Property molar

A useful feature of the partial molar properties is that the property of a mixture (subscript mix) can be written as the sum of the mole-weighted contributions of the partial molar properties of the components ... 

The systems of interest in chemical technology are usually comprised of fluids not appreciably influenced by surface, gravitational, electrical, or magnetic effects. For such homogeneous fluids, molar or specific volume, V, is observed to be a function of temperature, T, pressure, P, and composition. This observation leads to the basic postulate that macroscopic properties of homogeneous PPIT systems at internal equiUbrium can be expressed as functions of temperature, pressure, and composition only. Thus the internal energy and the entropy are functions of temperature, pressure, and composition. These molar or unit mass properties, represented by the symbols U, and S, are independent of system size and are intensive. Total system properties, J and S do depend on system size and are extensive. Thus, if the system contains n moles of fluid, = nAf, where Af is a molar property. Temperature... 

The derivatives in the summation are partial molar properties, denoted by Tf/ thus... 

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

Equation 163, written as = G- /-RT, clearly shows that In ( ) " is a partial molar property with respect to G /KT. MultipHcation of equation 175 by n and differentiation with respect to at constant T, P, and in accord with equation 116 yields, after reduction, equation 179 (constant T,x), where is the partial molar compressibiUty factor. This equation is the partial-property analogue of equation 178. 

Foi an ideal solution, G, = 0, and tlieiefoie 7 = 1- Compatison shows that equation 203 relates to exactiy as equation 163 relates ( ) to GG Moreover, just as ( ) is a partial property with respect to G /E.T, so In y is a partial property with respect to G /RT. Equation 116, the defining equation for a partial molar property, in this case becomes equation 204 ... 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

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. 

In some cases, reported data do not satisfy a consistency check, but these may be the only available data. In that case, it may be possible to smooth the data in order to obtain a set of partial molar quantities that is thermodynamically consistent. The procedure is simply to reconstmct the total molar property by a weighted mole fraction average of the n measured partial molar values and then recalculate normalised partial molar quantities. The new set should always be consistent. 

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. 

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 ... 

Pertinent examples on partial molar properties are presented in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed.. Sec. 10.3, McGraw-Hill, NewYonc, 1996). Gibbs/Duhem Equation Differentiation of Eq. (4-50) yields... 

Equation (4-47), which defines a partial molar property, provides a general means by which partial property values may be determined. However, for a.hinary solution an alternative method is useful. Equation (4-50) for a binaiy solution is... 

The partial molar property, other than the volume, of a constituent species in an ideal gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature hut at a pressure equal to its partial pressure in the mixture. 

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 ... 

Before describing these thermodynamic variables, we must talk about their properties. The variables are classified as intensive or extensive. Extensive variables depend upon the amount while intensive variables do not. Density is an example of an intensive variable. The density of an ice crystal in an iceberg is the same as the density of the entire iceberg. Volume, on the other hand, is an extensive variable. The volume of the ocean is very different from the volume of a drop of sea water. When we talk about an extensive thermodynamic variable Z we must be careful to specify the amount. This is usually done in terms of the molar property Zm, defined as... 

The third observation is that the equations apply as well when the extensive properties are replaced by the corresponding intensive molar properties.0 For... 

The relationships summarized in Table 3.1. expanded to include differences and molar properties, serve as the starting point for many useful thermodynamic calculations. An example is the calculation of AZ for a variety of processes in which p, V, and T are changed.e For any of the extensive variables Z = S, U, H, A or G, we can write... 

Hence, for a pure substance, the chemical potential is a measure of its molar Gibbs free energy. We next want to describe the chemical potential for a component in a mixture, but to do so, we first need to define and describe a quantity known as a partial molar property. 

To understand the significance of the partial molar property, we start with... 

Equation (5.16) can be integrated. We expect the partial molar properties to be functions of composition, and of temperature and pressure. For a system at constant temperature and constant pressure, the partial molar properties would be functions only of composition. We will start with an infinitesimal quantity of material, with the composition fixed by the initial amounts of each component present, and then increase the amounts of each component but always in that same fixed ratio so that the composition stays constant. When we do this. Z, stays constant, and the integration of equation... 

By either a direct integration in which Z is held constant, or by using Euler s theorem, we have accomplished the integration of equation (5.16), and are now prepared to understand the physical significance of the partial molar property. For a one-component system, Z = nZ, , where Zm is the molar property. Thus, Zm is the contribution to Z for a mole of substance, and the total Z is the molar Zm multiplied by the number of moles. For a two-component system, equation (5.17) gives... 

In summary, we have defined a partial molar property Z,- as... 

First, we note that all of the thermodynamic equations that we have derived for the total extensive variables apply to the partial molar properties. Thus, if... 

A similar proof can be used for applying any of our thermodynamic equations to partial molar properties. For example, if... 

Before leaving our discussion of partial molar properties, we want to emphasize that only the partial molar Gibbs free energy is equal to n,-. The chemical potential can be written as (cM/<9 ,)rv or (dH/dnj)s p H partial molar quantities for fi, into equations such as those given above. 

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... 

Obtaining partial molar properties involves the determination of the derivative... 

A method for determining partial molar properties, most often applied to electrolyte solutions, involves using the apparent molar property [Pg.222]

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