Distance variability

Note that this expression is equivalent to the barometric formula which gives the variation of atmospheric pressure ( c) with elevation (oc r). A first-order dependence on the distance variable holds in the barometric equation, since the acceleration is constant in this case. 

If the interval r is large compared with the time for a collision to be completed (but small compared with macroscopic times), then the arguments of the distribution functions are those appropriate to the positions and velocities before and after a binary collision. The integration over r2 may be replaced by one over the relative distance variable r2 — rx as noted in Section 1.7, collisions taking place during the time r occur in the volume g rbdbde, where g is the relative velocity, and (6,e) are the relative collision coordinates. Incomplete collisions, or motions involving periodic orbits take place in a volume independent of r when Avx(r) and Av2(r) refer to motion for which a collision does not take place (or to the force-field free portion of the... 

Thus the problem involves the two independent variables, time t and length Z. The distance variable can be eliminated by finite-differencing the reactor length into N equal-sized segments of length AZ such that N AZ equals L, where L is the total reactor length. 

This wave packet satisfies a partial differential equation, which will be used as the basis for the further development of a quantum theory. To find this differential equation, we first differentiate equation (2.2) twice with respect to the distance variable x to obtain... 

The first derivative U HRe) vanishes because the potential is a minimum at the distance R. The second derivative U XR ) is called the force constant for the diatomic molecule (see Section 4.1) and is given the symbol k. We also introduce the relative distance variable q, defined as... 

In some applications a function /(x, t), where x is a distance variable and t is the time, is represented as a Fourier integral of the form... 

The velocity of flow of catalyst along the kiln is used to transform the time variable into a distance variable. The catalyst linear velocity is given by... 

Apart from the distance variable x that Dunham used in his function V(x) for potential energy, other variables are amenable to production of term coefficients in symbolic form as functions of the corresponding coefficients in a power series of exactly the same form as in formula 16. Through any method to derive algebraic expressions for Dunham coefficients l j, the hamiltonian might have x as its distance variable, but after those expressions are produced they are convertible to contain coefficients of other variables possessing more convenient properties. To replace x, two defined variables are y [38],... 

We define r to be the distance variable in the radial direction and co to be the velocity of rotation, with co measured in radians/second. This means the velocity of a cylindrical layer of... 

This treatment leads to a system of stiff, second-order partial differential equations that can be solved numerically to yield both transient and steady-state concentration profiles within the layer (Caras et al., 1985a). Because the concentration profile changes most rapidly near the x = L boundary an ordinary finite-difference method does not yield a stable solution and is not applicable. Instead, it is necessary to transform the distance variable x into a dummy variable y using the relationship... 

Tubular Reactors. The simplest model of a tubular reactor, the plug-flow reactor at steady state is kinetically identical to a batch reactor. The time variable in the batch reactor is transformed into the distance variable by the velocity. An axial temperature gradient can be imposed on the tubular reactor as indicated by Gilles and Schuchmann (22) to obtain the same effects as a temperature program with time in a batch reactor. Even recycle with a plug flow reactor, treated by Kilkson (35) for stepwise addition without termination and condensation, could be duplicated in a batch reactor with holdback between batches. 

It is now assumed that the distance variable, p is also the variable in a harmonic cell potential with states (ln>, n = 0, 1,2,... and with energies, E = hco(n + 1/2). The zero-order states of the vibronic system are then the product states, (4 (x)ln>, n = 0, 1,.... The electronic ground state eigenvector is expanded about p = 0 as follows ... 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

Equation (10.38) is recognized as the Schrodinger equation (4.13) for the one-dimensional harmonic oscillator. In order for equation (10.38) to have the same eigenfunctions and eigenvalues as equation (4.13), the function Slq) must have the same asymptotic behavior as in (4.13). As the intemuclear distance R approaches infinity, the relative distance variable q also approaches infinity and the functions F(R) and S(q) = RF(R) must approach zero in order for the nuclear wave functions to be well-behaved. As 7 —> 0, which is equivalent to q —Re, the potential U(q becomes infinitely large, so that F(R) and S(q rapidly approach zero. Thus, the function S(q) approaches zero as q -Re and as Roo. The harmonic-oscillator eigenfunctions V W decrease rapidly in value as x increases from x = 0 and approach zero as X —> oo. They have essentially vanished at the value of x corresponding to q = —Re. Consequently, the functions S(iq in equation (10.38) and V ( ) in... 

Given an arbitrary set of basis functions (such as the roof functions used by Gillan, and Morriss and MacGowan), which do not form a complete set, we divide each y r) into two parts a part that can be expressed in terms of the basis functions (a coarse part), and the remainder which is orthogonal to the basis set (a fine part). In the following, the argument i refers to the discretized distance variable ri = iSr,i = 0,, ...N. This means that y (i) can be written as... 

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