Binary solution, chemical potential

Hence, the derivative of the solute chemical potential (or activity) with respect to solute concentration can be expressed in terms of a combination of number densities and particle number fluctuations or KBIs. The ability to express thermodynamic properties in terms of KBIs is the major strength of FST. This has been achieved without approximation and the relationship holds for any stable binary solution at any composition involving any type of components. Derivatives of other chemical potentials can be obtained by application of the GD equation, or by a simple interchange of indices. The same approach can be applied to the second expression in Equation 1.48, with a subsequent application of Equation 1.27, to provide chemical potential derivatives with respect to other concentration scales. 

In such a binary solution, the chemical potential of the solute and that of the solvent A/xg are related to the integral free energy of formation of the solution, AG per mole, containing a mole fraction Xp, of component A, and for component B, by the expression... 

Fig. 120.—The chemical potential of the solvent in a binary solution containing polymer at low concentrations vi). Curves have been calculated according to Eq. (XII-26) for a = 1000 and the values of dicated with each curve. ...

These simple expressions may also be obtained from the chemical potentials according to Eqs. (XII-26) and (XII-32) by appropriately changing subscripts and recalling that x in these equations represents the ratio of the molar volumes, which in the present case is unity. Owing to the identity of volume fractions with mole fractions in this case, Eqs. (18) and (19) are none other than the chemical potentials for a regular binary solution in which the heat of dilution can be expressed in the van Laar form. The critical conditions (see Eqs. 2)... 

Equations (4.7) and (4.8) may be solved numerically or graphically. The latter approach is illustrated in Figure 4.2 by using the Gibbs energy curves for the liquid and solid solutions of the binary system Si-Ge as an example. The chemical potentials of the two components of the solutions are given by eqs. (3.79) and (3.80) as... 

Fig. 54. Specific refractive index increments at constant composition (o) and constant chemical potential ( ) for solutions of nylon-6 in 2,2,3,3-tetrafluoropropanol/l-chlorophenol binary mixtures, is the volume fraction of l-chlorophenol and filled circles refer to the two pure single solvents161)...

Conway et described the most convenient conditions defining standard chemical potentials for adsorption with solvent displacement. First, for n = 1, the conditions are the same as in binary solution thermo-... 

The second law of thermodynamics dictates that ct is positive when Vp2 0 therefore, I>0. Hence, based on Equation Al-15, diffusion in binary solutions is always down the chemical potential (or activity) gradient. Comparing J2 in Equation Al-14 with Pick s law and assuming constant molar density p, we have... 

Diffusion in a binary system may also be determined by measurement of the intradiffusion coefficient (sometimes referred to as the self-diffusion coefficient), D. In the case of intradiffusion, no net flux of the bulk diffusant occurs the molecules undergo an exchange process. Measurements are usually carried out by using trace amounts of labelled components in a system free of any gradients in the chemical potential. The molecular movement of the solute is governed by frictional interactions between labelled solute and solvent, and labelled solute and unlabelled solute. 

For a binary polymer solution, the reciprocal of the osmotic compressibility 0n/0c at constant T and the solvent chemical potential p0 can be determined by sedimentation equilibrium through the relation [58,59] ... 

When = 0, then = 1 and the chemical potential Ac is that of the binary solid AC(s), Ac- This, of course, is also the Gibbs energy of AC(s). Since the restriction that be confined to values near unity is intended to apply to the end members of the solid solution, Ac as well as Ac are independent of also. Similarly, when = 1, then = 1 and Ac equals the Gibbs energy of BC(s), Ac-... 

Here 113 is completely defined by the variables Z and N3. For any given values of Z and N3, the reference of the activity coefficient will be chosen as the extremely dilute state (N3 = 0) of the given solute in a binary mixed solvent of the same composition Z. By the definition, the chemical potential of the reference state varies with Z. Hence, one obtains for the 1-1 salts... 

Grunwald and Bacarella (16) have shown that the rate of change of the standard chemical potential Go of a solute with water mole fraction Z in a binary solvent mixture can be expressed by the relation ... 

The attentive student may be concerned that the concept of ideal solution as introduced in Chapter 6 is possibly inconsistent with the usage of that term as defined in (7.45a-c). However, Sidebar 7.8 demonstrates that these definitions of solution ideality are in fact consistent. We may therefore regard chemical potential-based definitions of binary solution ideality [cf. (6.57)],... 

Defect thermodynamics is more complicated when applied to binary (or multi-component) compound crystals. For binary systems, there is one more independent thermodynamic variable to control. In the case of extended binary solid solutions, one would normally choose a composition variable for this purpose. For compounds with very narrow ranges of homogeneity (i.e., point defect concentrations), however, the composition is obviously not a convenient variable. The more natural choice is the chemical potential of one of the two components of the compound crystal. In practice one will often use the vapor pressure ( activity) of this component. 

The thermodynamic quantity 0y is a reduced standard-state chemical potential difference and is a function only of T, P, and the choice of standard state. The principal temperature dependence of the liquidus and solidus surfaces is contained in 0 j. The term is the ratio of the deviation from ideal-solution behavior in the liquid phase to that in the solid phase. This term is consistent with the notion that only the difference between the values of the Gibbs energy for the solid and liquid phases determines which equilibrium phases are present. Expressions for the limits of the quaternary phase diagram are easily obtained (e.g., for a ternary AJB C system, y = 1 and xD = 0 for a pseudobinary section, y = 1, xD = 0, and xc = 1/2 and for a binary AC system, x = y = xAC = 1 and xB = xD = 0). 

The chemical potential pB of species B in binary solution is found by the standard thermodynamic formula... 

The chemical potential of species B in a binary gas mixture above the solution is... 

These equations are used whenever we need an expression for the chemical potential of a strong electrolyte in solution. We have based the development only on a binary system. The equations are exactly the same when several strong electrolytes are present as solutes. In such cases the chemical potential of a given solute is a function of the molalities of all solutes through the mean activity coefficients. In general the reference state is defined as the solution in which the molality of all solutes is infinitesimally small. In special cases a mixed solvent consisting of the pure solvent and one or more solutes at a fixed molality may be used. The reference state in such cases is the infinitely dilute solution of all solutes except those whose concentrations are kept constant. Again, when two or more substances, pure or mixed, may be considered as solvents, a choice of solvent must be made and clearly stated. 

The methods for obtaining expressions for the chemical potential of a component that is a weak electrolyte in solution are the same as those used for strong electrolutes. For illustration we choose a binary system whose components are a weak electrolyte represented by the formula M2A and the solvent. We assume that the species are M +, MA , A2-, and M2A. We further assume that the species are in equilibrium with each other according to... 

A) When only one of the two components in a binary solution is volatile, the excess chemical potential of the volatile component can be determined by the methods that have been discussed. However, we require the values of the excess chemical potential of the other component or of the molar... 

The measurement of osmotic pressure and the determination of the excess chemical potential of a component by means of such measurements is representative of a system in which certain restrictions are applied. In this case the system is separated into two parts by means of a diathermic, rigid membrane that is permeable to only one of the components. For the purpose of discussion we consider the case in which the pure solvent is one phase and a binary solution is the other phase. The membrane is permeable only to the solvent. When a solute is added to a solvent at constant temperature and pressure, the chemical potential of the solvent is decreased. The pure solvent would then diffuse into such a solution when the two phases are separated by the semipermeable membrane but are at the same temperature and pressure. The chemical potential of the solvent in the solution can be... 

Here Ap [p = 1, excess chemical potential of the second component in the binary system composed of the second and third components at the composition equal to q. The reference state of the second component may be either the pure component or the infinitely dilute solution of the second component in the third component. In the limit of p=l, q/p becomes l/(k+ 1), where k = x2/x3 and, with the proper choice of components or... 

We consider only binary solutions in this discussion. The standard states of the two components are defined as the pure components, and the chemical potentials of the components are based on the molecular mass of the monomeric species. We designate the components by the subscripts 1 and 2 and the monomeric species by the subscripts A1 and B1, respectively. From the discussion given in Section 8.15 we know that the chemical potential of a substance considered in terms of the species present in a solution must be... 

Thorstenson and Plummer (1977), in an elegant theoretical discussion (see section on The Fundamental Problems), discussed the equilibrium criteria applicable to a system composed of a two-component solid that is a member of a binary solid solution and an aqueous phase, depending on whether the solid reacts with fixed or variable composition. Because of kinetic restrictions, a solid may react with a fixed composition, even though it is a member of a continuous solid solution. Thorstenson and Plummer refer to equilibrium between such a solid and an aqueous phase as stoichiometric saturation. Because the solid reacts with fixed composition (reacts congruently), the chemical potentials of individual components cannot be equated between phases the solid reacts thermodynamically as a one-component phase. The variance of the system is reduced from two to one and, according to Thorstenson and Plummer, the only equilibrium constraint is IAP g. calcite = Keq(x>- where Keq(x) is the equilibrium constant for the solid, a function of... 

In a binary solution, the Gibbs-Duhem relation [Eq. (15)] determines the variation of a partial molar property of one component in terms of the variation of the partial molar quantity of the other component. This relation is useful for obtaining chemical potentials in binary solutions when only one of the components has a measurable vapor pressure. Applying Eq. (15) to chemical potentials in a binary solution,... 

In contrast to a perfect solution, a solution is called an ideal solution, if Eq. 8.1 is valid for solute substances in the range of dilute concentrations only. Moreover, the unitary chemical potential p2(T,p) of solute substance 2 is not the same as the chemical potential p2( T,p) of solute 2 in the pure substance p2(T,p) p2(T,p) Henry s law. For the main constituent solvent, on the other hand, the unitary chemical potential p[( T,p) is normally set to be equal to f l p) in the ideal dilute solution p"(T,p) = p°(l p). The free enthalpy per mole of an ideal binary solution of solvent 1 and solute 2 is thus given by Eq. 8.10 ... 

As mentioned in section 8.1, the value of the unitary chemical potential pi depends on the choice of the reference system. There are two reference systems which are commonly used one is unsymmetrical and the other is symmetrical. In discussing the reference systems we shall for convenience limit ourselves to a binary solution. 

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