1. Electrolyte Solutions - Debye-Huckel Theory
Lecture 18
Electrolyte Solutions - Debye-Huckel Theory
Charge neutrality
Electrostatics effects
Charge distribution
Activity coefficient
Thermodynamics functions

2. Charge neutrality
When ionic material dissociates in the solvent, it has to stay charge neutral
Where zi is the charge of an ion and and ni is concentration
Considering equilibrium between ionic solid and solution, e.g., for NaCl
Which means that due to charge neutrality only the sum of the chemical potentials for anions and cations is defined, and not independent chemical potentials

3. Chemical potential We express chemical potential as
This definition is slightly different than before vs
But the difference can be absorbed into the definition of
For electrostatic problem one can split the activity coefficient into one due to short range interactions and one due to electrostatic (long range) interactions

4. Electrostatics
Electrostatic potential around a single charge in a medium with a dielectric constant
With many ions the potential will be modified by the time average potential due to other ions. It has to satisfy the Poisson equation
Where is the charge density around ion J

5. Charge distribution In spherical coordinates
The charge distribution is connected with potential via Boltzmann distribution
Where the sum is over all species. Expanding exponential and inserting to the top equation
Note that the first term in the expansion is zero due to charge neutrality

6. Charge distribution - II
Using charge neutrality condition Or
Where
Is so called ionic strength

7. Charge distribution - II
Equation
Has a physical solution
Where A is constant that can be evaluate requiring that total charge in the cloud around ion J is negative zJ leading to
Where a is the distance of minimum approach between cation and anion

8. Screening length
The inverse of is the Debye screening length over which the ion is neutralized by the cloud of other ions. Remembering that
One can see that the Debye length is long at high T and low ion concentrations. Large dielectric constant also promotes long Debye length as the interactions between ions are weaker. When the Debye length is ~ ion size the theory does not apply. Why?

9. Excess chemical potential
To find our excess chemical potential we can use so called charging process where initially neutral ion is charged for 0 to its final charge qJ=ezJ. We can calculate the work done during the charging process as
With
Upon integration

10. Activity coefficient With The activity coefficient is
Which for dilute solutions becomes

11. Excess Gibbs free energy
Where the last equality comes from

12. Osmotic pressure From math and thermodynamics Thus
Integrating both sides over dV from infinite volume to V

13. Osmotic pressure -2