When magnesium metal is added to a beaker of \(\mathrm{HCl}(a q),\) a gas is produced. Knowing that magnesium is oxidized and that hydrogen is reduced, write the balanced equation for the reaction. How many electrons are transferred in the balanced equation? What quantity of useful work can be obtained when Mg is added directly to the beaker of HCl? How can you harness this reaction to do useful work?

Since magnesium is oxidized and hydrogen is reduced, we can write the half-reactions for these processes: Oxidation (Mg to Mg²⁺): \[ \textrm{Mg} \rightarrow \textrm{Mg}^{2+} + 2e^{-} \] Reduction (H⁺ to H₂): \[ 2\textrm{H}^{+} + 2e^{-} \rightarrow \textrm{H}_{2} \]

Now, we have to combine the half-reactions to balance the overall equation. Since the number of electrons is the same in both half-reactions (2 electrons), we can just add them directly: \[ \textrm{Mg} + 2\textrm{H}^{+} \rightarrow \textrm{Mg}^{2+} + \textrm{H}_{2} \] Since H⁺ comes from the hydrochloric acid (HCl), we can also write the balanced equation as: \[ \textrm{Mg} + 2\textrm{HCl} \rightarrow \textrm{MgCl}_{2} + \textrm{H}_{2} \]

In the balanced equation, 2 electrons are transferred from magnesium to hydrogen ions. This can be seen in the half-reactions, where the oxidation half-reaction shows 2 electrons being lost by Mg atom, and the reduction half-reaction shows 2 electrons being gained by the 2 H⁺ ions.

The quantity of useful work that can be obtained from this reaction can be calculated using the Gibbs free energy (ΔG) and the Faraday constant (F). However, the problem does not provide us with the data such as temperature, concentration, or pressure needed for a precise calculation. In general, to obtain useful work from this reaction, one would typically set up a galvanic cell, where the redox reaction between Mg and HCl is separated by a salt bridge, and the electrons flow through an external circuit, doing work on it.

To harness this reaction to do useful work, we can create an electrochemical (galvanic) cell. The cell would consist of two half-cells: one containing magnesium metal as an electrode immersed in a magnesium salt solution \( (\textrm{Mg}^{2+}) \), and the other containing hydrogen gas and a platinum electrode immersed in a hydrochloric acid solution. The magnesium electrode serves as the anode, where oxidation occurs, while the hydrogen electrode serves as the cathode, where reduction occurs. A salt bridge connects the two half-cells to maintain electrical neutrality, allowing for the flow of ions. As the reaction proceeds, electrons are released at the magnesium electrode (anode) and travel through an external wire to the hydrogen electrode (cathode). The flow of electrons through the external circuit generates an electric current, which can be used to do useful work, such as powering a device or storing energy in a battery.