Partial molar properties calculation

Equation (4-49) is merely a special case of Eq. (4-48) however, Eq. (4-50) is a vital new relation. Known as the summahility equation, it provides for the calculation of solution properties from partial properties. Thus, a solution property apportioned according to the recipe of Eq. (4-47) may be recovered simply by adding the properties attributed to the individual species, each weighted oy its mole fraction in solution. The equations for partial molar properties are also valid for partial specific properties, in which case m replaces n and the x, are mass fractions. Equation (4-47) applied to the definitions of Eqs. (4-11) through (4-13) yields the partial-property relations ... 

Equation (5.23) is known as the Gibbs-Duhem equation. It relates the partial molar properties of the components in a mixture. Equation (5.23) can be used to calculate one partial molar property from the other. For example, solving for dZ gives... 

ANALYTIC METHODS FOR CALCULATION OF PARTIAL MOLAR PROPERTIES... 

The definition of a partial molar property, Eq. (11.2), provides the me-for calculation of partial properties from solution-property data. Implicit in definition is a second, equally important, equation that allows the calculation solution properties from knowledge of the partial properties. The derivation this second equation starts with the observation that the thermodynamic propertl of a homogeneous phase are functions of temperature, pressure, and the numb of moles of the individual species which comprise the phase. For thermodyna property M we may therefore write... 

The process of calculating the partial molar property of one component from the knowledge of partial molar property of the other component in a binary system, where the integral molar property changes continuously with the composition, is referred to as the Gibbs Duhem Integration. 

Although Eqs. 8.2-8 through 8.2-11 are well suited for calculations in which temperature, pressure, and the partial molar properties are the independent variables, it is usually more convenient to have T, P, and the mole fractions x-, as the independent variables. A change of variables is accomplished by realizing that for a C-component mixture there are only C — 1 independent mole fractions (since XLt Thus we... 

For more accurate calculations of the partial molar volume (or any other partial molar property), an analytical, rather than graphical, procedure is used. First, one fits the volume change on mixing, AmixY. with a polynomial in mole fraction, and then the necessary derivative is found analytically. Since AmixY must equal zero at X = 0 and xi = 1 (,X2 = 0), it is usually fit with a polynomial of the Redlich-Kister form ... 

To use classical Thermodynamics we need absolutely no knowledge of such things. We need only measure macroscopic properties of the bulk solution, such as its bulk composition, and from the.se we will be able to calculate molar properties of dissolved substances (partial molar properties) just as we do for pure substances. 

Conventional standard partial molar properties of aqueous species at 25°C and 1 bar, plus some of the coefficients required to calculate these properties at high T and P using the revised HKF model, and using 6 = 228 K and = 2600 bars. 

The first condition is true because equivalent with (5.60). The second equation is important because states that a property of a mixture may be calculated from partial molar properties weighted by molar fractions. By differentiation we get ... 

In practice, pure-component molar enthalpies are employed to approximate A/7rx. This approximation is exact for ideal solutions only, when partial molar properties reduce to pure-component molar properties. In general, one accounts for more than the making and breaking of chemical bonds in (3-35). Nonidealities such as heats of solution and ionic interactions are also accounted for when partial molar enthalpies are employed. Now, the first law of thermodynamics for open systems, which contains the total differential of specific enthalpy, is written in a form that allows one to calculate temperature profiles in a tubular reactor ... 

Partial molar properties are defined for any property that has an extensive form for example, volume, enthalpy, etc. They are intensive properties and as such, they are functions of pressure, temperature, and mol fractions. To see how partial molar properties can be useful, consider the following thought experiment a vessels that contains a mixture (for example, a solution of several components) is poured into another vessel B. We will calculate the enthalpy in vessel B as it builds up during this process. The differential of is given bveq. fo.Sl the process obviously takes place under constant pressure and constant temperature, therefore, dT=o and dP = o. This simplifies the differential to the form. 

Debenedetti, P. G. 1988. Fluctuation-based computer calculation of partial molar properties. 2. A numerically accurate method for the determination of partial molar energies and enthalpies. Journal of Chemical Physics. 88, 2681. 

Calculate the partial molar properties in a liquid at infinite dilution using the Porter approach. 

This equation fulfills the constraint that the property change of mixing for the pure components is zero. The Porter approach is now used to calculate the partial molar properties at infinite dilution of component 1 in component 2 (mf") and vice versa (rnf ). The corresponding properties of state of the pure components are named m and m2 ... 

The thermodynamic functions of the adsorbed phase are a function of pressure, temperature, and concentration of the adsorbed molecules on the surface of the solid so that it is essential to distinguish between the molar and partial molar properties. As the derivation of the thermodynamic functions is beyond the scope of this chapter, we shall only define the functions and discuss only the calculations of the enthalpy of adsorption or the heat of adsorption. 

Figures like Figure 6.8 are widely used for heat of mixing calculations. These have practicaly nothing to do with equilibrium, the subject of this book, but are related to partial molar properties, the subject of this chapter. 

The relationship between AHc or AHp and the true partial molar property AHy derived by Eq. (7) may be readily obtained from a consideration of the reference states involved in each calculation. For example, AHy is defined as... 

The method has been applied< in the calculation of the structural properties of the solutions of gaseous Brg in molten silver bromide and indicates that the partial molar properties are determined by the predominance of the Br3 -Br3 repulsions. 

Define a partial molar property and describe its role in determining the properties of mixtures. Calculate the value of a partial molar property for a species in a mixture from analytical and graphical methods. Apply the Gibbs-Duhem equation to relate the partial molar properties of different species. 

We have introduced a new type of property, the partial molar property. This property tells us about the contribution of a given species to the properties of a mixture. Our next question is How do we obtain values for these partial molar properties There are several ways in which to accomplish this task. In this section, we consider two examples of how we might calculate a partial molar property by analytical means when we have an equation that describes the total solution property or by graphical means from plots of total solution data. 

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