Rate constants steady state

Figure 4.1 Turnover of classical neurotransmitters. At normal rates of neuronal activity, endogenous stores of neurotransmitter are maintained at constant (steady-state) levels, indicating that the supply of new neurotransmitter (through synthesis) meets the demand (determined by release and metabolism). Consequently, the rate of the depletion (A) of the endogenous store of transmitter after inhibition of its synthesis indicates turnover rate and is described by the equation ...

The scheme in Fig. 5.5 indicates that the ligand, for example, oxalate, is adsorbed very fast in comparison to the dissolution reaction thus, adsorption equilibrium may be assumed. The surface chelate formed is able to weaken the original Al-oxygen bonds on the surface of the crystal lattice. The detachment of the oxalato-aluminum species is the slow and rate-determining step the initial sites are completely regenerated subsequent to the detachment step provided that the concentrations of the reactants are kept constant, steady state conditions with regard to the oxide surface species are established (Table 5.1). If, furthermore, the system is far from dissolution equilibrium, the back reaction can be neglected, and constant dissolution rates occur. 

Equation 3-22 for the polymerization rate is not directly usable because it contains a term for the concentration of radicals. Radical concentrations are difficult to measure quantitatively, since they are very low ( 10-8 M), and it is therefore desirable to eliminate [M- from Eq. 3-22. In order to do this, the steady-state assumption is made that the concentration of radicals increases initially, but almost instantaneously reaches a constant, steady-state value. The rate of change of the concentration of radicals quickly becomes and remains zero during the course of the polymerization. This is equivalent to stating that the rates of initiation Rj and termination R, of radicals are equal or... 

There are quite a few situations in which rates of transformation reactions of organic compounds are accelerated by reactive species that do not appear in the overall reaction equation. Such species, generally referred to as catalysts, are continuously regenerated that is, they are not consumed during the reaction. Examples of catalysts that we will discuss in the following chapters include reactive surface sites (Chapter 13), electron transfer mediators (Chapter 14), and, particularly enzymes, in the case of microbial transformations (Chapter 17). Consequently, in these cases the reaction cannot be characterized by a simple reaction order, that is, by a simple power law as used for the reactions discussed so far. Often in such situations, reaction kinetics are found to exhibit a gradual transition from first-order behavior at low compound concentration (the compound sees a constant steady-state concentration of the catalyst) to zero-order (i.e., constant term) behavior at high compound concentration (all reactive species are saturated ) ... 

At higher shear rates the steady-state viscosity o(cc)/y begins to depend on y. If a constant shear rate in this range is imposed, the shear stress rises beyond its steady state value, passes through a maximum before eventually approaching the steady state, (t(oo) = rj(y)y. The stress at the maximum o-(tm) grows in relation to ff(oo) as y increases. The time to reach the maximum tm is found to be inversely proportional to shear rate ... 

Accordingly, a flow-rate limited steady-state characterized by a constant concentration of silicon in solution, C< , and a constant leach rate... 

The overall (effective) reaction rate of a bimolecular reaction is, in general, determined by both the diffusion rate and the chemical reaction rate. A steady-state limit for the effective reaction rate is approached at long times and the rate constant takes a simple form. When the chemical reaction is very fast, the overall rate is determined by the diffusion rate, which is proportional to the diffusion constant. In the opposite limit where the chemical reaction is very slow, the overall rate is equal to the intrinsic rate of the chemical reaction. 

Y. Sano [107] described the influence of the film thickness, 8, on the drying course of water-moist polyimide films. In thick films (6 = 1 mm), the liquid-side diffusion plays an important role from the very beginning. The surface concentration quickly drops off to an equilibrium value and the temperature at the film surface increases to the drying air temperature, without reaching a constant steady-state goods temperature. A period of constant drying rate does not appear. 

Stage III Maximum Rate and Steady State. Definition. To express the overall rate of a sequence of reactions, a special mathematical treatment is often used, known as the steady state treatment. This is based on the assumption that the concentration of certain intermediate compounds or complexes is never large, that their concentration rises at the beginning of the reaction and soon reaches a constant (or steady) value, and that, at this point, the rate of change in the concentration, dc/dt, can be assumed to be zero. If the overall rate of reaction depends on the concentration of this intermediate, then the rate will have reached its maximum at this time. 

The concept of atmospheric lifetime is useful in discussions of the atmospheric degradation of anthropogenic molecules [5]. It can be defined in several ways. Most simply put it can be expressed as the turnover time, which is the atmospheric burden of a given species divided by its rate of emission, assuming a constant emission rate and steady state condition. Alternatively, it can be stated as the reciprocal of the pseudo first order rate constant (k ) for its removal ... 

It is experimentally observed that the number of growing chains remains approximately constant throughout the duration of most copolymerizations. In that case, the concentration of and are constant (steady state assumption), and the rate of conversion of M to is equal to the conversion of to M, so... 

During dissolution, mass of the mineral decreases and specihc surface area generally increases. Most researchers use the initial mineral mass and surface area to normalize reaction rate, but for experiments where the extent of reaction is large, the hnal surface area may be used to normalize the rate (Stillings and Brantley, 1995). Reactors are run until outlet concentration reaches a constant steady-state value. Dissolution rates are then reported with respect to solution chemistry as measured in the effluent. For example, measured rate is reported with respect to the outlet rather than inlet pH. 

Total body clearance (Clr) is defined as the theoretical total volume of blood, serum, or plasma completely cleared of drug per unit of time. It is usually expressed in units of mL/min, L/hr, mL/min/kg, or L/hr/kg. Like the elimination rate constant, CIt is the sum total of all the clearances contributed by each elimination route (i.e., Cfr = CIcr + CIcb + ClcM+ Clearance is a most important parameter, because it provides a better representation than does k of the body s abflity to eliminate a drug. In addition, Clj has more physiological meaning and is readily used to relate the dosing rate to steady-state concentration. 

At long times, the second, Cottrell-type, term decays to the point where its contribution to the overall current is negligible and then the currents tend to be a constant, steady-state value in which the rate of electrolysis equals the rate at which molecules diffuse to the electrode surface (Forster, 1994). 

This might be the dehydrogenation of a mixed feed of propane and n-butane, where the desired catalyst is selective for the K-butane dehydrogenation. Suppose that the temperature is constant and that both external and internal diffusion resistances affect the rate. At steady state, the rate (for the pellet, expressed per unit mass of catalyst) may be written in terms of either Eq. (10-1) or Eq. (11-44),... 

The column is initially solved at total reflux, constant overhead vapor rate, and steady-state conditions. At time t = 0 transient conditions develop as distillate drawoff begins. Numerical integration of the differential equations is performed for finite amounts of distillate draw-off, using initial values of the variables from the steady-state solution. 

A true steady state can be attained if, for example, the system is confined in a reaction vessel where a solution of A is continuously added to the system while some of the product is continuously removed at the same volume flow rate. Such steady states are by no means exceptional and occur often in living cells or chemical reactors. A steady state then lasts as long as the reaction conditions, including rates of inflow of reactant(s) and outflow of product(s), are kept constant. Also for other rate processes, e.g. involving mass or heat transfer, steady states are often achieved. 

Whereas the observed decay profile no longer is characterized by a single decay rate, the steady-state fluorescence intensity becomes dependent on both 7obs and fc>bs. The typical Stern-Volmer plot is no longer represented by equation 7a, but rather by equation 7b, where fcobs is defined by equation 6b, fc q is the bimolecular quenching rate constant, fco is the probe s mean excited-state unimolecular decay rate constant, fcobs is the mean observed decay rate constant, 70 is the distribution parameter of the Gaussian for the unimolecular decay, and 7obs is the distribution parameter for the observed unimolecular decay rate. 

The presteady-state burst will be followed by steady-state turnover at a rate given by cat The presteady-state burst of product formation will occur at a rate defined by the sum of the rates of the chemical reaction and product release. The amplitude is also a function of both rate constants, k2 and kj. Thus, the amplitude of the burst can be predicted from the rate of the burst and the rate of steady-state turnover. Although this model can account for burst kinetics, it is often inadequate due to the assumed irreversibility of the chemical reaction. The internal equilibrium arising from the reverse of the chemical reaction k-2) reduces the amplitude of the burst to less than predicted by Eq. (26). 

Evaluation of allows calculation of kp/kt from Eq. (6.84) and combining the latter with kplk obtained from Eq. (6.24) by measuring polymerization rates under steady-state conditions, the absolute rate constants kp and kt can be evaluated individually. Table 6.3 lists the kp and kt values and the corresponding activation energies for some common monomers. 

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