Question

Calculate the standard emf of a cell that uses the \(\mathrm{Mg} / \mathrm{Mg}^{2+}\) and \(\mathrm{Cu} / \mathrm{Cu}^{2+}\) half-cell reactions at \(25^{\circ} \mathrm{C}\). Write the equation for the cell reaction that occurs under standard-state conditions.

Short answer

The standard emf of the cell is 2.71 V and the cell reaction is \( \mathrm{Mg} + \mathrm{Cu}^{2+} \rightarrow \mathrm{Mg}^{2+} + \mathrm{Cu} \).

Step-by-step solution

Identify the Half-Reactions

The half-reactions for the Mg/Cu cell are as follows: for magnesium, the reduction reaction is \( \mathrm{Mg}^{2+} + 2\mathrm{e}^- \rightarrow \mathrm{Mg} \) and for copper, the reduction reaction is \( \mathrm{Cu}^{2+} + 2\mathrm{e}^- \rightarrow \mathrm{Cu} \).

Find Standard Reduction Potentials

Using a standard reduction potential table, find the standard reduction potentials: \( E^0(\mathrm{Mg}^{2+}/\mathrm{Mg}) = -2.37 \mathrm{V} \) and \( E^0(\mathrm{Cu}^{2+}/\mathrm{Cu}) = +0.34 \mathrm{V} \).

Determine the Anode and Cathode

In a galvanic cell, the anode is where oxidation occurs and has a more negative potential; the cathode is where reduction occurs. Thus, Mg is the anode and Cu is the cathode.

Calculate the Standard EMF

The standard EMF of the cell is calculated as the difference between the cathode and anode potentials. Thus, \( E^0_{cell} = E^0_{cathode} - E^0_{anode} = 0.34 \mathrm{V} - (-2.37 \mathrm{V}) = 2.71 \mathrm{V} \).

Write the Overall Cell Reaction

Combine the half-reactions to find the overall cell reaction: \( \mathrm{Mg} + \mathrm{Cu}^{2+} \rightarrow \mathrm{Mg}^{2+} + \mathrm{Cu} \).

Key concepts

Half-Reactions

To understand the concept of half-reactions, imagine breaking down a redox reaction into two parts. These parts involve either oxidation or reduction processes. Each half-reaction represents a type of chemical change that occurs in a galvanic cell. In our example of a magnesium-copper cell, half-reactions help us see how electrons are exchanged. For magnesium, the half-reaction is \[ \mathrm{Mg}^{2+} + 2\mathrm{e}^- \rightarrow \mathrm{Mg} \]This is a reduction half-reaction, meaning magnesium ions gain electrons to form magnesium metal.For copper, the half-reaction is\[ \mathrm{Cu}^{2+} + 2\mathrm{e}^- \rightarrow \mathrm{Cu} \]This is also a reduction half-reaction, where copper ions receive electrons to become copper metal. Understanding half-reactions is crucial because it helps in calculating the overall cell reaction and emf.

Reduction Potentials

Reduction potentials are like measuring the eagerness of a substance to gain electrons. They are usually found in tabulated form and expressed in volts (V). These values allow us to predict the direction of electron flow in a galvanic cell.For our specific example, the standard reduction potential of magnesium (\( E^0(\mathrm{Mg}^{2+}/\mathrm{Mg}) = -2.37 \, \mathrm{V} \)) indicates that it's less likely to gain electrons compared to copper (\( E^0(\mathrm{Cu}^{2+}/\mathrm{Cu}) = +0.34 \, \mathrm{V} \)). In general:

• A positive reduction potential means a substance is more likely to be reduced (gain electrons).
• A negative reduction potential indicates that the substance is less likely to be reduced.

Reduction potentials guide us in identifying cathodes and anodes in galvanic cells, crucial for calculating the standard emf.

Galvanic Cell

A galvanic cell, also known as a voltaic cell, is a device that generates electrical energy through spontaneous chemical reactions. It consists of two half-cells, each containing a solution and a metal electrode. In our example, one half-cell contains magnesium and its ions, while the other contains copper and its ions. The cell functions by utilizing the difference in the reduction potential between the two metals to move electrons through an external circuit. Key components of a galvanic cell:

• Anode: The electrode where oxidation occurs; electrons are released. Mg serves as the anode.
• Cathode: The electrode where reduction takes place; electrons are consumed. Cu serves as the cathode.
• Salt bridge: Helps maintain electrical neutrality by allowing ions to flow between half-cells.

Understanding how a galvanic cell works is essential for harnessing the energy released in chemical reactions.

Cell Reaction Equation

The cell reaction equation is the culmination of the individual half-reactions and shows the overall redox process taking place in a galvanic cell.For the Mg/Cu galvanic cell, the cell reaction can be written as follows:\[ \mathrm{Mg} + \mathrm{Cu}^{2+} \rightarrow \mathrm{Mg}^{2+} + \mathrm{Cu} \]This equation tells us that magnesium metal is oxidized to magnesium ions, while copper ions are reduced to copper metal.To create the cell reaction, follow these steps:

• Write the oxidation half-reaction (for the anode).
• Write the reduction half-reaction (for the cathode).
• Combine the half-reactions, ensuring the electrons are balanced.

The cell reaction equation is pivotal as it provides a comprehensive picture of the chemical changes and the flow of electrons within the cell.