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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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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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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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The cell potential is simply related to the free energy change for the reaction. In an electro chemical cell the system does work by transferring the electric charge through an external circuit. The free energy change $∆\mathrm{G}$ is equal to electrical work done.

$∆\mathrm{G} = {\mathrm{W}}_{\mathrm{electrical}}$

For reaction occurring in a electrochemical cell whose electrodes differ in potential by E_cell, the work done when amount of charge nF is transferred is given by
                   ${\mathrm{W}}_{\mathrm{elect}} = -{\mathrm{nFE}}_{\mathrm{cell}}$
∴                   $∆\mathrm{G} = -{\mathrm{nFE}}_{\mathrm{cell}}$
Under standard state conditions
ΔG° = – n FE°_cell.

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Equivalent conductivity : The conductivity of a volume (V) of a solution containing one equivalent of electrolyte placed between two electrodes separated by unit distance apart and of large enough area of cross-section to hold the entire volume (V) is called equivalent conductance. It is denoted by Λ_eq.
Λ_eq = k x V
where k = Specific conductance
V = Volume of solution containing one equivalent of electrolyte.

Molar conductivity : It is the conductivity of volume (V) of a solution containing 1 mole of a dissolved electrolyte place between two electrodes separated by unit distance apart and of enough area of cross-section to hold the entire volume V. It is denoted by Λ_m.
Λ_m= k x V = k / V where V = Volume of solution Containing 1 mole of electrolyte C = molarity of solution k = Specific conductivity.