When magnesium metal is added to a beaker of \(\mathrm{HCl}(\mathrm{aq})\), 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 \(\mathrm{Mg}\) is added directly to the beaker of \(\mathrm{HCl}\)? How can you harness this reaction to do useful work?

Initially, we will consider the half-reactions of the given chemical reaction. We know that magnesium is oxidized, which means it loses electrons, and hydrogen is reduced, meaning it gains electrons. Oxidation half-reaction: Mg (s) -> Mg^{2+} (\mathrm{aq}) + 2e^- Reduction half-reaction: 2H^+ (\mathrm{aq}) + 2e^- -> H_2(g)

Now, we will combine the half-reactions and write the balanced equation for the reaction. Mg (s) + 2H^+ (\mathrm{aq}) -> Mg^{2+} (\mathrm{aq}) + H_2(g)

The number of electrons transferred in the balanced equation can be determined from the half-reactions. In the oxidation half-reaction, magnesium loses 2 electrons. In the reduction half-reaction, 2 hydrogen ions gain 2 electrons. Therefore, 2 electrons are transferred in the balanced equation.

The useful work that can be obtained from a chemical reaction can be calculated using the Gibbs free energy change, which is related to the standard electrode potentials (E°) of the half-reactions. We can look up the standard electrode potentials for the half-reactions in a table: * Magnesium: Mg^{2+} (\mathrm{aq}) + 2e^- -> Mg (s) E° = -2.37 V * Hydrogen: 2H^+ (\mathrm{aq}) + 2e^- -> H_2(g) E° = 0.00 V

First, we calculate the standard cell potential for the magnesium-hydrogen redox reaction by subtracting the standard reduction potential of the anode (oxidation) from the standard reduction potential of the cathode (reduction): E°cell = E°cathode - E°anode = 0.00 V - (-2.37 V) = 2.37 V Now, we can calculate the Gibbs free energy change, ΔG°, using the following equation: ΔG° = -nFE°cell where, n = number of moles of electrons transferred (in this case, 2), F = Faraday's constant (96485 C/mol), and E°cell = standard cell potential (2.37 V). ΔG° = -2 * 96485 C/mol * 2.37 V ≈ -457 kJ/mol The quantity of useful work that can be obtained is equal to the negative of the Gibbs free energy change, which is approximately 457 kJ/mol.

To harness the energy from the reaction between magnesium and hydrochloric acid, the reaction can be set up in an electrochemical cell. Magnesium can act as the anode, and hydrogen ions from the HCl solution can act as the cathode. The electrons from the oxidized magnesium will flow through the external circuit and do useful work, while hydrogen gas is produced at the cathode.