ABSTRACT
When the electrostatic interactions between ions and counterions are relatively weak compared with their thermal energy, kBT (i.e. at low dilution), the distribution of ions and potential at any distance from any ion can be calculated. The effect on the energy of the system of the electrostatic interactions that result can then be quantified. This procedure results in the Debye-Hückel law:
which can generally be applied to solutions of I≤0.01 mol dm−3 (for which this assumption of relatively weak electrostatic interactions holds). A is a constant for a given solvent (0.509 for water at 298 K), B is a constant for a given ion in a given solvent (conveniently B is often approximately equal to 1 for most ions in water) and is the standard ionic strength (defined as being equal to 1 mol dm−3). This allows the mean activity coefficient, γ±, to be calculated for a salt MyXx, where
This expression simply emphasizes that changes in the activity coefficient of the cation cannot be calculated without changes in the activity coefficient of the anion, as anions cluster around cations and cations cluster around anions. If it is assumed that the effects of clustering on the two activity coefficients are approximately equal (as well as the parameter B for both ions), then the activity coefficient for an individual ion is given by:
At very low dilution (typically when I<0.001 mol dm−3), the first term in the denominator of the equation dominates the second term, and produces the simplified Debye-Hückel limiting law (which removes the ion dependent parameter B from the expression):
At higher values of I<1 mol dm−3, the assumption of weak interactions between ions made in deriving the Debye-Hückel expression becomes increasingly untenable. Activity coefficients in this regime generally fit the expression:
where C is an empirical parameter which can be adjusted to fit the data. This extra term is introduced to take account of the increasing importance of short-range ion-ion and ionsolvent forces and this equation is often called the Debye-Hückel extended law. The applicability of these equations to calculating mean activity coefficients is demonstrated in Fig. 1, where calculated values can be seen to fit experimentally determined values (see Topic E5) closely. For I>1 mol dm−3, where short-range interactions dominate and ions have increasingly incomplete solvation, theoretical calculation of activity coefficient data is notoriously unreliable.
