Conductivity

Instructor: Dr. Tamal Ghosh, Department of Chemistry, MNNIT, Allahabad

1
Electrolytes: Compound or species that conducts electricity at molten state or at ionised
state in solution.

Earlier Concepts

Arrhenious theory of Electrolytic Dissociation: (i) Electrolyte undergoes ionization to form

ions.
(ii) Equilibrium between ions and unionized
electrolyte molecules.

(A)As mentioned by Arrhenious in (i) and (ii), afterwards it was found that only (so
called) weak electrolytes follow these two.

(B)Some electrolytes act as strong electrolyte in one solvent, but behave as weak
electrolyte in some other solvents.

2
Electrical conductivity:
Ohm’s Law expresses the current, I, through a media of resistance R in the presence of a
potential difference V through the relation:
V = IR
The current is expressed in amperes (A), the resistance in ohms ( , with 1  = 1 V A-1), and the
potential difference in volts (V, with 1 V = 1 J C-1). By definition, the coulomb is defined through
1 A = 1 C s-1, with the ampere a fundamental unit in the SI.
V
Ohm’s law may be rearranged into: I = =GV
R
where G is the conductance and is reported in siemens, S (1 S = 1  -1 = 1 A V-1). In older
literature, conductance is reported in ‘reciprocal ohms’,  -1, denoted mho.

The conductance of a sample depends on its dimensions (as well as the temperature and
pressure). For a sample of length L and cross-section A, the conductance is:

A
G=
L
where  (kappa) is the conductivity, with units siemens per meter (S m-1).
3
The conductivity can be determined from the dimensions of the sample and current flowing
for a given potential difference from:

L I L
=G =
A V A

In practice, the conductivities of solutions are determined by calibration using a solution

of known conductivity.

A typical conductivity cell. The cell is made part of a ‘bridge’ and

its resistance is measured. The conductivity is normally
determined by comparison of its resistance to that of a solution
of known conductivity. An alternating current (AC) is used to
avoid the formation of decomposition products at the
electrodes.

4
Molar conductivity (𝒎 ):

• As number of charge carriers (ions) per unit volume increases with increasing
concentration, so  usually increases as electrolyte concentration increases.
• To get a measure of current carrying ability of a given amount of electrolyte, molar
conductivity ( 𝒎 ) was introduced.

𝒎 = where c is the molar concentration of the solute.
c
With molar concentration in mol dm-3, molar conductivity is expressed in S m-1 (mol dm-3)-1
which can be simplified to: S m2 mol-1
Specifically, the relation between units is: 1 S m-1 (mol dm-3)-1 = 1 mS m2 mol-1, where 1 mS =
10-3 S

• Molar conductivity is found to vary with concentration.

• One reason for this variation is that the number of ions in the solution might not be
proportional to the concentration of the electrolyte.
• For instance, the concentration of ions in a solution of a weak acid depends on the
concentration of the acid
in a complicated way and doubling the concentration of the acid added does not double
the number of ions.
• Secondly, because ions interact strongly with one another, the conductivity of a solution is
not exactly proportional to the number of ions present.
5
Strong and Weak Electrolytes:

The concentration dependence of molar conductivities

indicates that there are two classes of electrolyte.
• Strong Electrolyte: molar conductivity depends
only slightly on the molar concentration and, in
general, decreases slightly as the concentration is
increased.
• Weak Electrolyte: molar conductivity is normal at
concentrations close to zero but falls sharply to low
values as the concentration increases.

6
Strong Electrolytes:

Friedrich Kohlrausch showed that at low concentrations:

𝒎 = 𝟎𝒎 − K c1/2

This variation is called Kohlrausch’s law. The constant 𝟎𝒎 is the limiting molar
conductivity, the molar conductivity in the limit of zero concentration (when the ions
are effectively infinitely far apart and do not interact with one another). The constant K
is found to depend more on the stoichiometry of the electrolyte (that is, whether it is of
the form MA, or M2A, etc.) than on its specific identity.

Kohlrausch was able to establish experimentally that 𝟎𝒎 can be expressed as the sum
of contributions from its individual ions. If the limiting molar conductivity of the cations
is denoted + and that of the anions −, then his law of independent migration of ions
states that:

𝟎𝒎 = ++ + −−

where + and − are the numbers of cations and anions per formula unit of electrolyte
(for example, + = − = 1 for HCl, NaCl and CuSO4, but + = 1, − = 2 for MgCl2). This
implies that the ions migrate independently in the limit of zero concentration. 7
Outcome of Kohlrausch’s law of independent migration of ions:

1) Limiting molar conductivity (𝟎𝒎 ) of an electrolyte can be calculated if limiting ionic

conductivities are known.
𝟎𝒎 (BaCl2) = (Ba2+) + 2 x (Cl−) = (12.72 + 2 x 7.63) mS m2 mol-1 = 27.98 mS m2 mol-1

2) Limiting molar conductivity (𝟎𝒎 ) of a weak electrolyte or sparingly soluble salt can be
calculated from limiting molar conductivity of strong electrolytes using the above law.

𝟎𝒎 (CH3COOH) = (H+) + (CH3COO−)

= [(H+) + (Cl−)] + [(Na+) + (CH3COO−)] − [(Na+) + (Cl−)]

= 𝟎𝒎 (HCl) + 𝟎𝒎 (CH3COONa) − 𝟎𝒎 (NaCl)

= (42.6 + 9.1 − 12.65) mS m2 mol-1

= 39.05 mS m2 mol-1

8
Weak Electrolytes:

Weak electrolytes are not fully ionized in solution. The marked concentration
dependence of their molar conductivities arises from the displacement of the
equilibrium towards products at low molar concentrations.

The conductivity depends on the number of ions in the solution, and therefore on the
degree of ionization, , of the electrolyte. When referring to weak acids, we speak instead
of the degree of deprotonation. It is defined so that, the acid HA at a molar concentration
c, at equilibrium

If we ignore activity coefficients, the acidity constant, Ka, is approximately

… eq 1

from which it follows that: … eq 2 9

The acid is fully deprotonated at infinite dilution, and its molar conductivity is then 𝟎𝒎 .
Because only a fraction  is actually present as ions in the actual solution, the measured
molar conductivity 𝒎 is given by
𝒎 =  𝟎𝒎 … eq 3
with  given by eq 2.
Once we know Ka, we can use eq 2 and eq 3 to predict the concentration dependence of
the molar conductivity. The result agrees quite well with the experimental curve in Fig.
21.14. More usefully, we can use the concentration dependence of 𝒎 in measurements
of the limiting molar conductivity. We rearrange eq 1 into

… eq 4

using eq 3, we obtain Ostwald’s dilution law:

… eq 5

If 1/𝒎 is plotted against c𝒎 , then the

intercept at c = 0 will be 1/𝟎𝒎 . (Fig.
10
21.15).
The mobilities of ions:
• A cation responds to the application of the field by accelerating towards the negative
electrode and an anion responds by accelerating towards the positive electrode.
• However, this acceleration is short-lived.
• As the ion moves through the solvent it experiences a retarding force proportional to
its speed.
• The two forces act in opposite directions, and the ions quickly reach a terminal speed,
the drift speed, when the accelerating force is balanced by the viscous drag.
• The drift speed (s) of an ion is proportional to the strength of the applied field (E ). We
write
s=uE
where u is called the mobility of the ion.
ze
u= … eq 6 (e = charge of an electron)
6a
where z is charge number of an ion (absolute value),  is viscosity coefficient of the
solvent, a is radius of the ion.
The mobility of an ion determines the rate at which it can transport charge through a
solution and therefore its molar conductivity. The relation between two is
11

 + = z +u +F and  − = z− u− F

where z+ and z− are absolute charge number of cation and anion, u+ and u− are mobilities
of cation and anion and F is Faraday’s constant [F = 96485 C mol-1].
Mobilities of Group 1 (alkali metal) Cations:

Observed anomaly: The mobilities of the Group 1

cations increase down the group despite their
increasing radii.

The explanation is that the radius to use in eq 6 is

the hydrodynamic radius, the effective radius for
the migration of the ions taking into account the
entire object that moves.

When an ion migrates, it carries its hydrating water molecules with it, and as small ions are
most more extensively hydrated than large ions (because they give rise to a stronger electric
field in their vicinity), ions of small radius have a large hydrodynamic radius. Thus
hydrodynamic radius decreases down Group 1 because the extent of hydration decreases
with increasing ionic radius. 12
High mobility of the H+ in water:

It is believed that this high mobility reflects an entirely different mechanism for
conduction, the Grotthuss mechanism, in which the proton on one H2O molecule
migrates to its neighbour, the proton on that H2O molecule migrates to its neighbour,
and so on along a chain. The motion is therefore an effective motion of a proton, not the
actual motion of a single proton.

A simplified version of the Grotthuss

mechanism of proton conduction
through water. The proton leaving the
chain on the right is not the same as
the proton entering the chain on the
left.
13