Solvent Chemical Potentials from Phase Equilibria

Avap f is the difference of the molar enthalpies of the real gas and the liquid at the saturation vapor pressure of the liquid. 

Equations 12.1.13and 12.1.14 are two forms of the van t Hoff equation. They allow us to evaluate the standard molar reaction enthalpy of a reaction by a noncalorimetric method from the temperature dependence of In AT. For example, we can plot In K versus /T then according to Eq. 12.1.14, the slope of the curve at any value of l/T is equal to -AtH /R at the corresponding temperature T. 

A simple way to derive the equation for this last procedure is to substitute ArG° = 

Suppose we plot In K versus 1/ T. In a small temperature interval in which AjH° and ArS° are practically constant, the curve will appear linear. According to Eq. 12.1.15, the curve in this interval has a slope of —A H°/R, and the tangent to a point on the curve has its intercept at 1/ T=0 equal to A S°/R. 

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CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS 12.2 Solvent Chemical Potentials from Phase Equilibria... 

SOLVENT CHEMICAL POTENTIALS FROM PHASE EQUILIBRIA... 

This is an expression of Raoult s law which we have used previously. Freezing point depression. A solute which does not form solid solutions with the solvent and is therefore excluded from the solid phase lowers the freezing point of the solvent. It is the chemical potential of the solvent which is lowered by the solute, so the pure solvent reaches the same (lower) value at a lower temperature. At equilibrium... 

Another general type of behavior that occurs in polymer manufacture is shown in Figure 3. In many polymer processing operations, it is necessary to remove one or more solvents from the concentrated polymer at moderately low pressures. In such an instance, the phase equilibrium computation can be carried out if the chemical potential of the solvent in the polymer phase can be computed. Conditions of phase equilibrium require that the chemical potential of the solvent in the vapor phase be equal to that of the solvent in the liquid (polymer) phase. Note that the polymer is essentially involatile and is not present in the vapor phase. 

A three component system consisting of a solvent (0) and two further components (1 and 2) can be considered. The phase equilibrium between the solid (s) and liquid (1) phases is characterized by equality of the chemical potentials of a given component in the two phases. Supposing that the component are completely immiscible in the solid phase we obtain from the condition of equality of chemical potentials ... 

Passive sampling can be defined as any sampling technique based on the movement (by diffusion) of analyte molecules from the sampled medium to a receiving phase contained in a sampling device. This mass transfer process is driven by a difference in chemical potentials of the analyte in the two media. This process continues until equilibrium is reached in the system, or until the sampling process is stopped.14 Analytes are retained in a suitable medium within the device, known as a receiving or sorption phase. This can be a solvent, chemical reagent, absorbent, or... 

Liquid-liquid extraction is a process for separating the components of a liquid (the feed) by contact with a second liquid phase (the solvent). The process takes advantage of differences in the chemical properties of the feed components, such as differences in polarity and hydrophobic/hydrophilic character, to separate them. Stated more precisely, the transfer of components from one phase to the other is driven by a deviation from thermodynamic equilibrium, and the equilibrium state depends on the nature of the interactions between the feed components and the solvent phase. The potential for separating the feed components is determined by differences in these interactions. 

The fundamental mechanisms for solute mass transfer in liquid-liquid extraction involve molecular diffusion driven by a deviation from equilibrium. When a liquid feed is contacted with a liquid solvent, solute transfers from the interior of the feed phase across a liquid-liquid interface into the interior of the solvent phase. Transfer of solute will continue until the solute s chemical potential is the same in both phases and equilibrium is achieved. 

It is readily seen from Fig. 3.8 that for each value of x above Xc there are two different values of 2 at which the chemical potential of the solvent in the two phases is the same. This implies that solutions with concentrations defined by these two values of 2 can be in thermodynamic equilibrium for X > Xc- Moreover, it implies that a solution with an intermediate value of (j)2 will spontaneously separate into two stable liquid phases defined by these two concentrations this will occur with a concomitant decrease in the free energy. Such phenomena are in fact observed with solutions of flexible polymers for values of x above a critical value (Xc)- Thus, if X is increased by decreasing the temperature, an initially totally miscible system at higher temperatures is transformed to one of limited miscibility by lowering the temperature at some critical temperature Tc, incipient phase separation (as indicated by the onset of an opalescence) is encountered, followed by separation into distinct liquid phases at still lower temperatures. 

Here is the chemical potential of the volatile solvent in the vapor phase, ix[,. .. are the chemical potentials of the solvent in the various cups, filled with the samples, and the standard solutions, respectively. After establishment of the isopiestic equilibration, the solutions are reweighed. From the equilibrium concentrations, the activity coefficients can be calculated. 

The description of phase equilibria makes use of the partial molar free enthalpies, i, called also chemical potentials. For one-component phase equilibria the same formalism is used, just that the enthalpies, G, can be used directly. The first case treated is the freezing point lowering of component 1 (solvent) due to the presence of a component 2 (solute). It is assumed that there is complete solubility in the liquid phase (solution, s) and no solubility in the crystalline phase (c). The chemical potentials of the solvent in solution, crystals, and in the pure liquid (o) are shown in Fig. 2.26. At equilibrium, ft of component 1 must be equal in both phases as shown by Eq. (1). A similar set of equations can be written for component 2. By subtracting j,i° from both sides of Eq. (1), the more easily discussed mixing (left-hand side, LHS) and crystallization (right-hand side, RHS) are equated as Eq. (2). 

The next step towards the description of phase diagrams that include macromolecules is to change from the just discussed ideal solution to the real solutions, mentioned in Sect. 2.2.5. For this purpose one can look at the phase equilibrium between a solution and the corresponding vapor. The simplest case has a negligible vapor pressure for the second component, 2. The chemical potentials for the first component, 1, in solution and in the pure gas phase are written in Fig. 7.3 as Eqs. (1-3), following the discussion given in Fig. 2.26. For component 1, the chemical potential of the solvent, Pi°, is defined in Eq. (1) for its pure state, Xj = 1.0, but at its vapor pressure, Pi°, not at atmospheric pressure. It must then be written as... 

The procedure for the calculation of salt solubilities in aqueous solutions can be extended to organic solvents or solvent mixtures, starting from the condition that the fugacity (chemical potential) of a precipitated salt in phase equilibrium is identical in water, an organic solvent or the aqueous solution (r ee Figure 8.16). This means that the already available standard thermodynamic properties in the aqueous phase given in Table 8.1 can be used to determine the salt solubility in organic solvents or solvent mixtures [10]. 

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 ... 

The solvent molecules in the dilute phase have a higher (more negative ) chemical potential than those in the concentrated phase. This chemical potential difference causes a flow of solvent molecules from the dilute phase to the concentrated phase (the flow is proportional to - 3py9x). This is shown in figure VI - lb. This process continues until osmotic equilibrium has been reached, i.e. when the chemical potentials of the solvent molecules in both phases are equal (see figure VI - Ic) ... 

To find conditions for equilibrium, we imagine the solution to be divided into many thin curved volume elements at different distances from the axis of rotation as depicted in Fig. 9.12(b). We treat each volume element as a uiuform phase held at constant volume so that it is at a constant distance from the axis of rotation. The derivation is the same as the one used in the preceding section to obtain Eq. 9.8.1, and leads to the same conclusion in an equilibrium state the temperature and the chemical potential of each substance (solvent and solute) are uniform throughout the solution. 

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