Scaled-particle statistics

Theories more recent than the simple Langmuir (random-mixing) adsorption statistics and usually regarded as more realistic can handle the entropic terms in Equations 10 and 12. Two of these, which have been used for adsorbed ions at the mercury/electrolyte interface, are based on Flory-Huggins (12,13,25, 26) and scaled-particle statistics (31,... 

Reiss H 1977 Scaled particle theory of hard sphere fluids Statistical Mechanics and Statistical Methods in Theory and Application ed U Landman (New York Plenum) pp 99-140... 

Reiss, H., Scaled particle methods in the statistical thermodynamics of fluids, Adv. Chem. Phys. 1965, 9, 1-84... 

Systems (Friedmann). Scaled Particle Methods in the Statistical Thermodynamics 4 225... 

Statistical Thermodynamics of Fluids, Scaled Particle Methods in (Reiss). ... 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Until rigorous scale-up rules are determined, these cautionary rules are the state of the art for now. We offer a first step toward rigorous scaling rules by scaling particle surface velocities but caution that this work is only preliminary in nature. The best advice is to be cautious—understand the physics behind the problem and that statistics of the data collected. Remember that a fundamental understanding of the issues is still limited and luck is unlikely to be on your side, hence frustrating trial-and-error is still likely (and unfortunately) necessary to be employed. 

The essence of the scaled particle theory is that formation of a cavity in a fluid requires work. The theory for hard spheres has been well developed from statistical mechanics, and the work, W(R, p), can be calculated as follows ... 

The last section deals with the multi-site adsorption model of Nitta and his coworkers. In such model each adsorbate molecule is adsorbed onto n active sites and the adsorption is localised. For surfaces where mobile adsorption is possible, the approach using the scaled particle theory can be used in the statistical thermodynamics to obtain the required adsorption isotherm equation. This has been addressed by Nitta and co-workers and what to follow is the brief account of then-theory (Nitta et al., 1991). 

Efimov physics impacts not only the scattering properties of three identical bosons, but any three-body system in which at least two of the three pairwise two-body interactions are large. Interestingly, depending on the mass ratio of the different components and the particle statistics, the scaling factor can be much smaller than 22.7, facilitating the observation of successive Efimov resonances in future experiments [83]. 

H. Reiss, Scaled Particle Methods in the Statistical Thermodynamics of Fluids , Advances in Chemical Physics, vol. IX, ed. I. Prigogine, Interscience, 1965. 

There is naturally a wealth of publications on aspects of solvation and a comprehensive review would need a whole book. Hence, it is not practical to wade through all the developments in solvent effect theory, especially as other articles in this encyclopedia also deal with some aspects of solvation (see Related Articles at the end of this article). Instead, the focus will be on the methods used for the evaluation of the thermodynamics of cavity formation (TCF), which is a large part of solvation thermodynamics, and in particular on the application of the most successful statistical mechanical theory for this purpose, namely, the scaled particle theory (SPT) for hard sphere fluids (see Scaled Particle Theory). This article gives a brief introduction to the thermodynamic aspects of the solvation process, defines energy terms associated with solvation steps and presents a short review of statistical mechanical and empirical... 

Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. ... 

Particle size distribution is usually plotted on a log-probabiHty scale, which allows for quick evaluation of statistical parameters. Many naturally occurring and synthetic powders foUow a normal distribution, which gives a straight line when the log of the diameter is plotted against the percent occurrence. However, bimodal or other nonnormal distributions are also encountered in practice. 

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle energy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soHdity of matter is an effect of scale. 

Ideal and Nonrideal Solutions. Treatment of Solutions by Statistical Mechanics. A Solution Containing Diatomic Solute Particles. A Solution Containing Polyatomic Solute Particles. An Interstitial Solution. Review of Solutions in General. Quantities De-pendent on, and Quantities Independent of, the Composition of the Solution. Unitary Quantities and Cratic Quantities. Molality and Activities on the Molality Scale. 

Chemical dynamics and modeling were identified as important research frontiers in Chapter 4. They are critically important to the materials discussed in this chapter as well. At the molecular scale, important areas of investigation include studies of statistical mechaiucs, molecular and particle dynamics, dependence of molecular motion on intermolecular and interfacial forces, and kinetics of chemical processes and phase changes. 

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