Lattice statistics

If stationary phase interactions are negligible the lattice statistical thermodynamic model and the solvophobic model predict similar results. The strength of the lattice statistical thermodynamic model is that it can explain the shape selectivity observed for certain stationary phases and can accommodate silanophllic interactions. 

Laminar flow (LC) 514 Lattice statistical thermodynamic theory (LC) 401 Layers (TLC) 671 alumina 673 band broadening 662 bilayer 671 binders 671... 

History. Starting from the ID point statistics of Zernike and Prins [116] J. J. Hermans [128] designs various ID statistics of black and white rods. He applies these models to the SAXS curves of cellulose. Polydispersity of rod lengths is introduced by distribution functions, / , (,r)108. Hermans describes the loss of correlation along the series of rods by a convolution polynomial . One of Hermans lattice statistics is namedparacrystalby Hosemann [5,117]. Hosemann shows that the field of distorted structure is concisely treated by the methods of complex analysis. A controversial subject is Hosemann s extension of ID statistics to 3D [63,131,227,228],... 

Keywords Crystallization Lattice statistics Melting Monte Carlo simulations ... 

A weakening of the critical metal-oxygen bonds occurs as a consequence of the protonation of the oxide ions neighboring a surface metal center and imparting charge to the surface of the mineral lattice. The concentration (activity) of D should reflect that three of such oxide or hydroxide ions have to be protonated. If there is a certain numer of surface-adsorbed (bound) protons whose concentration (mol nr2) is much lower than the density of surface sites, S (mol 2), the probability of finding a metal center surrounded with three protonated oxide or hydroxide ions is proportional to (CJ/S)3. Thus, as has been derived from lattice statistics by Wieland et al. (1988), the activity of D is related to (C )3, and the rate of proton-promoted dissolution, Rh (mol nrr2 lr1), is proportional to the third power of the surface protonation ... 

Grant, D.M., Pugmire, R.J., Fletcher, T.H., and Kerstein, A.R., "A Chemical Model to Coal Deyolatilization Using Percolation Lattic Statistics , submitted for publication. 

The picture of the A2 problem given above has a striking resemblance to that obtained from the recent lattice theories [see, for example, Kurata (154)]1. This is encouraging, because the distribution-function method and the lattice statistics should converge to the same conclusion, at least on a semi-quantitative level. 

After the seminal work of Guggenheim on the quasichemical approximation of the lattice statistical-mechanical theory[l], various practical thermodynamic models such as excess Gibbs energies[2-3] and equations of state[4-5] were proposed. However, the quasichemical approximation of the Guggenheim combinatory yields exact solution only for pure fluid systems. Therefore one has to resort to numerical procedures to find the solution that is analytically applicable to real mixtures. Thus, in this study we present a new unified group contribution equation of state[GC-EOS] which is applicable for both pure or mixed state fluids with emphasis on the high pressure systems[6,7]. 

Nagle JF (1966) Lattice statistics of hydrogen bonded crystals. I. The residual entropy of ice. J Math Phys 7 1484-1491... 

The Oishi-Prausnitz model cannot be defined strictly as a lattice model. The combinatorial and residual terms in the original UNIFAC and UNIQUAC models can be derived from lattice statistics arguments similar to those used in deriving the other models discussed in this section. On the other hand, the free volume contribution to the Oishi-Prausnitz model is derived from the Flory equation of state discussed in the next section. Thus, the Oishi-Prausnitz model is a hybrid of the lattice-fluid and free volume approaches. 

In chapter 1.3 a number of examples of elaborations have already been given, mostly using lattice statistics. All of them Involve a "divide and rule" strategy, in that the system (i.e. the adsorbate) is subdivided into subsystems for which subsystem-partition functions can be formulated on the basis of an elementary physical model. For instance, in lattice theories of adsorption one adsorbed atom or molecule on a lattice site on the surface may be such a subsystem. In the simplest case the energy levels, occurring in the subsystem-partition function consist of a potential energy of attraction and a vibrational contribution, the latter of which can be directly obtained quantum mechanically. Having... 

At least in lattice statistics, pair interaction energies are usually accounted for by a parameter w, which is positive for repulsion and negative for attraction for m = 0 the situation reduces to the Langmuir case. 

In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

The r.h.s. of flg. 3.31 presents liquid-gas coexistence curves, of which curve I relates to the conditions of fig. 3.31a. Curve II, arises from somewhat improved lattice statistics. For curve I the chain is fully flexible, implying that each bond can bend back to coincide with the previous one. In statistical parlance it is said that the chain has no self-avoidance and obeys first-order Markov statistics. In curve II a second-order Markov approximation was used ) in which three consecutive bonds in the chain are forbidden to overlap and an energy difference of 1/kT is assigned to local sets of three that have a bend conformation. The figure demonstrates the extent of this variation T is reduced as a result of the loss of conform-... 

For an illustration of surface tensions under dynamic conditions, see ). Non-ionics at interfaces have edso been analyzed by lattice statistics ). 

Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Amer. Chem. Soc. 57,2680-2684(1935). Nagle, J. F. Lattice statistics of hydrogen bonded crystals. I. The residual entropy of ice. J. Math. Phys. 7, 1484-1491 (1966). 

Analytic combinatorial lattice statistics are used to calculate the packing and interactions of a molecule with the other molecules in the system. [The generalized lattice statistics used in this theory have been found to be very accurate (deviations less than 1%) compared (20,25) with Monte Carlo computer simulations in limiting cases presently amenable to such simulations.] Any continuum-space orientation of a molecule or molecular part or bond can be decomposed into its components parallel to the x, 2> And z axes of the system and then these components mapped onto the x, 2. and z axes of the SC lattice [see Figure 1(c)] in a manner analogous to normal coordinate analysis in, for example, molecular spectroscopy. 

Various continuum limits of the lattice description are taken (i.e., the number of Cl molecules Nm + , the number of C2 molecules Nm2 - °°, and the number of lattice sites M + ), such that the density and the other thermodynamic and molecular ordering variables can vary continuously for the system of molecules. (In these limits, the lattice statistics can treat molecules in which the number of segments per molecule is not an integer.)... 

Now is presented a short summary of how 0 and E in Qc are derived in this lattice theory. 0 - (n 0 ). where h refers to the component of the mixture. To determine 11, assume that first Nm Cl molecules (where Nm - °) and then Nm2 C2 molecules (where Nm2 - 00 ) are already on the lattice, and use analytic lattice combinatorial statistics to calculate the number of ways to place a Ch test molecule on the lattice. (These lattice statistics are invariant to the order in which the test molecule of each component is laid on the lattice.) 0 - (IIj gRhkiSFhki> > where Sihki is the... 

Lattice statistics semiconductors) with colloidal semiconductor... 

Combining concepts of surface coordination chemistry with established models of lattice statistics and activated complex theory, Wieland et al. (8) proposed a general rate expression for the proton-catalyzed dissolution of oxide minerals ... 

Gregory M. Martinez. August 1994. Lattice statistics for size, clustering and dissociation with example applications in vapor-hquid equilibria. Chem. Eng. Sci. 49 (15) 2423-2435. 

The polyhedral surface of Euler characteristic x = 2 used to specify the lattice-statistical problem is the surface of an L x L x L cube (i.e., a Cartesian shell) with local valency v — 3,4 (see below). The target molecule B is positioned at the centrosymmetric site of one face of the shell and the... 

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