Chemical potential of solvent and solute

For some kinds of hlgh-polvmer solutions, the chemical potentials of solvent and solute are given by the approximate equations... 

Thus, if a tangent is drawn to the AG =f (02) curve at a volume fraction of 02 = 02, then the extrapolation of this tangent to the A G" axis for values of 02 = 0 and 02 = 1 gives quantities from which the chemical potentials of solvent and solute, respectively, are obtained. 

As shown in Section 10.4.3, in weakly compressible liquid mixtures the temperature and the chemical potentials of solvent and solute contribute to the critical fluctuations directly, while the pressure could be treated as a nonordering field, ° ° whose influence only manifests itself through the... 

CHEMICAL POTENTIAL OF SOLVENT AND SOLUTE IN ELECTROLYTE SOLUTION... 

Solvent in Solution. We shall use the pure substance at the same temperature as the solution and at its equilibrium vapor pressure as the reference state for the component of a solution designated as the solvent. This choice of standard state is consistent with the limiting law for the activity of solvent given in Equation (16.2), where the limiting process leads to the solvent at its equilibrium vapor pressure. To relate the standard chemical potential of solvent in solution to the state that we defined for the pure liquid solvent, we need to use the relationship... 

We first consider the case of a two-component solution (biopolymer + solvent) over a moderately low range of biopolymer concentrations, i.e., C < 20 % wt/wt. The quantities pm x in the equations for the chemical potentials of solvent and biopolymer may be expressed as a power series in the biopolymer concentration, with some restriction on the required number of terms, depending on the steepness of the series convergence and the desired accuracy of the calculations (Prigogine and Defay, 1954). This approach is based on simplified equations for the chemical potentials of both components as a virial series in biopolymer concentration, as developed by Ogston (1962) at the level of approximation of just pairwise molecular interactions ... 

Furthermore, in many cases the effect of the third component, namely the solvent, is decisive. For example, the measured Henry s law constant for the system aromatic substance-HCl, only reflects the difference between the chemical potential of HCl in solution, and in the vapour... 

In the preceding chapters we considered Raoult s law and Henry s law, which are laws that describe the thermodynamic behavior of dilute solutions of nonelectrolytes these laws are strictly valid only in the limit of infinite dilution. They led to a simple linear dependence of the chemical potential on the logarithm of the mole fraction of solvent and solute, as in Equations (14.6) (Raoult s law) and (15.5) (Heiuy s law) or on the logarithm of the molality of the solute, as in Equation (15.11) (Hemy s law). These equations are of the same form as the equation derived for the dependence of the chemical potential of an ideal gas on the pressure [Equation (10.15)]. 

For ideal multicomponent systems, a simple linear relationship exists between the chemical potential fii) and the logarithm of the mole fraction of solvent and solute, respectively. 

When the concentration of a multicomponent system is expressed in terms of the molalities of the solutes, the expression for the chemical potential of the individual solutes and for the solvent are somewhat different. For dilute solutions the molality of a solute is approximately proportional to its mole fraction. (The molality, m, is the number of moles of solute per kilogram of solvent. When two or more substances, pure or mixed, may be considered as solvents, a choice of solvent must be clearly stated.) In conformity with Equation (8.68), we then express the chemical potential of a solute in a solution at a given temperature and pressure as... 

The definitions based on molarities, c, are very similar to those based on molalities, and again the solvent must be considered separately from the solutes. (The molarity is defined as the number of moles per liter of solution, and is dependent on the pressure and temperature.) Molarities, like the molalities, are used primarily for solutions for which the concentration ranges are limited. For dilute solutions the molarities of the solutes are approximately proportional to their mole fractions. We thus express the chemical potential of the fcth solute in solution at a given temperature and pressure as... 

These equations are used whenever we need an expression for the chemical potential of a strong electrolyte in solution. We have based the development only on a binary system. The equations are exactly the same when several strong electrolytes are present as solutes. In such cases the chemical potential of a given solute is a function of the molalities of all solutes through the mean activity coefficients. In general the reference state is defined as the solution in which the molality of all solutes is infinitesimally small. In special cases a mixed solvent consisting of the pure solvent and one or more solutes at a fixed molality may be used. The reference state in such cases is the infinitely dilute solution of all solutes except those whose concentrations are kept constant. Again, when two or more substances, pure or mixed, may be considered as solvents, a choice of solvent must be made and clearly stated. 

The low solubility of hydrocarbons and other mainly apolar substances in water has been ascribed phenomenologically to the hydrophobic interaction. The hydro-phobic free energy can be defined4 as the difference between the standard chemical potentials of an apolar solute at infinite dilution in a hydrocarbon solvent juhc and in water... 

We first take as a reference system an infinitely dilute solution of solute 2 in solvent 1. The chemical potentials of solvent 1 and solute 2, then, are given in the form of Eq. 8.13 for an ideal solution and in the form of Eq. 8.14 for a non-ideal solution ... 

The other choice is to define each unitary chemical potential if as being equal to the chemical potential p° in the pure state for both solvent 1 and solute 2 n (T, p) = tf(T, p). We then obtain Eqs. 8.15 and 8.16 for the chemical potentials of solvent 1 and solute 2 in both an ideal and a non-ideal solution ... 

Let us consider a semipermeable membrane separating a pure liquid solvent 1 from a solution containing solvent 1 and solute substances as shown in Fig. 8.2. The chemical potentials of solvent 1 in the pure solvent and in the solution, Uj and u,, are given by Eqs. 8.33 and 8.34, respectively ... 

To derive the pressure terms in the chemical potentials of solvents, solutes, and gases, we must rely on certain properties of partial derivatives as well as on commonly observed effects of pressure. To begin with, we will differentiate the chemical potential in Equation IV.9 with respect to P ... 

Section 2 brings the cluster development for the osmotic pressure. Section 3 generalizes the approach of Section 2 to distribution functions, including a new and simple derivation of the cluster expansion of the pair distribution function. Section 4 presents a new expression for the chemical potential of solvents in dilute solutions. Section 5 contains an application of our general solution theory to compact macromolecular molecules. Section 6 contains the second osmotic virial coefficient of flexible macromokcules, followed in Section 7 by concluding remarks. 

Let US suppose that the pressures applied to the two phases are p and p", then the chemical potential of solvent in the solution will be given, according to (20.8), by... 

Molalities or molarities of ions and non-ionic compounds, mol/kg or mol/L Fluid viscosity. Pa s, or chemical potential, J/mol Chemical potential of solute, J/mol Chemical potential of solvent, J/mol Chemical potential of the /th component, J/mol Number of pores Water flux, kg-water/s m ... 

Furthermore, in many cases the effect of the third component, namely the solvent, is decisive. For example, the measured Henry s law constant for the system aromatic substance-HCl, only reflects the difference between the chemical potential of HCl in solution, and in the vapour phase, p% (Kortiim and Vogel, 1955). The values obtained therefore do not permit a quantitative interpretation and only give qualitatively the relative order of the basicity of unsaturated compounds. This is also true for partition measurements between an acid and an organic phase, if in such a case the necessary thermodynamic assumptions have not been tested or established by separate investigations. 

The solvent in this case is a salt solution (solvent with small ions) which is in Donnan equilibrium with the polyelectrolyte solution, the index /xs means that the chemical potentials of all the solutes except that used in differentiation are constant, and K is given as [39]... 

When we looked at the solubility of naphthalene in various solvents (Section 6.4), we found that in benzene the actual solubility was close to the truly ideal value, as predicted on the basis of Raoult s Law, but in both hexane and methanol it was considerably lower. The chemical potential of the solid solute (and hence its activity in the solid state) is the same in all cases the activity of the naphthalene in solution must also be identical, for at equilibrium... 

A different and usually more practical way of overcoming the difficulty is to look at differences in chemical potentials of the same solute in different phases. Since the singular part of the chemical potential is the same in the two phases, we can perform an expansion similar to (7.7) but for the difference of the Gibbs free energy of the two phases, in which case the coefficient of the linear term is finite. For instance, if we have two solvents Wi and W2, we can write... 

A thermodynamic description of the process of micelle formation will include a description of both electrostatic and hydrophobic contributions to the overall Gibbs energy of the system. Hydrocarbons (e.g., dodecane) and water are not miscible the limited solubihty of hydrophobic species in water can be attributed to the hydrophobic effect. The hydrophobic Gibbs energy (or the transfer Gibbs energy) can be defined as the difference between the standard chemical potential of the hydrocarbon solute in water and a hydrocarbon solvent at infinite dilution [36-40]... 

In tMs chapter we consider those aspects of the interaction of solvent and solute that are most clearly physical in nature, setting aside the more chemical aspects until the next chapter. The intermolecular potential characteristic of nonpolar substances, the London or dispersion interaction, arises from the mutual, time-dependent polarization of the molecules. For two molecules weU separated in vacuo it is approximated by the London formula ... 

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