Partial molar quantities definition

From the definition of the partial molar quantities [Eq. (8.8)] we write... 

The Gibbs-Duhem equation also follows from the definition of partial molar quantities nid/Hi + r 2d 2 0. With the Gibbs-Duhem equation, d G/dc2 becomes... 

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. 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

From the definition of partial molar quantities [Equation (9.12)], Equation (9.26) can be written as... 

Any physical interpretation of a partial molar quantity must be consistent with its definition. It is simply the change of the property of the phase with a change of the number of moles of one component keeping the mole numbers of all the other components constant, in addition to the temperature and pressure. It is a property of the phase and not of the particular component. One physical concept of a partial molar quantities may be obtained by considering an infinite quantity of the phase. Then, the finite change of the property on the addition of 1 mole of the particular component of this infinite quantity of solution at constant temperature and pressure is numerically equal to the partial molar value of the property with respect to the component. 

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 shows that the chemical potential of a component is just its partial molar Gibbs free energy. Note that the definitions of the chemical potential in terms of other thermodynamic variables, given in Chapter 6, Eq. (8), are not partial molar quantities because pressure and temperature are not the variables held constant in these derivatives. 

The measurement of partial molar quantities will be illustrated with reference to partial molar volumes. We can measure absolute volumes of solution and, thus, can determine partial molar volumes directly from its definition ... 

Starting from the definition 5.22 we now establish several important properties of thermodynamic potentials (partial molar quantities of thermodynamic energy functions) for an ideal system of mixture. Differentiating G-H-TS with respect to n, with Tand p constant, we have pt = ht- Tsl and furthermore [d(jWf IT) / dT pn = (1 IT) (dp, / dT) - (p, / T1) = - [(r s, + pt) / T2] = -h,l T2. From this equation we obtain Eq. 5.34 for the partial molar enthalpy hf of a constituent i in an ideal mixture ... 

In an earlier section the free energy of a phase and the free energy of a total system were discussed generally in terms of the potentials (e.g., equation 48). With the definition of the chemical potential as a function of activity in hand, we will now consider the Gibbs energy of a system. In a similar fashion, the enthalpy and entropy of a system can be computed using the partial molar quantities and the mole numbers of each phase. 

Starting from the definition (7.1) we shall now establish several important properties of partial molar quantities for an ideal system. Taking first the partial molar enthalpy, we have, from (6.32),... 

Determination of Partial Molar Quantities Ir Direct Method. —In view of the definition of the partial molar property Qi as... 

Using the definitions of mean molar and partial molar quantities, Eqn (3.14) becomes... 

Because of the simplicity of the functions of state of the ideal gas, they serve well as models for other mixing experiments. Dilute solutions, for example, can be modeled as ideal gases with the empty space between the gas atoms being filled with a second component, the solvent. In this case, the ideal condition can be maintained as long as the overall interaction between solvent and solute is negligible. Deviations from the ideal mixing are treated by evaluation of the partial molar quantities, as illustrated on the example of volume, V, in Fig. 2.25. The first row of equations gives the definitions of the partial molar volumes and Vg and shows the addition... 

The equations that are commonly used to represent experimental data of (Z = G, y ) and p, are expressed as a function of Xj, whereas in Equation 4.14 derivatives with respect to are required. We need therefore to express than in a fnnction of X,. Taking into account the definition of the excess partial molar quantity, 7, as a function of the relationship between x, and the differentials of 7F- =f(X with respect to x, and of X with respect to and applying the treatment to one mole of mixture, after some substitutions and rearrangements, the diagonal elements p, can be expressed in a function of X and of four derivatives of the chemical potential of components 1 and 2,... 

Equations (67a)-(67d) show the particular role of partial molar Gibbs energies Gi. By their definition, quantities G/ are both partial molar quantities and chemical potentials as defined by the fundamental equation of thermodynamics ... 

In order to calculate the equOibrium composition of a system consisting of one or more phases in equilibrium with an aqueous solution of electrolytes, a review of the basic thermodynamic functions and the conditions of equilibrium is important, This is particularly true inasmuch as the study of aqueous solutions requires consideration of chemical and/or ionic reactions in the aqueous phase as well as a thermodynamic framework which is, for the most part, quite different from those definitions associated with nonelectrolytes. Therefore, in this section we will review the definition of the basic thermodynamic functions, the partial molar quantities, chemical potentials, conditions of equilibrium, activities, activity coefficients, standard states, and composition scales encountered in describing aqueous solutions. 

The thermodynamic analysis of solutions is facilitated by the introduction of quantities that measure how the extensive thermodynamic quantities (V, E, H, G,. ..) of the system depend on the state variables T, P, and nj. This leads to the definition of partial molar quantities where, if we let Y be any extensive thermodynamic property, we can define the partial molar value of Y for the ith component as ... 

The important Gibbs-Duhem relation between the partial molar quantities [5] is obtained by combination of the definition of Q in differential form,... 

A chemical potential is an intensive partial molar quantity. Its value does not depend on the sample mass since, by definition, a molar quantity concerns one mole. The chemical potential is also called the partial molar free enthalpy G, ... 

We may introduce similar definitions for partial molar quantities like the chemical potentials or partial molar volumes. For example... 

We divide by Avogadro s number to convert the partial molar Gibbs free energy to a molecular quantity, and the minus sign enters because the force and the gradient are in opposing directions. Recalling the definition of chemical potential [Eq. (8.13)], we write jUj + RT In aj = ii2 + RT In 7jC, where aj... 

The relative partial molar enthalpies of the species are obtained by using Eqs. (70) and (75) in Eq. (41). When the interaction coefficients linear functions of T as assumed here, these enthalpies can be written down directly from Eq. (70) since the partial derivatives defining them in Eq. (41) are all taken at constant values for the species mole fractions. Since the concept of excess quantities measures a quantity for a solution relative to its value in an ideal solution, all nonzero enthalpy quantities are excess. The total enthalpy of mixing is then the same as the excess enthalpy of mixing and a relative partial molar enthalpy is the same as the excess relative partial molar enthalpy. Therefore for brevity the adjective excess is not used here in connection with enthalpy quantities. By definition the relation between the relative partial molar entropy of species j, Sj, and the excess relative partial molar entropy sj is... 

Relations (4.283)-(4.286) in uniform mixture permit to express the partial specific thermodynamic quantities from extensive (4.283) as ([59], i.e. as an analogue of the molar classical definition [138, 141])... 

The derivative operator appearing in (3.4.5) is called the partial molar derivative, and the quantity F,- defined by (3.4.5) is called the partial molar F for component i. It is the partial molar property that can always be mole-fraction averaged to obtain the mixture property F. Note, however, that F is itself a property of the mixture, not a property of pure i partial molar properties depend on temperature, pressure, and composition. We emphasize that the definition (3.4.5) demands that F be extensive and that the properties held fixed can only be temperature, pressure, and all other mole numbers except N,. Partial molar properties are intensive state functions they may be either measurable or conceptual depending on the identity of F. 

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