Partial molar properties excess

It is convenient to think of the excess property as a mathematical operator that removes the ideal-solution part from a thermodynamic property. It is a linear operator and can be combined with other operators, such as the partial molar differentiation. Expressions that can be written between regular properties may be written for the excess and for the partial molar excess properties. For example, starting with the fundamental relationship... 

Partial molar excess properties can be defined analogously as E m = T x mf... 

The partial molar excess properties vary with composition. They can be derived directly from the curvature of the excess enthalpies or excess volumes as a function of the mole fraction. How the partial molar properties can be determined by the tangent line for the excess enthalpy and a composition of xi = 0.252 is shown in Figure 5.17. Using the partial molar excess values h, and the activity coefficient at a different temperature or pressure can be determined. But it has to be considered that these partial molar excess properties do not only depend on composition but also on temperature and pressure. 

Likewise, we can define a partial molar excess property by applying Equation (6.15) to the extensive excess property as follows ... 

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,... 

Partial molar properties play a central role in phase-equilibrium thermodynamics, and ii is convenient to broaden their definition to include partial molar residua) junctions and partial molar excess functions. Hence, we define, analogous to Eq- (1-2-5),... 

However, following Eq. 5.26 not the excess enthalpy but the partial molar excess enthalpy is the determining property to describe the temperature dependence of the activity coefficients. Depending on the curvature of h as a function of composition for positive (negative) values of h negative (positive) partial... 

Thermodynamic Properties of the Mixtures In those cases where F represents a molar thermodynamic property of the binary mixture of water and the cosolvent, the partial molar quantities of the components are of interest. Differentiation of Equation 3.43 with respect to the mole fractions yield the partial molar excess values. The excess partial molar value for water is ... 

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... 

Excess partial molar thermodynamic property of component i... 

Once we have this expression for g , the corresponding activity coefBcients for species a and b are given by the appropriate partial molar excess Gibbs energies via Equation (7.48). Applying the definition of a partial molar property to the excess Gibbs energy, we get ... 

All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding ... 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

In this chapter, we shall consider the methods by which values of partial molar quantities and excess molar quantities can be obtained from experimental data. Most of the methods are applicable to any thermodynamic property J, but special emphasis will be placed on the partial molar volume and the partial molar enthalpy, which are needed to determine the pressure and temperature coefficients of the chemical potential, and on the excess molar volume and the excess molar enthalpy, which are needed to determine the pressure and temperature coefficients of the excess Gibbs function. Furthermore, the volume is tangible and easy to visualize hence, it serves well in an initial exposition of partial molar quantities and excess molar quantities. 

Figure 6J Excess partial molar properties of jadeite component in NaAlSi206-KAlSi206 molten mixture. (A) Excess partial molar entropy. (B) Excess partial molar enthalpy. Reprinted from Eraser et al. (1983), Bulletin Mineralogique, 106, 111-117, with permission from Masson S.A., Paris, Erance.

Figure 17.5 Derived thermodynamic properties at T — 298.15 K and p = 0.1 MPa for (2Cic-CfiHi2 + X2n-CjHi4) (a) excess molar heat capacities obtained from the excess molar enthalpies (b) relative partial molar heat capacities obtained from the excess molar heat capacities (c) change of the excess molar volume with temperature obtained from the excess molar volumes and (d) change of the excess molar enthalpies with pressure obtained from the excess molar volumes.

Gibbs functions for a real salt solution and the corresponding ideal salt solution containing m2 moles of salt in a kilogram of solvent. GE can be calculated for many aqueous salt solutions from published values of 0 and y . In the same way, the corresponding excess enthalpy HE can be defined and this equals the apparent partial molar enthalpy. Thus the properties of salt solutions can be examined in plots of GE, HE, and T SE against m2, where SE is the... 

Molar or specific value, extensive tlrenrrodyiramic property Partial property, species i in solution Excess property == M — M "... 

They are used as industrial solvents for small- and large-scale separation processes, and they have unusual thermodynamic properties, which depend in a complicated manner on composition, pressure, and temperature for example, the excess molar enthalpy (fp-) of ethanol + water mixture against concentration exhibits three extrema in its dependence on composition at 333.15 K and 0.4 MPa. The thermodynamic behavior of these systems is particularly intricate in the water-rich region, as illustrated by the dependencies of the molar heat capacity and partial molar volume on composition. This sensitivity of the partial molar properties indicates that structural changes occur in the water-rich region of these mixtures. Of course, the unique structural properties of water are responsible for this behavior. ... 

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