Molality standard state, chemical potential

In summary, thermodynamic models of natural water systems require manipulation of chemical potential expressions in which three concentration scales may be involved mole fractions, partial pressures, and molalities. For aqueous solution species, we will use the moial scale for most solutes, with an infinite dilution reference state and a unit molality standard state (of unit activity), l or the case of nonpolar organic solutes, the pure liquid reference and standard states are used. Gaseous species will be described on the partial pressure (atm — bar) scale. Solids will be described using the mole fraction scale. Pure solids (and pure liquids) have jc, = 1, and hence p, = pf. 

The quantity /rf is the chemical potential of substance i in its molality standard state. This standard state is component i in a hypothetical solution with mi equal to m° (exactly 1 molkg ) and with Henry s law in the form of Eq. (6.2-11) valid at this molality. Again we specify a pressure of exactly 1 bar for this standard state. Since the standard state is a hypothetical solution, the actual 1-molal solution is not required to obey Henry s law. 

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. 

The value of k is constant because the standard chemical potentials in the two solvents are constants at a fixed temperature. Nemst s distribution law also can be stated in terms of molality. 

For solvents, 1, is equal to V because the standard state is the pure solvent, if we neglect the small effect of the difference between the vapor pressure of pure solvent and 1 bar. As the standard state for the solute is the hypothetical unit mole fraction state (Fig. 16.2) or the hypothetical 1-molal solution (Fig. 16.4), the chemical potential of the solute that follows Henry s law is given either by Equation (15.5) or Equation (15.11). In either case, because mole fraction and molality are not pressure dependent. 

At standard state, equation 8.211 obviously concerns standard state stable components—i.e., pure Fe metal at F = 25 °C, P = 1 bar, and a hypothetical one-molal Fe + solution referred to infinite dilution, at the same P and T conditions (cf section 8.4). The chemical potentials of components in reactions Fe2+(aqueous), ae-(aqueous)) SFC thosc of Standard State hence, by defini-tion, the activity of all components in reaction is 1—i.e.,... 

Here, p and m are the standard chemical potential and concentration (molal scale) of the /-component (z = 1 for solvent, z = 2 for biopolymer) A2 is the second virial coefficient (in molal scale units of cm /mol, i.e., taking the polymer molar mass into account) and m° is the standard-state molality for the polymer. 

Here is the chemical potential of i in its standard state and a, is its molal activity. This equation can be recast in the form known as the van t Hoff reaction isotherm ... 

The reference state of the electrolyte can now be defined in terms of thii equation. We use the infinitely dilute solution of the component in the solvent and let the mean activity coefficient go to unity as the molality or mean molality goes to zero. This definition fixes the standard state of the solute on the basis of Equation (8.184). We find later in this section that it is neither profitable nor convenient to express the chemical potential of the component in terms of its molality and activity. Moreover, we are not able to separate the individual quantities, and /i . Consequently, we arbitrarily define the standard chemical potential of the component by... 

Table 9.3. Standard chemical potential jj°, standard partial molar enthalpy h°, and standard partial molar entropy s,° for a few hydrated ions Standard state 101.3 kPa, 298 K, unit activity in molality scale.

It is of advantage to choose as the standard state of the undissociated part of the electrolyte its hypothetical unionized state in an ideal solution with the molality m = 1 (or molarity c = 1), and to consider as the standard state of the dissociated part of the substance its hypothetical completely ionized state in an ideal solution with the ion molality m+ = 1 and m — 1. If the chemical potential j.°ab corresponds to the first mentioned standard state, and the potential iA + + Xb to the second one, the difference in the standard free energy A0° between both states is expressed by the equation ... 

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. 

Also, it is customary to refer all thermodynamic properties to chemical potentials of all species, whether in the pure state or in solution, to their values under standard conditions. In that case the equilibrium constant will be designated, as before, by fCx and the pressure in the above equations is set at P = bar. Finally, it is possible to specify compositions in terms of molarity c, or molality m, leading to the specification of Kc and Km or Kc and Km - The resulting analysis becomes somewhat involved and will not be taken up here interested readers should read Section 3.7 for a full scale analysis of the treatment of nonideal solutions. 

In this equation, the following definition of the chemical potential of ion i in the hypothetical state of the pure ions dissolved in pure water to give an ideal solution at unit mean molality, under the standard state pressure has been used ... 

Randall, while Prigogine and Defay, in the French edition, adopt convention (c). These two usages have in effect been criticized by Guggenheim on the grounds that unless the standard pressure and standard temperature are universally recognized, then whenever, for example, the term standard is employed as in (c) some ambiguity exists unless the values of the standard pressure and temperature are stated. For this reason in the present translation the term standard has been restricted to (a) anymore detailed specification is given in full, for example the standard chemical potential at one atmosphere and at 600 °K jit (600 °K 1 atm.). The only implication in this is that the composition of the system is to be expressed in mole fractions. As will be seen in chap. XX, when other concentration units are used, a different superscript is employed e.g. p) for molalities and p) for molar... 

Because of the inconvenient nature of the standard state defined above, the concentration units used to describe the concentration dependence of the chemical potential are usually different. More convenient choices for concentration are molality and molarity. When the solution is dilute the relationship between mole fraction and molality is quite simple (see equation (1.2.3)). In terms of molality, the expression for the concentration dependence of the chemical potential of component B becomes... 

The concentration dependence of y is an important feature involved in the experimental and theoretical evaluation of electrolyte behavior. The chemical potential of the standard state, p x, is that for a hypothetical one-molal solution in which all real interactions are imagined to be absent (y =1.00). Thus, the... 

In solution we may write for the chemical potential p - pf -f Rrina (Section 7.2), where the standard state is one of unit molality which behaves as if it were at infinite dilution. Let us consider the simplified equilibrium... 

The pf is the chemical potential the solute would have in a 1 molal solution if that solution behaved according to the ideal dilute rule. This standard state is called the ideal solution of unit molality. It is a hypothetical state of a system. According to Eq. (16.20) the practical activity measures the chemical potential of the substance relative to the chemical potential in this hypothetical ideal solution of unit molality. Equation (16.20) is applicable to either volatile or in volatile solutes. 

We can now attempt to approach the problem of solvent basicity by another route the medium effect on the proton. The chemical potential of the proton in a solvent S can be expressed in two ways, depending on whether we take the standard state to be the hypothetical one-molal solution in S... 

Thus fl is a measure of the difference between the chemical potential of i in some equilibrium state, and the chemical potential of i in its standard state. When i is in its standard state, a = 1 and this difference is zero. As we have seen, the activity can take on several different forms, depending on whether we are using fiigacities, molalities, or mole fractions. Therefore the definition of the standard state will be different in each case, because a, = 1 will imply a different state for i in each case. 

The quantities g and are the chemical potentials of the solute in hypothetical reference states that are solutions of standard concentration and standard molality, respectively, in which B behaves as in an ideal-dilute solution. Section 9.7.1 will show that when the pressure is the standard pressure, these reference states are solute standard states. 

It is customary to use a molality basis for the reference and standard states of electrolyte solutes. This is the only basis used in this chapter, even when not expUcitly indicated for ions. The symbol, for instance, denotes the chemical potential of a cation in a standard state based on molality. 

Here ft and ifl are the chemical potentials of the cation and anion in solute reference states. Each reference state is defined as a hypothetical solution with the same temperature, pressure, and electric potential as the solution under consideration in this solution, the molality of the ion has the standard value m°, and the ion behaves according to Henry s law based on molality. y+ and y are single-ion activity coefficients on a molality basis. 

For ionic systems it has become customary to use the molality scale (mol/kg solvent). This scale has the advantage that the addition of another solute does not change the molality of a given solute. Values of AGf and AH for the formation of ions in water at T = 298.15 K are tabulated for the standard state of an ideal solution at a concentration of 1 mol/kg. This standard state is given the subscript ao. Thus the chemical potential or the activity of an ion is indicated by ao. The chemical potential of an ionized salt, Psait = v+p + v p, and the corresponding activity is denoted by the subscript ai. 

Thermodynamic Aspects of Solubility At equilibrium in a saturated solution, the chemical potential, or partial molal free energy, of the solute must be the same in the solution as in the solid phase. If we consider two different saturated solutions, there-fore, both in equilibrium with the same solid phase, the chemical potential of the solute must be the same in both. The chemical potential ( ) and activity (c) are related by the equation p — po — RT In o, where Po is the chemical potential of the substance in the standard state. Hence, if the same standard state is chosen for all the solutions considered, the activity of the solute must be the same in all. 

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... 

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