The equations are transcendental for the general case, and their solution has been discussed in several contexts [32-35]. One important issue is the treatment of the boundary condition at the surface as d is changed. Traditionally, the constant surface potential condition is used where po is constant however, it is equally plausible that ag is constant due to the behavior of charged sites on the surface. [Pg.181]

Equation (A2.1.21) includes, as a special case, the statement dS > 0 for adiabatic processes (for which Dq = 0) and, a fortiori, the same statement about processes that may occur in an isolated system (Dq = T)w = 0). If the universe is an isolated system (an assumption that, however plausible, is not yet subject to experimental verification), the first and second laws lead to the famous statement of Clausius The energy of the universe is constant the entropy of the universe tends always toward a maximum. [Pg.341]

Given this experimental result, it is plausible to assume (and is easily shown by statistical mechanics) that the chemical potential of a substance with partial pressure p. in an ideal-gas mixture is equal to that in the one-component ideal gas at pressure p = p. [Pg.358]

No one doubts the correctness of either of these statements of the third law and they are universally accepted as equivalent. Flowever, there seems to have been no completely satisfactory proof of their equivalence some additional, but very plausible, assumption appears necessary in making the coimection. [Pg.371]

If we wish to know the number of (VpV)-collisions that actually take place in this small time interval, we need to know exactly where each particle is located and then follow the motion of all the particles from time tto time t+ bt. In fact, this is what is done in computer simulated molecular dynamics. We wish to avoid this exact specification of the particle trajectories, and instead carry out a plausible argument for the computation of r To do this, Boltzmann made the following assumption, called the Stosszahlansatz, which we encountered already in the calculation of the mean free path [Pg.678]

This gives the total energy, which is also the kinetic energy in this case because the potential energy is zero within the box , m tenns of the electron density p x,y,z) = (NIL ). It therefore may be plausible to express kinetic energies in tenns of electron densities p(r), but it is by no means clear how to do so for real atoms and molecules with electron-nuclear and electron-electron interactions operative. [Pg.2181]

Often a degree of freedom moves very slowly for example, a heavy-atom coordinate. In that case, a plausible approach is to use a sudden approximation, i.e. fix that coordinate and do reduced dimensionality quantum-dynamics simulations on the remaining coordinates. A connnon application of this teclmique, in a three-dimensional case, is to fix the angle of approach to the target [120. 121] (see figure B3.4.14). [Pg.2311]

Unfortunately, this simple approach is not plausible numerically. The integral, as presented, will not converge, even for short times. The problem is that even trajectories which are wild , i.e. highly fluctuating, contribute. [Pg.2314]

We have in mind trajectory calculations in which the time step At is large and therefore the computed trajectory is unlikely to be the exact solution. Let Xnum. t) be the numerical solution as opposed to the true solution Xexact t)- A plausible estimate of the errors in X um t) can be obtained by plugging it back into the differential equation. [Pg.268]

Though by no means a complete theory, this is at least a reasonable explanation of the Knudsen minimum, and it then remains to explain why the minimum is not observed for flow through porous media. Pollard and Present attributed this to the limited length/diameter ratio of the channels in a typical porous medium and gave a plausible argument in favor of this view. [Pg.55]

Because of its free format, the input tile for TINKER is easier to construct by hand than the input tile for Program MM3. Place the hydrogen atoms at plausible disUitices from the carboti atoms to produce two input files for determining the [Pg.148]

N/L ). It therefore may be plausible to express kinetie energies in terms of eleetron densities p(r), but it is by no means elear how to do so for real atoms and moleeules with [Pg.501]

It ean be proven by showing that both sides of the identity obey the same differential equation. Here we will only demonstrate its plausibility by Taylor series expanding both sides [Pg.547]

Antithetical connections (the reversal of synthetic cleavages) and rearrangements are indicated by a con or rcarr on the double-lined arrow. Here it is always practical to draw right away the reagents instead of synthons. A plausible reaction mechanism may, of course, always be indicated. [Pg.195]

The rather low value obtained with the copper phthalocyanine, a low-energy solid (line (v)), is probably explicable by some reversible capillary condensation in the crevices of the aggregate, the effect of which would be to increase the uptake at a given relative pressure the plausibility of this explanation is supported by the fact that very low values of s, 1-47-1-77, were obtained with certain other phthalocyanines known to be meso-porous (cf. Chapter 3). [Pg.90]

The pores in question can represent only a small fraction of the pore system since the amount of enhanced adsorption is invariably small. Plausible models are solids composed of packed spheres, or of plate-like particles. In the former model, pendulate rings of liquid remain around points of contact of the spheres after evaporation of the majority of the condensate if the spheres are small enough this liquid will lie wholly within the range of the surface forces of the solid. In wedge-shaped pores, which are associated with plate-like particles, the residual liquid held in the apex of the wedge will also be under the influence of surface forces. [Pg.164]

The binomial distribution function is one of the most fundamental equations in statistics and finds several applications in this volume. To be sure that we appreciate its significance, we make the following observations about the plausibility of Eq. (1.21) [Pg.44]

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