Faraday explained that the decomposition of electrolytes by an electric current is governed by two laws.

First law: The amount of the substance liberated or deposited or dissolved at an electrode during electrolysis of an electrolyte is directly proportional to the quantity of electricity passing through the solution of electrolyte or melt. Mathematically Faraday’s first law is \(m \propto q\) or \(m = eq\) or \(m = ect\), where m is the mass of the substance liberated or deposited or dissolved, q is the quantity of electricity in coulomb, t is the time in seconds, C is the strength of current in amperes and e is the electrochemical equivalent of the ion or metal or molecule deposited or liberated or dissolved at the electrode. Electrochemical equivalent of a substance is the amount deposited or liberated or dissolved or underwent electrode reaction at an electrode by passing one ampere current for 1 s, i.e., 1 C. Chemical equivalent of a substance is the amount of substance deposited or liberated or dissolved or had undergone electrode reaction at an electrode during the passage of one Faraday of electricity during the electrolysis of electrolyte solution or melt.

\(\text{Chemical equivalent of an element or ion = }\frac{\text{Atomic weight}}{\text{Valency or charge of the ion}}\)

The electrochemical equivalent of an element is directly proportional to its chemical equivalent

\[e \propto E \text{ or }E = F.e\]

Unit of electrochemical equivalent is gram C^–1. One Faraday, i.e., 96,500 C is equal to the charge present on mole (6.023×10²³) [Avogadro’s number] of electrons or protons

\[m= \frac{ECt}{96500}\]

Second law of Faraday states that if same quantity of electricity is passed through different electrolyte solutions or melts, the amount of the different substances liberated or deposited or dissolved or had undergone reaction at electrode is directly proportional to their chemical equivalents.

\[\frac{W_1}{E_1} = \frac{W_2}{E_2} = \frac{W_3}{E_3}\]

The chemical equivalents depend on the number of electrons participated at the electrode reaction. The chemical equivalents or equivalent weights of NaCl, KCl, KBr, NaOH, etc., are equal to their molecular weights since only one electron take part in electrode reaction. The equivalent weights of other electrolytes depend on the number of electrons.

\[\text{Equivalent weight} =\frac{\text{Molecular weight}}{\text{Number of electrons involved in electrode reaction}}\]

The amount of electricity that can deposit \(108 g\) of silver from \(AgNO_3\) solution

1 Faraday

The correct answer is option 3. 1 Faraday.

Faraday's laws of electrolysis govern the relationship between the amount of substance produced or consumed during electrolysis and the quantity of electricity passed through the electrolyte.

According to Faraday's first law, the amount of substance (in moles) produced or consumed during electrolysis is directly proportional to the quantity of electricity (in coulombs) passed through the electrolyte. Mathematically, it can be expressed as:

\(\text{Amount of substance (in moles)} = \frac{\text{Quantity of electricity (in coulombs)}}{\text{Faraday's constant (in coulombs per mole)}} \)

Given that the molar mass of silver (\(Ag\)) is \(108 \, g/mol\), to deposit \(108 \, g\) of silver from \(AgNO_3\) solution, which corresponds to \(1 \, mol\) of silver, we need the charge carried by \(1 \, mole\) of electrons. This is equal to \(1 \, Faraday\) (\(F\)), which is approximately \(96485 \, C/mol\).

So, to deposit \(108 \, g\) of silver, we need \(1 \, Faraday\) of charge.

Therefore, the correct answer is: 1 Faraday