Introduction:

Conductometry signifies measuring the conductivity and conductometer evaluates the electrical conductivity of the ionic solutions.

This is completed by applying an electric field between the two electrodes. The ions wander in this field. The anions migrate to the anode and the cations to the cathode. In order to avoid conversions of substance and the formation of diffusion layers at the electrodes (that is, polarization), work is carried out by alternating voltage. The rule of thumb is that the frequency of the alternating voltage should be increased as the ion concentration rises. Modern conductometer automatically acclimatize the measuring frequency to the specific measuring conditions.

Electrical Resistance and Conductance:

Resistance (R):

The tendency of a material to impede the flow of current is termed as resistance. It is measured in ohms (Ω).

On the basis of Ohm's law, the resistance offered via a substance is directly proportional to its length (l), however inversely proportional to its cross sectional area (A).

R α l/A

In case of electrolytic solutions, 'l' symbolizes the distance between the two electrodes and 'A' is the cross sectional area of the electrodes.

The above equation can as well be represented as:

R = ρ (l/A)

Here, 'ρ' is the proportionality constant and is termed as specific resistance or resistivity. If A = 1 cm² and l = 1 cm, then R = ρ.

Thus, the specific resistance can be stated as follows:

Specific resistance or resistivity (ρ):

It is the resistance presented by a material or solution occupying one cm³ volume. 

This is measured in:

ohm. cm (in C.G.S system) or ohm. m (in S.I system). Its unit can be derived as:

ρ = R (A/l) = ohm. cm²/cm = ohm cm

Conductance (G):

This is the tendency of a material to let the flow of current via it. It is the reciprocal of resistance.

It is measured in ohm^-1 = mho = Siemens.

Conductance (G) = 1/Resistance (R)

By replacing, 'R' from equation:

G = (1/ρ). (A/l) = k. (A/l)

Here,

k = (1/ρ) = specific conductance or conductivity

From the equation:

k = G. (l/A)

When A = 1 cm² and l = 1 cm, then k = G

Thus, the specific conductance can be stated as:

Specific conductance or conductivity (k):

This is the conductance of a material or solution occupying one cm³ volume.

It is measured in: ohm^-1. cm^-1 = mho. cm^-1 (C.G.S system) or Siemens. m^-1 (S.I system)

The specific conductance based on the nature of substance or the electrolyte, and it rises with the increase in concentration of the electrolytic solution as the number of ions per unit area rises. The ratio of the distance between electrodes, 'l' to the cross sectional area, 'A' of the electrodes is termed as cell constant.

Cell constant (G*) = (l/A)

The cell constant can be found out by employing following relations that can be derived simply from expressions illustrated above.

G* = (l/A) = R/ρ = k/G = Rk = (1/Gρ)

Types of Substances based on Electrical Conductance:

Materials are divided into two kind based on the electrical conductivity:

1) Insulators:

The substances that resist the flow of electric current via them are termed as insulators. They don't have free electrons or freely moving charged particles.

Organic polymers (such as plastics), glass, diamond, quartz are merely however some illustrations.

2) Conductors: 

The substances that allow the flow of electricity via them with little resistance are termed as conductors.

Conductors are categorized into:

A) Metallic or electronic conductors: These are the conductors that conduct the electricity via the electrons, example: all metals, graphite and so on. In metallic conduction, no chemical reaction takes place throughout the conduction of electricity and conductivity reduces with the increase in temperature because of vibrational disturbances. 

B) Electrolyte: This is a substance that in aqueous solution or molten form lets electricity to pass via it and decomposes to oppositely charged ions throughout the procedure.  Example: NaCl, KCl, CH₃COOH, HCl and so on.

Free flow of ions in the direction of the oppositely charged electrodes takes place and all through conduction of electricity via electrolytes, oxidation takes place at anode while reduction takes place at cathode. The conductivity rises with the increase in temperature as the degree of ionization increases. The electrolytes experience dissociation to furnish ions either in the molten state or in aqueous solutions. The electrolytes are further categorized into two, based on the degree of ionization in water:

i) Strong electrolytes: Experience complete ionization in water. Example: NaCl, KCl, K₂SO₄, HCl, H₂SO₄, NaOH, NaNO₃ and so on.

ii) Weak electrolytes: Experience partial ionization in the water. Example: HF, CH₃COOH, NH₄OH, HCOOH and so on.

Non-electrolytes: The substances which don't furnish ions for electrical conduction are termed as non-electrolytes. Example: urea, glucose, sucrose and so on.

Equivalent Conductance and Molar Conductance:

Equivalent conductivity (Λ):

The conductance of that volume of solution having one equivalent of an electrolyte is termed as equivalent conductivity. This is represented by Λ.

Let us take the V cm³ of solution having one equivalent of an electrolyte. Its conductance is equivalent to equivalent conductance, Λ.

As well we are familiar that the conductance illustrated by 1 cm³ solution having this electrolyte is termed as specific conductance, k.

That is,

The conductance of V cm³ ....... Λ 

The conductance of 1 cm³ ........ k

Thus:

Λ = k.V           

We are familiar that the normality (N) of a solution is represented by the equation:

N = ne/V(in CC) x 1000

For the above electrolytic solution, number of equivalents, n_e = 1.

Therefore,

V (in CC) = 1000/N

By replacing the above value in the equation (Λ = k.V), we can now write:

Λ = k. (1000/N)

Units of Λ: = Ohm^-1. cm^-1/equivalents. cm^-3

                 = cm². ohm^-1. equiv^-1 = cm². mho. equiv^-1 or m². Siemens. equiv^-1

Molar conductivity (Λm or μ):

The conductance of that volume of solution having one mole of an electrolyte is termed as molar conductivity. It is represented by Λm or μ.

It is associated to specific conductance, k as:

μ = k.V  

Or μ = k. (1000/M)

Here, M is the molarity of the electrolytic solution.

Units of μ: = cm². ohm^-1. mol^-1 = cm². mho. mol^-1 or m². Siemens. mol^-1

The relation between equivalent conductance, Λ and molar conductance     can be represented by:

μ = Λ x equivalent factor of the electrolyte

The equivalent factor of the electrolyte is generally the net charge on either anions or cations represent in one formula unit of it. It might be equivalent to basicity in case of acids or equivalent to the acidity in case of bases.

The Factors affecting the Conductance of Electrolyte Solutions:

a) Temperature: The conductance of an electrolyte solution rises with the increase in temperature because of increase in the extent of ionization.

b) Nature of electrolyte: The strong electrolytes experience complete ionization and therefore exhibit higher conductivities as they furnish more number of ions.

While weak electrolytes experience the partial ionization and therefore exhibit comparatively low conductivities in their solutions.

c) Ionic size and mobility: The ionic mobility reduces with the increase in its size and therefore conductivity as well reduces. Example: In molten state, the conductivities of lithium salts are more as compare to those of cesium salts since the size of Li⁺ ion is smaller than that of Cs⁺ ion.

In aqueous solutions the degree of hydration influences the mobility of the ion, which in turn influences the conductivity. Heavily hydrated ions exhibit low conductance values as an outcome of the larger size. For example, in aqueous solutions Li⁺ ion having high charge density is heavily hydrated than Cs⁺ ion having low charge density. Therefore hydrated Li⁺ bigger than hydrated Cs⁺. As an outcome, lithium salts exhibit lower conductivities compared to those of cesium salts in water. 

Fig: Hydration of Lithium and Cesium ions in aqueous solution

d) The Viscosity:

The ionic mobility is decreased in more viscous solvents, that is, the simplicity at which ions migrate in aqueous solution reduces as viscosity increases. Therefore the conductivity reduces.

e) Concentration:

The specific conductance (k) rises with the increase in concentration of solution as the number of ions per unit volume rises. 

While, both the equivalent conductivity and molar conductance increase by the decrease in concentration (that is, upon dilution) as the degree of ionization increases. This is due to the concentration decreases; one can expect decrease in equivalent conductivity because of decrease in available number of ions per unit volume. Though the increase in volume (V) factor more than compensates this effect. The volume should be increased in order to get one equivalent of electrolyte as the concentration is reduced. Therefore the total effect is increase in equivalent conductivity.

Limiting equivalent conductivity (Λ_o):

The equivalent conductivity arrives a maximum value at some dilution and doesn't change upon further dilution (that is, by adding solvent further). This concentration is as well known as infinite dilution.

The equivalent conductivity at infinite dilution is termed as the limiting equivalent conductivity (Λ_o). At this dilution, the ionization of even the weak electrolyte is complete.

Though at infinite dilution (that is, whenever concentration approaches to zero) the conductivity of the solution is so low that it can't be measured correctly. Thus the limiting equivalent conductivity of the electrolyte is computed by employing Debye-Huckel-Onsagar equation as illustrated below. 

Conductance ratio (α):

The ratio of equivalent conductance at particular concentration, Λ_c to that at infinite dilution, Λ_o is termed as conductance ratio, 'α'.

α = Λ_c/Λ_o

For weak electrolytes, the 'α' is as well termed as degree of ionization.

This is possible to find out the equivalent conductivities of electrolytes in water at particular concentration by employing Debye-Huckel-Onsagar equation.

Λ_c = Λ_o - A√C

Here,

Λ_c = equivalent conductivity at particular concentration. 

Λ_o = equivalent conductivity at infinite dilution. 

c = concentration

A = a constant = [82.4/(DT)^1/2] + [(8.2 x 10⁵)/(DT)^3/2]Λ_o

D = Dipole moment of water

T = Absolute temperature

A straight line having negative slope is obtained whenever the equivalent conductivity values (Λ_c) of strong electrolytes are plotted against the square roots of different concentrations (√c). The equivalent conductivity at infinite dilution (Λ_o) can be found out by extending this straight line to zero concentration.

Fig: plot of equivalent conductance-square root of concentration

Though the equivalent conductivity of weak electrolytes rises steeply at extremely low concentrations (as illustrated in the above graph) and therefore their limiting values (Λ_o) can't be found out by extrapolating the Λ_c to zero concentration. 

Thus, Λ_o for weak electrolytes is obtained by employing Kohlrausch law of independent migration of ions, which is illustrated below.

Kohlrausch Law of Independent Migration of Ions:

In the year 1874, Kohlrausch formulated the law of independent migration of ions based on the experimental data of conductivities of different electrolytes. This law can be defined as follows:

At infinite dilution, the dissociation of electrolyte is complete and therefore each and every ion forms definite contribution to the equivalent conductivity of the electrolyte irrespective of the nature of other ions related with it. 

Thus the limiting equivalent conductivity of an electrolyte is the algebraic sum of limiting equivalent conductivities of its constituent ions.

That is, the limiting equivalent conductivity of the electrolyte, Λ_o^electrolyte

Λ_o^electrolyte = λ_o⁺ + λ_o^-

Here, λ_o⁺ and λ_o^- are the limiting equivalent conductivities of the cation and anion correspondingly. Though the Kohlrausch law can as well be defined in terms of molar conductivities as:

The limiting molar conductivity of the electrolyte is the sum of individual contributions of limiting molar conductivities of its constituent ions.

That is, the molar equivalent conductivity of an electrolyte, μ_o^electrolyte

μ_o^electrolyte = n₊μ_o⁺ + n_-μ_o^-

Here, μ_o⁺ and μ_o^- are the limiting molar conductivities of cation and anion correspondingly. And n⁺ and n^- are the stoichiometric numbers of positive and negative ions made for the period of the dissociation of electrolyte.

Experimental Basis and theoretical explanation of Kohlrausch Law:

Kohlrausch noticed that at infinite dilutions, the difference between the conductivities of sodium and potassium salts is constant irrespective of the related anions, as tabulated below. 

Salt pair   Conductivity (mho cm² equiv)      Difference

NaCl                    108.90                               21.20

KCl                      130.10

NaNO3                105.33                                21.17

KNO3                  126.50

NaBr                     111.10                               21.20

KBr                       132.30

The Kohlrausch argued that the constant difference in the conductivities of above pairs can be attributed to the fact that the mobility of sodium and potassium ions at infinite dilution is not affected by the nature of counter ions. The ions at such a low concentration migrate in the electric field as they are independent that is, they exhibit similar ionic conductance irrespective of the nature of counter ion.

Applications of Kohlrausch Law:

1) Computation of limiting conductivities of weak electrolytes: The Kohlrausch law can be employed to compute the limiting conductivities of weak electrolytes. Example: The computation of limiting equivalent conductance of acetic acid, a weak electrolyte is described below.

According to the Kohlrausch law, the limiting equivalent conductance values of CH₃COOH, CH₃COONa, HCl and NaCl can be represented as follows:

Λ_o^CH3COOH = λ_o^CH3COO- + λ_o^H+

Λ_o^CH3COONa = λ_o^CH3COO- + λ_o^Na+

Λ_o^HCl = λ_o^H+ + λ_o^Cl-

Λ_o^NaCl = λ_o^Na+ + λ_o^Cl-

Thus,

Λ_o^CH3COOH = Λ_o^CH3COONa + Λ_o^HCl - Λ_o^NaCl

2) Determination of degree of ionization (α) of weak electrolyte: The degree of ionization of weak electrolyte at a specific concentration is equivalent to the ratio of actual number of ions made because of partial ionization to the expected number of ions made on complete dissociation. 

α = Actual number of ions made because of partial dissociation/Expected number of particles made because of complete dissociation

As the conductance is proportional to the number of ions in the solution, the degree of ionization is equivalent to the conductance ratio as represented below.

α = Λ_C/Λ_o = Λ_C/(λ_o⁺ + λ_o^-)

Here,

Λ_c = equivalent conductivity at particular concentration. 

Λ_o = limiting equivalent conductivity.

λ_o⁺ = limiting equivalent conductivity of the cation.

λ_o^- = limiting equivalent conductivity of the anion.

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