Chemical potential pure solids

In this equation, e is the actual potential and f° is the standard potential, n is the number of electrons involved, and Q is a ratio of concentration terms. Q is equal to the ratio of concentrations of products to concentrations of reactants, each raised to the power corresponding to the coefficient in the balanced chemical equation. Pure solids and liquids and the solvent water are not included in Q their effective concentrations are assumed to be 1. Gas pressures in atmospheres are used instead of concentrations. For a general reaction... 

In this equation, X2 represents the mole fraction of naphthalene in the saturated solution in benzene. It is determined only by the chemical potential of solid naphthalene and of pure, supercooled liquid naphthalene. No property of the solvent (benzene) appears in Equation (14.45). Thus, we arrive at the conclusion that the solubility of naphthalene (in terms of mole fraction) is the same in all solvents with which it forms an ideal solution. Furthermore, nothing in the derivation of Equation (14.45) restricts its application to naphthalene. Hence, the solubility (in terms of mole fraction) of any specified solid is the same in all solvents with which it forms an ideal solution. 

The chemical potential of solid solutions, including impure minerals such as those in soils and rocks, is more difficult to define. Isomorphously substituted ions in a mineral change its activity and aqueous solubility from that of the pure mineral. Progress in defining solid activities has been slow. Soil minerals have often been assumed, by necessity, to he pure minerals and assumed to have activity = 1. This assumption is weak and is discussed later in this chapter. 

For precise measurements, diere is a slight correction for the effect of the slightly different pressure on the chemical potentials of the solid or of the components of the solution. More important, corrections must be made for the non-ideality of the pure gas and of the gaseous mixture. With these corrections, equation (A2.1.60) can be verified within experimental error. 

This is an expression of Raoult s law which we have used previously. Freezing point depression. A solute which does not form solid solutions with the solvent and is therefore excluded from the solid phase lowers the freezing point of the solvent. It is the chemical potential of the solvent which is lowered by the solute, so the pure solvent reaches the same (lower) value at a lower temperature. At equilibrium... 

The energy of a system can be changed by means of thermal energy or work energy, but a further possibility is to add or subtract moles of various substances to or from the system. The free energy of a pure substance depends upon its chemical nature, its quantity (AG is an extensive property), its state (solid, liquid or gas), and temperature and pressure. Gibbs called the partial molar free heat content (free energy) of the component of a system its chemical potential... 

If the coexisting liquid or solid phases are not pure, but solutions, equilibrium will be established when the chemical potential... 

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. 

In equilibrium the chemical potential must be equal in coexisting phases. The assumption is that the solid phase must consist of one component, water, whereas the liquid phase will be a mixture of water and salt. So the chemical potential for water in the solid phase fis is the chemical potential of the pure substance. However, in the liquid phase the water is diluted with the salt. Therefore the chemical potential of the water in liquid state must be corrected. X refers to the mole fraction of the solute, that is, salt or an organic substance. The equation is valid for small amounts of salt or additives in general ... 

A great many electrolytes have only limited solubility, which can be very low. If a solid electrolyte is added to a pure solvent in an amount greater than corresponds to its solubility, a heterogeneous system is formed in which equilibrium is established between the electrolyte ions in solution and in the solid phase. At constant temperature, this equilibrium can be described by the thermodynamic condition for equality of the chemical potentials of ions in the liquid and solid phases (under these conditions, cations and anions enter and leave the solid phase simultaneously, fulfilling the electroneutrality condition). In the liquid phase, the chemical potential of the ion is a function of its activity, while it is constant in the solid phase. If the formula unit of the electrolyte considered consists of v+ cations and v anions, then... 

The difference between the chemical potential of a pure and diluted ideal gas is simply given in terms of the logarithm of the mole fraction of the gas component. As we will see in the following sections this relationship between the chemical potential and composition is also valid for ideal solid and liquid solutions. 

Figure 5.18 Adding a chemical to a host (mixing) causes its chemical potential /i to decrease, thereby explaining why a melting-point temperature is a good test of purity. The heavy solid lines represent the chemical potential of the pure material and the thin lines are those of the host containing impurities...

Consider a dilute ideal solution of the solute B (which could be gaseous, liquid, or solid at the temperature in question) in the solvent A. Suppose that more concentrated solutions do not behave ideally and, in particular, the state of pure liquid B cannot be attained by going to more and more concentrated solutions (e.g., by removing A by volatilization). It is possible to define a standard chemical potential pertaining to a hypothetical standard state of the ideal infinitely dilute solution as the limit ... 

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. 

These conditions can be satisfied by drawing the common tangent to the G curves of M(O) and MO. As shown in Fig. 1.7, the chemical potentials of M and O for the M(O) phase with the composition x, are equal to those for the MO phase with the composition Xj, and the values correspond to MqMj and OgO, respectively. If the experimental conditions are similar to those described in Section 1.1, the solid phases must coexist with the gas phase. It may be adequate for the gas phase to be pure O2, because the vapour pressure of other species is very low in this case. The chemical potential of O for the gas phase is equal to OgO, which corresponds to the oxygen pressure. Thus we can understand the coexistence of the M(O) phase with Xj and the MO phase with X2 from the free energy change of composition. 

Let us now continue with our discussion of how to relate the chemical potential to measurable quantities. We have already seen that the chemical potential of a gaseous compound can be related to pressure. Since substances in both the liquid and solid phases also exert vapor pressures, Lewis reasoned that these pressures likewise reflected the escaping tendencies of these materials from their condensed phases (Fig. 3.9). He thereby extended this logic by defining the fugacities of pure liquids (including subcooled and superheated liquids, hence the subscript L ) and solids (subscript s ) as a function of their vapor pressures, pil ... 

If we also know the chemical potential of the compounds in the gas phase or in a solid phase, then we can also calculate the equilibria between the vapor and the liquid phase (VLE), or between solids and the liquid phase (SLE). If the vapor pressure pJap(T) of the pure compound i is known, the partial pressure of i in the gas phase is given by... 

The thermodynamic equations for the Gibbs energy, enthalpy, entropy, and chemical potential of pure liquids and solids, and for liquid and solid solutions, are developed in this chapter. The methods used and the equations developed are identical for both pure liquids and solids, and for liquid and solid solutions therefore, no distinction between these two states of aggregation is made. The basic concepts are the same as those for gases, but somewhat different methods are used between no single or common equation of state that is applicable to most liquids and solids has so far been developed. The thermodynamic relations for both single-component and multicomponent systems are developed. 

The difference in the chemical potential of the pure component in the liquid and solid phases can be evaluated in the same manner as was done in Section 10.10, because of the equality of the two chemical potentials at the melting point for the pressure P. The result is... 

The equation for the second component is the same with change of subscripts. The problem is to determine values of xt and zl at specified temperatures by the use of the two equations. We define the quantity A [T, P] as equal to both sides of Equation (10.198). This quantity is the change of the chemical potential on mixing for the liquid phase that has been defined previously however, for the solid phase the standard state is now the pure liquid component. A similar definition is made for Ap%[T, P], The conditions of equilibrium then become... 

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