Interactions in an electrolyte
Sähkökemian peruseet
KE-31.4100
Tanja Kallio
tanja.kallio@aalto.fi
C213
CH 2.4-2.5
Solvent – ion interactions
Solvent – ion interactions
W1 = discharging an ion
ze
ion
neutral
1
vacuum
3
2
ze
qdq
z 2e 2
w1    V (q )dq   

4

a
8 0 a
0
0
0
total
W2 = cavity formation +
surface tension
W2 ~ negligible
solvent
W3 = charging an molecule
w3 
z 2e 2
8 0 r a
z 2e2 
1
1  
wel  GIS  
80 a   r 
Experimental values for hydration energy
 /
Ghyd
 /
H hyd

Shyd
/
kJ mol1
kJ mol1
J mol1 K1
69
102
138
149
170
148
280
337
72
100
78
69
65
1055
475
365
295
275
250
285
160
130
1830
1505
1840
1980
4265
1090
530
415
330
305
280
325
215
205
1945
1600
1970
2115
4460
131
161
130
93
84
78
131
163
241
350
271
381
370
576
5,5
6,4
6,7
3,5
8,6
15,8
12,4
84,1
143,6
32,2
28,9
30,2
35
53
Anionit
–
F
–
Cl
–
Br
–
I
–
OH
–
NO3
133
181
196
220
133
465
340
315
275
430
510
365
335
290
520
156
94
78
55
180
4,3
23,3
30,2
41,7
0,2
179
300
310
95
34,5
ClO4
250
205
245
76
49,6
Kationit
Säde/pm
+
H
Li+
Na+
K+
Rb+
Cs+
NH4+
Me4N+
Et4N+
Mg2+
Ca2+
Fe2+
Ni2+
Fe3+
–

V m/
cm3 mol1
Ion – ion interactions
Debye lenght
Spatial distribution of ions around the central ion obeys
Boltzmann distribution
ci (r )  cib e  zi ( r )
(2.32)
Charge density around the central ion is obtained by
summarizing charge densities of all the ions
(r )   z i Fci 
i
 z ( r )
z i Fcib e i
i


z i Fcib 1 
i

z i F(r ) 
F 2 (r )
z i2 cib


RT 
RT
i
electroneutrality
first term of Taylor series
Dependence of potential on charge density is given by Poisson equation

 
0 r
2
F2
    ;  
 0  r RT
2
2
2
 z i2 cib
(2.34)
i
Debye length = thickness of the double layer
(2.33)
Falloff in the electrostatic potential
e  ( r  a ) 1
 (r ) 
4 0 r 1  a r
zc e
(2.36)
Debye-Hückel limiting law (1/2)
Electrostatic work done to move the central ion inside the
ion cloud
zc e
wion ion  N A  atm (a )dq ; atm (a )   (a )  V (a )  
0
potential distribution around
the central ion (2.36)
N A  z c e 2
wion  ion  
80  r 1  a

4 0 r 1  a
zc e
potential field crated by the
central ion at distance a (2.37)
(2.39)
When diluting the solution from concentration c1 to c2 (infinite dilute) work is done
a 
c 
 
wdil  RT ln 2   RT ln 2   RT ln 2   wosm  wion -ion
(2.40)
a
c

 1
 1
 1
By combining (2.39) and (2.40)
zi e2
wionion

ln  i  

RT
80  r kT 1  a
2 = 1 (infinite dilution)
activity coefficients origins from
electrostatic interactions between ions
(2.41)
V (r ) 
zc e 1
(2.37)
40  r r
Debye-Hückel limiting law (2/2)
Sifting to log system
log  i   zi2
A I
1  Ba I
A
; I
1
zi2ci

2 i
(2.42)
ion strength
Utilizing definition of mean activity:
log    z  z 
A I
1  Ba I
(2.43)
experimental
D-H law
D-H limiting law
1,8246 106
mol1/ 2 dm 3 / 2 K 3 / 2
3/ 2
 rT 
50,29 108
B
cm 1mol1/ 2 dm 3 / 2 K1/ 2
1/ 2
 rT 
Ionpairs
Equilibrium constants for ion assosiation/dissosiataion
Kd 
 A cA   B cB 
c AB

   c 1
± = 1 → Kd = 2c/(1  )
1 
Bjerrumin theory
Ions around the central ion obey Maxwell-Bolzman distribution
Potential profile immediately around the central ion obeys (2.37)
Hypothesis: ions form ion pair when distance is smaller than q
3
 z z e 2  b
 x  4 e x dx
K a  4000N A 
 40  r kT 

 2

Fouss theory
Ions must be in contact to form an ionpair
Probability of forming an ion pair depends on number of ions, solvent volume,
space occupied the species and electrostatic energy on the surface of the ion
1 
4
 1000 N A  a 3  e  E ( a ) / kT  K a
3

 2c
Super acids and Hammett acid function
very acidic acids  extension to the conventional pH
scale is needed
a weak indicator base B is added into the acid solution
B + H+  BH+
measurable
unknown
concentration depends on the pH of the
super aid
Hammet acid function is defined so that it becomes
equal to pH in ideally diluted aqueous solutions
H 0   log
cH  c B
c BH 
 log
aBH 
aB
  log(cH  )  log
equilibrium
measurable
constant for the
indicator acid
B
 BH
Hammett acid function H0 for 0.1 M HCl-solutions.
Abscissa: content of the organic component in mol-%

for super basis
BH + OH−(H2O)n  B− + (n + 1)H2O
H   pK w  log [OH  ]  (n  1) log [H 2O]
M.A. Paul and F.A. Long, Chem. Rev. 57 (1957) 1-45
Summary
Interaction in electrolyte solutions
solvent – ion interactions
ion – ion interactions
superacids
z 2e2 
1
1  
wel  GIS  
80 a   r 
pK d  log
[B]
 H0
[BH  ]
H   pK w  log [OH  ]  (n  1) log[H 2O]
ion
neutral
1
F2
 
 0 r RT
2
vacuum 2
3
solvent
 zi2cib
i
Debye length
Debye – Hückel law
log    z  z 
A I
1  Ba I