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Kohlrausch’s law of independent migration of ions states molar conductivity of an electrolyte at infinite dilution can be expressed as the sum of the contribution of individual ions. If molar conductivity of cations and anions are represented by λ^∞₊ and λ^∞_– respectively.

${\mathrm{\lambda }}_{\mathrm{m}}^{\infty } = {\mathrm{v}}_{+} {\mathrm{\lambda }}_{+}^{\infty }+{\mathrm{v}}_{-}{\mathrm{\lambda }}_{-}^{\infty }$

where v₊ and v_– are number of cations and anions per formula of electrolyte e.g.,

Λ^∞ CaCl₂ = λ^∞ (Ca²⁺) + 2 λ^∞ (CI^–)
Λ = KCl = λ^∞ (K⁺) + λ^∞ (CI^–)

Uses 1. It is used to find molar conductivity of weak electrolyte at infinite dilution which
cannot be obtained by extrapolation.

2. It is used to calculate degree of dissociation of weak electrolyte at a particular concentration.
Degree of dissociation $\left(\mathrm{\alpha }\right) = \frac{{\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{e}}}{{\mathrm{\Lambda }}_{\mathrm{m}}^{\infty }}.$
where Λ^e_m is molar conductivity of weak electrolyte at a particular concentration and Λ^e_mis molar conductivity of weak electrolyte at infinite dilution.

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The concentration of all species involved in the species involved in the electrode reaction is unity.This need not be always true.
Nernst shows that for the electrode reaction:
$M^{\mathrm{n}+}\left(\mathrm{aq}\right) + {\mathrm{ne}}^{-} \to \mathrm{M}\left(\mathrm{s}\right)$
the electrode potential at any concentration measured with respect to standard hydrogen electrode can be represented by:
${\mathrm{E}}_{\left(\raisebox{1ex}{${\mathrm{M}}^{\mathrm{n}+}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{M}$}\right.\right)} = {\mathrm{E}}_{\raisebox{1ex}{${\mathrm{M}}^{\mathrm{n}+}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{M}$}\right.}^{-} - \frac{\mathrm{RT}}{\mathrm{nF}}\mathrm{In} \frac{\left[\mathrm{M}\right]}{\left[{\mathrm{M}}^{\mathrm{n}+}\right]}$
but concentration of solid M is taken as unity as we have
${\mathrm{E}}_{\left(\raisebox{1ex}{${\mathrm{M}}^{\mathrm{n}+}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{M}$}\right.\right)} = {\mathrm{E}}_{\raisebox{1ex}{${\mathrm{M}}^{\mathrm{n}+}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{M}$}\right.}^{-} - \frac{\mathrm{RT}}{\mathrm{nF}}\mathrm{In} \frac{\left[1\right]}{\left[{\mathrm{M}}^{\mathrm{n}+}\right]}$

R is gas constant (8.314 JK–1 mol–1),
F is Faraday constant (96487 C mol^–1), T is temperature in kelvin and [Mⁿ⁺] is the concentration of the species, Mn

Let us take a electrode reaction
${\mathrm{Zn}}^{2+}+2{\mathrm{e}}^{-}\to \mathrm{Zn}$
The Nernst equation of this electrode
$\mathrm{E} = \mathrm{E}°-\frac{2.303 \mathrm{RT}}{\mathrm{nF}}\log \frac{{\mathrm{a}}_{\mathrm{product}}}{{\mathrm{a}}_{\mathrm{reactant}}}$
Instead of activity, we can take molar concentration.
$\mathrm{E} = \mathrm{E}°-\frac{0.05916}{\mathrm{n}}\log \frac{\left[\mathrm{Zn}\right]}{\left[{\mathrm{Zn}}^{2+}\right]}$
For pure solid and liquid molar concentration is taken as unity.
$\mathrm{E} = \mathrm{E}°-\frac{0.05916}{2}\log \frac{1}{\left[{\mathrm{Zn}}^{2+}\right]}$
Yes,  Nernst equation can be applied to the cell reaction.
                           $\mathrm{aA}+\mathrm{bB}\to \mathrm{cC}+\mathrm{dD}$

$\mathrm{E} = \mathrm{E}°-\frac{0.05916}{\mathrm{n}}\log \frac{{\left[\mathrm{C}\right]}^{\mathrm{c}}{\left[\mathrm{D}\right]}^{\mathrm{d}}}{{\left[\mathrm{A}\right]}^{\mathrm{a}} {\left[\mathrm{B}\right]}^{\mathrm{b}}}$

 

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Standard half-cell potential E° : If the ionic species have concentration of mol dm^–3and pressure of the gaseous species is 1 atm (101.325 KPa), than half-cell potential is called as standard half-cell potential. The temperature being 298 K.
Absolute value of half-cell potential cannot be determined experimentally. However, its value relative to reference electrode can be determined.
Reference electrode 11 given half-cell. E°_cell can be measured experimentally using a potentiometer.

Knowing the standard reduction potential of the reference electrode E°_half-cell can be found out.

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Faraday's first law. The amount of substance produced at an electrode during electrolysis is directly proportional to the quantity of electricity passed through the electrolyte.

                                   $\mathrm{W}\propto \mathrm{Q}$

or                               $\mathrm{W}\propto$It

or                                 W = ZIt

where W is the mass of substance produced at an electrode.
I is current in amperes
t is time in seconds for which current is passed.
Z is electro-chemical equivalent of substance.
Faraday's second law: It states that the masses of different substances liberated or dissolved by the same amount of electricity passed is directly proportional to their chemical equivalents. Or in other words “the same quantity of electricity will produce or dissolve chemically equivalent quantities of all substances.”
Mathematically,

$\frac{\mathrm{Mass} \mathrm{of} \mathrm{A} \mathrm{deposited}}{\mathrm{Mass} \mathrm{of} \mathrm{B} \mathrm{deposited}}=\frac{\mathrm{Equivalent} \mathrm{mass} \mathrm{of} \mathrm{A}}{\mathrm{Equivalent} \mathrm{mass} \mathrm{of} \mathrm{B}}$

Thus, we can say that the same quantity of electricity is required to produce one equivalent of any substance. It is called Faraday, F. It is equal to 96500 coulombs, and is equal to the charge on one mole of electrons.

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(i) Weak electrolytes : An electrolyte that ionizes partially in solution is called a weak electrolyte. The solution formed contains ions which are in equilibrium with un-ionised molecules, e.g., acetic acid dissolves in water to form H₃O⁺ and CH₃COO⁺ ion. The solution contains H₃O⁺ (hydronium ion), CH₃COO^– (acetate ion) and unionised CH₃COOH molecules.

The degree of ionisation of a weak electrolyte is much less than 1. These have low values of molar conductivities at high concentration. Degree of ionisation and molar conductivity both increases with dilution.
(ii) Strong electrolyte : An electrolyte which is almost completely ionised in solution is called a strong electrolyte. The degree of ionisation of a strong electrolyte is 1 or 100% (or nearly so). The solution formed contains ions which are in equilibrium with solid form of strong electrolyte.

Strong electrolyte	Weak electrolyte
1. These have higher molar conductivities at all concentrations. 2. λ°m values increase very slightly with dilution. 3. Degree of ionisation is very high at all concentration i.e., almost fully ionized. 4. Most of the salts like NaCl, KCl, NaNO3, BaCl2 and mineral acids like HCl, H2SO4, HNO3 and NaOH, KOH etc are common examples of strong electrolytes	1. These have much lower conductivities at high concentration. 2. λ°m values increase sharply with dilution. 3. Degree of ionisation is very low at high concentration and increases with dilution. 4. Salts like ammonium acetate, acetic acid, aq NH4OH, aqueous CO2 and organic acids and bases are common examples of weak electrolytes.