# (p.447) A18 Debye Limiting Law

# (p.447) A18 Debye Limiting Law

A strong electrolyte in aqueous solution is fully dissociated, but may not behave as though the concentration of individual ions is equal to the given stoichiometric concentration. On dissolution in water the ions in an electrolyte solution become hydrated, that is, they become attached, albeit loosely, to a number of water molecules in a *hydration sphere*, and some of the ionic charge is distributed over this sphere. Positive and negative ions hydrated ions are attracted one to the other electrostatically, and every hydrated ion may be regarded as being surrounded by oppositely-charged species, so forming an *ionic atmosphere*. The hydrated ions cluster and disperse dynamically but over a period of time, which is long compared to the lifetime of any cluster, there will be a certain fraction of the total stoichiometric concentration which is unavailable as free ions. This effect is expressed as the activity *a* of a species *i*:

where *c _{i}*, and

*f*, are, respectively, the concentration and activity coefficient of the

_{i}*i*th species. It is a further condition (Appendix A17) that:

In many thermodynamic arguments, particularly those involving electrolytes, it is necessary to know the value of *f* or *a*. Although single-ion activities cannot be measured, the Debye–Hückel theory of electrolytes leads to an equation for calculating the activity coefficient of a single ion [1]. An approximation derived from the Debye–Hückel theory, and usually termed the *Debye limiting equation* (aka Debye *limiting law*), permits the calculation of activity coefficients in solutions of concentration up to 0.01 molar for 1:1 electrolytes and 0.005 molar for 1:2 electrolytes:

where *z* is the numerical charge on an ion. The term *A* depends *inter alia* on the relative permittivity and the temperature of the solvent; for pure (conductivity) water at 298.15 K, *A* is 1.1734; *I* is the ionic strength of the solution, with concentration in mol dm^{−3}. Equation (A18.3) is often quoted in terms of log_{10} in which case *A*, for the given conditions, is $1.1734/\mathrm{ln}10,\text{}\text{or}\text{}\text{0.5096.}$ Extracting the units of *A* is an unnecessary complicating factor for the application of the limiting law approximation.

(p.448) For a generalized structure ${A}_{{z}_{+}}^{{\mathrm{\nu}}_{+}}{B}_{{z}_{-}}^{{\mathrm{\nu}}_{-}}:$

where

The ionic strength *I* is defined by

and includes *all* ionic components in a given electrolyte solution.

(p.449) References A18

Bibliography references:

[1] Debye PJW and Hückel E. *Phys. Z*. 1923; 24: 185.

[2] Davies CW. *Ion Association*. Butterworths, 1962.