Elimination rate equation

These rate equations can be expressed in terms of clearance values, or they can be expressed in terms of first-order rate equations if the rates follow linear kinetics. Several other useful PK parameters are defined based on these elimination rate equations in the following sections. 

For example, substitution of X (mass of drug in the body at time t) for Y in Eq. 1.8 yields the zero-order elimination rate equation ... 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

The number of independent rate equations is the same as the number of independent stoichiometric relations. In the present example. Reactions (1) and (2) are reversible reactions and are not independent. Accordingly, C,. and C, for example, can be eliminated from the equations for and which then become an integrable system. Usually only systems of linear differential equations with constant coefficients are solvable analytically. 

This assumption derives rate equations from which terms involving the concentrations of active centers can be eliminated. 

The differential rate equation is - dCf /dt = kCffiR, and the mass balanee equation is Ca i A = r r- Eliminating Ca between these equations and integrating gives the usual second-order integrated equation, whieh can be written in this form ... 

Kinetic studies at several temperatures followed by application of the Arrhenius equation as described constitutes the usual procedure for the measurement of activation parameters, but other methods have been described. Bunce et al. eliminate the rate constant between the Arrhenius equation and the integrated rate equation, obtaining an equation relating concentration to time and temperature. This is analyzed by nonlinear regression to extract the activation energy. Another approach is to program temperature as a function of time and to analyze the concentration-time data for the activation energy. This nonisothermal method is attractive because it is efficient, but its use is not widespread. ... 

The water elimination reactions of Co3(P04)2 8 H20 [838], zirconium phosphate [839] and both acid and basic gallium phosphates [840] are too complicated to make kinetic studies of more than empirical value. The decomposition of the double salt, Na3NiP3O10 12 H20 has been shown [593] to obey a composite rate equation comprised of two processes, one purely chemical and the other involving diffusion control, for which E = 38 and 49 kJ mole-1, respectively. There has been a thermodynamic study of CeP04 vaporization [841]. Decomposition of metal phosphites [842] involves oxidation and anion reorganization. 

These component balances are conceptually identical to a component balance written for a homogeneous system. Equation (1.6), but there is now a source term due to mass transfer across the interface. There are two equations (ODEs) and two primary unknowns, Og and a . The concentrations at the interface, a and a, are also unknown but can be found using the equilibrium relationship, Equation (11.4), and the equality of transfer rates. Equation (11.5). For membrane reactors. Equation (11.9) replaces Equation (11.4). Solution is possible whether or not Kjj is constant, but the case where it is constant allows a and a to be eliminated directly... 

The solution proceeds by first eliminating the variable t from the two rate equations. 

The process of eliminating the variable t from the rate equations, as in problem P2.02.14, does not lead to an analytically solvable problem, but a numerical solution is developed by ODE and is plotted. 

As the recycle ratio through a PFR is increased, changes in temperature and composition across the reactor itself become smaller. Eventually it can be regarded as a differential reactor with approximately constant temperature. Between the fresh inlet to the system and the product withdrawal, substantial differences will develop. The differential operation at virtually constant temperature thus eliminates the main objection to the PFR as a device for obtaining data from which a rate equation can be determined. 

With multiple rate controlling steps, a steady state is postulated, that is, all rates are equated to the overall rate. Equations for the individual steps are formulated in terms of variables such as interfacial concentrations and various coverages of the catalyst surface. Any such variables that are not measurable are eliminated in terms of measurable partial pressures and the rate, as well as various constants to be evaluated from the data. The solved problems deal with several cases for instance, P6.03.04 has two participants not in adsorptive equilibrium and P6.06.17 treats a process with five steps. 

The rate equations are linearized and the five linear equations are then solved by Gaussian elimination for the five constants. 

The values of y are tabulated with the data. There are 5 constants and S equations. The solution of the linear equatioons by Gaussian elimination results in the rate equation... 

Pai can ke eliminated from the rate equation in favor of pag. Then a cubic equation results. 

In the above equations, D0 represents the dose administered, Vd is the volume of distribution, K,d is the elimination rate constant, t is the dosing interval, and C0 is the plasma concentration at zero time point. It is assumed that the drug is absorbed completely (F = 1). 

We define the linear growth rate Vg as the linear velocity of displacement of a crystal face relative to some fixed point in the crystal. vg may be known as a function of c and c , derived from the theory of transport control, and as a function of c and cs as well, derived from the theory of surface control. Then c may be eliminated by equating the two mathematical expressions... 

From a consideration of either Eqs. (113) or (114) (K3), it is evident that a saddle point is predicted from the fitted rate equation. This could eliminate from consideration any kinetic models not capable of exhibiting such a saddle point, such as the generalized power function model of Eq. (1) and the several Hougen-Watson models so denoted in Table XVI. 

The exponents i and s in equations 15.13 and 15.14, referred to as the order of integration and overall crystal growth process, should not be confused with their more conventional use in chemical kinetics where they always refer to the power to which a concentration should be raised to give a factor proportional to the rate of an elementary reaction. As Mullin(3) points out, in crystallisation work, the exponent has no fundamental significance and cannot give any indication of the elemental species involved in the growth process. If i = 1 and s = 1, c, may be eliminated from equation 15.13 to give ... 

The elimination rate constant and half-life (h/2), the time taken for the drug concentration present in the circulation to decline to 50 % of the current value, are related by the equation ... 

Equation (6) is identical in form with Eq. (4). In fact, if 3 2, k-2, Eq. (6) reduces to Eq. (4). Although Eq. (5) is a more realistic mechanism compared with Eq. (1), especially when the rapid-equilibrium treatment is applied to the reversible reaction, the information obtainable from initial-rate studies of such unireactant system remains nevertheless the same Vi and K. This serves to justify the simplification used by the kineticist that is, the elimination of certain intermediates to maintain brevity of the rate equation (provided the mathematical form is unaltered). Thus, the forward reaction of an ordered Bi Bi mechanism is generally written as diagrammed below. 

Let us consider the general approach and nomenclature. First of all, we find it more convenient to deal with concentrations rather than conversions. Second, in examining product distribution the procedure is to eliminate the time variable by dividing one rate equation by another. We end up then with equations relating the rates of change of certain components with respect to other components of the systems. Such relationships are relatively easy to treat. Thus, we use two distinct analyses, one for determination of reactor size and the other for the study of product distribution. 

In general, taking the ratio of two rate equations eliminates the time variable and gives information on the product distribution. So dividing Eq. 34 by Eq. 32 we obtain the first-order linear differential equation... 

In these equations kei is the elimination rate constant and AUMC is the area under the first moment curve. A treatment of the statistical moment analysis is of course beyond the scope of this chapter and those concepts may not be very intuitive, but AUMC could be thought of, in a simplified way, as a measure of the concentration-time average of the time-concentration profile and AUC as a measure of the concentration average of the profile. Their ratio would yield MRT, a measure of the time average of the profile termed in fact mean residence time. Or, in other words, the time-concentration profile can be considered a statistical distribution curve and the AUC and MRT represent the zero and first moment with the latter being calculated from the ratio of AUMC and AUC. 

One possibility is to determine the activation energy from a detailed simulation of the experiment. While this is doable, we will here rewrite the rate equation for TPD to eliminate the rate and the coverage. 

There is a linear relationship between dose and plasma drug levels (i.e., linear or first-order pharmacokinetics) in normal and ultrarapid metabolizers. In these individuals, the earlier equation can be used to predict the daily dose needed to produce a specific plasma drug level once TDM has been done to estimate the patient s elimination rate. In poor metabolizers, TCAs follow nonlinear pharmacokinetics (i.e., disproportionate increases in plasma drug levels with dose increases) because they lack the CYP 2D6 and must use lower affinity enzymes to metabolize these drugs. 

Note The parameter pb is found in the continuity equation but is behavior eliminated when the rate equation is incorporated in its expression. ... 

Most drug elimination pathways will become saturated if the dose is high enough. When blood flow to an organ does not limit elimination (see below), the relation between elimination rate and concentration (C) is expressed mathematically in equation... 

---