Partial molar properties, definition

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 ... 

The partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

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... 

Let us consider a state property G of a system of one mole consisting ofN mole fraction of constituent and mole fraction of constituent B. Let G and Gg be the corresponding partial molar properties of A and B in the mixture or solution. Then by definition... 

Determination of Partial Molar Quantities Ir Direct Method. —In view of the definition of the partial molar property Qi as... 

Remembering the definition of enthalpy H = U + PV, we can introduce it as the partial molar properties of the reactive substance at the entrance and exit of the reactor. 

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),... 

Partial molar properties. Recall the definition of a partial molar property is... 

T is temperature, p is pressure, and p i is the chemical potential of component i in the mixtnre. The chemical potential is not a partial molar property of U because S and V are held constant during differentiation instead of T and p, based on the definition of /r in (29-3). The total differential of internal energy... 

The derivative operator appearing in (3.4.5) is called the partial molar derivative, and the quantity F,- defined by (3.4.5) is called the partial molar F for component i. It is the partial molar property that can always be mole-fraction averaged to obtain the mixture property F. Note, however, that F is itself a property of the mixture, not a property of pure i partial molar properties depend on temperature, pressure, and composition. We emphasize that the definition (3.4.5) demands that F be extensive and that the properties held fixed can only be temperature, pressure, and all other mole numbers except N,. Partial molar properties are intensive state functions they may be either measurable or conceptual depending on the identity of F. 

To obtain expressions for the partial molar properties of ideal solutions, we first determine the chemical potential. Using the ideal-solution fugacity (5.1.6) in the integrated definition of fugacity (4.3.12) we find... 

Excess partial molar properties are obtained by applying the partial molar operator on an excess property. Recall that the definition of the partial molar operator is... 

The Gibbs-Duhem equation is one of the most important relationships of mixture thermodynamics. A general form of the Gibbs-Duhem equation can be derived in combination with the general definition of the partial molar properties. The Gibbs-Duhem equation allows for the development of so-called consistency tests, which are a necessary but not sufficient condition for the correctness of experimental data. The total differential of the function... 

The properties of dissolved substances is discussed in terms of partial molar properties, the formal definition of which is... 

Our new derivative terms (9G/9n )j.p - look suspiciously like our definition of a partial molar property in Equation (2.1), and indeed they are partial molar Gibbs energies. They allow us to deal with compositional changes, and as such... 

Excess properties, the difference between the property in a real solution and in an ideal solution, are generally expressed as a relative or relative partial molar properties, such as the relative enthalpy, L, or relative partial molar enthalpy, L. The Gibbs energy is treated differently. The fact that Gj-p is a thermodynanoic potential leads naturally to the definition of a relative partial molar Gibbs energy (q. - /a°) which is not the difference from an ideal solution (/A — pL° is not zero even for an ideal solution) but the difference from a standard state, which in this chapter is a pure phase, but may also be some hypothetical state. The form of the equation relating q, - to composition then... 

Partial molar properties give information about the change of the total property due to addition of an infinitely small amount of substance of species i to the mixture. From eq 2.16 it becomes apparent that, by definition, the chemical... 

Our main reason for studying partial molar properties was to gain understanding of and facility in computing /r, = g,. However, by definition... 

We start with the definition and a brief description of partial molar properties (pmp) and then examine ... 

Notice that the derivatives are, according to the definition, Eq. 11.3.1, the partial molar properties of NM. Thus Eq.l 1.4.2 becomes ... 

The definitions above were based on a partial molar property. Hence they describe the contribution of species i to the solution. The fugacity and fugacity coefficient are given a hat instead of a bar to remind us that while they represent the contribution of species i in solution, they do not represent the mathematical definition of a partial molar property, that is. 

Many equations of state are explicit in P but not V (such as the cubic equations discussed in Section 4.3), so it is convenient to tiy to rewrite the second term in Equation (7.13) in terms of a derivative in P. Recall from the definition of a partial molar property ... 

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 ... 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

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