Partial molar property defined

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 ... 

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. 

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... 

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 ... 

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. 

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

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

In Chapter 5, we defined the partial molar property Z, and described how it could be used to determine the total thermodynamic property through the equation... 

Chapter 4 presents the Third Law, demonstrates its usefulness in generating absolute entropies, and describes its implications and limitations in real systems. Chapter 5 develops the concept of the chemical potential and its importance as a criterion for equilibrium. Partial molar properties are defined and described, and their relationship through the Gibbs-Duhem equation is presented. 

In open systems consisting of several components the thermodynamic properties of each component depend on the overall composition in addition to T and p. Chemical thermodynamics in such systems relies on the partial molar properties of the components. The partial molar Gibbs energy at constantp, Tand rij (eq. 1.77) has been given a special name due to its great importance the chemical potential. The corresponding partial molar enthalpy, entropy and volume under the same conditions are defined as... 

Because most chemical, biological, and geological processes occur at constant temperature and pressure, it is convenient to provide a special name for the partial derivatives of all thermodynamic properties with respect to mole number at constant pressure and temperature. They are called partial molar properties, and they are defined by the relationship... 

They also apply to the partial molar properties Z,-, a quantity that we will define and describe next. For example, it can be shown that... 

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

X, is the molar thermodynamic property of a pure component (adsorbate or adsorbent) and X, is the partial molar property of the component, 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. 

Partial molar properties were defined and discussed briefly in Sec. 11.1. Here show how they are related to one another. Recalling that /i, = Gu we may Eq. (10.2) as... 

This equation defines the partial molar volume of species i in solution. It is simply the volumetric response of the system to the addition at constant T and P of a differential amount of species i A partial molar property may be defined in like fashion for each extensive thermodynamic property. Letting M represent the molar value of such a property, we write the general defining equation for a partial molar property as... 

All three quantities are for the same T, F, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summabihty relation for these properties. 

The constituents of a solution are in fact intimately intermixed, and owi to molecular interactions cannot have private properties of their own. NevertheU they can have assigned property values, and partial molar properties, as define by Eq. (11.2), have all the characteristics of properties of the individual specie as they exist in solution. 

Tliis equation defines the partial molar property of species i in solution, where the generic symbol Mt may standfor the partial molar internal energy t/, the partial molar enthalpy //, the partial molar entropy 5,, the partial molar Gibbs energy G,, etc. It is a response function, representing the change of total prope ity n M due to additionat constant T and f of a differential amount of species i to a finite amount of solution. 

It is frequently more convenient to determine the partial specific properties, defined in terms of grams instead of moles, of the constituents of a solution, and then to multiply the results by the respective molecular weights to yield the partial molar properties. Any of the methods described above may be adapted for this purpose. The value of the property G or of AG per mole is replaced by the value per gram, and n or n, the mole fraction or number of moles, is replaced by the corresponding gram fraction or number of grams, respectively. 

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. 

In general for any extensive property X of a system, we define a partial molar property of component a as... 

Examples of partial molar properties include the partial molar enthalpy Ha, which is defined as... 

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 ... 

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),... 

It is possible to subdivide the properties used to describe a thermodynamic system (e.g., T, P, V,U,...) into two main classes termed intensive and extensive variables. This distinction is quite important since the two classes of variables are often treated in significantly different fashion. For present purposes, extensive properties are defined as those that depend on the mass of the system considered, such as volume and total energy content, indeed all the total system properties (Z) mentioned above. On the other hand, intensive properties do not depend on the mass of the system, an obvious example being density. For example, the density of two grams of water is the same as that of one gram at the same P, T, though the volume is double. Other common intensive variables include temperature, pressure, concentration, viscosity and all molar (Z) and partial molar (Z, defined below) quantities. ... 

Partial molar properties are defined by partial derivatives (equations 2.22,9.7), which does not provide a very easy route to understanding them. There is however a highly intuitive way of thinking about partial molar properties. We will use volume as an example because it is readily visualized, but all relations derived can be used equally well for any other state variable. 

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