# 4.3: Ionic Strength

The properties of electrolyte solutions can significantly deviate from the laws used to derive chemical potential of solutions. In nonelectrolyte solutions, the intermolecular forces are mostly comprised of weak Van der Waals interactions, which have a $$r^{-7}$$ dependence - as distance between molecules increases, the forces weaken by that distance raised to the 7th power. For practical purposes this can be considered ideal. In ionic solutions, however, there are significant electrostatic interactions between solute-solvent as well as solute-solute molecules. These electrostatic forces are governed by Coulomb's law, which has a $$r^{-2}$$ dependence - as distance between molecules increases, forces weaken by a factor of distance squared. Consequently, the behavior of an electrolyte solution deviates considerably from that of an ideal solution. Indeed, this is why, when solutions have a significant abundance of ions (as determined by judgment and experience) we utilize the activities of the individual components and not their concentrations to calculate deviations from ideal behavior. Examples in environmental engineering when we might need to use non-ideal solution approaches include when we are working with seawater, brackish water, brines from desalination processes or oil and gas exploration, and other solutions that result in high concentrations of electrolytes and charged solutes.

### The Debye-Hückel Limiting Law

In 1923, Peter Debye and Erich Hückel developed a theory that would allow us to calculate the mean ionic activity coefficient of the solution, $$\gamma_{\pm}$$, and could explain how the behavior of ions in solution contribute to this constant.

Assumptions of Debye-Hückel Theory

The Debye-Hückel theory is based on three assumptions of how ions act in solution:

1. Electrolytes completely dissociate into ions in solution.
2. Solutions of Electrolytes are very dilute, on the order of 0.01 M.
3. Each ion is surrounded by ions of the opposite charge, on average.

Debye and Hückel developed the following equation to calculate the mean ionic activity coefficient $$\gamma_{\pm}$$:

$\log_{10}\gamma_{\pm}=-\dfrac{1.824\times10^{6}}{(\varepsilon T)^{3/2}} z^2\sqrt{I} \label{1}$

where

• $$\varepsilon$$ is the dielectric constant of the medium, such as water for an aqueous solution,
• z is the charge of the cation or anion, respectively, and
• $$I$$ is the ionic strength of the solution.

The Equation $$\ref{1}$$ is known as the Debye-Hückel Limiting Law. The ionic strength is calculated using the following relation for all the ions in solution:

$I=\dfrac{1}{2}\sum_{i}C_{i}z_{i}^{2} \label{2}$

where $$C_{i}$$ and $$z_{i}$$ are the molar concentration and the charge of the ith ion in the electrolyte, respectively. Since most of the electrolyte solutions we study are aqueous $$(\varepsilon=78.54)$$ and have a temperature of 298 K, the Limiting Law in Equation \ref{1} reduces to

$\log_{10} \gamma_{\pm}=-0.509 z^2\sqrt{I} \label{3}$

The Debye-Hückel Limiting Law has been stated to be useful and valid for solutions with an ionic strength below 0.005 M.

### The Extended Debye-Hückel Limiting Law

A modified form of the above law takes into account additional spatial effects on the forces between between molecules, due to the sizes of the ions. This modified equation is:

$\log_{10}\gamma_{\pm}=-0.509 z^2\dfrac{\sqrt{I}}{1+Ba\sqrt{I}}$

where all previous parameters are the same, and a is the ion size parameter, an integer typically between 3 and 9. The extended D-H equation is stated to be useful and valid for solutions with an ionic strength below 0.1 M, and particularly when one compound dominates the ionic strength

### The Davies Equation

A final equation presented here for estimating activity coefficient is the Davies Equation, and it uses similar parameters and form as the two D-H models:

$\log_{10}\gamma_{\pm}=-0.509 z^2(\dfrac{\sqrt{I}}{1+\sqrt{I}}-0.3I)$

The Davies equation is stated to be useful at ionic strengths up to 0.5 M, making it a better choice than the D-H models for estimating gamma values in more concentrated solutions.

Additional models exist, such as the Pitzer Equation, also known as the Specific Interaction Model. However, we will stop our treatment of the topic at this point. Examples in class and in the homework will give you an opportunity to calculate I and gamma values.

## References

1. Atkins, P.W. Physical Chemistry. 5th Ed. New York: WH Freeman, 1994.
2. Chang, Raymond. Physical Chemistry for the Biosciences. Sausalito, California: University Science Books, 2005.