Derivation of the Activity Coefficient Equations

8An alternate approach is to add reversibly, an ion with no charge to the solution, and calculate the work needed in bringing the ion to full charge. Both methods lead to the same result for the activity coefficient. 

The electrical potential is obtained by solving the Poisson equation 

To solve the Poisson equation, we must express as a p function of the coordinates of the system and solve the resulting second-order differential equation to obtain ip(x,y, z), from which trci and hence, AG, p - p°, and 7 can be calculated. 

Solving the Poisson equation is not an easy thing to do. Assumptions must be made to relate, p to ib so that p can be eliminated from the equation. After this is done, approximations are required to simplify the resulting equation so that it can be solved. The assumptions used in the derivation limit the applicability of the resulting equations to low concentrations. 

The method attributed to Debye and Hiickel has been almost universally adopted by scientists. We will review the steps of their method and give the equations that are the end product of the derivation, but will leave it to others to present the mathematical details.6 

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