Partial molar properties Gibbs-Duhem equation

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. 

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 (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... 

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. 

The same type of polynomial formalism may also be applied to the partial molar enthalpy and entropy of the solute and converted into integral thermodynamic properties through use of the Gibbs-Duhem equation see Section 3.5. 

Equations for partial molar properties (constant composition) Gibbs-Duhem equations ... 

A review of chemical thermodynamics, especially as it relates to the properties of liquid solutions, has also been presented. Partial molar quantities such as the chemical potential are an important feature of the treatment of this subject. It is often the case that the activity and chemical potential of one quantity is relatively easy to determine directly by experiment, whereas that of another component is not. Under these circumstances, the change in chemical potential of one component can be related to that of another through the Gibbs-Duhem equation. This relationship and its use in estimating thermodynamic properties are extremely important in solution chemistry. 

Thus, the interrelationships provided by Eqs. 8.2-8 through 8.2-15 are really restrictions on the mixture equation of state. As such, these equations are important in minimizing the amount of experimental data necessary in evaluating the thermodynamic, properties of mixtures, in simplifying the description of multicomponent systems, and in testing the consistency of certain types of experimental data (see Chapter 10). Later in this chapter we show how the equations of change for mixtures and the Gibbs-Duhem equations provide a basis for the experimental determination of partial molar properties. 

Which is called the rsf Gibbs-Duhem equation. It shows that any extensive property g, for instance volume, heat capacity or energy of the solution, may be determined from its composition, if corresponding median partial molar values of its components are known. For instance, if the mole amount of individual components in the composition of a solution and their partial molar volumes are known, then the volume of the entire solution will be equal to the sum of their products. 

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... 

As a partial molar property, the chemical potential obeys the Gibbs-Duhem equation. This equation was derived inChapter o (see eg. Q.i2 ). Applied to the chemical potential it gives... 

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... 

At constant T and P, the Gibbs-Duhem equation simplifies to Si=i 0. Analogous to chemical potential are partial molar properties such as V,-, Hi, and S. Show that for all partial molar quantities such as V , Hi, and Si, at constant Tand Pone can also write... 

Equation 6.22 is true for any partial molar property, hi, st, and so on. Equation 6.22 has no common name when we write it for G, we find the Gibbs-Duhem equation ... 

The partial molar equation shows a unique and important relation between the partial molar properties in a mixture. When the differential of the partial molar equation is applied to the Gibbs energy, the result is the Gibbs-Duhem equation, which we will use in Chapter 9. 

The Gibbs-Duhem equation shows that as the cocenn-tration of one species of a mixture approaches zero, its chemical potential approaches minus infinity. This makes the chemical potential (partial molar Gibbs energy) an inconvenient working property for equilibrium calculations. For this reason we use the fugacity (Chapter 7) instead. 

This expression represents the Gibbs-Duhem equation and indicates that the intensive properties of the mixture temperature, pressure and partial molar properties, cannot vary independently. Restricted to constant T and P, Eq. 11.6.3 becomes ... 

We have seen in Chapter 11 that all partial molar properties must satisfy the Gibbs-Duhem equation, Eq. 11.6.3, which when applied to the molar excess Gibbs free energy yields (Problem 13.52) ... 

Define a partial molar property and describe its role in determining the properties of mixtures. Calculate the value of a partial molar property for a species in a mixture from analytical and graphical methods. Apply the Gibbs-Duhem equation to relate the partial molar properties of different species. 

The Gibbs—Duhem equation provides a very useful relationship between the partial molar properties of different species in a mixture. It results from mathematical manipulation of property relations. The approach is similar to that used in Chapter 5 to develop relationships between properties. The reason the Gibbs-Duhem equation is so useful is that it provides constraints between the partial molar properties of different species in a mixture. For example, in a binary mixture, if we know the values for a partial molar... 

To see the usefulness of the Gibbs-Duhem equation, lets examine the scenario where we wish to find the partial molar volume of species h in a binary solution when we know the partial molar volume of species a, Va, as a function of composition. If we apply Equation (6.19) to the property volume, we get ... 

Thus, if we have an expression for (or plot of) the partial molar volume of species a vs. mole fraction, we can apply this equation to get the corresponding expression for species b. The expressions for partial molar properties are not independent but rather constrained by the Gibbs-Duhem equation. Such an interelation makes sense from a molecular perspective. The partial molar properties are governed by how a species behaves in the mixture. We expect the partial molar properties of a and b to be related since it is the same... 

Use of the Gibbs-Duhem Equation to Relate Partial Molar Properties... 

Additionally, the pure species property, ki, is defined as the value of that property of species i as it exists as a pure species at the same T and P of the mixture. Values of a partial molar property for a species in a mixture can be calculated from an analytical expression by applying Equation (6.15) and by graphical methods, as illustrated in Figure 6.13. In the case of infinite dilution, species i becomes so dilute that a molecule of species i will not have any like species with which it interacts rather, it will interact only with unlike species. Additionally, partial molar properties of different species in a mixture can be related to one another by the Gibbs-Duhem equation ... 

In Section 6.3, we used the Gibbs—Duhem equation to provide a relationship between the partial molar properties of different species in a mixture. We can use this equation to relate the activity coefficients of different species in a mixture as well. We begin by writing Equation (6.19) in terms of partial molar Gibbs energy, that is, chemical potential ... 

The Gibbs-Duhem equation provides a general relation for the partial molar properties of different species in a mixture that must always be true. For example, we just saw how the activity coefficient of different species can be related to one another. In this section, we explore one way to use this interrelation to judge the quality of experimental data. The basic idea is to develop a way to see whether a set of data conform to the constraints posed by the Gibbs-Duhem equation. If the data reasonably match, we say they are thermodynamically consistent. On the other hand, data that do not conform to the Gibbs-Duhem equation are thermodynamically inconsistent and should be considered unreliable. The development that follows is based on the relation between activity coefficients in a binary mixture of species a and b. It serves as an example to this methodology there are several other ways that have been developed to apply this same type of idea. 

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