Solution statistical theories

An interesting question, which is closely related to the VPIE, is the deviation of isotopic mixtures from the ideal behavior. Isotopic mixtures, that is, mixtures of isotopic molecules (e.g., benzene and deuterated benzene), have long been considered as textbook examples of ideal solutions statistical theory predicts that mixtures of very similar species, in particular isotopes, will be ideal the only truly ideal solutions would thus involve isotopic species molecules which differ only by isotopic substitution... form ideal solutions except for isotope mixtures, ideal solutions will occur rather rarely we expect binary solutions to have ideal properties when the two components are isotopes of each other. ... 

Simkin B Ya and Sheikhet I I 1995 Quantum Ohemical and Statistical Theory of Solutions (London Ellis Horwood)... 

Spin orbitals, 258, 277, 279 Square well potential, in calculation of thermodynamic quantities of clathrates, 33 Stability of clathrates, 18 Stark effect, 378 Stark patterns, 377 Statistical mechanics base, clathrates, 5 Statistical model of solutions, 134 Statistical theory for clathrates, 10 Steam + quartz system, 99 Stereoregular polymers, 165 Stereospecificity, 166, 169 Steric hindrance, 376, 391 Steric repulsion, 75, 389, 390 Styrene methyl methacrylate polymer, 150... 

Systems like SFg [39, 40], HjO [41], CH3OH [41], and CBr4/C6Hi2 [42] have been examined using this technique. Three recent papers on ruthenium (11) tris-2, 2 -bipyridine, or [Ru (bpy)3] " [43], on photosynthetic O2 formation in biological systems [44], and on photoexcitation of NITPP — L2 [45] in solution also merit attention. Theoretical work advanced at the same time. Early approaches are due to Wilson et al. [46], whereas a statistical theory of time-resolved X-ray absorption was proposed by Mukamel et al. [47, 48]. This latter theory represents the counterpart of the X-ray diffraction theory developed in this chapter. 

Interpretive methods Involve modeling the retention surface (as opposed to the response surface) on the basis of experimental retention time data [478-480,485,525,541]. The model for the retention surface may be graphical or algebraic and based on mathematical or statistical theories. The retention surface is generally much simpler than the response surface and can be describe by an accurate model on the basis of a small number of experiments, typically 7 to 10. Solute recognition in all chromatograms is essential, however, and the accuracy of any predictions is dependent on the quality of the model. 

Fig. 4. A schematic two-dimensional illustration of the idea for an information theory model of hydrophobic hydration. Direct insertion of a solute of substantial size (the larger circle) will be impractical. For smaller solutes (the smaller circles) the situation is tractable a successful insertion is found, for example, in the upper panel on the right. For either the small or the large solute, statistical information can be collected that leads to reasonable but approximate models of the hydration free energy, Eq. (7). An important issue is that the solvent configurations (here, the point sets) are supplied by simulation or X-ray or neutron scattering experiments. Therefore, solvent structural assumptions can be avoided to some degree. The point set for the upper panel is obtained by pseudo-random-number generation so the correct inference would be of a Poisson distribution of points and = kTpv where v is the van der Waals volume of the solute. Quasi-random series were used for the bottom panel so those inferences should be different. See Pratt et al. (1999).

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

The simplest are statistical theories, where the input information is reduced to the distribution of units in different reaction states. The reaction state of a unit is defined by the number and type of bonds issuing from the unit. In a reacting system, the distribution fraction of units in different reaction states is a function of the reaction time (conversion) (cf. e.g. [7, 8, 29, 30] and can be obtained either experimentally (e.g. by NMR) or calculated by solution of a few simple kinetic differential equations. An example of reaction state distribution of an AB2 unit is... 

SOL.23.1. Prigogine, On the statistical theory of polymers and polymer solutions. Proceedings, International Conference on Theoretical Physics, Kyoto and Tokyo, Sept. 1953, pp. 398-399. 

Abstract—This paper is an analysis of measurements of the thermodynamic properties of aqueous solutions of non-electrolytes, which has been made in order to establish both the relative strength of different kinds of hydrogen bonds in such solutions and the correlation between bond-strengths and the phase-behaviour of the solutions. The thermodynamic properties are compared with the results of statistical theories of solutions and with the properties of more simple solutions. 

Although recent years have witnessed an impressive confluence of experiments and statistical theories, presently there is no comprehensive understanding of the interrelation between chemical sequences in synthetic copolymers and the conditions of synthesis. One has merely to glance at recent literature in polymer science and biophysics to realize that the problem of sequence-property relationship is by no means entirely solved. As always, in these circumstances, an alternative to analytical theories is computer simulations, which are designed to obtain a numerical answer without knowledge of an analytical solution. 

In the ideal macromolecular solution, the solute molecules consist of a large number, x, of segments, but each segment has size and attractive interactions similar to that of the solvent molecules. Several statistical theories have given as the entropy of mixing of such a solution from nA mol of solvent and nB mol of macromolecule... 

B. Ya. Simkin and 1.1. Sheikhet, Quantum-Chemical and Statistical Theory of Solutions. Computing Methods and their Application, Khimiya, Moscow, 1989. 

Statistical calculations provide a relatively simple alternative to the solution of classical or quantum-mechanical reaction dynamics by replacing the detailed dynamical calculations of the progress of a reaction with probabilities of the possible outcomes. However, statistical theories are only an appropriate means of describing certain reactions and it is not generally possible to identify suitable candidates in advance of experimental measurements. There are many statistical methods available which are distinguished by various ways of describing the reaction intermediate or the possible states of the reagents or products. 

We shall refer to equation (42) as the scaling property of the binding energy of heavy positive ions. Whereas in general we expect E to be a function of two variables Z and N, equation (42) shows that in the limit of applicability of statistical theory, the energy is a function which is a product of an explicit form Z7/3 times a function of the ratio NjZ. This is directly traceable to the properties shown in Figure 1, where a given solution of the dimensionless TF equation (10) is characterized only by this ratio. 

H. C. Longuet-Higgins, Faraday Soc. Discuss., 15,73 (1953). Solutions of Chain Molecules— A New Statistical Theory. 

To Buckingham et al. is due the theory of the Kerr effect in dilute solutions, whereas Kielich developed a statistical theory of the effect for multi-component systems of an arbitrary degree of concentration and showed the Kerr constant of real solutions to be a non-additive quantity. 

The retention times of analytes are controlled by the concentration(s) of the organic solvent(s) in the mobile phase. If a relatively small entropic contribution to the retention is neglected, theoretical considerations based either on the model of interaction indices [58], on the solubility parameter theory [51,52] or on the molecular statistical theory [57], lead to the derivation of a quadratic equation for the dependence of the logarithm of the retention factor of a solute. A, on the concentration of organic solvent. aqueous-organic mobile phase ... 

The link between GC quantities and the interacticm parameters of solution theories is readily established (39). In statistical theories of K>lution thermodynamics, the )lute activity is expressed as the sum of two terms, a combinatorial entropy and a noncombinatorial free energy of mixing. In the Flory-Hi ins approximation one has. 

Statistical theories of macromolecules in solutions have recently attracted considerable attention of theorists because of remarkable and wide-ranged properti of macromolecules, of their close connection to the theories of phase transitions in lattices, and relations to ferromagnetism and adsorption problems and of the discoveries in the structures and functions of DNA and other biological macromolecules. Needless to say, a great many papers and books have been pubUshed recently, but we confine our attention to statistical theories of macromolecules in solutions. In spite of the great number of papers in this field, however, the development of rigorous statistical theories of macromolecular solutions has been rather slow, and there have been presented many different approaches some of which have probably confused readers. Therefore, in this paper we aim at a rather unified and simplified theory of macromolecular solutions and at the same time we discuss some of the feattues of various other macromolecular solution theories and elucidate the present situation. In so doing we hope to attract attention of more theoretical chemists and physicists whose participation in this field is certainly needed. 

The statistical theories of macromol ular solutions may be roughly classified into two groups one may be called lattice theories and the other, just for distinction, gas theories. We shall put our standpoint in the latter type, but our aim will be clear if we start with the lattice theories. At the same time we will achieve the goal to give a brief aixount of the history of macromolecular solution theories. 

Summarizing, one can say that the lattice theories need improvement and compact macromolecules need more refined treatment. We shall develop in this paper a refined and unified theory of macromolecular solutions with special emphasis on dilute solutions. We shall put our standpoint on the general theory of solutions developed by McMillan and Mayer in 1945 and Kirkwood and Buff in 1951 (9). TTiese theories do not use the lattice model and are more natural for application especially to dilute solutions. The theories extend statistical theories on gases and this is the reason why we used the name gas theories (70) in the beginning of this Introduction. 

Another approach is to employ rigorous statistical thermodynamic theories. In this paper, the Kirkwood-Buff (KB) theory of solutions (fluctuation theory of solutions) is employed to analyze the thermodynamics of multicomponent mixtures, with the emphasis on quaternary mixtures. This theory connects the macroscopic properties of re-component solutions, such as the isothermal compressibility, the concentration deriva-... 

As we have already seen, the derivative of (1 - ) at the origin is finite in the case of a non-electrolyte, but infinite for an electrolyte. This behaviour of strong electrolytes is related to the long range electrostatic forces between the ions in the solution.J The statistical theory, in the form developed by Debye and Hiickel, leads to the following expression for the activity coefficient of an ion with a charge 2 , in a very dilute solution in which the ionic strength is I (cf. 27.38) ... 

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