Binary solution, partition function

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

The canonical ensemble partition function for a binary solution is — v/VT... 

The FCEM expressions for binary alloys were obtained using the Ising model Hamiltonian and an expansion of the partition function and free energy in terms of solute concentration [1,3,19]. The free energy of a binary alloy (A -solute, S - solvent) reads. 

In 1957 Hill introduced a binary solution theory based on an analysis of the semigrand partition function in which the pressure P, temperature T, and number of solvent particles, Nj, are held fixed (10, 11). In this section, we extend his derivation to a four-component system containing solvent (component 1), two polymers (components 2 and 3) and protein (component 4). The objective of the calculation is to derive expressions for the chemical potentials of all components. Later, by equating the chemical potentials of each component in each phase, we will determine the composition of each phase and hence the protein partition coefficient which is defined to be the ratio of protein compositions in the top and bottom phases. 

In the limit that the number of effective particles along the polymer diverges but the contour length and chain dimensions are held constant, one obtains the Edwards model of a polymer solution [9, 30]. Polymers are represented by random walks that interact via zero-ranged binary interactions of strength v. The partition function of an isolated chain is given by... 

Kosmas and Freed [41] presented another approach to scaling laws. Differing from the theory described above, it starts from the partition function for a solution of continuous chains which interact subject to the binary cluster approximation. For example, their theory derives for osmotic pressure (in three dimensions) a general scaling law which, in our notation, may be written... 

Binary systems. A typical plot of the log of partition ratios (k ) vs. % modifier is presented in Figure 2. Like solvent strength, chromatographic retention is a nonlinear function of mobile phase composition. However, plots of log k vs solvent strength (i.e., Ej ) are linear, for at least some solutes, as shown for phenols... 

The octanol/water partition coefficient is not the same as the ratio of a chemical s solubility in octanol to its solubility in water, because the organic and aqueous phases of the binary octanol/water system are not pure octanol and pure water. At equilibrium, the organic phase contains 2.3 moi/L of water, and the aqueous phase contains 4.5 x 10 mol/L of octanol (27). Moreover, Kow is often found to be a function of solute concentration for concentrations 0.01 mol/L. 

The thermodynamic state of a solute A in a dilute solution is usually stated in terms of thermodynamic standard functions of transfer AX (X = G, H, S, Cp) of A from a chosen standard state in a thermodynamic medium a to an another chosen standard state in a second thermodynamic medium g. The g phase may be constituted by a pure liquid or by a mixture of substances, the a phase analogously may be pure A (gas, liquid or solid) or A dissolved in another solvent C. When A is partitioned between the gas phase and a binary dilute aqueous solution the thermodynamic standard functions of transfer are called "thermodynamic standard functions of hydration"... 

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