Chemical potential solvent

A reverse osmosis membrane acts as the semipermeable barrier to flow ia the RO process, aHowiag selective passage of a particular species, usually water, while partially or completely retaining other species, ie, solutes such as salts. Chemical potential gradients across the membrane provide the driving forces for solute and solvent transport across the membrane. The solute chemical potential gradient, —is usually expressed ia terms of concentration the water (solvent) chemical potential gradient, —Afi, is usually expressed ia terms of pressure difference across the membrane. 

The isopiestic method is based upon the equality of the solvent chemical potentials and fugacities when solutions of different solutes, but the same solvent, are allowed to come to equilibrium together. A method in which a solute is allowed to establish an equilibrium distribution between two solvents has also been developed to determine activities of the solute, usually based on the Henry s law standard state. In this case, one brings together two immiscible solvents, A and B, adds a solute, and shakes the mixture to obtain two phases that are in equilibrium, a solution of the solute in A with composition. vA, and a solution of the solute in B with composition, a . 

Using standard thermodynamics, it can be shown that, at modest pressures, the equality of solvent chemical potential in both liquid and vapor phases can be transformed to... 

The solvent chemical potential is taken to be that given by the Flory-Huggins expression (18) ... 

The next problem is to find an expression for Asg. This entropy difference is a function of the particle volume fractions in the dispersion ( ) and in the floe (<(> ). As a first approximation, we assume that Ass is independent of the concentration and chain length of free polymer. This assumption is not necessarily true the floe structure, and thus < >f, may depend on the latter parameters because also the solvent chemical potential in the solution (affected by the presence of polymer) should be the same as that in the floe phase (determined by the high particle concentration). However, we assume that these effects will be small, and we take as a constant. 

For a binary polymer solution, the reciprocal of the osmotic compressibility 0n/0c at constant T and the solvent chemical potential p0 can be determined by sedimentation equilibrium through the relation [58,59] ... 

Here, A is the depletion layer thickness (assumed equal to the radius of gyration of the polymer, RG). H = r - 2a is the surface-to-surface particle separation, V ° is the molar volume of the solvent, and ji and ji are the solvent chemical potentials for the polymer solution and the pure solvent. It appears that the assumption A = RG is generally acceptable providing that the polymer solution is in the dilute concentration regime. At higher polymer concentrations, however, the value of A is reduced according to the relationship (Vincent, 1990) ... 

Furthermore, we note that the solvent chemical potential should be continuous... 

At the time of the conference, D.P. and K.D. discussed the possible peculiar shapes of the solvent chemical potential vs composition curves in swollen... 

Because variations in solvent chemical potential are generally much easier to determine experimentally (e.g., by osmotic pressure measurements, as described in Section 7.3.6), (6.37) gives the recipe for determining the more difficult solute from its Gibbs-Duhem dependence on other easily measured thermodynamic intensities. Equations such as (6.35)-(6.37) are sometimes referred to as Gibbs-Duhem equation(s), but they are really only special cases of (and thus less general than) the Gibbs-Duhem equation (6.34). 

Equation (A.26) is valid for all components, including the solvent. Because the chemical potential of the solvent is decreased by the presence of solute, the transfer of the solvent across the membrane in the direction of the solvent chemical potential gradient takes place (flow is from the dilute solution to the concentrated solution) with a concurrent increase of the pressure inside the more concentrated solution... 

The thermodynamic state of a polymer- solvent system is completely determined, as it was analized before, at fixed temperature and pressure by means of the interaction parameter g. This g is defined through the noncombinatorial part of the Gibbs mixing function, AGm- The more usual interaction parameter, x, is defined similarly but through the solvent chemical potential, A xi, derived from AGm-... 

Similarly, for a real solution the excess solvent chemical potential y, may be shown as a function of the osmotic coefficient, ([) ... 

Osmosis is the natural movement of a solvent through a semipermeable membrane into a solution of higher solute concentration, leading to equal solute concentrations on both sides of the membrane [93]. A semipermeable membrane can be crossed by solvent molecules but solute (ionic or high molecular weight compounds) permeation is impeded. Solvent migration from one side of the membrane to the opposite one takes place to render equal solute and solvent chemical potentials across the membrane. Osmotic pressure tt can be expressed by... 

The surface pressure of the adsorbed film can be calculated from the solvent chemical potential, which, due to its uniformity, leads to ... 

Next we integrate the pressure derivative of the pure solvent chemical potential (at zero solute molality). 

In this case the coefficients B are called the virial coefficients of the osmotic pressure. Note that these virial coefficients depend on both the temperature and the solvent activity XB, or the solvent chemical potential 1B = exp (fipB). 

The elastic response of the network contributes to the solvent chemical potential. Assuming isotropic dilation, which is typical in swelling experiments, is given by... 

To calculate the difference of the solvent chemical potentials in the vapour state, we use the equation... 

The solutions are also used to measure of the thermodynamic interactions between the polymer segments and solvent molecules. The latter is best discussed in terms of the virial coefficients. A.. The change of solvent chemical potential upon dissolution of polymer is given by ... 

In an osmotic pressure experiment, the additional pressure II exerted on the polymer solution results in the solvent chemical potential being rendered equal to that of the pure solvent at ambient pressure p ... 

Vapour-sorption experiments on different polymer plus solvent systems have shown that the elastic component of the solvent chemical potential exhibits a maximum, contrary to the phantom network theories or the Mooney-Rivlin equation. Furthermore, evidence has been found that the localisation and height of the maximum is dependent upon the nature of the diluent. 

V is solvent partial molar volume Pi is solvent chemical potential on the left of the membrane (initially solute-free environment)... 

Classical polymer solution thermodynamics often did not consider solvent activities or solvent activity coefficients but usually a dimensionless quantity, the so-called Flory-Huggins interaction parameter % is not only a function of temperature (and pressure), as was evident from its foundation, but it is also a function of composition and polymer molecular mass. As pointed out in many papers, it is more precise to call it %-function (what is in principle a residual solvent chemical potential function). Because of its widespread use and its possible sources of mistakes and misinterpretations, the necessary relations must be included here. Starting from Equation [4.4.1b], the difference between the chemical potentials of the solvent in the mixture and in the standard state belongs to the first... 

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