More

# Debye-Hückel Theory of Electrolytes

1. 1. Assumptions
2. 2. The Theory
3. 3. Sample Problem
4. 4. References
5. 5. Contributors

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 $$\frac{1}{r^7}$$ dependence, and 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 $$\frac{1}{r^{2}}$$ dependence. Consequently, the behavior of an electrolyte solution deviates considerably from that an ideal solution. Indeed, this is why we utilize the activity of the individual components and not the concentration to calculate deviations from ideal behavior. 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

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.

### The Theory

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

$log\gamma_{\pm}=-\dfrac{1.824\times10^{6}}{(\varepsilon T)^{3/2}}\mid z_{+}z_{-}\mid\sqrt{I}$

where $$\varepsilon$$ is the dielectric constant, $$z_{+}$$ and $$z_{-}$$ are the charges of the cation and anion, respectively, and I is a quantity called the ionic strength of the solution. The above equation is known as the Debye-Hückel Limiting Law. I is calculated by the following relation:

$$I=\frac{1}{2}\sum_{i}m_{i}z_{i}^{2}$$

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

$$log\gamma_{\pm}=-0.509\mid z_{+}z_{-}\mid\sqrt{I}$$.

### Sample Problem

To use the information we have now learned, let's calculate ionic strength, mean ionic activity coefficient $$\gamma_{\pm}$$, and the mean ionic molality $$m_{\pm}$$for a 0.02 molal solution of zinc chloride, $$ZnCl_{2}$$. Zinc chloride will dissolve as $$ZnCl_{2}\longrightarrow Zn^{2+}(aq)+2Cl^{-}(aq)$$. The concentrations of the zinc and chloride ions will then be 0.02 and 0.04 molal, respectively. First let's calculate the mean ionic molality. From reading Electrolyte Solutions, the mean ionic molality is defined as the average molality of the two ions:

$$m_{\pm}=(m_{+}^{\nu+}+m_{-}^{\nu-})^{\frac{1}{\nu}}$$,

where $$\nu$$ is the stoichiometric coefficient of the ions, and the total of the coefficients in the exponent. In our case, the mean ionic molality is

$$m_{\pm}=(m_{Zn}^{\nu(Zn)}+m_{Cl}^{\nu(Cl)})^{\frac{1}{\nu Zn+\nu Cl}}=[(.02)^{1}+(.04)^{2}]^{\frac{1}{3}} =(.02+.0016){}^{\frac{1}{3}}=(.0216){}^{\frac{1}{3}}=0.278$$.

To calculate the mean ionic activity coefficient, we first need the ionic strength of the solution from Equation (2):

$$I=\frac{1}{2}[(.02)(+2)^{2}+(.04)(-1)^{2}]=\frac{1}{2}(.08+.04)=.06$$.

Now we can use Equation (3) to calculate the activity coefficient:

$$log\gamma_{\pm}=-0.509\mid(+2)(-1)\mid\sqrt{.06}=(-.509)(2)(.245)=-0.250$$.

$$\gamma_{\pm}=10^{-.25}=.1627$$

### 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.

### Contributors

• Konstantin Malley (UCD)