Partial molar general relations

Partial Molar Properties Consider a homogeneous fluid solution comprised of any number of chemical species. For such a PVT system let the symbol M represent the molar (or unit-mass) value of any extensive thermodynamic property of the solution, where M may stand in turn for U, H, S, and so on. A total-system property is then nM, where n = Xi/i, and i is the index identifying chemical species. One might expect the solution propei fy M to be related solely to the properties M, of the pure chemical species which comprise the solution. However, no such generally vahd relation is known, and the connection must be establi ed experimentally for eveiy specific system. 

The partial molar quantity of a molar quantity Q, related to the component A, is generally written as 0A and is defined by ... 

Equations 116,117,121,122, and 124 are the general property relations between partial molar properties and solution properties. The symbol M may represent the molar value of any extensive thermodynamic property, for example, V, U, H, S, jA, or G. When M = if, the derivatives (dH/dT)p and (dH/dP)Tx are given by equations 75 and 79. Equations 121,122, and 124 then become the following ... 

For example, we may choose to as the average volume velocity, to = (c, V,) v,-. In more general terms, we may define to by X Pi vi > with Y Pi = 1 The s are weighting factors. If we formulate Eqn. (4.72) for two different reference velocities, to and to", and take into account the partial molar volumes (V,) which are not independent of each other (Gibbs-Duhem relation), we obtain after some algebraic rearrangements [H. Schmalzried (1981)] the quite general expression... 

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. 

This equation gives the relation between the change of enthalpy for this change of state and the partial molar enthalpies of the components relative to the chosen standard states.1 Equation (9.26) can easily be generalized to... 

All the general thermodynamic relations can be applied with minor symbolic modifications to the partial molar quantities ... 

Equation (15) is called the general Gibbs—Duhem relation. Because it tells us that there is a relation between the partial molar quantities of a solution, we will learn how to use it to determine a Xt when all other X/ il have been determined. (In a two-component system, knowing Asolvent determines Asolute.) This type of relationship is required by the phase rule because, at constant T, P, and c components, a single-phase system has only c — 1 degrees of freedom. 

The properties of solutions as represented by the symbol M may be on unit-mass basis as well as on a mole basis. The equations relating solutk properties are unchanged in form one merely replaces the various n s, represent ing moles, by m s, representing mass, and speaks of partial specific properti rather than of partial molar properties. In order to accommodate either, generally speak simply of partial properties. 

In this section, we investigate the relations between the macroscopic susceptibilities and the molecular polarizabilities. Consistent microscopic interpretations of many of the non-linear susceptibilities introduced in Section 2 will be given. Molar polarizabilities will be defined in analogy to the partial molar quantities (PMQ) known from chemical thermodynamics of multicomponent systems. The molar polarizabilities can be used as a consistent and general concept to describe virtually all linear and non-linear optical experiments on molecular media. First, these quantities will be explicitly derived for a number of NLO susceptibilities. Physical effects arising from will then be discussed very briefly, followed by a survey of experimental methods to determine second-order polarizabilities. 

This is a remarkable result. It states that the partial molar volume of the partial molar volume of a species is not related, in a simple manner, to the actual volume contributed by that species to the total volume of the system. We also note that in this particular example, the partial volume V(T, P, [Pg.108]

The method of derivation for H is generally useful for extensive properties to relate the effect of intensive properties on partial molar properties. 

With liquids, the requisite n and d measurements can obviously be made directly solids, in general, are examined in solution, and mixture formulae applied to the observations. If subscripts 1, 2, and 12 relate respectively to solvent, solute, and solution, and if concentrations are expressed as molar fractions /, and /2, or weight fractions wq and w2, the apparent partial molar or specific refractions (R2 or r2) can be extracted from equations (3) or (4), provided R1 or rx is invariant with concentration ... 

General equation relating the partial molar property to the pure component property and the property change on mixing... 

Finally, the analyses used here to-obtain expressions relating Vi-and Hi to AmixY and AmixM, respectively, are easily generalized, yielding the following for the partial molar property of any extensive function 0 ... 

It is possible to show that the criterion for chemical equilibrium developed here is also applicable to systems subject to constraints other than constant temperature and pressure (Problem 8.4). In fact, Eq. 8.8-1, like the phase equilibrium criterion of Eq. 8.7-9, is of general applicability. Of course, the difficulty that arises in using either of these equations is translating their simple form into a useful prescription for equilibrium calculations by relating the partial molar Gibbs energies to quantities of more direct interest, such as temperature, pressure, and mole fractions. This problem will be the focus of much of the rest of this book. 

Therefore, for a mixture in which the pure component and partial molar volumes are identical [i.e., V-, (T, P.x) =. v,-V,- (T, P) at all conditions], the fugacity of each species in the mixture is equal to its mole fraction times its pure-component fugacity evaluated at the same temperature and pressure as the /hixUire /, [T, P,x) — x fi T, P). However, if. as is generally the case, V-, Vj.-a hen /j and /, are related through the integral o er all pressures of the difference between the species partial molar and pure-component molar volumes. 

Besides (3.4.4), another attribute of partial molar properties, also derived in Appendix A, is that they obey a set of relations known as Gibbs-Duhem equations. For the generic extensive property F(T, P, N ), the general form of the Gibbs-Duhem equation is... 

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... 

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