Chemical potential from osmotic pressure

Osmosis is similar to diffusion in that the molecules move from a location of high chemical potential to one of low chemical potential. An osmotic pressure is generated in a colloidal solution when it is separated from its solvent by a barrier that is impermeable to the solute but is permeable to the solvent. The pure solvent will flow across the membrane, diluting the colloidal dispersion and, as the colloidal material cannot flow in the opposite direction, a pressure difference (osmotic pressure) will be created between the two compartments. Osmotic pressure is a colligative... 

The activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

Other thermodynamic parameters can be obtained from osmotic pressure. For example, the chemical potential of the solvent in the solution is given by -x/rtV,. From the foregoing discussion, it is evident that the thermo(% namic behavior of the dilute polymer solution depends on the following factors ... 

The KB inversion process involves the extraction of KBIs from the available experimental data. The experimental data required for this process—derivatives of the chemical potentials, partial molar volumes, and the isothermal compressibility—are all generally obtained as derivatives of various properties of the solution. Obtaining reliable derivatives can be challenging and will depend on the quality of the source data and the fitting function. Unfortunately, the experimental data often appear without a reliable statistical analysis of the errors involved, and hence the quality of the data is difficult to determine. Matteoli and Lepori have performed a fairly rigorous analysis of a series of binary mixtures and concluded that, for systems under ambient conditions, the quality of the resulting KBIs is primarily determined by the chemical potential data, followed by the partial molar volume data, whereas errors in the compressibility data have essentially no effect on the KBI values (Matteoli and Lepori 1984). Excess chemical potentials are typically obtained from partial pressure data, either isothermal or bubble point determinations, and from osmotic pressure or even electrochemical measurements. The particle number... 

From the chemical potential we may at once set down expressions for the activity ai of the solvent and for the osmotic pressure tz of the solution, using standard relations of thermodynamics. For the activity... 

Other thermodynamic functions may be derived from the partition function Q, or from the expression for the osmotic pressure. The chemical potential of the solvent in the solution (not to be confused with the excess chemical potential (mi —within a region of uniform segment expectancy, or density) is given, of course, by ... 

Osmosis is the passage of a pure solvent into a solution separated from it by a semipermeable membrane, which is permeable to the solvent but not to the polymeric solute. The osmotic pressure n is the pressure that must be applied to the solution in order to stop the flow. Equilibrium is reached when the chemical potential of the solvent is identical on either side of the membrane. The principle of a membrane osmometer is sketched in Figure 2. 

It is fortunate that theory has been extended to take into account selective interactions in multicomponent systems, and it is seen from Eq. (91) (which is the expression used for the plots in Fig. 42 b) that the intercept at infinite dilution of protein or other solute does give the reciprocal of its correct molecular weight M2. This procedure is a straightforward one whereby one specifies within the constant K [Eq. (24)] a specific refractive index increment (9n7dc2)TiM. The subscript (i (a shorter way of writing subscripts jUj and ju3) signifies that the increments are to be taken at constant chemical potential of all diffusible solutes, that is, the components other than the polymer. This constitutes the osmotic pressure condition whereby only the macromolecule (component-2) is non-diffusible through a semi-permeable membrane. The quantity... 

It is important to note that the concept of osmotic pressure is more general than suggested by the above experiment. In particular, one does not have to invoke the presence of a membrane (or even a concentration difference) to define osmotic pressure. The osmotic pressure, being a property of a solution, always exists and serves to counteract the tendency of the chemical potentials to equalize. It is not important how the differences in the chemical potential come about. The differences may arise due to other factors such as an electric field or gravity. For example, we see in Chapter 11 (Section 11.7a) how osmotic pressure plays a major role in giving rise to repulsion between electrical double layers here, the variation of the concentration in the electrical double layers arises from the electrostatic interaction between a charged surface and the ions in the solution. In Chapter 13 (Section 13.6b.3), we provide another example of the role of differences in osmotic pressures of a polymer solution in giving rise to an effective attractive force between colloidal particles suspended in the solution. 

Equation (13) reminds us that the chemical potential has its greatest value, p,, for a pure substance. Any value of a, less than unity will cause n, to be altered from by an amount RT In a which will be negative for a, < 1. Second, any pressure on a liquid that exceeds p°, increases n above This is seen from the combination of Equations (13) and (18). Thus consideration of the chemical potential of the solvent makes it clear how osmotic equilibrium comes about. The presence of a solute lowers the chemical potential of the solvent. This is offset by a positive pressure on the solution, the osmotic pressure 7r, so that the net chemical potential on the solution side of the membrane equals that of the pure solvent on the other side of the membrane. This is summarized by the expression... 

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). 

Let us now analyze osmotic flow from the viewpoint of the Gibbsian equilibration conditions (Chapter 5). Because volume V cannot be freely exchanged between chambers, the pressure does not equalize P(L) P(R). Similarly, solute nB cannot freely exchange, so /4P p However, solvent nA is free to exchange through the semipermeable membrane, so its chemical potential must equalize between left and right chambers ... 

Osmosis takes place when a solution and a solvent (or two solutions of different concentrations) are separated from each other by a semipermeable membrane - i.e. a membrane which is permeable to the solvent but not to the solute. The tendency to equalise chemical potentials (and, hence, concentrations) on either side of the membrane results in a net diffusion of solvent across the membrane. The counter-pressure necessary to balance this osmotic flow is termed the osmotic pressure. 

From the definition of chemical potential (Eq. 2.4) and the formal expression for osmotic pressure (Eq. 2.7), we can express the chemical potential of water (pw) as... 

Another AFM-based technique is chemical force microscopy (CFM) (Friedsam et al. 2004 Noy et al. 2003 Ortiz and Hadziioaimou 1999), where the AFM tip is functionalized with specific chemicals of interest, such as proteins or other food biopolymers, and can be used to probe the intermolecular interactions between food components. CFM combines chemical discrimination with the high spatial resolution of AFM by exploiting the forces between chemically derivatized AFM tips and the surface. The key interactions involved in food components include fundamental interactions such as van der Waals force, hydrogen bonding, electrostatic force, and elastic force arising from conformation entropy, and so on. (Dther interactions such as chemical bonding, depletion potential, capillary force, hydration force, hydrophobic/ hydrophobic force and osmotic pressure will also participate to affect the physical properties and phase behaviors of multicomponent food systems. Direct measurements of these inter- and intramolecular forces are of great interest because such forces dominate the behavior of different food systems. 

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... 

Aimar et al. [19] noted that in the UF of complex liquids, such as cheese whey, which contains proteins, salts and casein fragments, concentration polarization, and adsorption and cake formation play a role in flux behavior during crossflow filtration. They may induce osmotic pressure in the retentate side since the chemical potential of the solute-rich polarized layer is lower than that of the permeate, and therefore at equilibrium, a positive osmotic pressure develops in the retentate to equal that of the permeate. The smaller the solute, the greater is its contribution to the osmotic pressure of the liquid, so that in milk, lactose and the minerals have the biggest contribution to osmotic pressure. In skim milk or whey, the osmotic pressure is around 7 bar (700 kPa) and this must be exceeded in RO to commence permeation in UF, only the proteins contribute to the osmotic pressure, which increases exponentially with protein concentration [56]. In any case, a TMP greater than the osmotic pressure is required for solvent to flow from the retentate side to the permeate side. This leads to the reduction in the effectiveness of applied TMP as driving force to permeation. 

The thermoelectrical behaviour of many alloys is typical. Their thermoelectric potential is often much higher than that of the pure metals of which they are composed. These facts cannot be deduced from thermodynamics, which in general can tell us nothing new about constants which are characteristic of the chemical nature of substances. We must have recourse here to special theories, just as in the calculation of the osmotic pressure of solutions. We may mention that the electronic and molecular theories of R. Schenck I and A. Bernoulli J have done valuable service in this direction. 

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