Excess function partial molar

The relative partial molar enthalpies of the species are obtained by using Eqs. (70) and (75) in Eq. (41). When the interaction coefficients linear functions of T as assumed here, these enthalpies can be written down directly from Eq. (70) since the partial derivatives defining them in Eq. (41) are all taken at constant values for the species mole fractions. Since the concept of excess quantities measures a quantity for a solution relative to its value in an ideal solution, all nonzero enthalpy quantities are excess. The total enthalpy of mixing is then the same as the excess enthalpy of mixing and a relative partial molar enthalpy is the same as the excess relative partial molar enthalpy. Therefore for brevity the adjective excess is not used here in connection with enthalpy quantities. By definition the relation between the relative partial molar entropy of species j, Sj, and the excess relative partial molar entropy sj is... 

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. 

The excess molar volumes of 10-40 mol % methanol/C02 mixtures at 26°C as a function of pressure has been determined. The excess molar volumes varied with composition and pressure significant interaction between CO2 and methanol was noted from the observed excess molar volumes. To better characterize the interaction and its effect on analyte solubility, the partial molar volume of naphthalene at infinite dilution in liquid 10 and 40 mol % methanol/C02 mixtures was determined. The variation of the partial molar volume at infinite dilution with pressure correlated well with isothermal compressibility of the methanol/C02 mixtures (Souvignet and Olesik, 1995). 

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... 

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... 

On the basis of the above analysis it has been shown the partial molar quantities are easily obtained from intensive quantities like the molar volume when this quantity is plotted as a function of an intensive composition variable like the mole fraction. The plots in fig. 1.2 show that the molar volume is almost a linear function of the mole fraction of solute. If the curves in fig. 1.2 were actually perfect straight lines, the partial molar volumes would be constant independent of solution composition. Such a situation would arise if the solution were perfectly ideal. In reality, very few solutions are ideal, as will be seen from the discussion in the following section. In order to see more clearly the departure from ideality, one defines and calculates a quantity called the excess molar volume. This quantity is equal to the actual molar volume less the molar volume for the solution if it were ideal. The latter can be considered as the volume of the solution that would be found if the molecules of the two components form a solution without expansion or contraction. Thus, the ideal molar volume can be defined as... 

Thus, the excess functions (e.g., g , ftE, and sE) also reflect the contributions of interorolecular forces to mixture property tn. Partial molar property m, corresponding to molar mixture property m is defined in the usual way ... 

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),... 

So the remaining partial molar excess quantities can all be expressed as simple functions of the activity coefficient. 

Here Affi and dSi are the relative partial molar heats of dilution such that AGi=Affi-TASi=(fii—iJ.i. The use of a squared term in these definitions is simply recognition that the excess functions must involve two-body (and higher) interactions. Equations (3.31) and (3.32) when coupled with equation (3.30) yield... 

Flory and Krigbaum defined an enthalpy (Kj) parameter and an entropy of dilution ( /i) parameter such that the thermodynamic functions used to describe these long-range effects are given in terms of the excess partial molar quantities... 

We have an expression for excess molar volume in terms of mole fractions. To compute partial molar volumes, we need an expression for the actual excess volume as a function of moles. 

From first the excess function and then the partial molar Gibbs energy of gold can... 

Figure 3.14 Excess function of the partial molar Gibbs energy of silver in the silver-gold alloy plotted as function of y in Agj,Au. The solid line is a logarithmic fit of the experimental data.

With the data of the previous example, plot the excess volume as a function of the mol fraction of ethanol and determine the partial molar volumes of the two components at ethanol mol fraction of 0.4. 

An equation for the activity coefficient can be obtained by fitting an appropriate equation to the experimental data. Rather than fitting each individual activity coefficient to its own function, the preferred procedure is to fit the excess Gibbs energy as a function Xi. The activity coefficients are obtained from this fit by noting from eg. that In is in fact a partial molar property specifically, it is the partial molar... 

The principles of phase equilibrium do not apply to excess adsorption variables at high pressure where the excess adsorption passes throu a maximum. Under these conditions, the pressure is no longer a single-valued function of excess adsorption so that n cannot serve as an independent variable for the determination of partial molar quantities such as activity coefficients. Additional complications which arise at high pressure are (1) the selectivity for excess adsorption (S12 = (nf/j/i)/(n2/y2)) approaches infinity as nj — 0 and (2) the differential enthalpy of the ith component has a singularity at the pressure corresponding to maximum nf. For excess variables, the diffierential functions are undefined but the integral functions for enthalpy and entropy are smooth and well-behaved (1). 

The KB inversion process involves the extraction of KBIs from the available experimental data. The experimental data required for this process—derivatives of the chemical potentials, partial molar volumes, and the isothermal compressibility—are all generally obtained as derivatives of various properties of the solution. Obtaining reliable derivatives can be challenging and will depend on the quality of the source data and the fitting function. Unfortunately, the experimental data often appear without a reliable statistical analysis of the errors involved, and hence the quality of the data is difficult to determine. Matteoli and Lepori have performed a fairly rigorous analysis of a series of binary mixtures and concluded that, for systems under ambient conditions, the quality of the resulting KBIs is primarily determined by the chemical potential data, followed by the partial molar volume data, whereas errors in the compressibility data have essentially no effect on the KBI values (Matteoli and Lepori 1984). Excess chemical potentials are typically obtained from partial pressure data, either isothermal or bubble point determinations, and from osmotic pressure or even electrochemical measurements. The particle number... 

The need for accurate partial molar volumes of the components as functions of the composition can be met when the excess volume of the mixture, 1, is known. The KBIs are proportional to the partial molar volumes, and, hence, have the same relative accuracy as these quantities. If no partial molar volumes are known, then the approximations Va Va. V b Vb°, and XaVX+XbV b may be used for the calculation of the G . 

The equations that are commonly used to represent experimental data of (Z = G, y ) and p, are expressed as a function of Xj, whereas in Equation 4.14 derivatives with respect to are required. We need therefore to express than in a fnnction of X,. Taking into account the definition of the excess partial molar quantity, 7, as a function of the relationship between x, and the differentials of 7F- =f(X with respect to x, and of X with respect to and applying the treatment to one mole of mixture, after some substitutions and rearrangements, the diagonal elements p, can be expressed in a function of X and of four derivatives of the chemical potential of components 1 and 2,... 

Likewise, the expression for the excess partial molar quantity, Zf, as a function of x, is obtained as... 

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. 

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