If you like us, please share us on social media.
The latest UCD Hyperlibrary newsletter is now complete, check it out.

ChemWiki: The Dynamic Chemistry E-textbook > Physical Chemistry > Physical Properties of Matter > Solutions and Mixtures > Non-ideal Solutions > Debye-Hückel Theory

Copyright (c) 2006-2014 MindTouch Inc.

This file and accompanying files are licensed under the MindTouch Master Subscription Agreement (MSA).

At any time, you shall not, directly or indirectly: (i) sublicense, resell, rent, lease, distribute, market, commercialize or otherwise transfer rights or usage to: (a) the Software, (b) any modified version or derivative work of the Software created by you or for you, or (c) MindTouch Open Source (which includes all non-supported versions of MindTouch-developed software), for any purpose including timesharing or service bureau purposes; (ii) remove or alter any copyright, trademark or proprietary notice in the Software; (iii) transfer, use or export the Software in violation of any applicable laws or regulations of any government or governmental agency; (iv) use or run on any of your hardware, or have deployed for use, any production version of MindTouch Open Source; (v) use any of the Support Services, Error corrections, Updates or Upgrades, for the MindTouch Open Source software or for any Server for which Support Services are not then purchased as provided hereunder; or (vi) reverse engineer, decompile or modify any encrypted or encoded portion of the Software.

A complete copy of the MSA is available at http://www.mindtouch.com/msa

Debye-Hückel Theory

A solution is defined as a homogeneous mixture of two or more components existing in a single phase. In this description, the focus will be on liquid solutions because within the realm of biology and chemistry, liquid solutions play an important role in multiple processes. Without the existence of solutions, a cell would not be able to carry out glycolysis and other signaling cascades necessary for cell growth and development. Chemists, therefore, have studied the processes involved in solution chemistry in order to further the understanding of the solution chemistry in nature.


The mixing of solutions is driven by entropy, opposed to being driven by enthalpy. While an ideal gas by definition does not have interactions between particles, an ideal solution assumes there are interactions. Without the interactions, the solution would not be in a liquid phase. Rather, ideal solutions are defined as having an enthalpy of mixing or enthalpy of solution equal to zero (ΔHmixing or ΔHsolution = 0). This is because the interactions between two liquids, A-B, is the average of the A-A interactions and the B-B interactions. In an ideal solution the average A-A and B-B interactions are identical so there is no difference between the average A-B interactions and the A-A/B-B interactions.

Since in biology and chemistry the average interactions between A and B are not always equivalent to the interactions of A or B alone, the enthalpy of mixing is not zero. Consequently, a new term is used to describe the concentration of molecules in solution. Activity, \(a_1\), is the effective concentration that takes into account the deviation from ideal behavior, with the activity of an ideal solution equal to one.

An activity coefficient, \( \gamma_1\), is utilized to convert from the solute’s mole fraction, \(x_1\), (as a unit of concentration, mole fraction can be calculated from other concentration units like molarity, molality, or percent by weight) to activity, \(a_1\).

\( a_1=\gamma_1x_1\)

Debye-Hückel Formula

The Debye-Hückel formula is used to calculate the activity coefficient.

\( \log \gamma_\pm = - \displaystyle \frac{1.824 \times 10^6} { \left( \epsilon T \right)^{3/2}} | z_+ z_- | \sqrt I \)

This form of the Debye-Hückel equation is used if the solvent is water at 298 K.

\[ \log \gamma_\pm = - 0.509 | z_+ z_- | \sqrt I \]

\(\gamma_\pm\) mean ionic activity coefficent
\(z_+\) catonic charge of the electrolyte for \( \gamma_\pm \)
\(z_-\) anionic charge of the electrolyte for \( \gamma_\pm \)
\(I\) ionic strength
\(\epsilon\) relative dielectric constant for the solution
\(T\) temperature of the electrolyte solution
Example 1

Consider a solution of 0.01 M MgCl2 (aq) with an ionic strength of 0.030 M. What is the mean activity coefficient?


\( \log \gamma_\pm = - \displaystyle \frac{1.824 \times 10^6} { \left( \epsilon T \right)^{3/2}} | z_+ z_- | \sqrt I \)

\( \log \gamma_\pm = - 0.509 | z_+ z_- | \sqrt I \)

\( \log \gamma_\pm = - \displaystyle \frac{1.824 \times 10^6} { \left( 78.54 \cdot 298 \mathrm {K} \right)^{3/2}} | 2 \cdot 1 | \sqrt {0.0030} \)

\( \log \gamma_\pm = - 0.509 | 2 \cdot 1 | \sqrt {0.0030} \; m \)

 \( \gamma_\pm = 0.67 \)

 \( \gamma_\pm = 0.67 \)


  1. Chang, Raymond. Physical chemistry for the chemical and biological sciences. 3rd ed. Sausalito, Calif: University Science Books, 2000. Print.
  2. Ochiai, E-I. (1990) "Paradox of the Activity Coefficient \(\gamma_\pm\)." J. Chem. Educ. 67: 489.


  • Shirley Bradley (Hope College), Kent Kammermeier (Hope College)

You must to post a comment.
Last modified
09:26, 15 Jan 2014



(not set)
(not set)

Creative Commons License Unless otherwise noted, content in the UC Davis ChemWiki is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. Permissions beyond the scope of this license may be available at copyright@ucdavis.edu. Questions and concerns can be directed toward Prof. Delmar Larsen (dlarsen@ucdavis.edu), Founder and Director. Terms of Use