ENGR-1100 Introduction to Engineering Analysis

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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
qJezJ. 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