**Extensions of The Theory**

Warning: The notation in this section is (currently) different than in the rest of the article.

A number of approaches have been proposed to extend the validity of the law to concentration ranges as commonly encountered in chemistry

One such Extended Debye-Hückel Equation is given by:

where as its common logarithm is the activity coefficient, is the integer charge of the ion (1 for H+, 2 for Mg2+ etc.), is the ionic strength of the aqueous solution, and is the size or **effective diameter** of the ion in angstrom. The effective hydrated radius of the ion, a is the radius of the ion and its closely bound water molecules. Large ions and less highly charged ions bind water less tightly and have smaller hydrated radii than smaller, more highly charged ions. Typical values are 3Å for ions such as H+,Cl-,CN-, and HCOO-. The effective diameter for the hydronium ion is 9Å. and are constants with values of respectively 0.5085 and 0.3281 at 25°C in water .

The Extended Debye-Hückel Equation provides accurate results for μ ≤ 0.1 M. For solutions of greater ionic strengths, the Pitzer equations should be used. In these solutions the activity coefficient may actually increase with ionic strength.

The Debye-Hückel Equation cannot be used in the solutions of surfactants where the presence of micelles influences on the electrochemical properties of the system (even rough judgement overestimates γ for ~50%).

Read more about this topic: Debye–Hückel Equation

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