# The Free Energy of Charged Particles in an Electrolyte Solution

by Bryan Hassell AP 255 Fall 11

If we know the dependence of free energy of distance between charged particles and take the derivative of the free energy with respect to distance, we can find the force of interaction within a certain degree of approximation. There are several reasons why one would investigate the free energy as opposed to starting with the forces of interaction and the history of this debate is shown well in [1] and [2]. Here the authors of [1] explain that the matter of free energy of particles is one of independent interest and investigating the development of the calculations is extremely instructive thus meriting critical examination. In this investigation the previous errors found in determining the free energy of charged particles was found and two methods were indicated which resulted in the same expression for the forces of interaction found in earlier studies.

The first is purely thermodynamic and starts from an expression for the differential of free energy of a system of charged spheres submerged in an electrolyte solution. This method assumes that the surface of each sphere is an isopotential surface and each surface carries a charge $e_i$, which remains invariant (*this is later shown that combined requirements for isoptentiality and constancy of the total charge can only approximately be met). More rigorous though is assuming that each surface has a constant potential and the charge may change as the ionic atmospheres of the particles overlap as they approach each other. This is a case approximately correct if the potential is determined by adsorbed ions with a very high specific energy of adsorption and the free energy of the system would be:

$d\Phi = \sum \phi_ide_i - \sum (X_idx_i + Y_idy_i + Z_idz_i), (1)$

where $phi_i$ and $e_i$ are the potential and charge of the ith spherical particle, respectively; $x_i, y_i, z_i$ are the coordinates; $X-i, Y_i, Z_i$ are the components of the external forces acting on the particle. Here the terms of the charges of the ionic atmosphere are not introduced since the coordinates of the ions a dependent quantities. Integrating (1) with fixed positions of the particles we get:

$\Phi = \Phi_0 + \sum \int_{0}^{e_i}\phi_ide_i, (2)$

where $\phi_0$ is the free energy of corresponding to a state of the system with uncharged and non-interacting particles so that the free energy is independent of coordinates. The part of the free energy related to the forces of interaction, $F_e$, are then just the terms within the summation. The derivative of $F_e$ with respect to distance between particles gives the force of interaction between particles and the work of approach of particles from infinity is then:

$A = F_e - F_\infin = \sum \int_{0}^{e_i}(\phi_i-\phi_\infin) de_i. (3)$

For the sake of brevity some steps are excluded but will be described. The free energy $F_e$ can be expressed in terms of the electric fields and ionic atmospheres by differentiating $F_e$ with constant $\phi_i$ and $\epsilon$ and expressing the derivative through a normal surface derivative we get:

$dF_e = {\epsilon \over 4\pi}\int \phi d({\partial\phi \over \partial n})dS. (4)$

Transforming this using Green's Theorem and the Poisson Equation and integrating over the entire volume of the electrolyte we get:

$F_e = {\epsilon \over 8\pi}\int {|grad \phi|}^2 dv - \int\int_{0}^{\rho}\phi d\rho dv. (5)$

And we can take the dependence of $\rho$ on $\phi$ using the Poisson-Bolztmann equation and so the simplified equation for small potentials becomes:

$F_e = {\epsilon \over 8\pi}\int {|grad \phi|}^2 dv - {\kappa}^2\int \phi^2 dv. (6)$

This second term was introduced by Derjaguin [2] and was absent from other works and yielded a correct value for the forces of interaction. In [1] and [2] they also go on to show an extensive derivation of how the free energy of charged particles satisfies the Gibbs-Hemholtz equation and another way to come to equation (6) shown here by an artificial approach based on a hypothetical process of charging ions. This approach was used by Debye in calculating the electrical component of the free energy of strong electrolytes and an analogous approach was also done by Casimir.