Partial molar properties Gibbs energy

Once we have this expression for g , the corresponding activity coefBcients for species a and b are given by the appropriate partial molar excess Gibbs energies via Equation (7.48). Applying the definition of a partial molar property to the excess Gibbs energy, we get ... 

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. 

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. 

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. 

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... 

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... 

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. 

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. 

Gibbs-Duhem Relationship The partial molar properties of a multicomponent phase cannot be varied independently (the mole fractions, jc, = ,/E of the components total unity). For example, for the chemical potentials, /i, the Gibbs-Duhem relationship is En, dni = 0 (for details, see e.g., Atkirs, 1990 Blandamer, 1992 Denbigh, 1971). Similar constraints apply to the partial molar volumes, enthalpies, entropies, and heat capacities. For pure substances, the partial molar property is equal to the molar property. For example, the chemical potential of a pure solid or liquid is its energy per mole. For gaseous, liquid, or solid solutions, X, = X,(ny), that is, the chemical potentials and partial molar volumes of the species depend on the mole fractions. 

Since any partial molar property of a pure substance is simply the corresponding molar property, the chemical potential of a component i in pure form, denoted by p°, is evidently equal to the molar Gibbs free energy G° of pure component i at the same temperature and pressure. 

The chemical potential /z, is a partial molar property of the Gibbs free energy because, as illustrated by the mole number coefficient of G in equation (29-24r/), temperature, pressure, and all other mole numbers are held constant during differentiation with respect to Ni. 

One of the most important consequences of Euler s integral theorem, as applied to stability criteria and phase separation, is the expansion of the extensive Gibbs free energy of mixing for a multicomponent mixture in terms of partial molar properties. This result is employed to analyze chemical stability of a binary mixture. 

Note that the chemical potential G , defined by (3.2.23), has the structure of (3.4.5) that is, the chemical potential is the partial molar Gibbs energy. This is why we use the partial-molar notation for the chemical potential the notation reminds us that the chemical potential has mathematical and physical characteristics in common with other partial molar properties. For example, the integrated form of dG in (3.2.32) is consistent with the mole-fraction average (3.4.4) and the pure-fluid chemical potential (3.2.24) is consistent with (3.4.6) for the molar Gibbs energy. The chemical potential plays a central role in phase equilibria and chemical reaction equilibria therefore, we will need to know how G,- responds to changes of state. 

The most important partial molar property is the chemical potential. It is the partial molar Gibbs free energy and is given by... 

An equation for the activity coefficient can be obtained by fitting an appropriate equation to the experimental data. Rather than fitting each individual activity coefficient to its own function, the preferred procedure is to fit the excess Gibbs energy as a function Xi. The activity coefficients are obtained from this fit by noting from eg. that In is in fact a partial molar property specifically, it is the partial molar... 

The route to an activity coefficient is through an expression for the dimensionless excess Gibbs energy, g /RT, to which In y, is related as a partial molar property ... 

The chemical potential of component i is equal to the partial molar property of the Gibbs energy. 

Our new derivative terms (9G/9n )j.p - look suspiciously like our definition of a partial molar property in Equation (2.1), and indeed they are partial molar Gibbs energies. They allow us to deal with compositional changes, and as such... 

In this chapter, we discuss the various standard states used for the Gibbs energy and the activity. The standard state used for enthalpy, volume and heat capacity is quite different, and is discussed, along with a more detailed look at partial molar properties, in Chapter 10. 

We start with a fairly detailed look at the volumetric properties of solutions, because these are the most intuitive. Partial molar properties of the other state variables are the same in principle, but become more complicated in the case of enthalpy measurements because of its relative nature. The Gibbs energy is also a relative property, but is treated in quite a different way. 

Excess properties, the difference between the property in a real solution and in an ideal solution, are generally expressed as a relative or relative partial molar properties, such as the relative enthalpy, L, or relative partial molar enthalpy, L. The Gibbs energy is treated differently. The fact that Gj-p is a thermodynanoic potential leads naturally to the definition of a relative partial molar Gibbs energy (q. - /a°) which is not the difference from an ideal solution (/A — pL° is not zero even for an ideal solution) but the difference from a standard state, which in this chapter is a pure phase, but may also be some hypothetical state. The form of the equation relating q, - to composition then... 

Keywords Aqueous systems bibliography biochemical systems enthalpy data entropy data equilibrium data excess properties Gibbs energy data heat capacHy data partial molar properties review articles thermochemistry thermodynamics. 

Morton and Beckett, in an appendix to their book, present bond dissociation energies and standard electrode potentials. Brown, in his book, which is mainly concerned with the techniques of microcalorimetry, gives values of A/ff, AGf, S°, and C° (for 25 °C) for many biological substances, and partial molar properties for aqueous solutions. He also presents enthalpies and Gibbs energies of formation of adenosine phosphoric acid specimens and thermodynamic quantities for some reactions in solution. 

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