Surface, dividing

We will use the superscript a to denote surface quantities calculated on the preceding assumption that the bulk phases continue unchanged to an assumed mathematical dividing surface. For an arbitrary set of variations from equilibrium. 

Moreover, since F and are defined relative to an arbitrarily chosen dividing surface, it is possible in principle to place that surface so that F = 0 (this is discussed in more detail below), so that... 

An approach developed by Guggenheim [106] avoids the somewhat artificial concept of the Gibbs dividing surface by treating the surface region as a bulk phase whose upper and lower limits lie somewhere in the bulk phases not far from the interface. 

It has been pointed out [138] that algebraically equivalent expressions can be derived without invoking a surface solution model. Instead, surface excess as defined by the procedure of Gibbs is used, the dividing surface always being located so that the sum of the surface excess quantities equals a given constant value. This last is conveniently taken to be the maximum value of F. A somewhat related treatment was made by Handa and Mukeijee for the surface tension of mixtures of fluorocarbons and hydrocarbons [139]. 

Two alternative means around the difficulty have been used. One, due to Pethica [267] (but see also Alexander and Barnes [268]), is as follows. The Gibbs equation, Eq. III-80, for a three-component system at constant temperature and locating the dividing surface so that Fi is zero becomes... 

Fig. VII-4. Interactions across a dividing surface for a rare-gas crystal. (From Ref. 43.)...

Brunauer and co-workers [129, 130] found values of of 1310, 1180, and 386 ergs/cm for CaO, Ca(OH)2 and tobermorite (a calcium silicate hydrate). Jura and Garland [131] reported a value of 1040 ergs/cm for magnesium oxide. Patterson and coworkers [132] used fractionated sodium chloride particles prepared by a volatilization method to find that the surface contribution to the low-temperature heat capacity varied approximately in proportion to the area determined by gas adsorption. Questions of equilibrium arise in these and adsorption studies on finely divided surfaces as discussed in Section X-3. 

We suppose that the Gibbs dividing surface (see Section III-5) is located at the surface of the solid (with the implication that the solid itself is not soluble). It follows that the surface excess F, according to this definition, is given by (see Problem XI-9)... 

In the statistical description of ununolecular kinetics, known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory [4,7,8], it is assumed that complete IVR occurs on a timescale much shorter than that for the unimolecular reaction [9]. Furdiemiore, to identify states of the system as those for the reactant, a dividing surface [10], called a transition state, is placed at the potential energy barrier region of the potential energy surface. The assumption implicit m RRKM theory is described in the next section. 

In the vicinity of tire dividing surface, it is assumed that the Hamiltonian for the system may be separated into the two parts... 

The hypersurface fomied from variations in the system s coordinates and momenta at//(p, q) = /Tis the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -1. The assumption that the system s states are populated statistically means that the population density over the whole surface of the phase space is unifomi. Thus, the ratio of molecules at the dividing surface to the total molecules [dA(qi, p )/A]... 

B(A) is the probability of observing the system in state A, and B(B) is the probability of observing state B. In this model, the space is divided exactly into A and B. The dividing hyper-surface between the two is employed in Transition State Theory for rate calculations [19]. The identification of the dividing surface, which is usually assumed to depend on coordinates only, is a non-trivial task. Moreover, in principle, the dividing surface is a function of the whole phase space - coordinates and velocities, and therefore the exact calculation of it can be even more complex. Nevertheless, it is a crucial ingredient of the IVansition State Theory and variants of it. 

Surface Excess With a Gibbs dividing surface placed at the surface of the solid, the surface excess of component i, F (moVm"), is the amount per unit area of solid contained in the region near the surface, above that contained at the fluid-phase concentration far from the surface. This is depicted in two ways in Fig. 16-4. The quantity adsorbed per unit mass of adsorbent is... 

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6-9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is N-n(t), and the concentration of product states is N-pU). We assume that we can define the reactants and products as distinct macrostates that are separated by a transition state dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1). 

The original microscopic rate theory is the transition state theory (TST) [10-12]. This theory is based on two fundamental assumptions about the system dynamics. (1) There is a transition state dividing surface that separates the short-time intrastate dynamics from the long-time interstate dynamics. (2) Once the reactant gains sufficient energy in its reaction coordinate and crosses the transition state the system will lose energy and become deactivated product. That is, the reaction dynamics is activated crossing of the barrier, and every activated state will successfully react to fonn product. 

Given the foregoing assumptions, it is a simple matter to construct an expression for the transition state theory rate constant as the probability of (1) reaching the transition state dividing surface and (2) having a momenrnm along the reaction coordinate directed from reactant to product. Stated another way, is the equilibrium flux of reactant states across... 

The transition state theory rate constant can be constructed as follows. The total flux of trajectories across the transition state dividing surface will be equal to the rate of transition times the population of reactants at equilibrium N, or... 

We must be able to define the reaction coordinate along which the transition state theory dividing surface is defined. 

Equation (2.2) defines the statistically averaged flux of particles with energy E = P /2m -f V Q) and P > 0 across the dividing surface with Q =0. The step function 6 E — Vq) is introduced because the classical passage is possible only at > Vq. In classically forbidden regions, E < Vq, the barrier transparency is exponentially small and given by the well known WKB expression (see, e.g., Landau and Lifshitz [1981])... 

The identity of (3.97) and (3.98) means that the particle hits the product valley only having crossed the dividing surface x = x from left to right. If we were to use simply the step function 9(x— x ), we would be neglecting the recrossings of the dividing surface. 

Now we make the usual assumption in nonadiabatic transition theory that non-adiabaticity is essential only in the vicinity of the crossing point where e(Qc) = 0- Therefore, if the trajectory does not cross the dividing surface Q = Qc, its contribution to the path integral is to a good accuracy described by adiabatic approximation, i.e., e = ad Hence the real part of partition function, Zq is the same as in the adiabatic approximation. Then the rate constant may be written as... 

The functional B[(2(r)] actually depends only on the velocity dQ/dr at the moment when the non-adiabaticity region is crossed. If we take the path integral by the method of steepest descents, considering that the prefactor B[(2(r)] is much more weakly dependent on the realization of the path than Sad[Q(A]> we shall obtain the instanton trajectory for the adiabatic potential V a, then B[(2(t)] will have to be calculated for that trajectory. Since the instanton trajectory crosses the dividing surface twice, we finally have... 

The border between two three-dimensional atomic basins is a two-dimensional surface. Points on such dividing surfaces have the property that the gradient of the electron density is perpendicular to the normal vector of the surface, i.e. the radial part of the derivative of the electron density (the electronic flux ) is zero. 

Once the molecular volume has been divided, the electron density may be integrated within each of the atomic basins to give a net atomic charge. As the dividing surface is... 

Figure 9.2 Dividing surface between to atomic basins...

Fig. 23a,b. Snapshots of configurations showing a the nematic-isotropic b the smectic A-isotropic interfaces. The Gihhs dividing surfaces are indicated hy the dashed lines... 

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