Question

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?

Short answer

The balanced equation for the reaction between magnesium and hydrochloric acid is Mg (s) + 2H^+ (\mathrm{aq}) -> Mg^{2+} (\mathrm{aq}) + H_2(g), where 2 electrons are transferred. The useful work obtained from this reaction is approximately 457 kJ/mol. To harness this reaction for useful work, an electrochemical cell can be set up with magnesium as the anode and hydrogen ions as the cathode, allowing electrons to flow through the external circuit and perform work while producing hydrogen gas at the cathode.

Step-by-step solution

Write the half-reactions

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)

Combine half-reactions and write the balanced equation

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)

Determine the electrons transferred in the balanced equation

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.

Calculate the quantity of useful work

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

Calculate the standard cell potential and Gibbs free energy change

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.

Harnessing the reaction to do useful work

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.

Key concepts

Redox Reactions

Chemical reactions where one substance is oxidized and another is reduced are called redox reactions. In the scenario involving magnesium and hydrochloric acid, magnesium serves as the reducing agent. A reducing agent donates electrons, while the hydrogen ions from hydrochloric acid are oxidizing agents that accept electrons.
🔹 **Oxidation:** Magnesium (\[\text{Mg}\]) loses electrons and is oxidized to magnesium ions (\[\text{Mg}^{2+}\]).
🔹 **Reduction:** Hydrogen ions(\[\text{H}^+\]) gain electrons to form hydrogen gas (\[\text{H}_2\]).
The balanced equation for the reaction illustrates this exchange:\[\text{Mg (s) + 2H}^+ ext{(aq) → Mg}^{2+} ext{(aq) + H}_2 ext{(g)}\]. This clearly shows the oxidation of magnesium and the reduction of hydrogen ions. The number of electrons exchanged, in this case, is two.

Gibbs Free Energy

Gibbs free energy (\[\Delta G\]) is a measure of the usable energy derived from a chemical reaction. It helps predict whether the reaction will occur spontaneously. In our case, when magnesium reacts with hydrochloric acid, we calculate \[\Delta G\] using standard electrode potentials.
The formula to calculate Gibbs free energy change is:\[\Delta G = -nFE^{ ext{o}}_{ ext{cell}}\]. Here:

• \(n\) is the number of electrons transferred.
• \(F\) is Faraday's constant (96485 C/mol).
• \(E^{ ext{o}}_{ ext{cell}}\) is the standard cell potential.

Substituting values, we find \[\Delta G \approx \] -457 kJ/mol. This negative value suggests the reaction is spontaneous and capable of producing 457 kJ of work per mole.

Electrode Potentials

Electrode potentials determine the tendency of a chemical species to be reduced or oxidized. They are crucial in deriving the cell potential, which tells us the spontaneity and extent of a redox reaction. The standard electrode potential (\[E^{ ext{o}}\]) for magnesium oxidation is -2.37 V, making it a powerful reducing agent.
For hydrogen reduction, \[E^{ ext{o}}\] is defined as 0 Volts, serving as the baseline/reference. The formula for the cell potential is:\[E^{ ext{o}}_{ ext{cell}} = E^{ ext{o}}_{ ext{cathode}} - E^{ ext{o}}_{ ext{anode}}\]. Substituting the values gives us a standard cell potential of 2.37 V.
A positive \[E^{ ext{o}}_{ ext{cell}}\] indicates the reaction is capable of performing work when set up correctly.

Electrochemical Cells

Electrochemical cells convert chemical energy into electrical energy, which can effectively perform work. By physically separating the reduction and oxidation reactions to different compartments, the electron flow from magnesium (anode) to hydrogen ions (cathode) is channeled through a conductor.

• **Anode:** The site where oxidation happens. Here, magnesium loses electrons and becomes \[\text{Mg}^{2+}\].
• **Cathode:** Meanwhile, reduction takes place at the cathode where \[\text{H}^+\] ions gain electrons to form \[\text{H}_2\] gas.

This setup not only captures and converts the energy released during the reaction into usable electricity but also allows easy harnessing for external work, such as powering a small device.