Chemical potential solute

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. 

When it was recognized (31) that the SD model does not explain the negative solute rejections found for some organics, the extended solution—diffusion model was formulated. The SD model does not take into account possible pressure dependence of the solute chemical potential which, although negligible for inorganic salt solutions, can be important for organic solutes (28,29). 

We note that n in the summand of Eq. (20) annuls terms with n = 0 and permits the sum to start with n > 1. Thus, Eq. (20) can be rearranged to have an explicit leading factor of zi. This colligative property was noted above. Of course, a determination of Z establishes the solute chemical potential //1 we seek. We are motivated, therefore, to examine the coefficient multiplying zi. To that end, we bring forward the explicit extra factor in zj and rearrange Eq. (20) to obtain... 

In reverse osmosis both solvent and solute diffuse because of gradients in their chemical potentials. For the solvent there is no gradient of chemical potential at an osmotic pressure of x at applied pressures p greater than 7r, there is such a gradient that is proportional to the difference p — ir. To a first approximation, the gradient of the solute chemical potential is independent of p and depends on the difference between concentrations on opposite sides of the membrane. This leads to the result that the fraction of solute retained varies as [1 + const./(p — 7r)] 1. Verify that the following data for a reverse osmosis experiment with 0.1 M NaCl and a cellulose acetate membrane follow this relationship ... 

Since ideal conditions simplify calculations, an ideal gas at 1 atm pressure in the gas phase which is infinitely dilute in solution will be utilized. Then the total standard partial molar Gibbs free energy of solution (chemical potential), AG, can be directly related to KD, the distribution constant, by the expression... 

At equilibrium the solute chemical potential in the respective phases are equal nistat = u mob.(11) Therefore,... 

For example, if a Pt rod containing some Zn is placed in contact with an aqueous ZnCl2 solution, Zn+2 ions are transferred between the solution and the metal, until Eq. (36) holds. It takes only a minuscule transfer of Zn+2 to build up a very appreciable electrostatic potential, because electrons from the rod cannot transfer into the solution. Chemical potentials, which depend on concentrations, can therefore be calculated neglecting the ionic transfer. If the electrons could also... 

At equilibrium, the solvent and solute chemical potentials are equal, i.e.,... 

Note that the tree-energy variation as defined by the previous equations exactly corresponds to the change of the solute chemical potential along the reaction coordinates for a given fixed solute density, i.e., AA(iq) = A/u.e(iq) with /u.0 the solute standard chemical potential. 

From a series of transformations of Equation 1 we obtain a new partition function (T) whose independent variables are temperature, pressure, solvent mole number, and the chemical potentials of the solutes (components 2 and 3). These transformations consist of first creating a partition function with pressure rather than volume as an independent variable, and then using this result to create yet another partition function in which we have switched independent variables from solute mole numbers to solute chemical potentials. These operations are analogous to the Legendre transforms commonly employed in thermodynamics. 

The inverse of Equations 5 give activity as a power series in molality. Taking this inverse and collecting terms in like powers of the molalities up to first order we obtain Equations 6 which give the solute chemical potential as a power series in solute molality with osmotic virial coefficients that are functions of temperature and pressure. 

The relative supersaturation, ct = (C — C )jC, is proportional to the difference of the solute chemical potential between solution and crystal, RT ln(C/C ), for small values of (C — C )jC under the assumption of an ideal solution. For large supersaturation... 

In equation 2.14, the non-equilibrium solute chemical potential is calculated through the use of equation 2.12 and of an appropriate EoS for the polymer-penetrant system under consideration. The pseudo-equilibrium penetrant content in the polymer, can be easily calculated whenever the value of the pseudo-equilibrium polymer density Pp i is known. Such a quantity represents, obviously, a crucial input for the non-equiUbrium approach, since it labels the departure from equilibrium it must be given as a separate independent information, and cannot be calculated simply from temperature and pressure since it depends also on the thermomechanical history of the sample. 

Figure 10 Mechanisms of breakage of liquid films, (a) Fluctuation-wave-mechanism the film rupture results from growth of capillaty waves enhanced by attractive surface forces (92). (h) Pore-nudeation mechanism it is expected to be operative in very thin films, virtually representing two attached monolayers of amphiphilic molecules (99). (c) Solute-transport mechanism if a solute is transferred across the two surfaces of the liquid film due to gradients in the solute chemical potential, then Marangoni instability could appear and break the film...

Hence, the derivative of the solute chemical potential (or activity) with respect to solute concentration can be expressed in terms of a combination of number densities and particle number fluctuations or KBIs. The ability to express thermodynamic properties in terms of KBIs is the major strength of FST. This has been achieved without approximation and the relationship holds for any stable binary solution at any composition involving any type of components. Derivatives of other chemical potentials can be obtained by application of the GD equation, or by a simple interchange of indices. The same approach can be applied to the second expression in Equation 1.48, with a subsequent application of Equation 1.27, to provide chemical potential derivatives with respect to other concentration scales. 

Dialysis is a diffusion-based separation process that uses a semipermeable membrane to separate species by virtue of their different mobilities in the membrane. A feed solution, containing the solutes to be separated, flows on one side of the membrane while a solvem stream, the dialysate, flows on the other si (Fig. 21.1-1). Solute transport across the membrane occurs by diffusion driven by the difference in solute chemical potential between the two membrarre-solution interfaces. In practical dialysis devices, an obligatory transmembrane hydraulic pressure tray add an additional component of convective transport. Convective transport also may occur if one stream, usually the feed, is highly concentrated, thus giving rise to a transmembrarre osmotic gradient down which solvent will flow. In such circumstances, the description of solute transport becomes more complex since it must incorporate some function of the transmembrane fluid velocity. 

In the case of appreciable specific adsorption, i.e., if the adsorption is due largely to metal-/ interactions rather than lyophobic interactions with the solvent, changes of solvent will mainly modify filb Better solvents for i will lower the energy filb and diminish the magnitude of the standard free energy of adsorption. In cases where the adsorption arises mainly from lyophobic interactions, better solvation of / in the bulk will usually have a similar effect, since at the interface it is coordinated only by about half the number of solvent molecules that coordinate / in the bulk phase. Since for a given pure solute (chemical potential fil) in saturation equilibrium with its saturated solution ... 

Taking account of the fact tiiat in non-ideal solutions chemical potentials are written in terms of concentrations corrected by activity coefficients, to the standardization of composition should be added the condition that these coefficients approach unity, i.e. the activities become identical with the corresponding composition parameters (mole fractions, molarities, etc.). Only in these conditions is the standard potential fully defined. The extreme compositions with respect to the i-th component chosen xt 0 and a , - 1) are those at which the thermodynamic activity becomes identical with the respective concentration. 

By similar steps, combining Henry s law based on concentration or molality (Eqs. 9.4.17 and 9.4.18) with the relation/Lb = /t-B(8) + / 7 ln( //>°), we obtain for the solute chemical potential in the ideal-dilute range the equations... 

This equation shows how changes in the solute chemical potentials, due to a composition change at constant T and p, affect the chemical potential of the solvent. 

From Eqs. 9.2.46 and 9.2.50, the solute chemical potential is given by /t-b = C/b + pV — TS. In the dilute solution, we assume Ub and Ir are linear functions of xr as explained above. We also assume the dependence of 5b on xr is approximately the same as in an ideal mixture this is a prediction from statistical mechanics for a mixture in which aU molecules have similar sizes and shapes. Thus we expect the deviation of the chemical potential from ideal-dilute behavior, /tr = + RT InxB, can be... 

Let us take r as a reference position, such as the end of the centrifuge cell farthest from the axis of rotation. We define the standard chemical potential pL° g as the solute chenfical potential under standard state conditions on a concentration basis at this position. The solute chemical potential and activity at this position are related by... 

We see that the solute chemical potential in this case is the sum of the single-ion chemical potentials. 

By definition, the definition of an activity requires choosing a standard state. For example, for a solute i, the standard state can be chosen as being the state in which its concentration is C° (or its molality J°, or its mole fraction is x°,), the temperature is r, and the solution is ideal (recall that pressure exerts a very weak influence on the behavior of condensed phases). At concentration C°, (which is that in the standard state), the solute chemical potential is its standard chemical potential A°,. Hence, when the solution is ideal, the solute chemical potential (A, at concentration C, or at molality m, is given by the expressions... 

For real solutions, the solute chemical potentials are given by... 

---