Mixture partial molar properties

Fugacity coefficient of species i in a mixture Partial molar property of species i Solubility parameter of species i Volume fraction of species i in a mixture Surface area fraction of species i in a mixture Mole fraction of functional group m in a mixture Volume fraction of functional group m in a mixture Surface area fraction of functional group m in a mixture Henry s law constant of species i in a mixture (kPa)... 

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 ... 

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. 

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. 

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. 

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... 

Partial molar availability, 24 692 Partial molar entropy, of an ideal gas mixture, 24 673—674 Partial molar Gibbs energy, 24 672, 678 Partial molar properties, of mixtures, 24 667-668... 

As we will see, partial molar properties are of general application in the thermodynamics of mixtures and solutions. 

Figure 6J Excess partial molar properties of jadeite component in NaAlSi206-KAlSi206 molten mixture. (A) Excess partial molar entropy. (B) Excess partial molar enthalpy. Reprinted from Eraser et al. (1983), Bulletin Mineralogique, 106, 111-117, with permission from Masson S.A., Paris, Erance.

We will follow the IUPAC recommendation that surface properties per unit surface area be represented by the lower case (g = Gibbs free energy, u = energy, h = enthalpy, etc.) with a superscript.s designating that the property is for the surface. The quantities gs,us,hs... for the surface are in many ways comparable to molar properties (or partial molar properties for mixtures) in the bulk phase. 

It has been said chemists have solutions 3 Solutions are involved in so many chemical processes1 that we must have the mathematical tools to comfortably work with them, and thermodynamics provides many of these tools. Thermodynamic properties such as the chemical potential, partial molar properties, fugacities, and activities, provide the keys to unlock the description of mixtures. 

Partial molar properties take a special place in the thermodynamics of mixtures and phase equilibria. They are defined as... 

Equation (11.5) implies that a molar solution property is given as a sum its parts and that Mi is the molar property of species i as it exists in solutio This is a proper interpretation provided one understands that the defining equati for Mit Eq. (11.2), is an apportioning formula which arbitrarily assigns to eac species i a share of the mixture property, subject to the constraint of Eq. (11.5), The constituents of a solution are in fact intimately intermixed, and ov to molecular interactions cannot have private properties of their own. Neverthel they can have assigned property values, and partial molar properties, as defin by Eq. (11.2), have all the characteristics of properties of the individual speci as they exist in solution. 

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... 

They are used as industrial solvents for small- and large-scale separation processes, and they have unusual thermodynamic properties, which depend in a complicated manner on composition, pressure, and temperature for example, the excess molar enthalpy (fp-) of ethanol + water mixture against concentration exhibits three extrema in its dependence on composition at 333.15 K and 0.4 MPa. The thermodynamic behavior of these systems is particularly intricate in the water-rich region, as illustrated by the dependencies of the molar heat capacity and partial molar volume on composition. This sensitivity of the partial molar properties indicates that structural changes occur in the water-rich region of these mixtures. Of course, the unique structural properties of water are responsible for this behavior. ... 

Method of Intercepts.— The method of intercepts is useful in many instances, especially as it gives simultaneously the partial molar properties of both constituents of a binary mixture of any composition. Let a represent the mean value of a particular extensive property per mole of mixture, so that the observed value of the property O for the system is given by... 

The values of the mean molar property a for mixtures of various A compositions are plotted against 33 the mole fraction Ni, as shown in Fig. 33. Let 0 be the point at wUch the partial molar property is to be determined at 0 draw the tangent CD and the horizontal line EF, parallel to the base line AB. The slope of -CD is da/dNi, and so CE is equal to Ni(da/dNi) at 0. Since AE is the value of o at that point, it is evident from equation (42.9) that the distance AC gives the partial molar property 0 . In an exactly similar manner it can be shown that BD is equid to 0i for the mixture whose... 

For a general system, however, the volume, as well as other properties, is not additive. That is, the volume of a mixture is not equal to the sum of the volumes of the individual pure components. In this situation, it is not clear how to assign how much volume is occupied of each species. One logical manner to do this is through the use of partial molar properties. 

In this Chapter, we define partial molar properties and describe their application. We then discuss their relationship with the change of properties of a system on mixing. Finally, we examine the graphical representation of partial molar properties for binary mixtures. 

The value of any extensive property of a system is equal to the sum of the partial molar properties of each component multiplied by the amount of each component in the system. Therefore, we can divide the property of a mixture, such as the volume or enthalpy, between its individual components according to their partial molar properties. 

The relations derived in this section provide us a means to easily extract the partial molar properties of a system from a graph of the corresponding molar property with repect to composition. A generic plot of the variation of a molar property of a binary mixture with composition is given by the solid line in Fig. 5.1. When. 2 = 0, the system consists only of component 1, and the value of the molar property should be equal to the molar property of pure component 1 Xf. Likewise, when = 1, the system consists only of component 2, and tine value of X should be equal to X. The variation of a molar property of an unmixed system composed of pure 1 and pure 2 is given by the thick dashed line in Fig. 5.1. 

Most solutions do not exhibit ideal behavior, and the actual curve corresponding to the variation of the molar volume or enthalpy of the mixture deviates from a straight line (e.g., the solid line in Fig. 5.1). When the curve for the molar volume lies above the ideal mixture line, the system expands upon mixing when the curve lies below the line, the system contracts. In the case of the molar enthalpy, a curve that lies above the ideal mixture line corresponds to the system that absorbs heat (e.g., mixing lead bromide and water) a curve that lies below the line corresponds to the system releasing heat (e.g., mixing sulfuric acid and water). This non-ideal mixing in the case of the molar enthalpy is the principle used in cold packs and heat packs. We will develop mathematical models to describe non-ideal mixtures. We use partial molar properties in more detail later. 

Since it is unlikely that enough information will be available for any mixing property to obtain A6>mix as an explicit function of mole fractions or species mole numbers for ternary, quaternary, etc. mixtures, it is not surprising that there is little information on partial molar properties in such systems. 

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 ... 

That is, a molar property of a mixture is the mole-fraction-weighted sum of its constituent partial molar properties. The partial molar property m of species I in solution becomes equal to molar properly m, of pure 1 in the appropriate limit ... 

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