Although the thermodynamic behavior of species in solution is ultimately tied to their chemical potentials, a connection between this property and the concentration of the component is needed. This connection is made via a quantity called the activity of a solution species. The activity is a measure of the thermodynamic "strength" of a component in a solution compared to that of the pure substance; the purer, the stronger. As an example, when alcohol is mixed with water its effectiveness is reduced1.
The link between chemical potential and concentration consists of two parts. In the first part, the chemical potential and the activity of a solution component are related by the definition of the activity a^
The definition has been chosen so that the activity tends to unity for pure i; that is, ^ = gi, the molar free energy of pure i. Activity varies monotonically with concentration. Therefore, when component i approaches infinite dilution ai ^ 0 and ^ ^ - ot. This inconvenient behavior of the chemical potential at zero concentration is avoided by using the activity in practical thermodynamic calculations involving species in solution. Another reason for the choice of the mathematical form of the relation between and ai embodied in Eq (7.29) is that the activity is directly measurable as the ratio of the equilibrium pressure exerted by a component in solution to the vapor pressure of the pure substance. This important connection is discussed in Chap. 8.
The second part of the relation between chemical potential and concentration is the definition of the activity coefficient as the ratio of the activity to the mole fraction:
The activity coefficient has the following properties:
• it is unity for a pure substance (i.e., when xi = 1)
• it approaches a constant value as xi ^ 0
• it can be either greater than or less than unity
• it is unity for all concentrations if the solution is ideal
• it is a function of solution composition and temperature.
The reason that yA ^ 1 in an A-B binary solution dilute in component B is because A molecules are surrounded mainly by other A molecules; the interactions are predominantly of the A-A type, so component A behaves as if it were pure. This limiting
1 Fanciers of single-malt scotches recognize this fact by imbibing it straight, undiluted by water or ice.
behavior is called Raoult's law. At the other extreme, the activity coefficient of A in a solution dilute in A approaches a constant value characteristic of the A-B intermolecular interactions. This behavior is termed Henry's law. A more detailed description of Raoult's and Henry's laws is presented in Chap. 8.
A useful connection between the activity coefficients of species in a solution is obtained by eliminating ai between Eqs (7.29) and (7.30) and substituting the resulting equation into the Gibbs-Duhem equation, Eq (7.28). This procedure yields2:
This equation is particularly useful for two-component (A-B) solutions, where it becomes:
Problem 7.7 shows how this equation can be used to assess the validity of formulas for hex. In an equally important application, the above equation can be integrated to give the Gibbs free energy analog of Eq (7.19) for the enthalpy:
JxB ln YA0
where yA0 is the limiting value of the activity coefficient of A as xA approaches zero. In this limit yB approaches unity, which accounts for the zero lower limit of the integral of dlnyB. This extraordinary relation permits the activity coefficient of B to be computed from the measurement of the activity coefficient of A over a composition range starting from pure B. Figure 7.6 shows the graphical implementation of Eq (7.32). The curve intersects the horizontal axis at a value -lnyA0 and asymptotically approaches an ordinate value of infinity as lnyA ^ 0 (i.e., as xA ^ 1). The shaded area in this plot is lnyB at the solution composition xA.
Problem 7.11 is an example of the use of Eq (7.32). Other exercises dealing with either Eqs (7.31) or (7.32) are problems 7.8, 7.11, 7.12, 7.14 and 7.15.
This result makes use of: Zxidlnxi = Zxi(1/xi)dxi = Zdxi = dZxi = 0
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