Chemical Potential in Homogeneous Mixtures

To provide an example, we will now look at how the chemical potential fi of water depends upon its mole fraction x in various mixtures (Fig. 13.1). 

On the right, all curves show the same slope RT in the vicinity of x= 1. On the left, they all approach negative infinity. The similar initial part of the curves on the right side is a consequence of indirect mass action which is the same for all substances (see Sect. 12.3). The similarities and differences for varying mixtures become even clearer when plotted logarithmically (Fig. 13.2). 

In this representation, all curves end up on the left side in parallel straight fines with the slope RT. In Sect. 6.5 we set up the following equation to describe mass action [Eq. (6.28)]  

A logarithmic plot where fi is chosen to be the ordinate and In x the abscissa, should result in a straight line with a slope of RT  

A mixture is called ideal if this relation is not only valid for low mole fractions but in the whole range 0 x 1 and in particular for xq = 1. If we write again for the chemical potential of the pure substance, the equation above simplifies to (cf. Sect. 12.3)  

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Chemical Potentials in Homogeneous Mixtures the Gibbs-Duhem Equation... 

The concept of substance activity was derived by Gilbert N. Lewis in 1907 from the laws of equilibrium thermodynamics and is described in detail in the text entitled Thermodynamics and the Free Energy of Chemical Substances by Lewis and Randell (1923). In a homogeneous mixture, each component has a chemical potential (jjl), which describes how much the free energy changes per mole of substance added to the system. The chemical potential of water (pw) in a solution is given by... 

The necessity of introducing a combinatorial contribution to the chemical potential is a result of the neglect of size effects in the thermodynamics of pairwise interacting surface models. It also appears in lattice models that do not allow for a realistic representation of molecular sizes and are often simplified to models of equally sized lattice objects. The task of the combinatorial contribution is to represent the chemical potential of virtually homogeneous interacting objects of different size in 1 mol of a liquid mixture of a given composition with respect to the size and shape of the molecules. 

Equation 5.21 shows the interrelationship among the chemical potentials of the constituent substances in a homogeneous mixture and is often used for the determination of the chemical potential of solute constituents in solutions. 

The chemical potential of a substance i in a homogeneous mixture depends on the temperature, pressure, and concentrations of constituent substances, p, = p,(T,p,xl , ) whereas, that of a pure substance is a function of temperature and pressure only. As mentioned in the foregoing chapters, the mixing of substances causes an increase in entropy of the system and hence changes the chemical potentials of the substances... 

In this section we would like to deal with the kinetics of the liquid-liquid phase separation in polymer mixtures and the reverse phenomenon, the isothermal phase dissolution. Let us consider a blend which exhibits LCST behavior and which is initially in the one-phase region. If the temperature is raised setting the initially homogeneous system into the two-phase region then concentration fluctuations become unstable and phase separation starts. The driving force for this process is provided by the gradient of the chemical potential. The kinetics of phase dissolution, on the other hand, can be studied when phase-separated structures are transferred into the one-phase region below the LCST. 

Again, the primary phase particles of the required substance modifica tion (material precursors) are usually very small. When seeds of the synthe sized phase are used, these primary particles are identical in size to the seeds. In the homogeneous liquid solutions or gas mixtures, the size ofpri mary particles is determined by the nucleation processes. The small size of the primary phase particles can influence considerably the chemical poten tial of the phase to be formed. For example, in the case of spherical parti cles, the chemical potential is determined by equation (1.5). Hence, the equilibrium partial pressure, p, of the saturated vapor or concentration, c of the saturated solution of the substance—for example, of the synthe sized one component phase—is determined by the Kelvin Thomson equation... 

A proper exposition of the subject rests on the following cardinal principle Let q represent any composition variable which specifies the makeup of a uniform solution. In what follows we let q stand for mole fraction x, molarity c, or molality m then the chemical potential of species i in the homogeneous mixture shall be given by the expression... 

The second scheme, which is more generally used, involves a hybrid procedure patterned after the methodology of Section 2.11. Here one distinguishes between pure condensed phases, indexed by the symbol s, and components forming homogeneous mixtures, indexed by the symbol j. For the pure condensed phases one adopts Eq. (3.6.2) in the specification of the chemical potential for species in solution it is conventional to introduce Eq. (3.5.21). The equilibrium condition for the reaction = 0 is now specified by... 

In a homogeneous mixture each component i has a chemical potential pt, defined as the partial molar free energy of that component (i.e., the change in Gibbs energy per mole of component ti added, for addition of an infinitesimally small amount). It is given by... 

Thus the chemical potential corresponds to the change in Gibbs energy of a homogeneous multicomponent system on the introduction of an infinitesimal amount of a component into the mixture at constant p, T and constant amounts of the other eomponents. 

The analysis of chemical equilibrium is based on the concept of chemical potential (Chapter 5). Consider a homogeneous gaseous system. The chemical potential fij of a component j in a mixture is given by ... 

We shall consider a homogeneous gas phase of r components and r -f 1 degrees of freedom (x,. . . , Xr-i,p,T). In order to compute the chemical potential of component i in the mixture, the notion of the partial pressure of component i will be introduced. The partial pressure of component i in the mixture, is defined as the pressure exerted by pure i in equilibrium with the mixture through a membrane permeable to i alone. 

In order to describe these kinds of processes in the usual way, it is practical to assign an amount of substance and a chemical potential to a portion of a homogeneous mixture, too. The sum of the amotmts n, b. c. of the pure substances A, B, C,... that make up the homogeneous mixture M equals the amotmt of substance M of the mixture ... 

There is, however, a fundamental difference While in the case of homogeneous mixtures, the chemical potentials of the components are different in their mixed and unmixed state, //a, i b, He - - always have the same values in heterogeneous mixtures whether A, B, C,. .. are present in their mixed or unmixed states. In order to differentiate the chemical potential of a heterogeneous mixture from the chemical potential of a homogeneous mixture, we will label it with the index fW. 

Fig. 13.5 The (average) chemical potential as a function of the composition of a heterogeneous mixture (solid line) (For comparison, the dotted curve for a homogeneous mixture is also included in the graphic.).

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