The difference in the free energy of a substance (Gi) in a non-standard state relative to its free energy in a defined standard state (Gi0) is derived from its activity (a;):
The activity of the substance is related to its concentration through the relation:
where Yi is the activity coefficient of the substance in its non-standard state, and [Ci] is its concentration.
If we consider the general reaction:
then the thermodynamic equilibrium constant would be given by:
Substitution of equation (2) into equation (4) for the various species yields:
It is common practice to define an equilibrium constant in terms of species concentrations:
The relationship between the concentration-based equilibrium constant and the thermodynamic equilibrium constant is therefore:
C YAa YBb
It may be noted that in an ideal solution where all ions are completely non-interacting (a situation realized only under conditions of high dilution), the activity coefficient term would equal unity, and then K would be equal to KC. The ionic strength of a solution is defined as:
where Ci is the concentration (in units of moles/liter) of the ith ion, and zi is its charge. For example, the ionic strength of a 0.1M solution of KCl is 0.1, and the ionic strength of a 0.1M solution of K2SO4 equals 0.3.
According to the Debye-Hiickel theory, the activity coefficient of the ith ion in a solution having an ionic strength of is given by:
where A is a constant equal to 0.51, B is a constant equal to 3.3 x 107, and d is the size of the hydrated ion. For many univalent ions, d is approximately 3 A, so equation (9) simplifies to:
Finally, in extremely dilute solutions, the quantity p1/2 in the denominator is negligible relative to unity, and equation (10) further simplifies to the expression known as the Limiting Law:
For an uncharged molecule, the activity exhibits a different dependence on ionic strength:
where k is a constant. For simple carboxylic acids, k will fall within the range of -0.05 to +0.20. The value of this constant depends on the characteristics of the uncharged molecule as well as the specific ions in the solution that contribute to the overall ionic strength.
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