here, k are Arrhenius overall rate constants defined by... 

Here, k is the rate constant defined by the Arrhenius equation, c is the extent of vulcanization or cure, and n is the order of the reaction. 

Here ns = n(r, oo) is the stationary pair distribution that obeys Eq. (3.39) and ki = kj(oc) is the stationary (Markovian) rate constant defined through Ko in Eq. (3.14). Since at short Xd the distribution (3.307) simply reproduces the shape of the ionization rate W/(r), it is the same for all theories, but this identity is violated in the opposite limit of large xD. 

As mentioned previously, the task of model-based data fitting for a given matrix Y is to determine the best rate constants defining the matrix C, as well as the best molar absorptivities collected in the matrix A. The quality of the fit is represented by the matrix of residuals, R = Y - C x A. Assuming white noise, i.e., normally distributed noise of constant standard deviation, the sum of the squares, ssq, of all elements is statistically the best measure to be minimized. This is generally called a least-squares fit. 

It is of interest to see how the information on the magnetic properties of the individual radicals relates to the spin-chemicaUy observed spin relaxation in the Ru-chromo-phore linked radical pairs. As detailed in Ref. 23, the relaxation rate constant defined in Scheme 10.1 can be extracted from the magnetic field dependence of the radical pair recombination kinetics. The pertinent data points for DCA-POZ and DCA-PSZ are shown in Fig. 10.5 together with the theoretical field dependence of and its various contributions. For DCA-PTZ, no significant deviation from the DCA-POZ case is observed. 

The substitution effect within the amino is caused by steric and electronic effects. In the ideal case of equal reactivity of amine hydrogens, the ratio q = k /k = 1/2 because the primary amine has two hydrogens and the secondary amine only one. Sometimes, the factor 2 appears in front of the l.h.s. of Eq. (30) because primary amine is bifunctional The rate constant defined in this way kj = k /2 and for the ideal case one has kj/k = 1. 

Increased attention has been focused on vibrational, rotational, and translational nonequilibria in reacting systems as well. To account for these nonequilibrium effects, it is becoming increasingly traditional to express specific reaction-rate constants in terms of sums or integrals of reaction cross-sections over states or energy levels of the reactants involved [3], [11]. This approach helps to relate the microscopic and macroscopic aspects of rate processes and facilitates the use of fundamental experimental information, such as that obtained from molecular-beam studies [57], in calculation of macroscopic rate constants. Proceeding from measurements at the molecular level to obtain the rate constant defined in equation (4) remains a large and ambitious task. 

It should be remarked here that Eq. 73 defines a dimensionless tunneling probability, not a tunneling rate or rate constant. Above, we considered a wave function extended over two metal phases we did not consider a measurable tunneling rate. It can be expected that the lifetime of an electron in phase A before tunneling into phase B is proportional to l/T. In theories of electrochemical electron transfer, tunneling rate constants defined as... 

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