Chemical potential condensed ideal solution

We begin our specification of the chemical potential of nonideal solutions in the condensed state that is based on the canonical formulation of ideal solutions, introduced in Section 2.5,... 

A solution is defined as a condensed phase (liquid or solid) containing several substances. The main substance of the solution is called solvent and the other constituent substances dissolved in the solvent are solutes. Solutions are classified into ideal solutions and non-ideal solutions. For an ideal solution the chemical potential of a constituent substance i is given by ... 

Ben-Naim and Marcus [1] discussed a process termed solvation that applies to a particle of a (non-ionic ) substance transferring from its isolated state in the gas phase into a liquid irrespective of the concentration. The particle would then be surrounded by solvent molecules only in an ideally dilute solution (infinite dilution), or by solvent molecules as well as by molecules of its own kind at any arbitrary mole fraction with regard to the solvent, and by molecules identical with itself only on condensation into its own liquid. The interactions involved and their thermodynamics are aU covered by the same concept of solvation. The solvation process of a solute S is defined [1] as the transfer of a particle of S Ifom a fixed position in the (ideal) gas phase (superscript G) to a fixed position in a liquid (superscript L) at a given temperature T and pressure P. Statistical mechanics specifies the chemical potential of S in the ideal gas phase as ... 

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... 

A problem of obvious importance is the determination of the chemical potential of constituents that form a liquid or solid solution. We proceed by analogy to Eq. (2.4.15) for the ideal gas mixture. This objective is sensible, at least for ideal solutions defined below, because the different constituents in an ideal condensed phase do not interact, so they form an analog to the ideal gas mixture for which the partial pressure F,- constimtes the independent variable. The corresponding composition variable is the mole fraction x,. The solutions must be sufficiently dilute for the ideal Uquid model, discussed below, to apply. 

The functional form of equation 7.51 is so convenient that it is also preserved in practice for condensed phases, even if a formal derivation is no longer possible in general. Consider an ideal liquid solution in which Xi is the mole fraction of component i. The form of the chemical potential is then... 

In the liquid phase, just as in the vapor phase, we need to choose a suitable reference state with a corresponding reference chemical potential and reference fugacity to complete the definition provided by Equation (7.3). We then adjust for the difference between the reference phase and the real system. However, while there is an obvious reference case for gases—the ideal gas—there is no single suitable choice for the liquid phase. There are two common choices for the reference state (1) the Lewis/Randall rule and (2) Henry s law. The choice of reference state often depends on the system. Both these reference states are limiting cases that result from a natural idealization for condensed phases the ideal solution. 

Parameter A can be chosen to be any pointer of variation within the system for example, a torsional rotation to be followed in small angular steps or, more radically, the change of a solute molecule into a solvent molecule, of a hydroxyl group into a methyl group, or the actual transmutation of a reactant into a product in a chemical reaction, provided a suitable hamiltonian is available. In principle one could chose the starting state as the ideal gas, use equation 9.25 to calculate the exact free energy by statistical mechanics, and then use 9.26 or 9.27 to turn on the intermolecular potential and obtain the value of the free energy of a condensed phase. 

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