Equation describing kinetics

Polymerization rate expression. The equations describing kinetics of free-radical polymerization steps contain a term for the concentration of radicals [M ], which exists at very low concentration ( 10 M) and thus... 

Empirical equations, describing kinetics of the change of PETP characteristics have been found in the same work ... 

When a detailed chemical description is not required, a limited set of a few stoichiometric equations can be included into the scheme just to describe the rate of heat evolution and change of total number of gas species in the system. Chemically oversimplified models of this kind are widely used, for instance, to describe heat-transfer and to optimize thermal regimes in reactors (see, e.g., Fukuhara and Igarashi, 2005 Kolios et al., 2001). A similar approach is used to describe the fuel combustion and corresponding dynamic phenomena in engines of different types simplified equations describing kinetic features are solved together with complex equations of heat- and mass-transfer and fluid dynamics (Frolov et al., 1997 Williams, 1997). 

Of course, in most living organisms, the reactions are taking place in a much more complicated environment than the laboratory flask. The cell is divided up into numerous compartments, reactions take place on membranes or adsorbed to other molecules or surfaces, and so forth. In such circumstances the equations describing kinetic processes become very complex. Nevertheless, the basic principles described here will still apply. 

The simplest continuum equation describing kinetic roughening is based on a Langevin equatirm approach and has been introduced by Edwards and Wilkinson (EW) [39] ... 

The rate equation describing the kinetics of this reaction is... 

Many industrial crystallizers operate in a weU-mixed or nearly weU-mixed manner, and the equations derived above can be used to describe their performance. Furthermore, the simplicity of the equations describing an MSMPR crystallizer make experimental equipment configured to meet the assumptions lea ding to equation 44 useful in determining nucleation and growth kinetics in systems of interest. 

Surfactants aid dewatering of filter cakes after the cakes have formed and have very Httle observed effect on the rate of cake formation. Equations describing the effect of a surfactant show that dewatering is enhanced by lowering the capillary pressure of water in the cake rather than by a kinetic effect. The amount of residual water in a filter cake is related to the capillary forces hoi ding the Hquids in the cake. Laplace s equation relates the capillary pressure (P ) to surface tension (cj), contact angle of air and Hquid on the soHd (9) which is a measure of wettabiHty, and capillary radius (r ), or a similar measure appHcable to filter cakes. 

The production of hydrocarbons using traditional Fischer-Tropsch catalysts is governed by chain growth or polymerization kinetics. The equation describing the production of hydrocarbons, commonly referred to as the Anderson-Schulz-Flory equation, is ... 

There is no general explicit mathematical treatment of complicated rate equations. In Section 3.1 we describe kinetic schemes that lead to closed-form integrated rate equations of practical utility. Section 3.2 treats many further approaches, both experimental and mathematical, to these complicated systems. The chapter concludes with comments on the development of a kinetic scheme for a complex reaction. 

The Arrhenius equation is best viewed as an empirical relationship that describes kinetic data very well. It is commonly applied in the linearized form... 

Each of the processes shown in Figure 2.8 can be described by a Michaelis-Menten type of biochemical reaction, a standard generalized mathematical equation describing the interaction of a substrate with an enzyme. Michaelis and Men ten realized in 1913 that the kinetics of enzyme reactions differed from the kinetics of conventional... 

Sometimes the time variable is eliminated from the set of differential equations describing the kinetics of the coupled system, e.g. by dividing... 

In deriving the kinetic equation describing the arrival of various ionic species at the cathode, it is assumed that the primary species N2 + is formed at the central wire at a constant rate, and during its passage in the direction x perpendicular to the axis its concentration is modified by various reactions. In this treatment both ion diffusion and ion-ion or electron-ion recombination processes are neglected because the geometry of the discharge tube and the presence of an electric field would... 

To solve the kinetics for the most general case, in which, for example, we allow partial pressures to vary with time, we need the full set of differential equations describing the coverage of all species participating in the reaction ... 

In the following table, given on the next page, we summarize the methods used to obtain the necessary equations describing nuclei growth and the kinetics of growth. 

The kinetic and polarization equations described in Sections 6.1 and 6.2 have been derived under the assumption that the component concentrations do not change during the reaction. Therefore, the current density appearing in these equations is the kinetic current density 4. Similarly, the current density appearing in the equations of Section 6.3 is the diffusion current density When the two types of polarization are effective simultaneously, the real current density i (Fig. 6.6, curve 3) will be smaller than current densities and ij (Fig. 6.6, curves 1 and 2) for a given value of polarization. 

The kinetic equations describing the joint effects of activation and concentration polarization are very complex and we shall consider only the the case of a simple first-order reaction of the type (6.2) proceeding in the presence in the solntion of an excess of a foreign electrolyte. To simplify the appearance of these equations (which even in this case are very cnmbersome), in this section we use a more compact notation that contains two new kinetic parameters ... 

The kinetic equation describing the relationship between reaction rate and temperature, pressure, and concentrations is a function of the type ... 

In case of adsorption of acceptor particles the kinetic equation describing the rate of change in density of ASS occupied by electrons can be written as... 

On the basis of consideration of this model one can easily obtain expressions describing kinetics of the change in electric conductivity of sintered partially reduced oxide adsorbent during development of direct and inverse reaction described by expression (2.17). The consumption of superstoichiometric metal atoms which takes place during adsorption of O2 is described by equation... 

The kinetic equation describing the change in concentration of conductivity electrons and consequently the time dependence of electric conductivity has the following shape... 

The second and third terms in equation (2.86) describe kinetics of annihilation of radicals corresponding to above processes developing in the first and second order of magnitude with respect to chemisorbed radicals. Note that the rate constant A" may be dependent on concentration of free radicals in volume. 

Thus, the rigorous solution of kinetic equation describing the change in electric conductivity of a semiconductor during adsorption of radicals enables one to deduce that information on concentration of radicals in ambient volume can be obtained measuring both the stationary values of electric conductivity attained over a certain period of time after activation of the radical source and from the measurements of initial rates in change of electric conductivity during desactivation or activation of the radical flux incident on the surface of adsorbent, i.e. 

The kernel (26) and the absorbing probability (27) are controlled by the rate constants of the elementary reactions of chain propagation kap and monomer concentrations Ma(x) at the moment r. These latter are obtainable by solving the set of kinetic equations describing in terms of the ideal kinetic model the alteration with time of concentrations of monomers Ma and reactive centers Ra. 

In the gas phase, the reaction of O- with NH3 and hydrocarbons occurs with a collision frequency close to unity.43 Steady-state conditions for both NH3(s) and C5- ) were assumed and the transient electrophilic species O 5- the oxidant, the oxide 02 (a) species poisoning the reaction.44 The estimate of the surface lifetime of the 0 (s) species was 10 8 s under the reaction conditions of 298 K and low pressure ( 10 r Torr). The kinetic model used was subsequently examined more quantitatively by computer modelling the kinetics and solving the relevant differential equations describing the above... 

Equation (3) is in the form of a differential equation describing a first-order kinetic process, and, as a result, drug absorption generally adheres to first-order kinetics. The rate of absorption should increase directly with an increase in drug concentration in the GI fluids. 

A kinetic equation describing urine data can be developed as follows. If... 

Exponential rate expressions are also useful in deriving kinetic equations because they can be substituted into differential equations, which can then be integrated. For example, from Scheme 2 the differential equation describing the rate of appearance of unchanged drug in urine may be written ... 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

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