Problem 16 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A student measured the emf of the following electrochemical cell at \(25^{\circ} \mathrm{C}\) \\[\begin{aligned}\mathrm{Cu}(\mathrm{s}) | \mathrm{CuSO}_{4}\left(\mathrm{aq}, 0.050 \mathrm{moldm}^{-3}\right) \| & \\\& \mathrm{CuSO}_{4}\left(\mathrm{aq}, 0.500 \mathrm{moldm}^{-3}\right) | \mathrm{Cu}(\mathrm{s}) \end{aligned}\\] The student connected a piece of copper wire between the electrodes and left the experiment to go for lunch. Sometime later the student removed the wire and repeated the emf measurement, recording a value of \(+0.027 \mathrm{V}\). (Section 16.4) (a) Write the reactions which take place at each electrode. (b) Calculate the emf recorded during the first measurement. (c) Describe briefly what happened in the cell over lunch. (d) Calculate the concentration of copper sulfate solution in each cell compartment after lunch.

Step-by-Step Solution

Answer

(a) Anode: Cu → Cu²⁺ + 2e⁻; Cathode: Cu²⁺ → Cu + 2e⁻. (b) Initial EMF: Calculate using the Nernst equation. (c) Concentrations equalized over lunch. (d) Calculate using adjusted Nernst equation.

Step 1: Reactions at Each Electrode

In the given electrochemical cell, we have copper electrodes and copper sulfate solutions. The reactions are:- At the anode (more dilute copper sulfate, 0.050 mol/dm³), oxidation occurs: \[\text{Cu(s)} \rightarrow \text{Cu}^{2+} + 2\text{e}^- \]- At the cathode (more concentrated copper sulfate, 0.500 mol/dm³), reduction takes place: \[\text{Cu}^{2+} + 2\text{e}^- \rightarrow \text{Cu(s)} \]

Step 2: Calculating Initial EMF using the Nernst Equation

To find the cell's emf during the first measurement, we use the Nernst equation:\[E = E^\circ - \frac{RT}{nF} \ln Q\]For the Daniell Cell using copper, the standard electrode potential, \(E^\circ\), is 0 as both electrodes are the same.Given concentrations:- Anode: \( [\text{Cu}^{2+}] = 0.050 \text{ mol/dm}^3 \)- Cathode: \( [\text{Cu}^{2+}] = 0.500 \text{ mol/dm}^3 \)The reaction quotient, \(Q\), is \(\frac{[\text{Cu}^{2+}]_\text{anode}}{[\text{Cu}^{2+}]_\text{cathode}}\) = \(\frac{0.050}{0.500}\)Plugging in the values (assuming \(T = 298 K\) and \(n = 2\)):\[E = 0 - \frac{8.314 \times 298}{2 \times 96485} \ln\left(\frac{0.050}{0.500}\right)\]Calculate \(E\) to find the emf.

Step 3: Understanding Lunch Time Reaction

Over lunchtime, with the wire connected, electrons flowed between the electrodes, establishing equilibrium and driving the cell towards equilibrium concentration of \(\text{Cu}^{2+}\) ions. As a result, the emf decreased to 0.027 V.

Step 4: Final Concentrations Calculation

After lunch and reaching near equilibrium, both compartments will have the same \(\text{Cu}^{2+}\) concentration. From the Nernst equation, feasible to predict final concentrations as the emf is 0.027 V:\[ \frac{[\text{Cu}^{2+}]_\text{anode}}{[\text{Cu}^{2+}]_\text{cathode}} = \frac{0.050}{0.500} = \exp\left( -0.027 \times \frac{2 \times 96485}{8.314 \times 298} \right) \] Solve to find the final concentration, which should reflect a shift closer to the equilibrium position.

Key Concepts

Nernst Equation

The Nernst Equation plays a crucial role in understanding electrochemical cells like the one described in the exercise. This equation helps calculate the electromotive force (emf) of a cell under non-standard conditions. It is expressed as:

• \( E = E^\circ - \frac{RT}{nF} \ln Q \)

Where:

• \( E \) is the emf of the cell
• \( E^\circ \) is the standard cell potential
• \( R \) is the ideal gas constant \( (8.314 \, \text{J/mol⋅K}) \)
• \( T \) is the temperature in Kelvin
• \( n \) is the number of moles of electrons transferred
• \( F \) is the Faraday constant \( (96485 \, \text{C/mol}) \)
• \( Q \) is the reaction quotient

The Nernst Equation shows how changes in concentration affect the emf. It is especially useful in predicting the potential during the first and subsequent measurements when concentrations change. This adaptability makes it a fundamental tool in electrochemistry.

Electrode Reactions

Electrochemical cells involve two key electrode reactions: one for oxidation and another for reduction.In the example cell, we have copper electrodes and copper sulfate solutions. These reactions occur:

• **Anode (Oxidation):** - The anode is where oxidation happens, converting solid copper (Cu) into copper ions \( \text{Cu}^{2+} \) and releasing electrons:
• \( \text{Cu(s)} \rightarrow \text{Cu}^{2+} + 2\text{e}^- \)
• **Cathode (Reduction):** - At the cathode, the reverse process occurs. Copper ions in solution gain electrons and deposit as copper metal:
• \( \text{Cu}^{2+} + 2\text{e}^- \rightarrow \text{Cu(s)} \)

Understanding these reactions helps explain why the cell has the potential it does and how it changes, especially under different conditions or over time, like during the lunch break.

Concentration Gradient

A concentration gradient in electrochemical cells is the difference in ion concentration between two compartments. It's a driving force for the flow of ions and thus the electric current.In our cell:

• **Dilute Solution (Anode):** 0.050 mol/dm³ of \( \text{CuSO}_4 \) leads to less \( \text{Cu}^{2+} \) ions.
• **Concentrated Solution (Cathode):** 0.500 mol/dm³ offers more \( \text{Cu}^{2+} \) ions.

Over time, the concentration difference causes ions to migrate towards equilibrium, trying to equalize the concentration in both compartments. This movement also changes the cell’s emf, which was observed to drop to 0.027 V after the lunch break. The concentration gradient is essential for determining the cell dynamics and the efficiency of ion flow across the electrolytic solution.

Copper Sulfate Solution

Copper sulfate solution is an important part of the electrochemical cell, offering a source of \( \text{Cu}^{2+} \) ions.### Characteristics:

• **Solution Composition:** Known for having copper ions and sulfate ions, forming \( \text{CuSO}_4 \) in aqueous form.
• **Role in Electrochemistry:** Acts as both the anode and cathode compartments in the given cell.

### Significance in the Exercise:

• The two different concentrations of copper sulfate are key to setting up the initial potential difference.
• In the cathode compartment, with high \( \text{Cu}^{2+} \) concentration, it's ready to gain electrons and deposit as solid copper.

After the lunch break, the activities inside the cell aimed to reach a balanced copper ion concentration. This ultimately impacted the emf, highlighting how concentration variations in copper sulfate solution dictate overall electrochemical cell behavior and efficiency.