Ideal solution, standard chemical potential

In an ideal solution, the chemical potential of species i is calculated from the standard chemical potential the gas constant R, the Kelvin temperature T, and the mole fraction jc, as ... 

In a general case of a mixture, no component takes preference and the standard state is that of the pure component. In solutions, however, one component, termed the solvent, is treated differently from the others, called solutes. Dilute solutions occupy a special position, as the solvent is present in a large excess. The quantities pertaining to the solvent are denoted by the subscript 0 and those of the solute by the subscript 1. For >0 and x0-+ 1, Po = Po and P — kxxx. Equation (1.1.5) is again valid for the chemical potentials of both components. The standard chemical potential of the solvent is defined in the same way as the standard chemical potential of the component of an ideal mixture, the standard state being that of the pure solvent. The standard chemical potential of the dissolved component jU is the chemical potential of that pure component in the physically unattainable state corresponding to linear extrapolation of the behaviour of this component according to Henry s law up to point xx = 1 at the temperature of the mixture T and at pressure p = kx, which is the proportionality constant of Henry s law. 

For a solution of a non-volatile substance (e.g. a solid) in a liquid the vapour pressure of the solute can be neglected. The reference state for such a substance is usually its very dilute solution—in the limiting case an infinitely dilute solution—which has identical properties with an ideal solution and is thus useful, especially for introducing activity coefficients (see Sections 1.1.4 and 1.3). The standard chemical potential of such a solute is defined as... 

The activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

For solutions obeying Henry s law, as for ideal solutions, and for solutions of ideal gases, the chemical potential is a linear function of the logarithm of the composition variable, and the standard chemical potential depends on the choice of composition variable. The chemical potential is, of course, independent of our choice of standard state and composition measure. 

In what follows we shall always write A, = fC. We assume that the ligand is provided from either an ideal gas phase or an ideal dilute solution. Hence, A, is related to the standard chemical potential and is independent of the concentration C. On the other hand, for the nonideal phase, A will in general depend on concentration C. A first-order dependence on C is discussed in Appendix D. Note also that A, is a dimensionless quantity. Therefore, any units used for concentration C must be the same as for (Aq) . 

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 ... 

Although Pb tends to - °o as Xb tends to 0 (and In Xb also tends to - °o), the difference on the right-hand side of Eq. (2.18) tends to the finite quantity pi, the standard chemical potential of B. At infinite dilution (practically, at high dilution) of B in the solvent A, particles (molecules, ions) of B have in their surroundings only molecules of A, but not other particles of B, with which to interact. Their surroundings are thus a constant environment of A, independent of the actual concentration of B or of the eventual presence of other solutes, C, D, all at high dilution. The standard chemical potential of the solute in an ideal dilute solution thus describes the solute-solvent interactions exclusively. 

For a solute in solution. For a solute in a liquid or solid solution the standard state is referenced to the ideal dilute behaviour of the solute. It is the (hypothetical) state of solute B at the standard molality m, standard pressure and exhibiting infinitely diluted solution behaviour. The standard chemical potential is defined as... 

Sometimes (amount) concentration c is used as a variable in place of molality m both of the above equations then have c in place of m throughout. Occasionally mole fraction x is used in place of m both of the above equations then have x in place of m throughout, and x = 1. Although the standard state of a solute is always referenced to ideal dilute behaviour, the definition of the standard state and the value of the standard chemical potential g are different depending on whether molality m, concentration c, or mole fraction x is used as a variable. 

The quantity of primary interest in our thermodynamic construction is the partial molar Gihhs free energy or chemical potential of the solute in solution. This chemical potential depends on the solution conditions the temperature, pressure, and solution composition. A standard thermodynamic analysis of equilibrium concludes that the chemical potential in a local region of a system is independent of spatial position. The ideal and excess contributions to the chemical potential determine the driving forces for chemical equilibrium, solute partitioning, and conformational equilibrium. This section introduces results that will be the object of the following portions of the chapter, and gives an initial discussion of those expected results. 

Carnot s equations, 146-147 Carnot s theorem, 142-143 Chemical potential, 298, 302, 303 as equilibrium criterion, 298-299, 503 for ideal gas, 302 for ideal solution, 303 Chemical reaction equilibrium constant for, 504-516 equilibrium conversion of, 518-528, 533-542 heat effects of, 116-133 reaction coordinate for, 497-501 reversible, 41-42, 505-507 standard property changes for, 125, 505 stoichiometry, 497-501... 

The chemical potential of component i will be equal to the standard chemical potential when the mole fraction of component i equals 1. The standard state is thus pure component i. If species i is an ion, the standard state is the hypothetical state of pure ion i. In a real solution, an extra term, RT In is added to the chemical potential in order to account for the deviation from ideality ... 

Solvent activity coefficients are defined (Parker, 1966) such that °y< reflects the change in the standard chemical potential fi of a solute, i (hypothetically ideal, in respect to Henry s Law, unimolar solution), on transfer from an arbitrarily chosen reference solvent (i.e. the standard... 

It should be noted that gives the standard chemical potential for the ideally dilute solute in a hypothetical system in which the mole fraction of B is unity. This is obviously a fictitious state which is impossible in reality but whose properties are obtained by extrapolating the Henry s law line to Xb = 1 (see fig. 1.12). When Henry s law is not obeyed, an activity coefficient 73 introduced so that the product Yb h b is equal to the vapor pressure Pb- The activity of the dilute component Ub is defined to be 73 3- Thus, the general expression for the concentration dependence of pb becomes... 

In the case of pure solids such as Ag and AgCl the chemical potential is identical to the standard chemical potential at 25°C and 1 bar pressure. For solutions, the standard state of the solute is unit activity at the same temperature and pressure. In the case of electrolytes as solutes, the activity is defined on the concentration (molarity) scale, and the standard state is the hypothetical ideal state of unit molarity for which the activity coefficient ye is unity. Under these circumstances, the activity of the solvent, which does not appear explicitly in equation (9.2.9), is also unity to a good approximation when the solvent is water. For gases the standard state is a pressure of 1 bar (10 Pa) at 25°C. In the older literature the standard pressure was 1 atm (101,325 Pa). In data compilations appearing after 1982, the standard state of 1 bar and 25°C is always used for gases [G3]. 

This law is readily derived from thermodynamics, assuming both solutions to be ideally dilute. At equilibrium we must have for the solute (2) that fi2,a — /F./i- Although the standard chemical potential of the pure solute /1° is, of course, the same, the apparent standard chemical potential fiQ (see Fig. 2.2) will generally be different. We thus have... 

A multicomponent solution is ideal only if (10.43) is satisfied by every component. A solution, in general, approaches ideality as it becomes more and more dilute in all but one component (the solvent). The standard chemical potential p is the chemical potential of pure species i(jq = 1) at the same temperature and pressure as the solution under discussion. Note that in general p is a function of both T and p but does not depend on the chemical composition of the solution. 

The activity coefficient y, is in general a function of pressure and temperature together with the mole fractions of all substances in solution. For an ideal solution = 1. The standard chemical potential /x is defined as the chemical potential at the hypothetical state for which Yi 1 and jc, —1 (Denbigh, 1981). The product of the mole fraction jr, of a solution component and its activity coefficient y, is defined as the activity, a of the component... 

In actual practice, the change in the standard chemical potential (Ap°) is evaluated as the measure of the tendency of the solute to move from the solution to the fiber (i.e., the relative affinity of the substance for the fiber relative to the solution phase). This parameter is generally called the affinity and is usually expressed as ion affinities instead of as molecular affinities (see Table 5-18). One expression that describes the standard chemical potential, if the ion forms ideal solutions, is... 

If the solutions are assumed to be ideal (i.e., the activity coefficients are equal to unity), then the difference in the standard chemical potential between a surfactant in an aggregate of size n and a free surfactant, — can be written as... 

Figure 4.5 shows the change in the standard chemical potential (=standard molar Gibbs energy, Dtr uc) for the transfer of hydrocarbons from their own medium to water. This change may be established by studying the equilibrium between pure hydrocarbon (HC) and hydrocarbon dissolved in water (mole fraction HC in water, Xgc). At equilibrium, and for an ideal solution of HC in water. 

By definition, the definition of an activity requires choosing a standard state. For example, for a solute i, the standard state can be chosen as being the state in which its concentration is C° (or its molality J°, or its mole fraction is x°,), the temperature is r, and the solution is ideal (recall that pressure exerts a very weak influence on the behavior of condensed phases). At concentration C°, (which is that in the standard state), the solute chemical potential is its standard chemical potential A°,. Hence, when the solution is ideal, the solute chemical potential (A, at concentration C, or at molality m, is given by the expressions... 

In the above we have assumed all the solutions phase species, such as or Cr, behave ideally . Accordingly concentrations were used to relate chemical potentials and standard chemical potentials— for example as in eqn (1.30). In reality, as will be explained and emphasised in Chapters 2 and 4, the assumption of ideality is unlikely to be true for the case of electrolyte solutions. It is then necessary to use activities rather than concentrations. An ion, i, has a chemical potential,... 

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