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CHAPTER 7
SOLUTION OF ELECTOLYTES
Electrochemistry is concerned with:
1.
The properties of solutions of electrolytes (e.g. conductivity,
association and dissociation)
2.
Processes that occur at electrodes when they immersed in
solutions of electrolytes and applications.

A very great deal can be learned about the behavior of ions in
water and other solvents by investigations of electrical effects.

Measurements of conductivities of aqueous solutions at various
emf lead to understanding of:
-
the extent to which substances are ionized (degree of
dissociation) in water
-
the association of ions with surrounding water
molecules and the way in which ions move in water.
(It is convenient to begin our study of electrochemistry with a brief
survey of the development of concepts related to electricity):
1
Electrical Units:
Depends on the permittivity (dielectric constant) of the medium:
1.
The electrostatic force F between two changes Q1 and
Q2 separated by a distance r in:
1) In vacuum having a permittivity (dielectric constant) o:
F  Q1Q2 / r 2
F
where,
Q1Q2
4o r 2
 o  8.854 x10 12 C 2 J 1m 1
(When Q1 and Q2 are the same F is repulsive and when
they are different F is attractive)
2) In a medium having a permittivity :
F
where,
Q1Q2
4or 2
r = o 
F 
or
Q1Q2
4r r 2
(relative permittivity)
Unit of charge Q is coulomb C
Unit of distance r is meter m
Unit of force F is Neuton N which is Jm-1
2
2.
The electric field E at any point is the force exerted on a
unit change (1C) at that point. The field strength at a distance r
from a change Q, in a medium of dialectic constant , is thus:
E
Q
4or 2
F
or
Q
4r r 2
The SI unit of field strength is NC-1, since:
Joule = Volt x coulomb = Newton x meter and N = J m-1
 N C-1 = J m-1 C-1 = V C m-1 C-1 = V m-1
(The SI unit of field strength E is usually expresses as volt
per meter Vm-1)
3.
The electric potential :
The field strength E at a distance r from a charge Q is given as:
E
d
dr
and the electric potential is thus given as:
 d  E  dr

and
Qr
Q

4o r r 2 4r r
(The SI unit of the electric potential  is the volt V)
3
7.1. Faraday's Laws of Electrolysis:

Faraday's work was the first to suggest a relationship
between matter and electricity:
1.
The mass of an element produced at an
electrode is proportional to the quantity of
electricity Q passed through the liquid:
Q  It
(A s = C)
(Ampere x second = coulomb)
2.
The mass of an element librated at an electrode is
proportional to the equivalent weight of the element:
M   e  M
Charge of one electron = 1.602 x 10-19 C
Charge of one mole of electrons
= 1.602 x 10-19 C x 6.022 x 1023
= 96472 C
~ 96500 C (Farday's constant)
For the reaction:
Ag   e  Ag
1 mol of electrons or 96500 C will produce 1 mol of Ag
H   e  1 H 2
2
For the reaction:
1mol of electrons or 96500 C produce 0.5 mol of H2
4
Example 1:
An aqueous solution of gold (III) nitrate Au(NO3)3 was
electrolyzed with a current of 0.0250 A, until 1.20 g of Au (atomic
mass 197.0 g mol-1) had been deposited at the cathode calculate:
a) the quantity of electricity Q passed
b) the duration t of the experiment
c) the volume V of O2 at STP librated at the anode.
Answer:
a) at cathode:
Au 3  3e   Au
1 / 3 Au 3  e   1 / 3 Au
or
1 mol of e- produces 1/3 mol of Au
or
96500 C produces 1/3 mol of Au

Q (?) produces (1.2g/mol / 197 g/mol) of Au
Q
1.2molx96500C 3x1.2 x96500C

 1.76 x103 C  1.76 x103 ( AS )
197
197 x( 1 )mol
3
Q  It
b)
or
Q 1.76 x103 As
t 
 7.04 x104 s
I
0.025 A
5
Example 1 (continue):
c) At anode:
2H 2 O  O2  4H   4e 
1 H 2 O  1 O2  H   e 
2
4
or

1 mol of e- will librate 1/4 mol of O2
or
96500 C will librates 1/4 mol of O2

1.76x103 C will librates

nO2
(?) nO2
1.76 x103 Cx( 1 )mol
4

 4.56 x103 molO2
96500C
But, volume of 1 mol of any gas at STP occupies 22.4 dm3

VO2 = 4.56x10-3 mol x 22.4 dm3 mol-1 = 0.102 dm3
(Faraday's work on electrolysis was of great importance and
applications)
6
7.2. Molar Conductivity of Solutions:

Important information about the nature of solutions has been
provided by measurements of their conductivities.

Solutions can be classified into:
1.
Nonelectrolytes: do not dissociate into ions and have
no (or low) conductivities e.g. sucrose.
2.
Electrolytes: form ions in solutions and have high
conductivites. They can be classified into:
a. Strong electrolytes are completely dissociated
e.g. solutions of NaCl and HCl:
NaCl  Na+ + Cl-
and
HCl  H+ + Cl-
b. Weak electrolytes are partially dissociated
e.g. solutions of organic acids(acetic acid):
CH3COOH  Na+ + Cl
We measure conductivity by measuring electrical resistance.

Further insight into these matters is provided by studies of the way
in which electrical conductivities of solutions vary with the
concentration of solute.
7
According to Ohm's Law:
R
Where,

V
I
R
is electrical resistance of material
(ohm Ω)
V
is electric potential difference (volt V)
I
is electric current strength (ampere A)
The reciprocal of the resistance is the electrical conductance G (SI
unit is siemen S = Ω-1)
G 

1
R
Conductance (cm-1)
For a material of length (l) and cross–sectional area A:
G  A
G 
1
l
Gk
A
l
Conductance
k G
l
A
Conductivity (Ω-1cm-1)

The proportionality constant k (which is the conductance of a unit
cube) is known as the conductivity.

For the particular case of a solution of an electrolyte, it is know as
the electrolytic conductivity (the SI unit is Ω-1m-1 or S m-1, the more
commonly used unit is Ω-1cm-1 or S cm-1
8

Kohlrausch introduced the concept of equivalent conductivity.
Dividing the electrolytic conductivity k by concentration C gives
molar conductivity Λ.
k G
l
A
(Ω-1cm-1)
  k /C
(Ω-1cm2mol-1)
Example:
The electrolytic conductivity of a 0.1 M solution of acetic acid was
found to be 5.3 x 10-4 Ω-1cm-1. Calculate the molar conductivity of the
solution.
Answer:
 5.3x104  1cm 1 5.3x104  1cm 1
1
1
1
 


5
.
3

cm
mol
C
0.1moldm 3
0.1x103 molcm 3

The molar conductivity slightly decreases as the concentration
increased in /C curve in all cases.

Two patterns can be distinguished in the /C figure. The
extrapolation of the curve back to zero concentration gives a
quantity known as Λo the molar conductivity at infinite dilution.
9
7.3 Weak Electrolytes:
1.
Arrhenius theory (1884):
For the dissociation of an electrolyte A B into A+ and B-:
AB  A   B 
The degree of dissociation  is given as:


o
Van't Haff Factor i is given as:
i  1  v 
where,
 = number of ions.
10
2.
Ostwald's Dilution Low (1888):

In 1888 the ideas of Arrhenius were expressed quantitatively
by the Russian–German Physical Chemist Friedrich W.
Ostwalld in terms of a dilution law.

Consider the weak electrolyte AB, suppose that an amount (n)
of the electrolyte is present in a volume (V) and that reaction
of dissociation is ():
AB  A   B 
initial amount
n
change in amount
equil. amount
equil. conc.
0
0
n
n
n
(n - n)
n
n
n(1 - )
n
n
(n/V) (1 - )
C(1 - )
(n/V) (n/V)
C
C
The equilibrium constant K is then given as:
[ A ] [ B  ] (n  / V )(n  / V ) n  2
K


[ AB]
[n(1 ) / V ]
V (1 )

Since,

K  C  2 /(1 )


o
(degree of dissociation)
C ( /  o ) 2
K
1  ( /  o )
11
Ostwald's Dilution Law
7.4 Strong Electrolytes:

J. J. Van Laar (1894) pointed out that the strong electrostatic forces
present in an ionic solution must have an important effect on the
behavior of ions.
Debye Hückel Theory (1923):

According to the theory there is a mutual interference between
ions in solution that becomes more pronounced as the
concentration increases.

Because of the strong attractive forces between ions of opposite
signs, the arrangement of ions in solution is not completely
random.

There is small degree of order and on the average near to any
positive ion there are more negative ions than positive ones,
whereas for a negative ion there are more positive ions than the
negative ones.

The average number of opposite sign ions around a given ion is
called the ionic atmosphere.

The thickness of the ionic atmosphere is defined as:
1   o T 

  e 2 i ci z i2 L 
1/ 2
12
The thickness of ionic atmosphere:
  o T 


  e 2 i ci z i2 L 
1/ 2
1
where,
o
is permittivity of vacuum

is permittivity of medium

is Boltzmann constant
e is electronic charge
c is ionic concentration
z is ionic charge
L is Avogadroa’s constant
The term
where,
 i ci z i2
I 
gives rise to ionic strength I
1
 i ci zi2
2
13
Example:
Estimate the thickness of the ionic atmosphere for a solution of:
a) 0.01 M NaCl
b) 0.01 M ZnCl2
Both in water at 25oC, with  = 78
Answer:
a)
NaCl
0.01M
 cz
2
i i i
1

 Na+
Cl-
+
0.01 M
0.01 M
 (0.01x12 )  (0.01x12 )  0.02moldm3
 3.04 x10  9
( JN 1m)1 / 2
Since
J = Kgm2s-2 and N = Kgms-2 or N = Jm-1

(JN-1m)1/2 = (m2)1/2 = m

1

a)
 3.04 x10 9 m  3.04nm
ZnCl2
0.01M
 cz
2
i i i
 Zn2+
+
0.01 M
2Cl0.02 M
 (0.01x22 )  (0.02x12 )  0.06moldm3
1
 5.5610 9 m  5.56nm

14
It is important to note that the thickness is:
- Inversely proportion to ionic concentration and charge:
1

1

1
ci


1
z
- Directly proportion to dielectric constant:
1

1

  1/ 2
 T 1/ 2
(See Table 7.1 at page 268)
__________________________________________
Molar concentration
0.1M
0.01M
Univalent
0.962nm
3.04nm
Bivalent
0.481nm
1.52nm
_________________________________________
15
7.5 Independent Migration of Ions:

In principle the plots of molar concentration  against
concentration c can be extrapolated back to zero concentration to
give the o value. In practice this extrapolation can only
satisfactorily by made with strong electrolytes. With weak
electrolytes there is a strong dependence of  on c at low
concentrations and therefore the extrapolation do not lead to
reliable o values.

In 1875 Kohlrausch made a number of determinations of the  o
values and observed that each ion makes its own contribution to
the value conductivity, irrespective of the nature of the other ion
with which it is associated. This behavior was explained in terms
of Kohlrausch's law of independent migration of ions. In other
words:
 o  o  o
where,
o and o are the ion conductivities of a cation
and an anion, respectively at infinite dilution.
16
Ionic Mobilities:
It is not possible from the o values alone to determine the

individual
o and o values.
The   and   values are proportional to the speeds with which
the ions move under standard conditions. The mobility of an ion u,
is defined as the speed with which the ion moves under a unit.
o


o

o
 u
o  Fu
where, F is Faraday's constant and
u is the mobility of a cation or an anion

The ionic mobility can be measure by applying voltage of a
potential gradient V to the system:
Speed
and

V
Speed = uV
The SI unit of ionic mobility is m2V-1s-1, but it is more common to
use the unit cm2 V-1 s-1.
Example:
The mobility of a sodium ion in water at 25oC is 5.19x10-4 cm2V-1s-1
calculate the molar conductivity of the sodium ion.
Answer:

o
Na 
 Fu Na 
cm 2
 96500 Asmol x5.19 x10
 50.1AV 1cm 2 mol 1  50.1 1cm 2 mol 1
Vs
1
4
17
7.6 Transport Numbers:

The transport numbers (or migration number) is the fraction of the
current carried by each ion present in solution.

Consider the dissociation of the weak electrolyte
AaBb:
AaBb  aAz+ + bBz
The transport numbers of +ve and –ve ions, respectively are:
t 
but
I
I  I
I+  u+
= aFz+u+
 t 
and
and
I-  u-
az  u 
az  u   bz  u 
u
t

  u u


t 
and
for electrical neutrality of solution: az+
and
= bFz-u-
and
aFz  u 
and
aFz  u   bFz  u 
t 
I
I  I
t 
bz  u 
az  u   bz  u 
= bz-
and
t+ + t- = 1
18
t 
bFz  u 
aFz  u   bFz  u 
t 
u
u  u
(1)
(2)
but

or
+  u+
and
-  u -
+ =Fz+ u+
and
- =Fz- u-
u+ = + /Fz+
and
u- = - /Fz-
(3)
from equation (3) into equation (1) for u+ and u- :
t 

 / Fz 
 / Fz    / Fz 
t 
and
 / Fz 
 / Fz    / Fz  (4)
az+ = bz-
but

z+ = (b/a)z-
(5)
from equation (5) into equation (4) for z+ :
 /(b / a) z 
t

   /(b / a) z   / z




t 
or
t 
and
a  / b
a  / b    / z 
and
 / z 
 /( a / b) z    / z 
t 
 / z 
a  / b    / z 
multiplying by b:
a 
or
t 
but
 = a+ + b-

t 
a   b 
and
t 
b 
a   b 
(7)
a 

and
19
t 
b 


t 
a 

and
For univalent electrolyte:
t 


a=b=1


t 
b 
and
t 


If t+ and t- can be measured over arrange of concentrations, the
values at zero concentration, t+ and t-, can be obtained by
extrapolation.
20

Two experimental methods have been employed for the
determination of transport numbers.
1) Hittorf method:
2) Moving Boundary Method:
1) Hittorf method:

The solution of MX electrolyte
to be electrolyzed is placed in
the cell, and small current is
passed between the electrodes
for a short period:

When 96500 C of electricity passes through the solution,
96500 t+C are carried by M+ ions and 96500 t-C are carried
by X- ions

Solution then is run out through
the stopcocks, and the samples
are analyzed for concentration
change.

Then:
amount lost from anode compartment
amount lost from cathode compartment
 t
 t
From which t+ and t- can be determined.
where,
t  t  1
21
2) Moving Boundary Method:
(e.g. Determination of transport number of hydrogen ion)
t   u / u  u
The distances a a' and b b' are
proportional to the ionic velocities:
Therefore,
t 
bb'
aa 'bb'
and
t  1  t
22
7.7
Ion Conductivities:

The transport numbers extrapolated to infinite dilution are:
t 
o
o
o  t   o

Once a single
o
and
t 
o
(1)
and
 o  o  o
(2)
and
o  t   o
(3)
o
or
o
value has been determined a complete set
can be calculated from the available
7.3 at page 281.

 o values as shown in table
An important use for individual ion conductivities is in
determining the  o values for weak electrolytes (  o can be
obtained from the extrapolation to zero concentration)

e.g. For acetic acid:
 o  o  o  (349.8  50.9) 1cm 2 mol 1  390.7 1cm 2 mol 1

What is about of
o
or
o
are not in table 3:
23

When
o
or
o
of an electrolyte are not available, an alternative
way is used to avoid the necessity of splitting the  o values into
the individual ionic contributions.

For example the molar conductivity  o of any electrolyte MA can
be expressed as:
 o (MA)   o (MCl )   o ( NaA)   o ( NaCl)
where, MCl, NaA and NaCl are all strong electrolytes and their  o
values are readily obtained by extrapolates and therefore permit the
calculation of  o for MA.
e.g: For acetic acid:
 o (CH 3CooH )   o ( HCl )   o (CH 3COONa )   o ( NaCl)


This procedure is also useful for highly insoluble salts.
To calculate the mobility u divide
  Fu
  Fu
o
and
u 
and
u 
24
by F.

F

F

Summary of important equations:
 o  o   
u
t 
u  u
(Kohlransch Law)
and
u
t 
u  u
o  u 
and
o  u 
o  Fu
and
o  Fu
t 
o
o
and
t 
Speed(+) = u+v (for +ve ion)
Speed(-) = u-v (for –ve ion)
25
o
o
Ionic Solvation:

The magnitudes of the individual ion conductivities (Table 7.3) are
of considerable interest. There is no simple dependence of the
conductivity on the size of the ion. For example K+ moves faster
than
either
Li+
or
Na+
o
( K   73.5 , Na   50.1 and
o
oLi  38.6 ).


The explanation is that Li+ because of its small size, becomes
strongly attached to about four surrounding water molecules by
ion-dipole and other forces, so that when the current passes it is
Li( H 2 O) 4 and not Li+ that moves. With Na+ the binding of water
molecules is less strong, and with K+ it is still weaker.

Generally, the smaller the ion, the greater the binding between the
ion and water molecule, because the water molecule can approach
the ion more closely
26
Mobilities of Hydrogen ( H+) and Hydroxide (OH-) ions:

Of particular interest is the very high conductivity of the hydrogen
and the hydroxide ions (H   349.8
o
and
o
OH

 198.6)

The hydrogen ion has an abnormally high conductivity in a number
of hydroxylic solvents such as water, methanol, and ethanol, but it
behaves more normally in non-hydroxylic solvents such as
nitrobenzene and liquid ammonia.

There is a powerful electrostatic attraction between a water
molecule and the proton:
H   H 2 O  H 3O 

The equilibrium lies very much to the right. Then:
H 3O   H 2 O  H 2 O  H 3O 

Protons, although not free in solution, can be pass from water
molecule to another freely.

Similar types of mechanisms have been proposed for the
hydroxylic solvents. In methyl alcohol, for example:
CH 3  OH  H   CH 3  O  H 2

Then
CH 3  O  H 2  CH 3  OH  CH 3  OH  CH 3  O  H 2

For hydroxide ion:
OH   H 2O  H 2O  OH 
27
Ionic mobilities and diffusion coefficients:

Ions also migrate in the absence of an electric field of there is a
concentration gradient. The migration of a substance when there is
a concentration gradient is known as diffusion. The tendency of a
substance to move in a concentration gradient is measured in terms
of a diffusion coefficient D. In 1988 W. H. Nernest showed that:
Du
D
where,

kTu
Q
Q is the ionic charge and u is the ionic mobility
An important relationship between molar conductivity and
viscosity was discovered in 1906 by P. Walden:
Λo  = constant
28
Walden's Rule
7.8 Thermodynamics of Ions:

The convention standard Enthalpies, Gibbs, Energies and
Entropies of formation and Hydration of individual ions at 25oC
are calculated based on setting the values of H+ as zero (see table
7.4 at page 74).

However, owing to the large number of interactions involved, the
theoretical treatment of an ion in aqueous solution is very difficult.

The following absolute values are generally agreed to be
approximately correct for the proton:
ΔhydHo = -1090.8
KJ/mol
ΔhydSo = -131.8
KJ/mol
ΔhydGo = -1051.4
KJ/mol
(See example 7.5)
29
7.10 Activity Coefficients:

To account for the effect of electrostatic interactions between ions,
especially at high concentrations of solution, the term activity is
used:

The activity ai of an ion i is proportional to its concentration ci:
ai  ci
ai = ici
(1)
where,  is called activity coefficient (for ideal solution of very
dilute concentration  = 1).
Debye-Hückel Limiting Law:
log  i   zi2  I
(2)
where,  is a quantity depends on properties such as  and T, when
water is solvent at 25oC,  equals to 0.51 mol-1/2 dm3/2
I, is called the ionic strength of solution and is given as:
1
I  i ci z i2
(3)
2

Experimentally we can not measure the activity coefficient or indeed
any thermodynamic property of a single ion. To overcome this difficulty
we define a mean activity coefficient :
 (
   )
    .  
(4)
where, + and - are the number of +ve and –ve ions produced
by the electrolyte.
30
By taking logarithms:
(     ) log      log      log  
(5)
From equation 2 into equation 5 for logi:
(     ) log      ( z 2  I )    ( z 2  I )
(     ) log      I (  z 2    z 2 )
 2 z 2  2 z 2
(     ) log      I (

)


For electrical neutrality:
  z    z
or

 2 z2   2 z2
(     ) log      I ( 2 z 2 )(
1

1
)
 
  
(     ) log      I ( 2 z 2 )( 
)
  
  z 2
log      I (
)

but
  z    z

  z_

  z

log      | z  z  | I
and for aqueous solution at 25oC:
log    0.51 | z  z  | I
31
(Debye-Huckel limiting law)
Example calculations of ionic strength I:
1)
For a uni-univalent electrolyte:
e.g: 1 M salution of NaCl
c+ = 1 M
and
c- = 1 M
z+ = 1
and
z- = 1
1
2)
 ci zi 
2
2
1
(1  1)  1M
2
For a uni-bivalent electrolyte
e.g: 1 M K2 SO4
c+ = 2 M
and
c- = 1 M
z+ = 1
and
z- = 1
1
3)
cz
2
i i
2
1
 [( 2) x(1)  (1) x(4)  3M
2
For a uni-trivalent electrolyte
e.g: 1 M Na3PO4
c+ = 3 M
and
c- = 1 M
z+ = 1
and
z- = 3
1
1
2
c
z

[(3) x(1)  (1) x(9)]  6M
2 i i
2
32
Deviations from the Debye-Huckel Limiting Law DHLL:

Experimentally the DHLL is found to apply satisfactory only at
extremely low concentrations, at higher concentrations there are
very significant deviations.

Various theories have been put formed to explain these deviations
from the DHLL, but here they can be considered only briefly.
1) One faction that has been neglected in the development of the
equations is the fact that the ions occupy space and they are not
a point changes with no restrictions on how closely they can
come together. If the theory is modified in such a way that the
centers of the ions cannot approach one another closer than the
distance a , the expression for i becomes:
2
z B I
log  i   i 1
1  aB I
(1)
And the corresponding equation for
log    

is:
zzB I
1  aB1 I
(2)
2) The orientation of solvent molecules by the ionic atmosphere
was necessary to account for and Huckel showed that:
log    
zzB I
1  aB
1
33
I
 CI
(3)
7.11 Ionic Equilibria and Calculation of Activity Coefficients:

Equilibrium is usually established very rapidly between ionic
species in solution. In this section an account will be given of the
more important ways in which activity coefficients can be
determined when equilibrium are established in solution.
1)
Activity coefficients from equilibrium constant measurement:

Consider the dissociation of acetic acid:
CH 3 COOH  H   CH 3COO 
initial activity
change in activity
equil. activity
or
a
a
0
a
a  a  
a(1 )  
K

0
aH aCH COO


a



3
CH 3COOH
 c.  c
c 2 2
K
.  .  
. 
c(1 )
(1 )
where,
 (
   )
    .  
(mean activity coefficient)
u 1
(for a dilute solution) and
c  [ H  ]  [CH 3COO  ]
34
 c.  c
c 2 2
K
.  .  
. 
c(1 )
(1 )
By taking logarithm:
 c 2 
  log K  2 log  
log 
 1  
where, c is the concentration and  is the degree of ionization, which
can be determined from conductivity measurements.
  /  o
1
I   ci zi2
2
For uni-univalent electrolyte:
1
1
I  [c (1) 2  c (1) 2 ]  [c  12  c  12 ]  c
2
2
35
2) Activity coefficients from solubility product measurement:

Consider the dissociation of AgCl solution:
AgCl( s)  Ag   Cl 
K s  a Ag  .aCl   [ Ag  ] [Cl  ]    
K s  [ Ag  ] [Cl  ]  2
(1)
K s  S 2  2
or
log K s  2 log S  2 log  
where,   is given by DHLL and S is the solubility
log     2  2  B I
(2)
(See problems 19 and 20)
In general for springly soluble uni-univalent salt AB:
AB  A   B 
K sp  a A aB   [ A ] [ B  ] 2  S.S 2  S 2 . 
where,
[A+] = [B-] = S
By taking logarithm:
log K sp  2 log S  2 log  
or
log S 
1
log K sp  log  
2
36
7.13 The Donnan Equilibrium:

A mater of considerable importance is the ionic equilibrium that
exists between two solutions separated by a membrane.
1)
An equilibrium established when all ions can diffuse
through the membrane:
[Na1+]
[Na2+]
C1
C2
[Cl1-]
[Cl2-]
C1
C2
A very simple case is when solutions of sodium chloride of
volume 1 dm3, are separated by the membrane at equilibrium:
G  GNa   GCl   0
[ Na  ]2
[Cl  ]2
 RT ln
 Rt ln
0


[ Na ]1
[Cl ]1
[ Na  ]2
[Cl  ]2
 ln
 ln
0
[ Na  ]1
[Cl  ]1
[ Na  ]2
[Cl  ]1
 ln
 ln
0
[ Na  ]1
[Cl  ]2
[ Na  ]2
[Cl  ]2
 ln
 ln
[ Na  ]1
[Cl  ]1
[ Na  ]2 [Cl  ]2  [ Na  ]1[Cl  ]1
37
(1)
(2)
or
[ Na  ]2 [Cl  ]2
1
[ Na  ]1[Cl  ]1
(3)
For electrical neutrality:
[ Na  ]1  [Cl  ]1
and
[ Na  ]2  [Cl  ]2
[ Na  ]1  [Cl  ]1 = [ Na  ]2  [Cl  ]2
(In other words, at equilibrium we have equal concentration of
the electrolyte an each side)
38
2)
An equilibrium established when not all ions can diffuse
through the membrane:

Complications arise with ions that are too large to diffuse
through the membrane, and the diffusible ions then reach a
special type of equilibrium, known as the Donnan
equilibrium.

For example, suppose that initially have or the left hand side
Na+ ions and non diffusible P- anions, on the right hand side
there are Na+ and Cl- ions:
Na+
[Na1+]
[Na2+]
[Na1+]
[Na2+]
(C1+X)
(C2-X)
C1
C2
[P-]
C1
[Cl1-]
[Cl2-]
X
(C2-X)
[P-]
[Cl2-]
C1
C2
ClAfter equilibrium
Before equilibrium

Since there are no Cl- ions on the left-hand side, spontaneous
diffusion of Cl- ions from right to left will occur.

Since there must be always electrical neutrality on each side
of the membrane, an equal number of Na+ ions must also
diffuse form right to left.
39
Equation (3) is still obeyed even if the system contains a nondiffusible ions addition to Na+ and Cl- ions.
[ Na  ]2 [Cl  ]2
1
[ Na  ]1[Cl  ]1
(3)
Therefore,
[ Na  ]2 [Cl  ]2
1
[ Na  ]1[Cl  ]1
(c2  x)2
1
(c1  x) x
or
and
(c2  x) 2  (c1  x) x
c22
x
c1  2c2
(Equilibria of the Donnan type are relevant to many types of
biological systems, the theory is particularly important with
reference to the passage of ions across the membranes of the never
fibers)
40

Summary of important equations:
 o  o   
u
t 
u  u
o  u 
o  Fu
t 
(Kohlransch Law)
and
and
o  u 
and
o  Fu
o
o
u
t 
u  u
and
t 
o
o
Speed(+) = u+v (for +ve ion)
Speed(-) = u-v (for –ve ion)
log      | z  z  | I
(Debye-Huckel limiting law)
41