Electrochemistry : Molar & Equivalent Conductivity, Kohlrausch's law, Galvanic Cell, Electrode Potential, Nerst Equation, Electrolysis, Faraday's Law

Conductivity of Electrolytic Solutions

Resistance (R) : Resistance of a column of eletrolyte solution is proportional to distance between the two electrodes, l, and inversely proportional to its area of corss section A.

Conductance (G) : Conductance measure rate of flow of charge in a metallic wire or electrolytic solution. It is receprocal of resistance.

Conductivity & Resistivity

Resistivity $\rho =\frac{RA}{l}\text{SI unit =}\Omega m$

Cell constant $\frac{l}{A}=\kappa \cdot R\text{SI unit =}{\mathrm{m}}^{-1}$

Conductivity $\mathrm{\kappa }=\frac{1}{\rho }=\frac{l}{RA}=\frac{\text{Cell constant}}{R}\text{SI unit =}\mathrm{S}{\mathrm{m}}^{-1}$

Factors affecting Conductivity

• Conductivity ∝ number of ions
  Na₂S > NaCl

• Conductivity ∝ size of ion in gaseous state

• $\mathrm{Conductivity}\propto \frac{1}{\text{size of ion in hydrated state}}$
  ${\mathrm{Cs}}^{+}>{\mathrm{Rb}}^{+}>{\mathrm{K}}^{+}>{\mathrm{Na}}^{+}$
• $\mathrm{Conductivity}\propto \frac{1}{\mathrm{Charge}}$
  ${\mathrm{Na}}^{+}>{\mathrm{Mg}}^{+2}>{\mathrm{Al}}^{+3}>{\mathrm{Zr}}^{+4}$

• Conductivity ∝ Acidic strength or Basic strength
  HCOOH > CH₃COOH

• Conductivity of H⁺ and OH^- are exceptionally high due to Grothus effect
  H⁺ > OH^- > other ions

• Conductivity ∝ Temperature

• $\mathrm{Conductivity}\propto \frac{1}{\mathrm{Viscosity}}$

Molar Conductivity

Molar Conductivity = ${\mathrm{\Lambda }}_{\mathrm{m}}=\frac{\mathrm{\kappa }}{\mathrm{C}}$

Here, the units of
$\begin{array}{lcr}\kappa & =& \mathrm{S}{\mathrm{m}}^{-1}\\ \mathrm{C}& =& \mathrm{mol}{\mathrm{m}}^{-3}\\ {\Lambda }_m& =& \mathrm{S}{\mathrm{m}}^2{\mathrm{mol}}^{-1}\end{array}$

But in chemistry, we are generally given with concentration(C) in the units of $\mathrm{mol}{\mathrm{L}}^{-1}$ so,

Molar Conductivity = ${\mathrm{\Lambda }}_{\mathrm{m}}=\frac{\mathrm{\kappa }1000}{\mathrm{M}}$ , to convert $\mathrm{mol}{\mathrm{L}}^{-1}$  into $\mathrm{mol}{\mathrm{cm}}^{-3}$

Molar Conductivity = ${\mathrm{\Lambda }}_{\mathrm{m}}=\frac{\mathrm{\kappa }}{1000\mathrm{M}}$ to convert $\mathrm{mol}{\mathrm{L}}^{-1}$  into $\mathrm{mol}{\mathrm{m}}^{-3}$

Equivalent Conductivity

Equivalent Conductivity = ${\mathrm{\Lambda }}_{\mathrm{eq}}=\frac{{\mathrm{\Lambda }}_{\mathrm{m}}}{\mathrm{n}-\mathrm{factor}}$

Units = $\frac{\mathrm{S}{\mathrm{cm}}^2}{\mathrm{gram}\mathrm{eq}}$

Effect of Dilution on Conductivity

On dilution:

• No of ions increase
• distance between ions increase
• Molar conductivity also increases

• $\mathrm{AB}\to {\mathrm{A}}^{+}+{\mathrm{B}}^{-}={\left({\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{\infty }}\right)}_{\mathrm{AB}}={\left({\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{\infty }}\right)}_{{\mathrm{A}}^{+}}+{\left({\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{\infty }}\right)}_{{\mathrm{B}}^{-}}$
• Value of molar conductivity is maximum and constant, does not depend on parent electrolyte (weak or strong)
• ${\left({\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{\infty }}{\mathrm{CH}}_3{\mathrm{COO}}^{-}\right)}_{\mathrm{from}{\mathrm{CH}}_3\mathrm{COOH}}={\left({\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{\infty }}{\mathrm{CH}}_3{\mathrm{COO}}^{-}\right)}_{\mathrm{from}{\mathrm{CH}}_3\mathrm{COONa}}$
• Ions migrate independently

Examples:

(i.) ${\mathrm{CH}}_3\mathrm{COOH}\leftrightharpoons {\mathrm{CH}}_3{\mathrm{COO}}^{-}+{\mathrm{H}}^{+}$

(ii.) ${\mathrm{A}}_{\mathrm{x}}{\mathrm{B}}_{\mathrm{y}}\leftrightharpoons {\mathrm{xA}}^{+\mathrm{y}}+{\mathrm{yB}}^{-\mathrm{x}}$

(b.) Strong Electrolyte

• ${\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{v}}$ = molar conductivity at V volume
• ${\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{\infty }}$ = molar conductivity at infinite dilution

α and K_a in terms of Conductivity

α = degree of dissociation

K_a = dissociation constant

Electrochemical Process

Process : Electrical energy ⇋ Chemical energy

Cell : A device in which the electrochemical process take place.

Types of Cell :

Electrochemical cell has mainly two components, electrodes (anode & cathode) + salt bridge.

Electrodes

or

Anode : $\mathrm{Zn}\to {\mathrm{Zn}}^{+2}+2{\mathrm{e}}^{-}\to \text{Oxidation}\to {\mathrm{E}}_{\mathrm{Zn}\mathrm{\mid }{\mathrm{Zn}}^{+2}}^{\mathrm{o}}$

Cathode : ${\mathrm{Cu}}^{+2}+2{\mathrm{e}}^{-}\to \mathrm{Cu}\to \text{Reduction}\to {\mathrm{E}}_{{\mathrm{Cu}}^{+2}\mathrm{\mid }\mathrm{Cu}}^{\mathrm{o}}$

Salt Bridge

• Connect anode and cathode half cell
• prevent liquid-liquid junction potential
• Contains strong electrolyte
• Does not participate in cell reaction (Oxidation or Reduction)
• Mobility of cation and anion of strong electrolyte are almost equal

Electrode Potential (E)

Potential difference developed at the interface of electrode and electrolyte.

Oxidation Potential ↑ tendency to get itself oxidised ↑ Reducing power ↑

Reduction Potential ↑ tendency to get itself reduced ↑ Oxidising power ↑

If electrode potential (E) is measured at 298K temperature and 1M concentration of electrolyte, it is said to be as
Standard electrode potential (E^o).

For a particular element

|Oxidation potential| = |Reduction potential|

| E^o_op | = | E^o_rp |

E^o_Cell = (E^o_op)_anode + (E^o_rp)_cathode

Gibbs Free Energy and Electrode Potential

or

• n = Number of e- transfered
• F = Faraday’s constant
• 1F = 96500C
• E = Electrode potential
  ${\mathrm{E}}_{\mathrm{cell}},{\mathrm{E}}_{\mathrm{op}},{\mathrm{E}}_{\mathrm{rp}}$
• E^o = Electrode potential
  ${\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}},{\mathrm{E}}_{\mathrm{op}}^{\mathrm{o}},{\mathrm{E}}_{\mathrm{rp}}^{\mathrm{o}}$

Example :

${\mathrm{E}}_{\mathrm{cell}}={\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}-\frac{\mathrm{RT}}{\mathrm{nF}}ln{\mathrm{Q}}_{\mathrm{c}}$  Applicable in all temperature Range

Similar for oxidation and reduction half cell, replace E_cell with E_op or E_rp

After putting the values of
R = 8.314 joule mol^-1 K^-1
F = 96500 Coulomb
ln = 2.303 log
at a temperature T = 298K or 25^oC

${\mathrm{E}}_{\mathrm{cell}}={\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}-\frac{0.0591}{\mathrm{n}}log{\mathrm{Q}}_{\mathrm{c}}$

Example :

Anode half-cell : ${\mathrm{Zn}}_{\left(\mathrm{s}\right)}\to {\mathrm{Zn}}_{\left(\mathrm{aq}\right)}^{+2}+2{\mathrm{e}}^{-}{\mathrm{Q}}_{\mathrm{c}}=\left[{\mathrm{Zn}}^{+2}\right]$

Cathode half-cell : ${\mathrm{Cu}}_{\left(\mathrm{aq}\right)}^{+2}+2{\mathrm{e}}^{-}\to {\mathrm{Cu}}_{\left(\mathrm{s}\right)}{\mathrm{Q}}_{\mathrm{c}}=\frac{1}{\left[{\mathrm{Cu}}^{+2}\right]}$

Overall Cell : ${\mathrm{Zn}}_{\left(\mathrm{s}\right)}+{\mathrm{Cu}}_{\left(\mathrm{aq}\right)}^{+2} \to {\mathrm{Cu}}_{\left(\mathrm{s}\right)}+{\mathrm{Zn}}_{\left(\mathrm{aq}\right)}^{+2}{\mathrm{Q}}_{\mathrm{c}}=\frac{\left[{\mathrm{Zn}}^{+2}\right]}{\left[{\mathrm{Cu}}^{+2}\right]}$

or

Equilibrium Constant and Electrode Potential

at equilibrium Q_c = K_eq and ΔG = 0, hence E_cell = 0 (∵ ΔG = -nFE_cell)

or

${\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}=\frac{0.0591}{\mathrm{n}}log{\mathrm{K}}_{\mathrm{eq}}$  at 298K

Concentration Cell

A concentration cell is comprised of two half-cells with the same electrodes, but differing in concentrations of electrolyte solution.

A concentration cell dilutes the more concentrated (cathode) solution and concentrate the more dilute (anode) solution, creating a voltage as the cell reaches an equilibrium.

Anode half-cell : $\mathrm{Zn}\to \underset{\left[0.10\mathrm{M}\right]}{{\mathrm{Zn}}^{+2}}+2{\mathrm{e}}^{-}$

Cathode half-cell : $\underset{\left[1.0\mathrm{M}\right]}{{\mathrm{Zn}}^{+2}}+2{\mathrm{e}}^{-}\to \mathrm{Zn}$

This voltage is achieved by transferring the electrons from the cell with the lower concentration (anode) to the cell with the higher concentration (cathode).

Overall Cell : $\underset{\left[1.0\mathrm{M}\right]}{{\mathrm{Zn}}^{+2}}\to \underset{\left[0.10\mathrm{M}\right]}{{\mathrm{Zn}}^{+2}}{\mathrm{Q}}_{\mathrm{c}}=\frac{\left[0.10\right]}{\left[1.0\right]}$

E^o_cell of a concentration cell = 0, because the electrodes are identical. At standard condition i.e. 1M concentration and 298K

${\mathrm{Zn}}_{\left(\mathrm{s}\right)}\to \underset{\left[1.0\mathrm{M}\right]}{{\mathrm{Zn}}^{+2}}+2{\mathrm{e}}^{-}{\mathrm{E}}_{\mathrm{op}}^{\mathrm{o}}=+0.76$

$\underset{\left[1.0\mathrm{M}\right]}{{\mathrm{Zn}}^{+2}}+2{\mathrm{e}}^{-}\to {\mathrm{Zn}}_{\left(\mathrm{s}\right)}{\mathrm{E}}_{\mathrm{rp}}^{\mathrm{o}}=-0.76$

$\begin{array}{rclcl}{\mathrm{E}}_{\mathrm{cell}}^{\mathrm{o}}& =& {\mathrm{E}}_{\mathrm{op}}^{\mathrm{o}}& +& {\mathrm{E}}_{\mathrm{rp}}^{\mathrm{o}}\\ & =& \left(+0.76\right)& +& \left(-0.76\right)\\ & =& 0.76& -& 0.76\\ & =& 0& & \end{array}$

When the cell reaches to equilibrium condition, concentration of both electrodes becomes equal, and E_cell = 0

Electrolysis

It is process which uses electrical energy (DC current) to derive non-spontaneous chemical reaction.

This process is used to derive the reaction in opposite direction in a electrochemical cell in order to recharge it after discharging.

It is also used for qualitative and quantative analysis of electrochemical cell reaction.

Faraday’s Law of Electrolysis for quantitative analysis

Faraday’s 1st Law

Mass deposited (m) ∝ Charge (Q)

m = ZQ = ZIt $\because \mathrm{I}=\frac{\mathrm{Q}}{\mathrm{t}}\text{and}\mathrm{Z}=\frac{\mathrm{Molar}\mathrm{mass}}{\mathrm{n}\mathrm{F}}$

Faraday’s 2nd Law

Mass deposited (m) ∝ Equivalent mass (E)

Examples :

If x Liter of 0.2M CuSO₄ is reduced to Cu by a current of 0.965A in 1 hour, then value of x is?

Total charge (Q) = 0.965 x 60 x 60 = 0.965 x 36000 Coulomb

1 Faraday (F) = 96500 Coulomb

Since all the 0.2M CuSO₄ has been reduced to Cu solid, so

Initial moles of CuSO₄ = Final moles of Cu solid