Mixture, Molar Gibbs energy

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

The partial molar Gibbs energy of mixing of a component i in a non-ideal mixture can in general be expressed in terms of activity coefficients 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... 

When no distinction between the solvent and the solute in a liquid mixture is made, then nonideal mixtures can still be described by means of an expression similar to Eq. (2.16), but with the addition of a term that is the excess molar Gibbs energy of mixing, AGab (see also section 2.2). Thus the Gibbs energy per mole of mixture is ... 

The best-known example is the partial molar Gibbs energy, better known as the earlier-mentioned thermodynamic potential p. The thermodynamic potential of component i in a homogeneous mixture is... 

Frame 37 considered the thermodynamics of mixing. The total molar Gibbs energy, Gm of a binary liquid mixture, is given (in terms of the amounts, a and nB, of the two components) by equation (37.9), Frame 37 ... 

Table 8 Activities, Activity Coefficients and Excess Partial Molar Gibbs Energies for Agl in Agl + AgBr and Agl + AgCl Mixtures at 600°C... 

In this expression, ° Ga is the standard (° signifies the value s P= atm) molar Gibbs energy of the pure component A and xa is the mole fraction of component A. The Gibbs energy of the mechanical mixture serves as a reference state for the properties of a solution, in which there is chemical mixing between components on an atomic or molecular level. 

The partial molar Gibbs energy or chemical potential of species i in an ideal gas mixture is given by Eq. (4-195), written as... 

The molar Gibbs energy of the mixture, Gmix. equals the sum... 

By Equation (4.121), the partial molar Gibbs energy of component i represents the contribution of i to the total Gibbs energy of the mixture. 

Lowercase roman letters usually denote molar properties of a phase. Thus, g, A. s, and v are the molar Gibbs energy, molar enthalpy, molar entropy, and molar volume. Whan it is essentia] to distinguish between a molar property of a mixture nod that of a pure component, we identify the pure-component property by a subscript. For example, ft, is the molar enthalpy of pure i. Total properties are usually designated by capilal letters, Thus H is the total enthalpy of a mixture it is related to the molar mixture enihelpy A by H nh. where n is the total number of moles in the mixture. 

From these equations we see that the partial molar Gibbs energy assumes special importance in mixtures, as the molar Gibbs energy does in pure fluids. 

Experimental values for some of the partial molar quantities can be obtained from laboratory measurements on mixtures. In particular, mixture density measurements can be used to obtain partial molar volumes, and heat-of-mixing data yield information on partial molar enthalpies. Both of these measurements are considered here. In Chapter 10 phase equilibrium measurements that provide information on the partial molar Gibbs energy of a component in a mixture are discussed. Once the partial molar enthalpy and partial molar Gibbs energy are known at the anie temperature, the partial molar entropy can be computed from the relation S-, = (G — H-,)/T. 

Therefore, at equilibrium, the fugacity of each species must be the same in the two phases. Since this result follows directly from Eq. 8.7-10, it may be substituted for it. Furthermore, since we can make estimates for the fugacity of a species in a mixture in a more direct fashion than for partial molar Gibbs energies, it is more convenient to use Eq. 9.2-11 as the basis for phase equilibrium calculations. 

The difficulty with this description is that Ga and Gb are not separately measurable, because, as a result of Eq. 9.10-3, it is not possible to vary the number of moles of cations holding the number of moles of anions fixed, or vice versa. (Even in mixed electrolyte solutions, that is, solutions of several electrolytes, the condition of overall electrical neutrality makes it impossible to vary the number of only one ionic species.) To maintain the present thermodynamic description of mixtures and, in particular, the concept of the partial molar Gibbs energy, we instead consider a single electrolyte solu-,tion to be a three-component system solvent, undissociated electrolyte, and dissociated electrolyte. Letting Nab,d be the moles of dissociated electrolyte, we then have... 

Figure 11.2-5 The molar Gibbs energy of ideal (A = 0) and nonideal (A 0) binary mixtures if phase separation does not occur (solid line) and when phase separation occurs (dashed line).

Since the system is dosed and nonreacting, the number of moles of each sp>ecies is fixed. Therefore, in the analysis here we will consider the molar Gibbs energy of the mixture G, rather than C. 

The use of this equation is more complicated than using the energy balance because the partial molar Gibbs energy contains both the activity coefficient and the mole fraction of each component (which in the case of biomass is really a complex mixture) that is. 

Chemical potential - For a mixture of substances, the chemical potential of constituent B is defined as the partial derivative of the Gibbs energy G with respect to the amount (number of moles) of B, with temperature, pressure, and amounts of all other constituents held constant. Also called partial molar Gibbs energy. [2]... 

By Eq. (11.10), the p of a pure substance is simply the molar Gibbs energy for this reason the symbol p was introduced for molar Gibbs energy in Section 10.8. In mixtures, pj is the partial molar Gibbs energy of the substance i. 

In Eq. (11.14a) we interpret Pi(T,p) as the chemical potential of the pure species i in the same state of aggregation as the mixture that is, in a liquid mixture, Pi(T, p) is the chemical potential, or molar Gibbs energy, of pure liquid i at temperature T and pressure p, and Xi is the mole fraction of i in the liquid mixture. We will introduce particular empirical evidence to justify this generalization in Chapter 13. 

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