# Electrical Transport

**Conductance (π)**

Any substance that allows an electric current to pass through is called conductor and the capacity to conduct electricity is called the conductivity or conductance of a conductor. Conductor offers to resistance to flow the electric current.

So, the conductance is resiprocal to resistance (R)

π = 1/R

Unit of π = 1/ohm = ohm

^{−1}= mhos

SI Unit is Siemen (S)

**Specific or Electrolytic Conductance (κ)**

The resiprocal of specific resistance (ρ) is called specific conductance.

We know that-

R ∝ l/a

where, l is length of the wire and a is cross sectional area of the wire

or, R = ρ. l/a

or, 1/ρ = 1/R . l/a

or, κ = π . l/a

**Unit of κ**= ohm

^{−1}. cm/cm

^{2}

ohm

^{−1}. cm

^{−1}

Its S.I. unit is Sm

^{−1}

when, l = 1 and a = 1

then, κ = π

So, specific conductance is the conductance of a conductor of unit length and unit cross sectional area. For electrolytic solutions, the conductance of one cc of the solution is called its specific conductance.

**Equivalent Conductance (Λ)**

It is denoted by capital lambda(Λ) and is conductance of V cc of the solution containing one gm-equivalent of an electrolyte. So, it is the product of Specific or Electrolytic Conductance (κ) and the volume of the solution (V) in cc containing one-gm equivalent of the electrolyte.

or, Λ = κ . V

If the concentration of a solution is C g-equivalent/liter, then-

C g-equivalent is present in 1000cc of the solution.

so, 1 g-equivalent is present in 1000cc/C of the solution.

or, Volume in cc containing one gm-equivalent = 1000/C

so, Λ = 1000.κ/C

**Unit of Λ**= κ/C = ohm

^{−1}. cm

^{−1}/eq-cm

^{−3}

or, ohm

^{−1}. cm

^{2}eq

^{−1}

**Molecular Conductance (μ)**

It is denoted by capital mu(μ) and is conductance of V cc of the solution containing one mole of an electrolyte. So, it is the product of Specific or Electrolytic Conductance (κ) and the volume of the solution (V) in cc containing one mole of the electrolyte.

or, μ = κ . V

If the concentration of a solution is C mole/liter, then-

C mole is present in 1000cc of the solution.

so, 1 mole is present in 1000cc/C of the solution.

or, Volume in cc containing one mole = 1000/C

so, μ = 1000.κ/C

When concentration approaches zero, the molar conductivity is known as

**limiting molar conductivity**and is represented by E

^{o}

_{m}.

**Unit of μ**= κ/C = ohm

^{−1}. cm

^{−2}mol

^{−1}

or, Sm

^{2}. cm

^{2}mol

^{−1}

**Relationship between molar conductivity and Concentration**

For strong electrolyte, molar conductivity increases with dilution and can be represented by the equation-

μ = E

^{o}

_{m}− AC

^{1/2}

where A is a constant depending upon the type of the electrolyte, the nature of the solvent and the temperature.

The equation is called Debye Huckel-Onsager equation and is found to hold good at low concentrations.

If we plot μ against c

^{1/2}, we obtain a straight line with intercept equal to E

^{o}

_{m}and slope equal to '–A'.

**Cell Constant**

We know that the resistance (R) is given by-

R ∝ l/a

or, R = ρ. l/a

or, 1/ρ = 1/R . l/a

or, κ = π . l/a

where, ρ is specific conductance, l is length of wire, a is cross sectional area of the wire κ is specific conductance and π is conductance of conductor.

The ratio of l to a is called cell constant.

Hence, cell constant = κ / π = κ . R

**Unit of cell constant:**

l/a = cm/cm

^{2}= cm

^{−1}

**Effect of dilution on conductivity**

The specific conductance depends on the number of ions present per cc of the solution. Though degree of dissociation increases on dilution but the number of ions per cc decreases. So, the specific conductance decreases on dilution.

The equibvalent conductance is the product of specific conductance and volume of the solution containing one gm-equivalent of the electrolyte.

Λ = κ . V

As the decreasing κ value is more than compensated by the increasing V value, hence, Λ increases. on dilution.

**Kohlrausch's Law**

In 1974, Kohlrausch formulated the law of independent migration of ions based on the experimental data of conductivities aof various electrolytes and stated that-

*The equivalent conductivity of an electrolyte at infinite dilution (Λ*.

_{o}) is the sum of the ionic conductivities of their cations and anionsΛ

_{o}= λ

_{+}+ λ

_{−}

where- λ

_{+}and λ

_{−}are cationic and anionic conductivities at infinite dilution respectively.

Kohlrausch's Law can also be states in terms of molar conductivities as-

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

Thus, molar equivalent conductivity of an electrolyte-

μ

_{o}= n

_{+}μ

_{o}

^{+}+ n

_{−}μ

_{o}

^{−}

where, n

_{+}and n

_{−}are the coefficients of positive and negative ions formed during the dissociation of electrolytes and μ

_{o}

^{+}and μ

_{o}

^{−}are limiting molar conductivities of cations and anions respectively.

**Applications:**

Useful in

calculating equivalent conductivity at infinite dilution.

Calculation of degree of dissociation of an electrolyte.

Calculation of solubility of sparingly soluble salt,

Calculation of ionic product of water.

**Q. Calculate the limiting molar conductivity of CH**

_{3}COOH. The molar conductivities of CH_{3}COONa, HCl and NaCl at infinite dilution are 90.1 S.cm^{2}.mol^{-1}, 426.16 S.cm^{2}.mol^{-1}and 126.45 S.cm^{2}.mol^{-1}respectively.Given-

λ

_{CH3COONa}= 90.1 S.cm

^{2}.mol

^{-1}

λ

_{HCl}=426.16 S.cm

^{2}.mol

^{-1}

λ

_{NaCl}=126.45 S.cm

^{2}.mol

^{-1}

According to Kohlrausch law-

λ

_{CH3COOH}= λ

_{CH3COONa}+ λ

_{HCl}– λ

_{NaCl}

or, λ

_{CH3COOH}= 90.1 + 426.16 – 126.45

or, λ

_{CH3COOH}= 390.71 S.cm

^{2}.mol

^{-1}

So, the limiting molar conductivity of CH

_{3}COOH is, 390.71S.cm

^{2}.mol

^{-1}.

**Q. From the given molar conductivities at infinite dilution, calculate λ**

λ

λ

λ

_{m}for NH_{4}OH.λ

_{m}for Ba(OH)_{2}= 457.6 ohm^{-1}cm^{2}mol^{-1}λ

_{m}for Ba(Cl)_{2}= 240.6 ohm^{-1}cm^{2}mol^{-1}λ

_{m}for NH_{4}Cl = 129.8 ohm^{-1}cm^{2}mol^{-1}Answer: 238.3 ohm

^{-1}cm

^{2}mol

^{-1}

**Transport Number or Transference Number or Hittorf Number**

The fraction of the total current carried by an ion in an electrolytic solution is known as transport number. It is denoted by letter 't'. For cation it is represented as t

_{+}and for anion it resepresented as t

_{–}.

Sum of transport numbers of different ions is always one. It is unitless quantity.

t

_{+}+ t

_{−}= 1

The value of ionic conductivities or mobilities are proportional to speeds of the ions

i.e. λ

_{+}∝ U and λ

_{−}∝ V

or, λ

_{+}= F.U

and λ

_{−}= F.V

Transport number of Cation (t

_{+}) = U/(U + V)

or, t

_{+}= λ

_{+}/(λ

_{+}+ λ

_{-})

or, t

_{+}= λ

_{+}/Λ

or, Λ = t

_{+}/λ

_{+}

Similarly,

Transport number of Cation (t

_{−}) = V/(U + V)

or, t

_{−}= λ

_{−}/(λ

_{+}+ λ

_{-})

or, t

_{−}= λ

_{−}/Λ

or, Λ = t

_{−}/λ

_{−}

where U and V are speed of cations and anions respectively.

Transport number can be determined by Hittorf's method, moving boundary method, emf method and from ionic mobility.

**Factors affecting transport number**

**Temperature**

On increasing temperature, the transport number of cation and anion approches closely to 0.5. It means that the transport number of an ion, if less than 0.5 at the room temperature increases and if greater than 0.5 at room temperature decreases with rise in temperature.

**Concentration of the electrolyte**

Transport number generally varies with the concentration of the electrolyte.

**Nature of the other ions present in solution**

The transport number of anion depends upon the speed of the anion and the cation and vice versa.

**Hydration of ion**

Transport number increases when degree of hydration decreases.

**Ionic Mobility**

The velocity of an ion under a unit potential gradient or field strength is called ionic mobility.

Therefore, ionic mobility = velocity of the ion/potential gradient or field strength.

The velocity of an ion in a solution depends on the nature of the ion, concentration of the solution, temperature, and the applied potential gradient. It is related to the ionic conductance of the solution.

The limiting value of ionic mobility is obtained at infinite dilution when the interionic attraction is totally absent.

From Kohlrausch law at infinite dilution-

Λ

_{o}= λ

_{+}+ λ

_{−}

λ

_{+}and λ

_{−}are cataionic and anionic conductance respectively at infinite dilution.

The value of ionic conductance or mobilities are proportional to speed of the ions.

λ

_{+}∝ u and λ

_{−}∝ v

or, λ

_{+}= F u and λ

_{−}= F v

So, the transport number of cation and anion are-

t

_{+}= u/(u+v) = λ

_{+}/(λ

_{+}+ λ

_{−})

or, t

_{+}= λ

_{+}/Λ

_{o}

or, Λ

_{o}= λ

_{+}/t

_{+}

t

_{−}= v/(u+v) = λ

_{−}/(λ

_{+}+ λ

_{−})

or, t

_{−}= λ

_{−}/Λ

_{o}

or, Λ

_{o}= λ

_{+}/t

_{−}

**Unit**

metre

^{2}s

^{-1}volt

^{-1}

**Electrmotive Force (emf)**

emf stands for electromotive force. A galvanic cell consists of two electrodes each having its own potential. The difference of potential between these two electrodes of a cell causes a current to flow from electrode of higher potential to the electrode of lower potential is called emf of the cell.

emf is expressed in volts. Greater the emf (i.e. potential difference between two electrode) greater the electricity flow and greater is the tendency of the cell redox to occur.

Potential of a cell assembled of two electrodes can be determined from the two individual electrode potentials using this formula-

ΔV

_{cell}= E

_{red,cathode}− E

_{red,anode}

or, ΔV

_{cell}= E

_{red,cathode}+ E

_{oxy,anode}

**Electrode Potential or Half Cell Potential**

Electrode potential is the electromotive force of a galvanic cell built from a standard reference electrode and another electrode to be characterized. The electrode potential has its origin in the potential difference developed at the interface between the electrode and the electrolyte.

When a metal(M) is dipped in a solution containing its own ion(M

^{+}), a potential is developed between them. This is called

**Single Electrode Potential(E)**. It is not possible to measure accurately the absolute value of single electrode potential directly. Only the difference in potential between two electrodes can be measured experimentally with the help of reference electrode whose potential is already known. Cell potential is measured by the following formula-

E

_{Cell}= E

_{Cathode}+ E

_{Anode}

emf of single electrode potential is-

E

_{M/M+n}= E

^{o}

_{M/M+n}+ (RT/nF)ln a

When a = 1 then E = E

^{o}. So, Standard electrode potential is the single electrode potential at unit activity.

M ⇌ M

^{+n}+ ne single elecrode potential for the above equilibrium-

E

_{M/M+n}= E

^{o}

_{M/M+n}+ (RT/nF)ln [M]/[M

^{+n}]

or, E

_{M/M+n}= E

^{o}

_{M/M+n}+ (RT/nF)ln 1/[M

^{+n}] (as [M] = 1)

or, E

_{M/M+n}= E

^{o}

_{M/M+n}− (RT/nF)ln [M

^{+n}]

or, E

_{M/M+n}= E

^{o}

_{M/M+n}− (0.0591/n)log [M

^{+n}]

This is the general expression for electrode potential.

**Oxidation Potential and Reduction Potential**

The tendency of an electrode to lose electrons is called oxidation potential while tendency of electrode to gain electrons is called reduction potential.

**Standard Electrode Potential**

The potential of a half-reaction measured against the Standard Hydrogen Electrode under standard conditions is called the standard electrode potential for that half-reaction.

Standard conditions are-

Temperature = 298K

Pessure = 1atm

Concentration of the electrolyte = 1M.

Example-

Consider Zn and Hydrogen electrode

Zn ⇌ Zn

^{2+}+ 2e

H

_{2}⇌ 2H

^{+}+ 2e

When these two electrodes (two equilibria) are brought into electrical contact using an external wire and a salt bridge, the electrons will be pushed from the zinc equilibrium (electrode) to the hydrogen equilbrium (electrode) with a force of - 0.76V (the negative sign simply indicates the direction of flow - from zinc to hydrogen ions). So, the standard electrode potential of Zn is −0.76 volts and the overall reaction is-

Zn + 2H

^{+}→ Zn

^{2+}+ H

_{2}

**Uses of Standard Electrode Potentials**

Uses of standard electrode potentials are given below –

1. It is used to measure relative strengths of various oxidants and reductants.

2. It is used to calculate standard cell potential.

3. It is used to predict possible reactions.

4. Prediction of equilibrium in the reaction.

**Standard Hydrogen Electrode (SHE)**

Standard Hydrogen Electrode is a reference electrode consists of a container, containing solution kept at 298K.

A wire containing Platinum electrode coated with platinum black is immersed in the solution.

Pure hydrogen gas is bubbled in the solution at 1bar pressure.

The potential of standard hydrogen electrode is taken as zero volt at all temperatures.

Standard hydrogen electrode may act as anode or cathode depending upon the nature of the other electrode.

If its acts as anode, the oxidation reaction taking place is

H

_{2}(gas) → 2H

^{+}(aq) + 2e

If it acts as cathode then the reduction half reaction occurring is

2H

^{+}(aq) + 2e → H

_{2}(gas)

**Calomel Electrode**

It is a secondary electrode. It has a glass tube fitted with two side tubes. One side tube is used to fill saturated KCl solution and other is connected to another electrode. At the bottom of the main tube, an extra pure Hg is kept which is an contact with Hg

_{2}Cl

_{2}. A pt-wire is sealed into a glass tube for making electrical contact with the external circuit.

Reduction occurs when it is combined with hydrogen electrode. Its potential depends on the Concentration of KCl solution. Its standard electrode potential is 0.2415V. It can act as an anode or cathode depending on the electrode potential of the coupled electrode. The calomel electrode has following cell reaction-

2Hg + 2Cl

^{−}→ Hg

_{2}Cl

_{2}+ 2e (Oxidation)

Hg

_{2}Cl

_{2}+ 2e → 2Hg + 2Cl

^{−}(Reduction)

**Electrochemical Series**

Electrochemical series describes the arrangement of elements in order of their increasing electrode potential values. By measuring the potentials of various electrodes versus standard hydrogen electrode (SHE), a series of standard electrode potentials has been established.

Electrodes with positive E° values for reduction half reaction act as cathodes versus SHE, while those with negative E° values of reduction half reactions behave as anodes versus SHE.

The negative sign of standard reduction potential indicates that an electrode when joined with SHE acts as anode and oxidation occurs on this electrode. For example, standard reduction potential of zinc is -0.76 volt. When zinc electrode is joined with SHE, it acts as anode (-ve electrode) i.e., oxidation occurs on this electrode.

Similarly, the +ve sign of standard reduction potential indicates that the electrode when joined with SHE acts as cathode and reduction occurs on this electrode.

On moving down the series-

Reduction Potential Decreases

Oxidation Poitential Increases

Strength of Oxidizing Agent Decreases

Strength of Reducing Agent Increases

Reactivity of Metals Increases

Reactivity of Nonmetals Decreases