Free energy partial molar

The equations developed for the partial molar volume and enthalpy can be generalized to all state variables, but given the importance of the Gibbs free energy function, it will be convenient to have some of these equations in their free energy form here. 

First we note that because of this importance, the partial molar free energy is given its own name (chemical potential) and symbol (/it). Equation (9.7) then becomes, when Z becomes G, 

The other difference is that at both ends, the curve for G of the solution is asymptotic to the vertical axes, which is not the case for the volume curve. This means that at Xb = 0, //b = oo. This is a consequence of the functional relationship between fii and Xi to be explored in succeeding chapters. 

Readers will see many instances in the literature of numbers being assigned to /x or to G, e.g. /xa = -1234 Jmol In all such cases, the /x or G referred to is an abbreviation for a A/x or AG, often by considering the free energy of elements to be zero (which, of course, they are not). 

Since Zi is homogeneous in the zeroth degree in ni, n2. etc., Euler s theorem (Chapter 2) gives the Gibbs-Duhem equation  

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If only one mole of the gas is considered, the quantity G is called the chemical potential, /x, or the partial molar free energy. In this case n = 1, and equation 20.197 becomes... 

Equations for the Standard Free Energy of Formation and Partial Molar Free Energies of Atomic Oxygen for Plutonium Oxides (1600-2150 K), cal/mol... 

These expressions comprise the nonideal terms in the previous equations for the chemical potential, Eqs. (30) and (31 ). They may therefore be regarded as the excess relative partial molar free energy, or chemical potential, frequently used in the treatment of solutions of nonelectrolytesi.e, the chemical potential in excess (algebraically) of the ideal contribution, which is —RTV2/M in dilute solutions. 

Mu M2, Ma , Chemical potential (i.e., partial molar free energy) of solvent, polymer (all species collectively), x-mer, and a polymer unit, respectively. 

The terms GA and GB are formally defined as the partial molar free energies of A and B respectively in the solution. 

One must recognize that the partial molar free energy is not a measurable quantity because the absolute value of the free energy is not measurable. However, the relative partial molar free energy, AGa, defined as follows, is measurable... 

Besides the partial molar and the relative partial molar free energies of the components, some other important thermodynamic properties are the partial molar and the relative partial molar enthalpies and entropies. The partial molar enthalpy and entropy of the component A are defined by... 

The relative partial molar free energy of A is written in terms of the integral molar free energy of the solution as... 

The relationships presented thus far for partial, integral and relative partial molar free energies are applicable in a similar manner to entropy, enthalpy and also volume. 

G, = partial molar free energy of Component i (kJkmol-1)... 

Note that the partial molar free energy is used to designate the molar free energy of the individual component as it exists in the mixture. This is necessary because, except in ideal systems, the properties of a mixture are not additive properties of the pure components. As a result, G = JT N, G, for a mixture. The free energy change for reactants and products at standard conditions is given by ... 

The partial molar free energy of A in a given phase, jiA, can be determined from... 

Equations (12) and (13) indicate that an accurate description of the free energy -composition relationship for a particular phase, allows the shift in the deposition potential of each constituent in that phase to be calculated from the respective partial molar free energies. Although such a simple treatment is rather attractive, one should... 

We begin with a discussion of the vapor pressure isotope effect (VPIE). To do so we compare the equilibria between condensed and vapor phase for samples of two isotopomers. At equilibrium, condensed(c) = vapor(v), the partial molar free energies, a(v), and p,(c), of the two phases are equal this, in fact, is the thermodynamic... 

As is customary we select the ideal vapor at unit pressure, P°, as the standard state. The partial molar free energy (chemical potential) of the vapor, p,(v), is... 

B and C are second and third virial coefficients. The partial molar free energy of the vapor phase neglecting higher order terms is thus... 

In Equation 5.5 pa°(v) is the standard state partial molar free energy of the vapor, P is the observed vapor pressure, and BP is a first order correction for vapor nonideality. 

Thermodynamic analysis of VPIE data for separated isotopes (see, e.g., Figs. 7.18a and b) proceeds by equating partial molar free energies in condensed and vapor phases for each isotope. Using Equations 5.1, 5.5 and 5.6 one obtains... 

Here x and x are isotopomer mole fractions in the binary mixture. Remembering x = 1 — x, differentiating to obtain partial molar free energies (and using the thermodynamic relations p,E(V) = AE(V) — x (dAE(V)/dx ) and xE (V) = AE(V) + x (dAE(V)/dx ) one finds expressions for the excess partial molar free energies, xE(V) and il (V). In the high dilution limit, an important case of practical interest, the excess chemical potential of the trace isotopomer, say the unprimed one, is... 

At temperatures well below UCST, solubilities of hydrocarbons in water or water in hydrocarbons drop to very low values. The solutions are very nearly ideal in the Henry s law sense, and the isotope effects on solubility can be directly interpreted as the isotope effect on the standard state partial molar free energy of transfer from the Raoult s law standard state to the Henry s law standard state. Good examples include the aqueous solutions of benzene, cyclohexane, toluene,... 

Concentrations of aqueous electrolyte solutions are conventionally expressed using the aquamolality scale (L = moles salt per 55.508 mol solvent (l,000g for H20)). Some typical solubilities (298.15K) are listed in Table 5.13. Almost all salts are less soluble in D20 than in H20. For those salts whose solubility increases with temperature, which is the ordinary behaviour, the isotope effects decrease with temperature. Writing the standard state partial molar free energy of pure solid salt as Pxsalt) and its standard state in solution as p, (HorD) we have on comparing the saturated solutions in H20 and D20,... 

The integral extends from LH to Ld. Equation 5.47b demonstrates that the solvent solubility IE offers a convenient way to determine the IE on the standard state partial molar free energy for the salt provided the concentration dependence of its activity coefficient in one solvent, most likely H20, is available at high concentration. 

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