Theory of dilute electrolyte solutions and ionized gases
-
Upload
alexander-barton -
Category
Documents
-
view
232 -
download
0
description
Transcript of Theory of dilute electrolyte solutions and ionized gases
![Page 1: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/1.jpg)
1
Theory of dilute electrolyte solutions and ionized
gases
![Page 2: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/2.jpg)
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
ar
ar )(
rqqru jiij
e
![Page 3: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/3.jpg)
3
Charged particles in a gas
expand the exponential
the integral on the right diverges, so B2 is infinite
![Page 4: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/4.jpg)
4
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
![Page 5: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/5.jpg)
5
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)
![Page 6: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/6.jpg)
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
![Page 7: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/7.jpg)
7
Activity coefficients of electrolyte solutions
New reference state (Henry’s law reference state) based on the infinitelydiluted limit :
![Page 8: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/8.jpg)
8
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)
![Page 9: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/9.jpg)
9
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:
![Page 10: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/10.jpg)
10
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
![Page 11: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/11.jpg)
11
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
![Page 12: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/12.jpg)
12
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
![Page 13: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/13.jpg)
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
![Page 14: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/14.jpg)
14
Debye-Huckel theory
![Page 15: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/15.jpg)
15
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
![Page 16: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/16.jpg)
16
Debye-Huckel theory
and the average electrostatic potential
![Page 17: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/17.jpg)
17
also:
![Page 18: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/18.jpg)
18
Debye-Huckel theory
![Page 19: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/19.jpg)
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 computethe evolution of A as the system is charged.
![Page 20: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/20.jpg)
20
Debye-Huckel theorythe average total electrostatic potential
taking the Laplacian and using Poisson’s equation
![Page 21: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/21.jpg)
21
Debye-Huckel theorythe average charge density provided by ion 1 at the origin can be related to the rdf:
therefore
using spherical coordinates:
![Page 22: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/22.jpg)
22
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
![Page 23: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/23.jpg)
23
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:
![Page 24: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/24.jpg)
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
![Page 25: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/25.jpg)
25
Poisson-Boltzmann equation
and keeping only the first term
and because of charge neutrality
linearized PB equation
![Page 26: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/26.jpg)
26
Linearized PB equation
defining:
general solution:
![Page 27: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/27.jpg)
27
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
![Page 28: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/28.jpg)
28
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
![Page 29: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/29.jpg)
29
electrostatic interaction energy between ion i and ion j
Coulomb potentialat short distances
shielded potential for longer distances
![Page 30: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/30.jpg)
30
total charge density
![Page 31: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/31.jpg)
31
total charge in an spherical shell surrounding ion i
rmax surrounding any ion where the charge is a maximum
![Page 32: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/32.jpg)
32
Debye length
measure of the ion atmosphere around a central ion
total charge outside an ion i
![Page 33: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/33.jpg)
33
thermodynamic properties
to integrate:
![Page 34: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/34.jpg)
34
thermodynamic properties
![Page 35: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/35.jpg)
35
chemical potential
electrostatic activitycoefficient
![Page 36: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/36.jpg)
36
Mean activity coefficient
because the activity coefficients of anions and cations are not independent of each other
After some algebra
![Page 37: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/37.jpg)
37
Debye-Huckel activity coefficient
in the limit of very low ionic strength K 0
Debye-Huckel limiting law
![Page 38: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/38.jpg)
38
Activity coefficients of various salts as a function of molarity
DH
DH
DH solid line: experimental
dashed line:
![Page 39: Theory of dilute electrolyte solutions and ionized gases](https://reader036.fdocuments.us/reader036/viewer/2022062306/5a4d1b297f8b9ab0599986c0/html5/thumbnails/39.jpg)
39
Mean molar activity coefficient for HCl in water
DH