Question

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 \(\mathrm{Mg}\) 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 metal and hydrochloric acid is \(Mg + 2HCl \rightarrow MgCl_2 + H_2\). In this equation, 2 electrons are transferred, and the useful work that can be obtained from this reaction under standard conditions is -455.26 kJ/mol. To harness this reaction for useful work, a galvanic cell can be created with a magnesium anode and a hydrogen cathode, converting the chemical energy of the redox reaction into electrical energy to power a circuit or charge a battery.

Step-by-step solution

Write the unbalanced half-reactions

First, we need to write the unbalanced half-reactions for the reduction of hydrogen ions and the oxidation of magnesium metal. Reduction half-reaction: \(H^+ + e^- \rightarrow H\) Oxidation half-reaction: \(Mg \rightarrow Mg^{2+} + 2e^-\)

Balance the half-reactions

Now we need to balance these half-reactions by recognizing that magnesium loses 2 electrons and hydrogen gains 2 electrons in the reaction. Reduction half-reaction: \(2H^+ + 2e^- \rightarrow H_2\) Oxidation half-reaction: \(Mg \rightarrow Mg^{2+} + 2e^-\)

Combine the half-reactions

Let's combine the half-reactions to find the balanced overall reaction equation. Since the number of electrons transferred is the same in both half-reactions, they can be combined directly: Overall reaction: \(Mg + 2H^+ \rightarrow Mg^{2+} + H_2\) We can add the chloride ions back into the overall reaction to show the balanced equation involving aqueous hydrochloric acid: Balanced equation: \(Mg + 2HCl \rightarrow MgCl_2 + H_2\)

Find the number of electrons transferred

The number of electrons transferred can be found by looking at the balanced half-reactions. In the oxidation half-reaction, magnesium loses 2 electrons: \(Mg \rightarrow Mg^{2+} + 2e^-\) So, 2 electrons are transferred in the balanced equation.

Calculate the useful work obtained

To calculate the useful work obtained from the reaction, we need to know the Gibbs free energy change of the reaction, ΔG. The formula to calculate the work done under standard conditions is: \(W_{max} = -nFE\) Where, \(W_{max}\) = useful work obtained \(n\) = number of moles of electrons transferred (2 moles) \(F\) = Faraday's constant (96485 C/mol) \(E\) = standard cell potential We need to find the standard cell potential for the magnesium-hydrogen reaction. Looking up the standard reduction potentials, we find: \(E^0_{H^+}/_{H_2} = 0 V\) \(E^0_{Mg^{2+}}/_{Mg} = -2.37 V\) Since magnesium is being oxidized, the standard potential for magnesium should be reversed: \(-E^0_{Mg}/_{Mg^{2+}} = 2.37 V\) The total standard cell potential is the sum of the two standard potentials: \(E = E^0_{cathode} - E^0_{anode} = -E^0_{Mg}/_{Mg^{2+}} + E^0_{H^+}/_{H_2} = 2.37 V\) Now we can plug this value into the equation for useful work: \(W_{max} = -nFE = -(2)(96485)(2.37)\) \(W_{max} = -455.26 kJ/mol\) Note that the negative sign indicates that the reaction is spontaneous and will produce work.

Harnessing the reaction for useful work

One possible way to harness the reaction of magnesium with hydrochloric acid to do useful work is to create a galvanic (voltaic) cell, which converts the chemical energy of the redox reaction into electrical energy. The cell would have a magnesium anode and a hydrogen cathode. The reaction between magnesium and hydrochloric acid would produce a voltage across the cell, which can be used to power an electrical circuit or charge a battery.

Key concepts

Half-Reactions

In a redox reaction, both oxidation and reduction processes occur simultaneously. To understand these processes, we use half-reactions. Each half-reaction represents either the gain or loss of electrons. In the given problem, we have two half-reactions:

• Reduction half-reaction: This represents the gain of electrons. For the given reaction, the hydrogen ions (\(H^+\)) gain electrons to become hydrogen gas (\(H_2\)).
• Oxidation half-reaction: This illustrates the loss of electrons. In this case, magnesium (\(Mg\)) loses electrons to form magnesium ions (\(Mg^{2+}\)).

Half-reactions are crucial because they allow us to see how electrons are transferred between reactants. With these equations, we can balance the overall chemical reaction by ensuring that the number of electrons lost equals the number of electrons gained. This is key in figuring out the total electron transfer that occurs.

Electron Transfer

Electron transfer is at the heart of redox reactions. In this context, transfer refers to the movement of electrons from one reactant to another. In our example, when magnesium is added to hydrochloric acid, electrons move from magnesium to hydrogen ions. This movement is how the redox process occurs.

Determining the number of electrons involved in the transfer helps to balance the reaction. For the problem at hand, 2 electrons are transferred in the balanced equation. This calculation comes from observing the balanced half-reactions:

• The oxidation half-reaction shows 2 electrons released by magnesium.
• The reduction half-reaction shows 2 electrons are required by hydrogen ions to form hydrogen gas.

Understanding electron transfer helps in measuring the redox process's spontaneity, which can also influence the energy calculations in reactions.

Gibbs Free Energy

Gibbs free energy (\(\Delta G\)) is an important concept when dealing with redox reactions. It tells us about the spontaneity and the maximum useful work that can be done by the system. In the context of the problem, \(\Delta G\) can determine how much work can be produced when magnesium reacts with hydrochloric acid.

The formula to find the useful work involves:

• \(n\) — the number of moles of electrons transferred (in this case, 2 moles)
• \(F\) — Faraday's constant (96485 \(C/mol\)
• \(E\) — standard cell potential

This calculation, \(W_{max} = -nFE\), gives the maximum work, which is negative in spontaneous reactions, showing that energy is released. For the magnesium-hydrochloric acid reaction, this value comes out to be \(-455.26 \text{kJ/mol}\), suggesting significant energy output from this redox process.

Galvanic Cells

Galvanic cells, also known as voltaic cells, convert chemical energy from redox reactions into electrical energy. They're like simple batteries, using spontaneous redox reactions to create voltage. In the exercise example, magnesium acts as the anode, and hydrogen is the cathode in a potential galvanic cell.

Here's how it works:

• The anode (magnesium) undergoes oxidation and releases electrons.
• The cathode (hydrogen ions) undergoes reduction and gains electrons.

A galvanic cell harnesses this electron flow to perform work, such as powering a device or charging a battery. The overall cell reaction — the magnesium reacting with hydrochloric acid — naturally generates voltage over the galvanic cell. This setup exemplifies how chemical energy can be efficiently converted into a useful form of energy, helping power various processes and devices.