Free energy excess partial molar

As already pointed out, Yu is 1 if a compound forms an ideal solution. In this rather rare case, the term RTkiyu, which we denote as partial molar excess free energy of compound i in solution t, Gpe, is 0. This means that the difference between the chemical potential of the compound in solution and its chemical potential in the reference state is only due to the different concentration of the compound i in the two states. The term R In xtf=S 1 expresses the partial molar entropy of ideal mixing (a purely statistical term) when diluting the compound from its pure liquid (xiL =1) into a solvent that consists of otherwise like molecules. 

Equilibrium Vaporization. The cesium release results presented in this chapter may also be used to demonstrate our earlier conclusion that equilbirium vaporization represents the upper limit for the fractional fission-product release as a function of sodium vaporization. Figure 6 shows three cesium release curves. Curve A was calculated from the Rayleigh Equation in conjunction with the partial molar excess free energy of mixing of infinitely dilute cesium—sodium solutions reported... 

We also see that the excess free enthalpy GE is differentiated with respect to the temperature and the number of moles of the solution to give the excess entropy SF and the partial molar excess free energy of mixing RTlnyi as follows ... 

From thermodynamic considerations, the equilibrium partitioning of an organic compound in a solvent/water system is determined by the difference between the partial molar excess free energy (AG% ) of the compound in the organic solvent (o) and in the water phase (w) with molar volumes and V, respectively (Schwarzenbach, Gschwend and Imboden, 1993) ... 

The gas-liquid chromatography is a convenient technique for studying the thermodynamic properties of liquid crystals and liquid crystalline solutions. The basis for such applications is the following relation between the activity coefficient 7 and the partial molar excess free energy Gf of the solute at infinite dilution... 

Nagasawa and Takahashi, 1972). More specifically, the value of the second virial coefficient determines the excess chemical potential, juE (also known as the excess partial molar Gibbs free energy), which characterizes the formation of biopolymer-solvent and biopolymer-biopolymer pair contacts ... 

The partial molar excess Gibbs free energy of the methylene group can be used as a means of characterizing stationary phases. From rearrangement of Equation 11.46, an expression for the standard partial molar Gibbs free energy can be obtained. 

The quantity of primary interest in our thermodynamic construction is the partial molar Gihhs free energy or chemical potential of the solute in solution. This chemical potential depends on the solution conditions the temperature, pressure, and solution composition. A standard thermodynamic analysis of equilibrium concludes that the chemical potential in a local region of a system is independent of spatial position. The ideal and excess contributions to the chemical potential determine the driving forces for chemical equilibrium, solute partitioning, and conformational equilibrium. This section introduces results that will be the object of the following portions of the chapter, and gives an initial discussion of those expected results. 

We apply the partial molar operator and notice that on the left-hand side we obtain the chemical potential of component in the solution. The first term on the right-hand side is the chemical potential of the component in ideal solution, and the second term is the excess partial molar Gibbs free energy ... 

We return to the definition of the activity coefficient and solve for the excess partial molar Gibbs free energy ... 

Sato, T., Chiba, A., and Nozaki, R. (1999). Dynamical aspects of mixing schemes in ethanol-water mixtures in terms of the excess partial molar activation free energy, enthalpy, and entropy of the dielectric relaxation process. J. Chem. Phys., 110, 2508-2521. 

Here, R is the gas constant, Mj is the solvent s molecular weight, and P 22 are the saturated vapour pressure and second virial coefficient of the pure solvent at temperature T, respectively. The required vapour pressures and virial coefficients can be calculated or estimated from known relationships [473-477]. Other treatments for the determination of y , which include a term to correct for the free volume contribution due to differences in the size of the solute and solvent, have also been used [478, 479]. The temperature dependence of the resulting solute activity coefficient is related to the infinite dilmion partial molar excess Gibbs energy (AG ) through Eq. (9). 

The partial molar excess Gibbs free energy and partial molar excess enthalpy of mixing are defined by the following equations ... 

For an ideal solution, the total standard partial molar excess Gibbs free energy, AGT, equals the ideal contribution, AG, and AGp = O. In other words, there is no interaction occurring between the solute and the stationary phase. When a nonpolar mole-, cule such as an alkane is chosen as the solute, AG will increase as the polarity of the stationary phase increases. There-—c... 

Of course AG° will not be the same value for every alkane, especially as the polarity of the stationary phase increases. So it becomes necessary to define a unit of the alkane, the methylene group, on which to base comparisons. Thus, the partial molar excess Gibbs free energy of the methylene group, AGg(cH2>, is defined by... 

Based on Eq. (11), the partial molar excess Gibbs free energy will result in Eq. (32). 

An application of continuum solvation calculations that has not been extensively studied is the effect of temperature. A straightforward way to determine the solvation free energy at different temperatures is to use the known temperature dependence of the solvent properties (dielectric constant, ionization potential, refractive index, and density of the solvent) and do an ab initio solvation calculation at each temperature. Elcock and McCammon (1997) studied the solvation of amino acids in water from 5 to 100°C and found that the scale factor a should increase with temperature to describe correctly the temperature dependence of the solvation free energy. Tawa and Pratt (1995) examined the equilibrium ionization of liquid water and drew similar conclusions. An alternative way to study temperature effect is through the enthalpy of solvation. The temperature dependence of is related to the partial molar excess enthalpy at infinite dilution,... 

The excess partial molar entropy, enthalpy, and volume can now be obtained from familiar partial derivatives of free energy as follows ... 

Because in the west of China some salt lake brines contain abundant boron and lithium, in which solute-solvent and solute-solute interactions are complex, studies on the ihermochemical properties for the systems related with the brines are essential to understand the effects of temperature on excess free energies and solubility, and to build a thermodynamic model that can be applied for prediction of the properties. Yin et al. [43] measured the enthalpies of dilution for aqueous Li2B407 solutions from 0.0212 to 2.1530 mol/kg at 298.15 K. The relative apparent molar enthalpies and relative partial molar enthalpies of the solvent and solute were also calculated, and the thermodynamic properties of the complex aqueous solutions were represented by a modified Pitzer ion-interaction model. 

The excess activation free energy AG, enthalpy Aff, and entropy A5, and their partial molar quantities for alcohols AG, A//, and A5 were also calculated for the dominating processes with T, in methanol, ethanol, and 1-propanol water mixtures [67-69] and are depicted in Figure 6.6. These thermodynamic quantities were calculated according to the Eyring transition state theory [93]. Based on the curves in Figure 6.6 characteristic molar fraction values (0.30,0.18, and 0.14 for methanol, ethanol, and 1-propanol water mixtures, respectively) were also reported above and below which the behavior of the partial molar excess activation quantities... 

Such quantities depend on partial derivatives of the excess free energy with respect to molar fractions (cf. 1.8.2) which are much altered by the order-disorder effects. Especially the problem of the nature of the singularity at the critical temperature requires an exact solution of the combinatorial problem. These problems which are still awaiting a satisfactory general solution, will not be studied in this book. 

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