First-order rate processes

Any initiator which decomposes by a first-order rate process can, therefore, be characterized by the two parameters k and E. ... 

A simple way to model the lag phase is to suppose that the maximum growth rate fimax evolves to its final value by a first-order rate process jUmax = Moo[l — exp(—af)]. Repeat Example 12.7 using a=lh. Compare your results for X, S, and p with those of Example 12.7. Make the comparison at the end of the exponential phase. 

Heckel proposed that a correlation exists between yield strength and an empirically determined constant, K, which is a measure of the ability of the compact to deform [28,112]. He discovered that, indeed, K is inversely proportional to yield strength. Further, he derived an equation expressing the relationship between the density of a compact and the compressional force applied. This relationship is based on the assumption that decreasing void space (i.e., decreasing porosity) of a compact follows a first order rate process ... 

Advection. Advection to and from a compartment of volume V at flow rate G m3/h by, for example, air or water flow can be expressed as a pseudo first order rate process with a rate constant k equal to G/V. The rate is also given by... 

The First-Order Kinetic Model. Karickhoff (1, 68) has proposed a two-compartment equilibrium-kinetic model for describing the solute uptake or release by a sediment. This model is based on the assumption that two types of sorption sites exist labile sites, S, which are in equilibrium with bulk aqueous solution, and hindered sites, Sjj, which are controlled by a slow first-order rate process. Conceptually, sorption according to this model can be considered either as a two-stage process ... 

The two forms, with protein from Themiste zostericola, undergo spontaneous spectral changes (28, 31) and complete loss of EPR signal (30) by a first-order rate process, with a rate constant independent of protein concentration. This change is ascribed to a remarkable intramolecular disproportionation process within the octamer ... 

An important question concerning energy trapping is whether its kinetics are limited substantially by (a) exciton diffusion from the antenna to RCs or (b) electron transfer reactions which occur within the RC itself. The former is known as the diffusion limited model while the latter is trap limited. For many years PSII was considered to be diffusion limited, due mainly to the extensive kinetic modelling studies of Butler and coworkers [232,233] in which this hypothesis was assumed. More recently this point of view has been strongly contested by Holzwarth and coworkers [230,234,235] who have convincingly analyzed the main open RC PSII fluorescence decay components (200-300 ps, 500-600 ps for PSII with outer plus inner antenna) in terms of exciton dynamics within a system of first order rate processes. A similar analysis has also been presented to explain the two PSII photovoltage rise components (300 ps, 500 ps)... 

The diffusion-controlled, extraction kinetics of A, therefore, can be described as a pseudo-first-order rate process with apparent rate constants... 

Fig. 3. Graphical representations of a first-order rate process.

An instructive example is the physiological variable serum creatinine. Creatinine is an endogenous metabolite formed from, and thus reflecting, muscle mass. Total body muscle mass is sufficiently constant to render measurement of serum creatinine useful for assessing actual renal function. The serum value of creatinine (R) is namely dependent on the continuous (zero-order) input of creatinine into the blood (A in) and its renal elimination rate, which is a first-order rate process (A out x ) In case of an extensive muscle breakdown, kin will temporarily increase. It may also be permanently low, for example in old age when muscle mass is reduced. Likewise, creatinine clearance may decrease for various reasons, described by a decrease in A out- An increase in creatinine clearance may occur as well, for example following recovery from renal disease. According to pharmacodynamic indirect response models. 

The statement is true. Passive diffusion is a first-order rate process as it is dependent on the concentration of the chemical. In contrast, active transport is a zero-order process as it is not dependent on the concentration. 

At resonance, the rf field B, causes a spin transfer from the lower to the upper level. The equilibrium distribution of the spins in the static field B0 is disturbed. Following any disruption, the nuclear spins relax to be in equilibrium with their surroundings (the lattice ), which, of course, includes B0. This relaxation is assumed to be a first-order rate process with a rate constant l/ T, characterizing each kind of nucleus. Tx is called the spin-lattice relaxation time. It covers a range of about 10-4 to 104 s. 

Longitudinal and transverse relaxations have been assumed by Bloch et al. [6] to be first-order rate processes. Following this assumption, the increase of Mz to M0 and the decay of Mx and My to zero may be expressed in terms of spin-lattice and spin-spin relaxation times, T, and T2 ... 

If we let K = (D Sa Pc/d), then, since A is present in the equation, n must equal 1, so we have a first-order rate process. Fick s law of diffusion, which is important for quantitating rates of absorption, distribution, and elimination, is thus the basis for using first-order kinetics in most pharmacokinetic models. 

Immediately on entering the body, a chemical begins changing location, concentration, or chemical identity. It may be transported independently by several components of the circulatory system, absorbed by various tissues, or stored the chemical may effect an action, be detoxified, or be activated the parent compound or its metabo-lite(s) may react with body constituents, be stored, or be eliminated—to name some of the more important actions. Each of these processes may be described by rate constants similar to those described earlier in our discussion of first-order rate processes that are associated with toxicant absorption, distribution, and elimination and occur... 

This type of design procedure is based on two additional assumptions which may not always be true in practice. First, it is assumed that equilibrium is completely attained in each stage. In practice this is usually not the case because mass transfer is a first-order rate process [Eq. (8)] and complete equilibrium is only reached asymptotically. A second factor is that in many types of continuous countercurrent equipment, some reverse flow (backmixing) occurs for example in Fig. 5b, a small portion of stream R2 may find its way back into stage 1. 

Many drugs undergo complex in vitro drug degradations and biotransformations in the body (i.e., pharmacokinetics). The approaches to solve the rate equations described so far (i.e., analytical method) cannot handle complex rate processes without some difficulty. The Laplace transform method is a simple method for solving ordinary linear differential equations. Although the Laplace transform method has been used for more complex applications in physics, engineering, and other research areas, here it will be applied to ordinary differential equations of first-order rate processes. 

The transition-state theory. Molecules colliding or possessing sufficient energy can combine to form unstable intermediates, known as activated complexes. These activated complexes transiently exist and are spontaneously converted to products in a first-order rate process. They are also constantly in equilibrium with the reactants. Thus, reactions can be written ... 

Since the fundamental rate equation of the diffusion layer model has the typical form of a first-order rate process (5.1), using (5.4) and (5.14), the MDT is found equal to the reciprocal of the rate constant k ... 

To determine the effect of acid catalysed decomposition of NADH on the electrochemical response in our experiments, the decrease in oxidation current for NADH was recorded as a function of time. The results of this experiment were compared with the decrease in NADH concentration as spectrophotometrically determined. The rates of decrease of the current and the concentration of NADH are both first-order and occur on similar timescales (Fig. 2.14). Analysis of the data for the two experiments provide first-order rate constants of 1.68 and 1.16 x 10-4 s-1 for the electrochemical and spectrophotometric measurements, respectively. The small difference between these two constants can be explained by the additional consumption of NADH by reaction at the electrode during the electrochemical measurement. This electrochemical process is also a first-order rate process, and the extent of the effect can be determined by using the treatment of Hitchman and Albery [50] for electrolysis using a rotating disc electrode. The results are consistent with the observed difference in the two rate constants. 

Because of the factor e k Equation III.4 indicates that y decays exponentially with time for a first-order rate process (e.g., Fig. 4-11). Moreover, y(r) decreases to He of its initial value [y(0)] when t satisfies the following relation ... 

Kinetics of growth. In order to derive the correct version of Equation (75) using the Reynolds transport theorem, the kinetics of growth needs to be discussed. Let [Z] be the concentration of mixed population of microorganisms ntiUzing an organic waste. The rate of increase of [Z] fits the first order rate process as follows ... 

Under most conditions the initial rate, Uo, of the reaction is directly proportional to enzyme concentration (8). In assays to determine the amount of enzyme in a sample the initial substrate concentration should be at least 10 times so that the reaction is zero order with respect to substrate concentration (Equation 3). At substrate concentrations less than 0.1 Km the reaction follows a first-order rate process with the rate directly proportional to substrate concentration. Enzyme rate assays to determine the amount of a compound as substrate in a sample should be run under these conditions. At substrate concentrations greater than 0.1 Km and less than 10 Km the reaction follows a mixed-order process intermediate between first and zero order. 

Rao, 1997) G increased from an initial value, G q, to a maximum plateau value, that increased with the proportion of starch in the mixture. A slight decrease in G was observed during the later part of aging, probably due to weakening of starch granules. The rise in G up to the maximum value followed an apparent first order rate process, Equation 4.58. 

Array formulas can simplify worksheets, as illustrated in the following four examples, where absorbance values from a first-order rate process are fitted to the equation A calc = Aoe. The worksheet calculates the sum of squares of residuals, which was minimized by changing Aq and k to obtain the least-squares best fit of the calculated absorbance to the experimental values. The values of Aq and k obtained in this way were 0.85503 and 0.49537, respectively. 

Fig. 5), the vaginal uptake of both alkanols and alka-noic acids also follows a first-order rate process and is dependent on the drug concentration in the vaginal fluid. The results agree well with a physical model that has a hydrodynamic diffusion layer in series with the mucosal membrane, that consists of two parallel pathways a lipoidal pathway and an aqueous pore pathway (Fig. 6). Immediately behind the mucosa (serosal side) a perfect sink is maintained by hemoperfusion. 

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