Lecture 18 Electrolyte Solutions - Debye-Huckel Theory

Charge neutrality Electrostatics effects Charge distribution Activity coefficient Thermodynamics functions

Charge neutralityWhen 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

€

zini = 0i

∑

€

μNasd + μCl

sd ⇔ μNasol + μCl

sol

Chemical potentialWe 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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μ jsol = μ j

0( )

sol+ kT lnn j + kT lnγ

€

kT ln n j

€

kT ln X j

€

μ j0

( )sol

€

μ jsol = μ j

0( )

sol+ kT lnn j + kT lnγ + kT lnγ E

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 averaged potential due to other ions. It has to satisfy the Poisson equation

Where is the charge density around ion J

€

ϕ J (r) =qJ

εr

€

∇2ϕ J (r) =−4π

ερ e (r)

€

ρe (r)

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

€

1

r

d2 rϕ J (r)( )dr2

=−4π

ερ e (r)

€

ρe (r) = e zini

i

∑ e−zi eϕ J (r) / kT

€

1

r

d2 rϕ J (r)( )dr2

=−4πe

εzini

i

∑ (1− zieϕ J (r) /kT)

Charge distribution - II Using charge neutrality condition

Or

Where

Is so called ionic strength

€

1

r

d2 rϕ J (r)( )dr2

=4πe2

εkTnizi

2

i

∑ ϕ J (r)

€

1

r

d2 rϕ J (r)( )dr2

= κ 2ϕ J (r)

€

κ 2 =4πe2

εkTnizi

2

i

∑

Charge distribution - II Equation

Has a physical solution

Where A is constant that can be evaluated 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

€

1

r

d2 rϕ J (r)( )dr2

= κ 2ϕ J (r)

€

ϕ J (r) =Ae−κr

r

€

ϕ J (r) =zJe

ε

eκa

1+ κa

e−κr

r

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?

€

κ 2 =4πe2

εkTnizi

2

i

∑€

κ

Excess chemical potential

To find our excess chemical potential we can use so called charging process where initially neutral ion is charged from 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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Wel =0

1

∫ dr4πr2

a

∞

∫d ξzi( )eρ J (r)

r

€

ρJ (r) =−eκ 2ξzJe

κa

4πrε 1+ κa( )e−κr

€

μJex = Wel = −

zJ2e2

2ε

κ

1+ κa

Activity coefficientWith

The activity coefficient is

Which for dilute solutions becomes

€

μJex = Wel = −

zJ2e2

2ε

κ

1+ κa

€

lnγ JE =

μ Jex

kT= −

zJ2e2

2εkT

κ

1+ κa

€

lnγ JE = −

zJ2e2κ

2εkT

€

κa → 0

Excess Gibbs free energy

Where the last equality comes from

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Gex = μ iex

i

∑ N i = −e2κ

2εzi

2

i

∑ N i = −Ve 2κ

2εzi

2

i

∑ ni = −Vκ 3

8πkT

€

κ 2 =4πe2

εkTnizi

2

i

∑

Osmotic pressureFrom math and thermodynamics

Thus

Integrating both sides over dV from infinite volume to V

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∂G

∂V

⎛

⎝ ⎜

⎞

⎠ ⎟N ,T

=∂G

∂P

⎛

⎝ ⎜

⎞

⎠ ⎟N ,T

∂P

∂V

⎛

⎝ ⎜

⎞

⎠ ⎟N ,T

= V∂P

∂V

⎛

⎝ ⎜

⎞

⎠ ⎟N ,T

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∂P

∂V

⎛

⎝ ⎜

⎞

⎠ ⎟N ,T

=1

V

∂G

∂V

⎛

⎝ ⎜

⎞

⎠ ⎟N ,T

€

∂P

∂V ⎛ ⎝

⎞ ⎠N ,T

=∞

V

∫1

V

∂G

∂V ⎛ ⎝

⎞ ⎠N ,T∞

V

∫ = P(V ) − P(∞) = P(V )

Osmotic pressure -2