Chemical Potentials and Equilibrium Relationships

In these next frames we return to two topics mentioned earlier, those of chemical potential (Frames 5, 27, 28 and 29) and equilibria (Frame 14). 

1 Equalisation of Chemical Potentials for Components in Multiple Phases at Equilibrium 

In Frame 29 we established two principles which form the basis of how we describe the stability of phases in equilibrium with each other. 

In this system only one component is present (no need to specify i) and according to equation (35.1) and the principle in section 35.1 above, at equilibrium  

The above applies to all components present in a thermodynamic system 

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The concept of chemical potentials, the equilibrium criterion involving chemical potentials, and the various relationships derived from it (including the Gibbs phase rule derived in Chapter 5) can be used to explain the effect of pressure and temperature on phase equilibria in both a qualitative and quantitive way. 

Chapter 4 presents the Third Law, demonstrates its usefulness in generating absolute entropies, and describes its implications and limitations in real systems. Chapter 5 develops the concept of the chemical potential and its importance as a criterion for equilibrium. Partial molar properties are defined and described, and their relationship through the Gibbs-Duhem equation is presented. 

The Equilibrium Constant. The relationship between the reaction free energy or the affinity for each chemical reaction and the composition of a closed system is readily obtained by applying Equation 14 for AG to the relationships between chemical potential, and the activity, af ... 

The fundamental result is eq. (10.6). which equates the chemical potentials of a component in every phase the component is present. Equations (io.iq)-(io.2o) express the equilibrium criterion in terms of fugacity (or fugacity coefficient) and represent equivalent forms of eq. (10.6). Another important result is a relationship between chemical potential and fugacity ... 

This equation relates the chemical potential and fugacity at two states at the same temperature. This relationship allows us to use fugacity interchangeably with chemical potential to solve problems in phase equilibrium. 

Chemical potentials are central for an understanding of material/phase equilibrium and phase stability. FST can be used to study metastable phase and phase instabilities. However, the vast majority of the studies using the FST of solutions involve a single stable phase with multiple components. Here, we are concerned with the relationships among the chemical potentials, and their derivatives, and the local solution distributions. Thermodynamically, from Equations 1.2 and 1.3, we have ... 

The fugacity bears the same relationship to the chemical potential for real fluids as does the partial pressure for ideal gases. Because of the direct relationship between chemical potential and fugacity, the condition for equilibrium expressed by Equation 1.9 is equivalent to the equality of component fugacities in the phases ... 

A transformation, at constant temperature and pressure, occurs so that the substance transfers from the state of higher to lower chemical potential until equilibrium is attained. In the case of dyeing, if the chemical potential of the dye in solution is higher than that in the fibre, the dye will transfer to the fibre. As the chemical potential in the solution falls, that in the fibre increases. At equilibrium, the chemical potential of the dye in the fibre is equal to the chemical potential of the dye in the solution, leading to the relationships shown in Eq. 2.8 ... 

Thus, for each component, its chemical potential is the same in all phases that are in equilibrium. We will see below that the relationships involving the pressure and temperature variations of the chemical potential that we have developed earlier will be helpful in explaining the effect of these variables on phase equilibria. 

Barton and Toulmin (1964) have derived a relationship between fs2, temperature, and the Ag content of electrum in equilibrium with argentite, using the equation of White et al. (1957) for the chemical potential of Ag in electrum in combination with the equation... 

Equating the chemical potential to zero gives a relationship between the equilibrium degree of swelling and the molecular weight Mc. The relation for Mc ph is obtained for a tetrafunctional phantom network model as... 

Equilibrium values of numbers Na of monomer molecules M in a globule can obviously be found from the condition of the equilibrium of the values of the chemical potentials of these molecules inside, iia, and outside, // , this globule. The explicit expressions for //1 and /x2 are obtained by differentiation of the relationship (Eq. 65), complemented by the incompressibility condition ip + cp = i, with respect to Mi and M2, respectively. With these expressions at hand, the set of two equations, (/j,i = //p /j2 = pu ), for the calculation of Mi and N2 has been derived [51]. Having these quantities calculated, it is easy... 

The most fundamental manner of demonstrating the relationship between sorbed water vapor and a solid is the water sorption-desorption isotherm. The water sorption-desorption isotherm describes the relationship between the equilibrium amount of water vapor sorbed to a solid (usually expressed as amount per unit mass or per unit surface area of solid) and the thermodynamic quantity, water activity (aw), at constant temperature and pressure. At equilibrium the chemical potential of water sorbed to the solid must equal the chemical potential of water in the vapor phase. Water activity in the vapor phase is related to chemical potential by... 

Quantitative structure-chemical reactivity relationships (QSRR). Chemical reactivities involve the formation and/or cleavage of chemical bonds. Examples of chemical reactivity data are equilibrium constants, rate constants, polarographic half wave potentials and oxidation-reduction potentials. 

This exposition has been greatly simplified. At equilibrium, the sums of the electrochemical potentials, p, within each of the two half cells comprising the overall cell are the same, and p is related to the chemical potential n a by the relationship p = n + nF. The occurrence of a potential E at the electrode is a manifestation of the difference in electric field, A0 between the electrodes and their respective couples in solution, as a function of their separation distances. 

The condition of Equation (13.7) can be met only if p,j = p,n, which is the condition of transfer equilibrium between phases. Or, to put the argument differently, if the chemical potentials (escaping tendencies) of a substance in two phases differ, spontaneous transfer will occur from the phase of higher chemical potential to the phase of lower chemical potential, with a decrease in the Gibbs function of the system, until the chemical potentials are equal (see Section 10.5). For each component present in aU p phases, (p 1) equations of the form of Equation (13.7) provide constraints at transfer equilibrium. Furthermore, an equation of the form of Equation (13.7) can be written for each one of the C components in the system in transfer equUibrium between any two phases. Thus, C(p — 1) independent relationships among the chemical potentials can be written. As chemical potentials are functions of the mole fractions at constant temperamre and pressure, C(p — 1) relationships exist among the mole fractions. If we sum the independent relationships for temperature. 

Solvent in Solution. We shall use the pure substance at the same temperature as the solution and at its equilibrium vapor pressure as the reference state for the component of a solution designated as the solvent. This choice of standard state is consistent with the limiting law for the activity of solvent given in Equation (16.2), where the limiting process leads to the solvent at its equilibrium vapor pressure. To relate the standard chemical potential of solvent in solution to the state that we defined for the pure liquid solvent, we need to use the relationship... 

Significance of Activity Coefficients. While we typically focus our attention on the analytical concentration of reactant(s) and product(s) for a given chemical process, the thermodynamic concept of equilibrium depends on the chemical potential of a species. This is shown by the following relationship... 

T is temperature, P is pressure, and / is the fugacity of the component. In Equation 3 subscript k refers to each component of the system. In the present discussion the fugacity 42) is employed in preference to the chemical potential 21). Earlier in the history of the petroleum industry, Raoult s 55) and Dalton s laws were applied to equilibrium at pressures considerably above that of the atmosphere. These relationships, which assume perfect gas laws and additive volumes in the gas phase and zero volume for the liquid phase, prove to be of practical utility only at low pressures. Henry s law was found to be a useful approximation only for gases which were of low solubility and at reduced pressures less than unity. 

This relationship constitutes the basic definition of the activity. If the solution behaves ideally, a, =x, and Equation (18) define Raoult s law. Those four solution properties that we know as the colligative properties are all based on Equation (12) in each, solvent in solution is in equilibrium with pure solvent in another phase and has the same chemical potential in both phases. This can be solvent vapor in equilibrium with solvent in solution (as in vapor pressure lowering and boiling point elevation) or solvent in solution in equilibrium with pure, solid solvent (as in freezing point depression). Equation (12) also applies to osmotic equilibrium as shown in Figure 3.2. 

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