Chemical potential concentrated solutions

Possible driving forces for solute flux can be enumerated as a linear combination of gradient contributions [Eq. (20)] to solute potential across the membrane barrier (see Part I of this volume). These transbarrier gradients include chemical potential (concentration gradient-driven diffusion), hydrostatic potential (pressure gradient-driven convection), electrical potential (ion gradient-driven cotransport), osmotic potential (osmotic pressure-driven convection), and chemical potential modified by chemical or biochemical reaction. 

In this case the unitary value of the chemical potential of solute substance i can be estimated, as mentioned above, by extrapolating the chemical potential of dilute constituent i to xt = 1 from the dilute concentration range in which the linear relation of Eq. 5.22 holds. 

Usnally the solution-diffusion model is used to explain the flow of a solvent throngh an RO membrane.1 This explanation holds that solvent dissolves in the membrane material then diffuses across it in response lo a chemical potential gradient. Solute is presumed to pass throngh (he membrane by diffusion driven by he solute concentration difference across cbe membrane. The explanation implies that solute retention Is proportional to flux, in Ibe absence of other effects,1... 

The amount of substance present in the micellar state, cmjc = mnmic / NA may exceed the concentration of it in the molecular solution by several orders of magnitude. The micelles thus play a role of a reservoir (a depot) which allows one to keep the surfactant concentration (and chemical potential) in solution constant, in cases when surfactant is consumed, e.g. in the processes of sol, emulsion and suspension stabilization in detergent formulations, etc. (see Chapter VIII). A combination of high surface activity with the possibility for one to prepare micellar surfactant solutions with high substance content (despite the low true solubility of surfactants) allows for a the broad use of micelle-forming surfactants in various applications. 

When there is concentration difference of solutes across an ion exchange membrane (driving force difference of chemical potential) the solute diffuses through the membrane. Thus, a diffusion potential corresponding to the concentration gradient is generated across the membrane. The flux of i through the membrane, J is expressed by the Nemst-Planck equation as... 

The chemical potential of solutions of nonelectrolytes can always be written in terms of a series of positive integral powers of the concentration... 

Consider now equation (3.3.39b). It relates the interfacial concentration of solute 1 to the bulk concentration of solute 1 through the dependence of interfacial tension on the bulk chemical potential of solute 1. Remember, we need the equilibrium criterion so that we can relate the... 

This type of diffusion can also be described in terms of the solute s energy or, more exactly, in terms of its chemical potential. The solute s chemical potential does not change across the membrane s interface, because equilibrium exists there. Moreover, this potential, which drops smoothly with concentration, as shown in Fig. 2.2-2(c), is the driving force responsible for the diffusion. The exact role of this driving force is discussed more completely in Sections 6.3 and 7.2. 

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. 

Solution—Diffusion Model. In the solution—diffusion model, it is assumed that (/) the RO membrane has a homogeneous, nonporous surface layer (2) both the solute and solvent dissolve in this layer and then each diffuses across it (J) solute and solvent diffusion is uncoupled and each is the result of the particular material s chemical potential gradient across the membrane and (4) the gradients are the result of concentration and pressure differences across the membrane (26,30). The driving force for water transport is primarily a result of the net transmembrane pressure difference and can be represented by equation 5 ... 

For the solute flux, it is assumed that chemical potential difference owing to pressure is negligible. Thus the driving force is almost entirely a result of concentration differences. The solute flux, J), is defined as in equation 6 ... 

Expressions of supersaturation can then be formulated as follows (/) the difference between the chemical potential of the system and the chemical potential of at saturation, ji — ji where the chemical potential is a function of both temperature and concentration (2) the difference between the solute concentration and the concentration at equiUbrium, c — c (J) the difference between the system temperature at equiUbrium and the actual temperature,... 

Rate of Diffusion. Diffusion is the process by which molecules are transported from one part of a system to another as a result of random molecular motion. This eventually leads to an equalization of chemical potential and concentration throughout the system, and in the case of dyeing an equihbrium between dye in the fiber and dye in the dyebath. In dyeing there are three stages to diffusion diffusion of dye through the bulk solution of the dyebath to the fiber surface, diffusion through this surface, and diffusion of dye from the surface into the body of the fiber to allow for more dye to diffuse through the surface layer. These processes have been summarized elsewhere (9). 

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

Beyond the CMC, surfactants which are added to the solution thus form micelles which are in equilibrium with the free surfactants. This explains why Xi and level off at that concentration. Note that even though it is called critical, the CMC is not related to a phase transition. Therefore, it is not defined unambiguously. In the simulations, some authors identify it with the concentration where more than half of the surfactants are assembled into aggregates [114] others determine the intersection point of linear fits to the low concentration and the high concentration regime, either plotting the free surfactant concentration vs the total surfactant concentration [115], or plotting the surfactant chemical potential vs ln( ) [119]. 

Electropolishing techniques utilise anodic potentials and currents to aid dissolution and passivation and thus to promote the polishing process in solutions akin to those used in chemical polishing. The solutions have the same basic constitution with three mechanistic requirements—oxidant (A), contaminater (B) and diffusion layer promoter (C) —but, by using anodic currents, less concentrated acid solutions can be used and an additional variable for process flexibility and control is available. 

Previous Considerations have been confined to the effect of pressure and concentration upon coverage, but in an electrochemical equilibrium the activity and chemical potentials of the species adsorbing at the interface will also be a function of the potential difference A4>. For a solution containing unit activity of the species the effective pressure of the species at the interface is given by... 

In the case of ions in solution, and of gases, the chemical potential will depend upon concentration and pressure, respectively. For ions in solution the standard chemical potential of the hydrogen ion, at the temperature and pressure under consideration, is given an arbitrary value of zero at a specified concentration... 

By assuming that fi o the same as that for pure water, which is unlikely in this concentrated solution, and by substituting for the chemical potentials in equation 20.228, Pourbaix has calculated that... 

Example 2 Finely divided gold is agitated with an oxygen-saturated alkali-cyanide solution of pH = 12 containing 10 moldm of CN . Calculate the concentration of Au(CN)2 ions in solution at 25°C when the reaction is at equilibrium. The standard chemical potentials of the species involved (in kJ) are CN, 165 HjO, —237 Au(CN)f, 244 OH", —157. Assume that fl jo = 1, and take 2-303RT = 5710. 

The subscripts 1,2,3 refer to the main solvent, the polymer, and the solvent added, respectively. The meanings of the other symbols are n refractive index m molarity of respective component in solvent 1 C the concentration in g cm"3 of the solution V the partial specific volume p the chemical potential M molecular weight (for the polymer per residue). The surscript ° indicates infinite dilution of the polymer. 

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