Problem 1 - An Introduction to Materials Engineering and Science for Chemical and Materials Engineers

Question

One-half of an electrochemical cell consists of a pure nickel electrode in a solution of \(\mathrm{Ni}^{2+}\) ions; the other is a cadmium electrode immersed in a \(\mathrm{Cd}^{2+}\) solution. If the cell is a standard one, write the spontaneous overall reaction and calculate the voltage that is generated.

Step-by-Step Solution

Answer

The spontaneous overall reaction is \( \text{Ni}^{2+} + \text{Cd} \rightarrow \text{Ni} + \text{Cd}^{2+} \), and the cell potential is \( 0.17 \, \text{V} \).

Step 1: Identify the Half-Reactions

The nickel half-reaction is \( ext{Ni}^{2+} + 2e^- \rightarrow ext{Ni} \). The cadmium half-reaction is \( ext{Cd}^{2+} + 2e^- \rightarrow ext{Cd} \). Write these down to understand the transfer of electrons in the electrochemical cell.

Step 2: Determine Standard Reduction Potentials

The standard reduction potential \( E^0 \) for the nickel half-reaction \( ext{Ni}^{2+} + 2e^- \rightarrow ext{Ni} \) is \(-0.23 \, \text{V}\). For the cadmium half-reaction \( ext{Cd}^{2+} + 2e^- \rightarrow ext{Cd} \), it is \(-0.40 \, \text{V}\). These values are determined from standard tables.

Step 3: Reverse the Less Positive Half-Reaction

Since a spontaneous reaction is desired, reverse the cadmium half-reaction to undergo oxidation. The reaction becomes \( ext{Cd} \rightarrow ext{Cd}^{2+} + 2e^- \) and the transformation of half-reaction to oxidation changes the sign of the potential to \( +0.40 \, \text{V}\).

Step 4: Write the Spontaneous Overall Reaction

Combine the two half-reactions (nickel reduction and cadmium oxidation). The overall reaction is \( ext{Ni}^{2+} + ext{Cd} \rightarrow ext{Ni} + ext{Cd}^{2+} \).

Step 5: Calculate the Standard Cell Potential

The cell potential \( E^0_{cell} \) is calculated by adding the reduction potential of nickel and the oxidation potential of cadmium: \( E^0_{cell} = E^0_{Ni^{2+}/Ni} - E^0_{Cd/Cd^{2+}} = -0.23 \, \text{V} + 0.40 \, \text{V} = 0.17 \, \text{V}\). This positive value confirms the reaction is spontaneous.

Key Concepts

Standard Reduction Potentials

Standard reduction potentials are crucial in determining the direction and spontaneity of electrochemical reactions. Each half-reaction has an associated standard reduction potential, denoted as \( E^0 \), measured under standard conditions. This involves 1 M concentration for aqueous species, 1 atm for gases, and pure solids or liquids. To find these potentials, scientists refer to a table that lists values relative to the standard hydrogen electrode, which is assigned a potential of 0.00 V. When comparing these values:

• A more positive \( E^0 \) means a greater tendency for the substance to gain electrons and be reduced.
• An electrode with a more negative \( E^0 \) is less likely to capture electrons and more likely to lose electrons, which makes it favorable to act as the anode in an electrochemical cell.

In our example with nickel and cadmium, nickel has a higher \( E^0 \) than cadmium, indicating nickel's ease of reduction. Understanding these potentials helps in predicting cell behavior and calculating the cell voltage.

Half-Reactions

Half-reactions represent the separate processes happening at the anode and cathode in an electrochemical cell. These reactions individually show the oxidation or reduction taking place, and when combined, they give the full redox (reduction-oxidation) reaction.

• The oxidation half-reaction involves the loss of electrons. In our case, cadmium undergoes oxidation by turning from \( \text{Cd} \) to \( \text{Cd}^{2+} + 2e^- \).
• The reduction half-reaction involves the gain of electrons. Nickel ions \( (\text{Ni}^{2+}) \) gain electrons to form metallic nickel \( \text{Ni} \).

Understanding each half-reaction is essential for determining the cell's net reaction and is crucial for calculating the cell voltage and analyzing cell efficiency.

Spontaneous Reactions

A spontaneous reaction is a chemical reaction that occurs naturally under given circumstances without external energy. In electrochemical cells, spontaneity is determined by the cell potential \( E^0_{cell} \).

• If \( E^0_{cell} \) is positive, the overall cell reaction is spontaneous.
• If \( E^0_{cell} \) is negative, the reaction is non-spontaneous under standard conditions.

For the nickel-cadmium cell, reversing the less positive half-reaction (cadmium) creates a scenario, where the overall potential difference is actually plus \( 0.17 \, \text{V} \), thus confirming that the reaction will proceed on its own. This is why calculating and comparing the potentials of involved half-reactions is a key step in understanding electrochemical cell operations.