In the electrolysis of an aqueous \(\mathrm{AgNO}_{3}\) solution, \(0.67 \mathrm{~g}\) of \(\mathrm{Ag}\) is deposited after a certain period of time. (a) Write the half-reaction for the reduction of \(\mathrm{Ag}^{+}\). (b) What is the probable oxidation halfreaction? (c) Calculate the quantity of electricity used, in coulombs.

The reduction half-reaction in electrolysis is fundamental to understanding how chemicals gain electrons. During the electrolysis of silver nitrate (AgNO₃), silver ions (\texttt{Ag}⁺) in the solution receive electrons and are reduced to solid silver (\texttt{Ag}). In simple terms, reduction is the gain of electrons. The half-reaction for the reduction of silver ions to silver metal is expressed as follows:
\[Ag^{+}(aq) + e^- \rightarrow Ag(s)\]
In this equation, one electron (\texttt{e}^-) is added to a silver ion, which leads to the formation of a silver atom, depicted as solid silver (\texttt{Ag}(s)) on the electrode.

In contrast to the reduction half-reaction, the oxidation half-reaction involves the loss of electrons from a species. For the silver nitrate electrolysis scenario, water molecules undergo oxidation. Oxidation often occurs at the anode in an electrolytic cell.
\[2H_2O(l) - 4e^- \rightarrow O_2(g) + 4H^+(aq)\]
This equation represents the oxidation of water. Two water molecules (\texttt{H}₂\texttt{O}) lose a total of four electrons (\texttt{e}^-) and are transformed into oxygen gas (\texttt{O}₂(g)) and four hydrogen ions (\texttt{H}⁺(aq)). The released electrons travel through the circuit to the cathode, where they are used to reduce the silver ions.

Faraday's law of electrolysis quantifies the relationship between the amount of substance that undergoes electrolysis and the total electric charge that passes through the electrolyte. It states that the amount of substance deposited or dissolved at an electrode is directly proportional to the quantity of electricity (\texttt{Q}) that passes through the electrolyte.
This law is represented by the equation:
\[Q=n \times F\]
where \(n\) is the number of moles of the substance, and \(F\) is the Faraday constant (approximately 96,500 coulombs per mole of electrons). By understanding Faraday's law, students can predict the outcome of the electrolysis process and calculate the exact amount of a substance that will be deposited at an electrode based on the electric current applied.

The quantity of electricity consumed during electrolysis can be calculated using the mass of the substance deposited and Faraday's law. This calculation is important for practical applications such as electroplating and electrorefining.
For example, in the case of silver deposition, one can calculate the number of moles of silver (\texttt{n}) with the equation:
\[n= \frac{m}{M}\]
where \(m\) is the mass of silver deposited, and \(M\) is the molar mass of silver. Once the number of moles of silver is found, the quantity of electricity (\texttt{Q}) in coulombs can be calculated with Faraday's law:
\[Q=n \times F\]
Understanding this calculation allows students to evaluate the efficiency of electrolysis processes and to design systems with the capacity to produce desired quantities of a given substance.