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**Electrolyte Solutions - Debye-Huckel Theory**

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

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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

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**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

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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

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**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

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**Charge distribution - II**

Using charge neutrality condition Or Where Is so called ionic strength

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**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

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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?

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**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

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**Activity coefficient With The activity coefficient is**

Which for dilute solutions becomes

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**Excess Gibbs free energy**

Where the last equality comes from

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**Osmotic pressure From math and thermodynamics Thus**

Integrating both sides over dV from infinite volume to V

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Osmotic pressure -2

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31 Polyelectrolyte Chains at Finite Concentrations Counterion Condensation N=187, f=1/3, LJ =1.5, u=3 c 3 = 1.5 10 -4 c 3 = 1.5 10 -2.

31 Polyelectrolyte Chains at Finite Concentrations Counterion Condensation N=187, f=1/3, LJ =1.5, u=3 c 3 = 1.5 10 -4 c 3 = 1.5 10 -2.

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