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In 1923, Peter Debye and Erich Hückel published a paper in which they derived a simple equation that could be used to quantitatively predict mean activity coefficients of electrolytes in dilute aqueous solutions [1]. The equation soon became one of the most widely used equations in analytical and physical chemistry and was soon adapted also for calculation of many other thermodynamical and transport properties in electrolyte solutions.
Despite its widespread use, the Debye-Hückel equation is still not well understood, as evidenced not only by physically incorrect derivations of the Debye-Hückel equation in numerous respected textbooks of physical chemistry and electrochemistry [2-4], but also in the classic treatise on the thermodynamics of polyelectrolyte solutions [5, 6].
Here we show that the activity coefficient of a single ion in the Debye-Hückel equation can be easily determined from the electric potential in the solution, while the energy involved in attractive and repulsive interactions within the model ionic solution can be determined simultaneously. The change in Gibbs free energy can be evaluated by a two-step procedure, where the first step refers to the elucidation of the attractive electrostatic interactions and the second step refers to the repulsive interactions. The absolute value of the attractive potential energy is exactly twice that of the repulsive one. With this new insight into the Debye-Hückel equation, it is possible to better understand the thermodynamics of electrolyte solutions.
References:
[1] P. Debye and E. Hückel: Zur Theorie der Elektrolyte. Phys. Z. 1923, 24, 185-206.
[2] P. W. Atkins and J. De Paula: Atkins' physical chemistry. Oxford University Press, Oxford, 2010.
[3] D. A. MacInnes: The principles of electrochemistry. Reinhold Publishing Corporation, New York, 1939.
[4] J. O. M. Bockris and A. K. N. Reddy: Modern electrochemistry. Ionics. Plenum Press, New York, 2nd edn., 1998.
[5] A. Katchalsky, S. Lifson and J. Mazur: The electrostatic free energy of polyelectrolyte solutions. 1. Randomly kinked macromolecules. J. Polym. Sci. 1953, 11, 409-423.
[6] R. A. Marcus: Calculation of Thermodynamic Properties of Polyelectrolytes. J. Chem. Phys. 1955, 23, 1057-1068.
