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Theory of dilute electrolyte solutions and ionized

gases

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Electrolyte solutions and plasmas• They have very long range interactions• Consider a simple Coulombic potential

• e is the temperature dependent dielectric constant e= er x e0

ar

ar )(

rqqru jiij

e

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Charged particles in a gas

expand the exponential

the integral on the right diverges, so B2 is infinite

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Charged particles in a gasHowever, as the distance between particles increases shielding may exist due to the presence of other charged particles between them, thus theactual range is shorter than that predicted by 1/r. This is the basis ofDebye-Huckel theory

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Charged particles in a gasIn ionized gases, the system contains ions and electrons; in an electrolyte (liquid) solution it contains ions and solvent

We will define systems where ions are treated atomistically and solvent is a continuum. We will calculate properties based on the PMF

Goal: derive expressions for activity coefficients of ions in solution

So far:

reference state: pure component (mi0 pure component chemical potential)

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Activity coefficients of electrolyte solutions

reference state: pure component (mi0 pure component chemical potential)

gi = 1 for the pure component limit and departs from 1 as the solution is diluted

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Activity coefficients of electrolyte solutions

New reference state (Henry’s law reference state) based on the infinitelydiluted limit :

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Other reference states

based on molality (Mi), number of moles of solute per kg of solvent;mi

0 is the chemical potential of the species in a hypothetical 1molal solution

in solutions of neutral molecules, the Henry’s law activity coefficient is 1for very diluted solutions;but in electrolyte solutions the deviations are large; for example

for a solution of NaCl in a a0.01 molal aqueous solution (mole fraction ofsolute of 1x10-4)

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this theory is valid for ionized gases (ions and electrons): e =1and for electrolyte solutions (cations and anions) : e is based on the solvent

Balance of charges:

For N initial undissociated molecules in a volume V, charge neutrality requires:

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Debye-Huckel theoryions are treated microscopically and solvent as a continuum. Issue: when the separation between particles is small, the molecular size of the solvent is important;for this reason the model applies to dilute solutions (large separations between particles) and not to concentrated solutions

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Debye-Huckel theoryModel is based on electrostatics.The electrostatic potential due to a set of point charges qi at positions ri’ ina continuum dielectric medium is:

if instead of a set of point charges there is a continuous charge distribution

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taking the laplacian derivative of this expression:

from electrostatic potential theory

therefore we get Poisson equationso given a charge distribution functionwe can calculate the potential function solving Poisson equation with boundary conditions

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Solving Poisson equation for various charge distributions

lets assume that we know

and h2(r) with f2(r);

for a charge distribution that is the sum of h1 + h2, the solution is f1 + f2

superposition principle

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Debye-Huckel theory

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Debye-Huckel theoryconsider an ion located at position vector r1 taken as the origin of the coordinate system r1(0,0,0); the electrostatic potential at this point is

and the total electrostatic potential is

and the electrostatic potential energy is

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Debye-Huckel theory

and the average electrostatic potential

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also:

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Debye-Huckel theory

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Debye-Huckel theory

So, if we can obtain the average electrostatic potential acting on ion j by all the other ions in the system as a function of T and ion density, we can computethe evolution of A as the system is charged.

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Debye-Huckel theorythe average total electrostatic potential

taking the Laplacian and using Poisson’s equation

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Debye-Huckel theorythe average charge density provided by ion 1 at the origin can be related to the rdf:

therefore

using spherical coordinates:

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Debye-Huckel theory

the solution to this equation can be considered in two regions: one is a hard core

and the solution is:

a/2 is theradius of the sphere

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Debye-Huckel theoryevaluating the integration constants:

potential due to the central ionpotential due to the charge distribution external to the sphere ofradius r

for the 2nd region, r >a, we solve:

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Debye-Huckel theory

for the 2nd region, r >a, we solve:

the pmf is the result of the interaction of ion i with all the ions

Poisson-Boltzmann equation

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Poisson-Boltzmann equation

and keeping only the first term

and because of charge neutrality

linearized PB equation

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Linearized PB equation

defining:

general solution:

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Linearized PB equationat infinite distances the EP vanishes, then C4 is 0

and

But the EP has to be the same at the boundary between regions and the derivative has to be continuous

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Linearized PB equationBut the EP has to be the same at the boundary between regions and the derivative has to be continuous

and the EP for all the other ions other than ion 1

dependence on temperature and density through parameter K

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electrostatic interaction energy between ion i and ion j

Coulomb potentialat short distances

shielded potential for longer distances

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total charge density

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total charge in an spherical shell surrounding ion i

rmax surrounding any ion where the charge is a maximum

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Debye length

measure of the ion atmosphere around a central ion

total charge outside an ion i

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thermodynamic properties

to integrate:

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thermodynamic properties

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chemical potential

electrostatic activitycoefficient

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Mean activity coefficient

because the activity coefficients of anions and cations are not independent of each other

After some algebra

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Debye-Huckel activity coefficient

in the limit of very low ionic strength K 0

Debye-Huckel limiting law

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Activity coefficients of various salts as a function of molarity

DH

DH

DH solid line: experimental

dashed line:

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Mean molar activity coefficient for HCl in water

DH