Conductivity, Ionic Mobilities, Transport Number
Conducting media
Insulators: there is a small amount of charge carrying particle
Metals:
electrons are the charge carrying particles.
Electrolyte solutions:
ions are the charge carrying particles.
Semiconductors: band structure determines the number of charge carriers.
l
A
R
Resistance:
 - specific resistivity
l – length of wire
A – cross sectional area of wire
G
Conductance:
Conductivity:

Molar conductivity:
1
R
1
Ω-1 = S (Siemens)
unit: m-1 Ω-1 = S m-1

Λm 

unit: if c is given in mol m-3,
c
than S m2 mol-1
Λm is concentration and temperature dependent, Λm = f(c,T),.
In general, increasing the number of charge carriers increases the
conductivity.
Number of ions present in a solution of unit volume (concentration) depends on
the concentration in a complicated way. → weak electrolytes
The Coulomb interactions prevents the molar conductivity vs. concentration
function to show ideal behaviour. → strong electrolytes.
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Figure 1. The variation of the molar conductivity m with the concentration of
the electrolyte.
Measuring conductance: conductivity cell
Noble metal electrodes → electrolysis
alternating current → polarization
Strong electrolytes
Kohlrausch’s law:
Λm  Λm0  k  c1/ 2
1.
describes the concentration dependence of molar conductivity . When a straight
line is fit to the points of m vs. c1/2 function for a strong electrolyte the intercept
0
gives Λm ,which is the limiting molar conductivity the molar conductivity in the
limit of zero concentration. Constant k depends principally on the stoichiometry
of salt, rather than specific identity.
In the condition: c → 0,
Λm  Λm0 , the distance of ions in the
solution is so large, that there is no interaction between them.
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Figure 2. The Λm vs. c1/ 2 function. The circles represent experimenal points,
while the solid lines are theoretical lines calculated with the Debye-HückelOnsager theory ( Λm  Λm0  P  Q  Λm0  c1/ 2 ).
Independent migration of ions
Λm        
2.
λ+ and λ- are the limiting molar conductivity of cations and anions respectively,
  and   are the numbers of cations and anions per formula
     for NaCl, CuSO4
   1,    2 for BaCl2.
Each ion is assumed to make its own contribution to molar conductivity,
irrespective the nature of other ion with which it is associated.
Table. Limiting ionic conductivities of sodium and potassium salts in water at
298 K, λ / mS m2 mol-1
Electrolyte
Electrolyte
Difference
Λm0
Λm0
KCl
149.79
NaCl
126.39
23.4
KI
150.31
NaI
126.88
23.4
½ K2SO4
153.48
½ Na2SO4
130.1
23.4
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The difference between the Λm0 values of sodium and potassium salts is
irrespective of the anions, as it can be seen from Table 1
Weak electrolytes
A part of the amount of dissolved substance appears in the solution as ions,
according to the ionization equilibrium.
van’t Hoff’s experimental findings for osmotic pressure, π measurements in
weak electrolytes:
  m0  R  T
π ,  i  m0  R  T
3.
where m0 is the concentration in mol/kg determined from mass measurement.
The effective concentration, i  m0 is usually greater than m0, while i>1. The
number of ionic and non-ionic particles present in a solution is given by
i  m0    m0  1     m0
The molecule dissociates ν ions, and α is the degree of ionization (1 > α > 0).
Example: weak acid.
CH3COOH ↔ H+ + CH3COOAcetic acid dissociates into two ionic species, therefore ν = 2, and
i  m0  2  m0  1    m0  m0   1
van’t Hoff found α to be proportional the ratio of molar conductivities,

Λm
Λm0
4.
When m0 → 0, then α = 1, therefore Λm0  Λm . In dilute solution of weak
electrolytes the ionization is complete.
Eq. 4 is used to determine equilibrium constant for ionization processes.
Data can be calculated from conductivity measurements.
For general case, a weak acid dissociates
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HAc ↔ H+ + AcThe thermodinamic equilibrium constant:
a   aH 
Ka  Ac
aHAc
For dilute solutions (c < 10-4 molal) concentrations can be used instead of
activities.
The equilibrium concentrations for a weak acid dissociation,
[H+] = [Ac-] = α∙c, and [HAc] = (1 – α)∙c.
Ka 
a Ac  a
H
aHAc
c   c 
2

 c
c  1   
1
5.
Equation 5. is a quadratic equation in α,
c   2    Ka  Ka  0
Transforming Eq. 5.
1  c 
1

1  c 
2
Ka
, and divided by α

Ka
Using Eq. 4 we get, one of the forms of Ostwald’s dilution law
Λm0

 1 c  0 m
Λm
 m  Ka
or in a more applicable form
1
1

 0  c  0 2m
Λm  m
 m   K a
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6.
5
Plotting the data for Eq. 6. in a linearized form
Figure 3. Graph used to determine 0m from the intercept, and Ka from the slope.
Intercept =
Slope =
1
0m
1
0m 2  Ka
Ionic Mobilities, Transport (transference) Number
Electric force field directs migrating ions parallel to the force field lines and
accelerates them toward the electrodes. A counter force prevents their speed to
exceed a limit. This is the friction force resisting the relative motion of fluid
layers.
The condition for mechanical equilibrium of transporting ion,
F friction  F field
1.
In a short time after the electric field has been switched on, the ions, e.g. the
cations migrate with a constant speed, s toward the negative electrode.
F friction  s  6   r 
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F field  zeE  ze

l
A potential difference Δφ at a distance l produces E electric field which make
spherical ions with radius, r move against friction, that is caused mainly by
solvent molecules.
When net force is zero (see Eq. 1.) the migration speed, s, at a migrating ion
travels can be given as
s
zeE
6r
As η, the viscosity of liquid or the radius of migrating ion increases the speed
lowers.
The hydrodynamic radius should be taken into account. For alkali metal ions (Li
– Cs) the hydrodynamic radius of Li+ is the greatest, though its radius without
hydration shell is the smallest. The surface charge density of Li+ is the greatest
in the first column of periodic system.
Migration speed can be referred to unit field:
u
s
ze

E 6r
2.
the quantity u is called ionic mobility. The ion mobility is independent of the
magnitude of electric field.
Table 1. Ionic mobilities in water at 298 K, u / 10-8 m2 s-1 V-1
H+
Na+
OHCl-
36.23
5.19
Mobility as a function of conductivity λ = f(u)
Definitions
Electroneutrality principle, example: H2SO4
formula
charge number
charge
ion concentration
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ν+ = 2
z+ = 1
z+e
c∙ν+
ν- = 1
z- = 2
z -e
c∙ν7
20.64
7,91
Electroneutrality:
z+ ν+ = z- ν-
An experimental cell is set in which a strong electrolyte of concentration c is
filled. The noble electrode metal pair is polarized at a potential Δφ which
maintains an electric field, E (V/cm).
Figure 4. In the calculation of the current, all the cations within a distance s+·Δt
(i.e. those in volume A·s+·Δt) will pass through the area A. Likewise anions…
The ion flux:
J ion 
N
A  t
3.
The number of ions, N pass through area, A at a time Δt producing ion flux.
Each of the positive or negative ion concentration is chosen the same flux can be
observed (electroneutrality).
Ion concentration:
c∙ν+
Number density,
N
V
c∙ν- , in general: c∙ν
can also be given in terms of ion concentration. If nion = ν·n, than
(introducing NA the Avogadro number)
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N NA N
n

 N A  ion  N A   c
V NAV
V
N
 N A    c
V
N
 N A    c
V
4.
Volume can be given by speed s
V  s  t  A
5.
From Eqs. 3. 4. and 5.
stA cN A
 s cN A
At
J ion 
Number of ions number of charges conversion:
J ch arg e  J ion ze
J ch arg e  s cN A  ze  s c  zF
Introducing:
J ch arg e  u
s  uE , and E 
(NA e = F, the Faraday constant)

l

c  zF
l
Current can be given as: I  J ch arg e A
I u

c  zF  A
l
6.
Current can also be given from Ohm’s law:
I


 G  
A
R
l
7.
Comparing Eq. 6. and 7. we get
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  u c  zF
Defining individual molar conductivity:


 u  zF
c
8.
Eq. 8. tells us the relation between a theoretical quantity, mobility of an ion and
an experimentally determined quantity, molar conductivity.
In the limit of zero concentration
Λm0  F z  u  z u 
9.
Eq. 9. simplifies for electrolytes with z+ = z-, i.e. z : z electrolytes
(KCl with z = 1, CuSO4 with z =2).
Λm0  zF u  u 
Transport number, t
It is the fraction of the current carried by each ion that is present in solution
t
Ii
I
10.
i 1
The sum of transport numbers in a solution should give one,
 t   t  1
where  t and  t are the sum of the transport number for cations and anions
respectively.
When a solution contains c1 concentration of NaCl and c2 concentration of
KNO3 the transport number of Na+ can be given as
t Na 
uNa Na z Na c1 FA 
FA 
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
l

 uNa Na z Na c1  uCl Cl zCl c1  uK K z K c2  uNO  NO z NO c2 
l
3
10
3
3
We used Eq. 6. for substituting the current caused by different ions. This long
equation is simplified by the following facts:
νNa = 1, zNa = 1, νK = 1, zK = 1, νCl = 1, zCl = 1, νNO3 = 1, zNO3 = 1
t Na 
uNa c1
c1  uNa  uCl   c2  uK  uNO
3

11.
When we have a single z : z electrolyte with mobilities u+ and u-, the transport
number of anion is
t 
u
u  u
This equation can also be given in molar conductivity representation
t 

  
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