Differential first-order

The differential first-order equation to represent the rate of degradation of the parent chemical was... 

In order to prove this point let us compare (11.6), (11.7), (11.10), and (11.13) with (11.14), (11.15), (11.16), and (11.17). These equations are in the Cauchy form of the differential first-order equations, but can be arranged into matrix form for the typical state variable representation of the dynamics of the system. Here we will... 

The initial slope can be found indirectly, by first deriving the pulse response function of the differentiated first-order function. In the Laplace domain, differentiation corresponds to multiplying by s. 

Eq.(2.1) represents a differential first-order equation in which unique unknown magnitude is speed of a corpuscle v, and argument—a time t. Speed of substance of a master phase in all space points is necessary known. In the capacity of initial data, except the size and properties of a corpuscle, its mle during the initial moment of a time is set. It is underlined also that should occur at collision of a corpuscle with a wall or with other corpuscle. For performance of calculation the members containing are transferred to the left-hand side of Eq. (2.1). Speed and a corpuscle position during each subsequent moment of a time is defined by a numerical integration on atime with some step At with all other members of Eq.(2.1). 

Written in matrix notation, the system of first-order differential equations, (A3.4.139) takes the fomi... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

Since this agrees with the first Bom differential cross section for (in)elastic scattering, Femii s Rule 2 is therefore valid to first order in the interaction F. 

Equation (B2.4.13) is a pair of first-order differential equations, so its fonnal solution is given by equation (B2.4.14)), in which exp() means the exponential of a matrix. 

Here, dj = cos(y,j) and sy = sin(Yy). The three angles are obtained by solving the following three coupled first-order differential equations, which follow from Eq. (19) [36,84,85] ... 

Equation (26) is a set of partial first-order differential equations. Each component of the Curl forms an equation and this equation may or may not be coupled to the other equations. In general, the number of equations is equal to the number of components of the Curl equations. At this stage, to solve this set of equation in its most general case seems to be a fomiidable task. 

The adiabatic-to-diabatic transfomiation matrix, Ap, fulfills the following first-order differential vector equation [see Eq. (19)] ... 

In Section V.B, we discussed to some extent the 3x3 adiabatic-to-diabatic transformation matrix A(= for a tri-state system. This matrix was expressed in terms of three (Euler-type) angles Y,y,r = 1,2,3 [see Eq. (81)], which fulfill a set of three coupled, first-order, differential equations [see Eq. (82)]. 

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

On subsciCuLlng (12.49) into uhe dynamical equations we may expand each term in powers of the perturbations and retain only terms of the zeroth and first orders. The terms of order zero can then be eliminated by subtracting the steady state equations, and what remains is a set of linear partial differential equations in the perturbations. Thus equations (12.46) and (12.47) yield the following pair of linearized perturbation equations... 

In the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection tenns (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called selective or inconsistent upwinding. Selective upwinding can be interpreted as the introduction of an artificial diffusion in addition to the physical diffusion to the weighted residual statement of the differential equation. This improves the stability of the scheme but the accuracy of the solution declines. 

Time derivatives in expansion (2.113) can now be substituted using the differential equation (2.112) (Donea, 1984 The first order time derivative in expansion (2.113) is substituted using Equation (2.112) as... 

The remaining terms in equation set (4.125) are identical to their counterparts derived for the steady-state case (given as Equations (4.55) to (4.60)). By application of the 9 time-stepping method, described in Chapter 2, Section 2.5, to the set of first-order ordinary differential equations (4.125) the working equations of the solution scheme are obtained. The general form of tliese equations will be identical to Equation (2.111) in Chapter 2,... 

Notiee that H is a seeond order differential operator in the spaee of the thirty-nine eartesian eoordinates that deseribe the positions of the ten eleetrons and three nuelei. It is a seeond order operator beeause the momenta appear in the kinetie energy as pj and pa, and the quantum meehanieal operator for eaeh momentum p = -ih d/dq is of first order. 

The net result is that we now have two first-order differential equations of the eigenvalue form ... 

Direct-Computation Rate Methods Rate methods for analyzing kinetic data are based on the differential form of the rate law. The rate of a reaction at time f, (rate)f, is determined from the slope of a curve showing the change in concentration for a reactant or product as a function of time (Figure 13.5). For a reaction that is first-order, or pseudo-first-order in analyte, the rate at time f is given as... 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

This expression describes the variation of the pressure-temperature coordinates of a first-order transition in terms of the changes in S and V which occur there. The Clapeyron equation cannot be applied to a second-order transition (subscript 2), because ASj and AVj are zero and their ratio is undefined for the second-order case. However, we may apply L Hopital s rule to both the numerator and denominator of the right-hand side of Eq. (4.47) to establish the limiting value of dp/dTj. In this procedure we may differentiate either with respect to p. 

The relationship of this type of model to a tme differential analysis has been discussed for the case of linear equiHbrium and first-order kinetics (74,75). A minor extension of this work leads to the foUowing relations for a bed section in which dow rates of soHd and Hquid are constant. For the number of theoretical trays,... 

The solution of equation 16 is a decreasing, simple exponential where = k ([A ] + [P ]) + k. The perturbation approach generates small deviations in concentrations that permit use of the linearized differential equation and is another instance of pseudo-first-order behavior. Measurements over a range of [A ] + [T ] allow the kineticist to plot against that quantity and determine / ftom the slope and from the intercept. 

Much of the language used for empirical rate laws can also be appHed to the differential equations associated with each step of a mechanism. Equation 23b is first order in each of I and C and second order overall. Equation 23a implies that one must consider both the forward reaction and the reverse reaction. The forward reaction is second order overall the reverse reaction is first order in [I. Additional language is used for mechanisms that should never be apphed to empirical rate laws. The second equation is said to describe a bimolecular mechanism. A bimolecular mechanism implies a second-order differential equation however, a second-order empirical rate law does not guarantee a bimolecular mechanism. A mechanism may be bimolecular in one component, for example 2A I. 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Whichever the type, a differential equation is said to be of /ith order if it involves derivatives of order n but no higher. The equation in the first example is of first order and that in the second example of second order. The degree of a differential equation is the power to which the derivative of the highest order is raised after the equation has been cleared of fractions and radicals in the dependent variable and its derivatives. 

Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M x, y) dx + N x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = f f x) dx + c, which may or may not be expressible in simpler form. 

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