How to Derive the Debye Huckel Onsager Equation for Electrolyte Solutions
In this article, we will show you how to derive the Debye Huckel Onsager equation, which is a generalization of the Debye Huckel limiting law for electrolyte solutions. The Debye Huckel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength[^2^]. The Debye Huckel Onsager equation extends this law to account for the effects of ion mobility and electrical conductivity[^1^].
The derivation of the Debye Huckel Onsager equation involves the following steps:
Assume that the electrolyte solution is composed of spherical ions that obey the Boltzmann distribution and interact with each other through Coulomb forces.
Apply the Poisson equation to relate the electric potential and the charge density in the solution.
Introduce the Debye screening length, which is a measure of how far the electric field of an ion is screened by the surrounding ions.
Linearize the Poisson equation by assuming that the electric potential is small compared to the thermal energy.
Solve the linearized Poisson equation to obtain the Debye Huckel potential, which is an approximation of the electric potential around an ion.
Use the Nernst Einstein relation to relate the electrical conductivity and the ion mobility in the solution.
Use the Onsager reciprocal relations, which express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium[^3^], to relate the ion mobility and the activity coefficient.
Combine these relations to obtain the Debye Huckel Onsager equation, which expresses the activity coefficient as a function of ionic strength, ion charge, ion mobility, and electrical conductivity.
where $\\gamma_i$ is the activity coefficient of ion species $i$, $z_i$ is the charge number of ion species $i$, $e$ is the elementary charge, $\\epsilon_0$ is the vacuum permittivity, $\\epsilon_r$ is the relative permittivity of the solvent, $k_B$ is
the Boltzmann constant, $T$ is
the absolute temperature, $\\kappa$ is
the inverse of
the Debye screening length, and $\\Lambda$ is
the Onsager limiting equivalent conductivity.
The Debye Huckel Onsager equation is valid for moderately concentrated electrolyte solutions (up to about 0.1 mol/L) and can be used to calculate various thermodynamic properties such as osmotic pressure, chemical potential, and solubility product[^1^].
If you want to learn more about the derivation of
the Debye Huckel Onsager equation, you can download a PDF file with detailed steps and explanations from this link: Debye Huckel Onsager Equation Derivation PDF 35. aa16f39245