1. Theory of dilute electrolyte solutions and ionized gases

2. 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

3. Charged particles in a gas
expand the exponential
the integral on the right diverges, so B2 is infinite

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

5. Charged particles in a gas
In 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)

6. 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

7. Activity coefficients of electrolyte solutions
New reference state (Henry’s law reference state) based on the infinitely
diluted limit :

8. Other reference states
based on molality (Mi), number of moles of solute per kg of solvent;
mi0 is the chemical potential of the species in a hypothetical 1molal solution
in solutions of neutral molecules, the Henry’s law activity coefficient is 1
for 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 of
solute of 1x10-4)

9. this theory is valid for ionized gases (ions and electrons): e =1
and 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:

10. Debye-Huckel theory
ions 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

11. Debye-Huckel theory Model is based on electrostatics.
The electrostatic potential due to a set of point charges qi at positions ri’ in
a continuum dielectric medium is:
if instead of a set of point charges there is a continuous charge distribution

12. taking the laplacian derivative of this expression:
from electrostatic potential theory
therefore we get Poisson equation
so given a charge distribution function
we can calculate the potential
function solving Poisson equation
with boundary conditions

13. 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

14. Debye-Huckel theory

15. Debye-Huckel theory
consider 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

16. Debye-Huckel theory
and the average electrostatic potential

17. also:

18. Debye-Huckel theory

19. 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 compute
the evolution of A as the system is charged.

20. Debye-Huckel theory the average total electrostatic potential
taking the Laplacian and using Poisson’s equation

21. Debye-Huckel theory
the average charge density provided by ion 1 at the origin can be related to the rdf:
therefore
using spherical coordinates:

22. Debye-Huckel theory
the solution to this equation can be considered in two regions: one is a hard core
a/2 is the
radius of the sphere
and the solution is:

23. Debye-Huckel theory evaluating the integration constants:
potential due to the
charge distribution
external to the sphere of
radius r
potential due to the central ion
for the 2nd region, r >a, we solve:

24. 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

25. Poisson-Boltzmann equation
and keeping only the first term
and because of charge neutrality
linearized PB equation

26. Linearized PB equation
defining:
general solution:

27. Linearized PB equation
at 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

28. Linearized PB equation
But 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

29. electrostatic interaction energy between ion i and ion j
Coulomb potential
at short distances
shielded potential for longer distances

30. total charge density

31. total charge in an spherical shell surrounding ion i
rmax surrounding any ion where the charge is a maximum

32. Debye length measure of the ion atmosphere around a central ion
total charge outside an ion i

33. thermodynamic properties
to integrate:

34. thermodynamic properties

35. chemical potential
electrostatic activity
coefficient

36. Mean activity coefficient
because the activity coefficients of anions and cations are not
independent of each other
After some algebra

37. Debye-Huckel activity coefficient
in the limit of very low ionic strength K 0
Debye-Huckel limiting law

38. Activity coefficients of various salts as a function of molarityDH
solid line: experimental DH
dashed line: DH

39. Mean molar activity coefficient for HCl in waterDH