A galvanic cell using \(\mathrm{Mg} / \mathrm{Mg}^{2+}\) and \(\mathrm{Cu} / \mathrm{Cu}^{2+}\) half-cell's operates under standard-state conditions at \(25^{\circ} \mathrm{C}\) and each compartment has a volume of \(218 \mathrm{~mL}\). The cell delivers 0.22 A for \(31.6 \mathrm{~h}\). (a) How many grams of \(\mathrm{Cu}\) are deposited? (b) What is the \(\left[\mathrm{Cu}^{2+}\right]\) remaining?

Faraday's laws of electrolysis are critical to understanding how electroplating and batteries work. The first law states that the mass of a substance altered at an electrode during electrolysis is proportional to the quantity of electric charge passed through the circuit. This is quantitatively expressed as \( m = (Q/F) \times M \), where \( m \) is the mass of substance altered, \( Q \) is the total electric charge, \( F \) is Faraday's constant (approximately 96485 C/mol), and \( M \) is the molar mass of the substance.

This means, if you know the current and the duration of electrolysis, you can calculate how much substance is deposited on an electrode. In the context of a galvanic cell, where a current is produced because of a spontaneous electrochemical reaction, Faraday's laws help us determine the quantity of metal deposited from the electrolyte onto an electrode over time. Inspecting this fundamental principle guides students to accurately predict and calculate the outcomes of electrochemical processes.

An electrochemical reaction involves the movement of electrons from one substance to another. In a galvanic, or voltaic, cell, this process happens spontaneously and results in the generation of electrical energy. The cell consists of two half-cells: one where oxidation occurs (loss of electrons) and another where reduction happens (gain of electrons). These half-cells are connected by a salt bridge that maintains electrical neutrality.

The overall reaction in a galvanic cell can be understood through the individual reactions at the anode and cathode. For instance, in the case of a \( Mg/Cu^{2+} \) cell, \( Mg \) is oxidized to \( Mg^{2+} \) at the anode, while \( Cu^{2+} \) ions are reduced to solid \( Cu \) at the cathode. Since these reactions involve the transfer of electrons, they can be quantified via the flow of current. As current passes through the cell, it quantifies electron flow which, in turn, indicates how much of a material has undergone electrochemical change.

Molarity (or molar concentration) is a measurement of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution, and its unit is moles per liter (mol/L). The formula for molarity is given by \( M = n/V \), where \( n \) is the number of moles of the solute, and \( V \) is the volume of the solution in liters.

This concentration calculation is crucial in chemistry as it allows us to predict how substances will react with one another in a solution. When dealing with a galvanic cell, after an electrochemical reaction has progressed, the molarity of the electrolyte will change as ions are deposited as solids on the electrodes. This is precisely why after copper is deposited on the cathode, the concentration of \( Cu^{2+} \) ions in the solution decreases. To determine the remaining concentration, we subtract the moles of \( Cu^{2+} \) consumed in the reaction from the initial moles present in the solution, then we use the molarity formula with the new number of moles and the same volume to find the final concentration.