Problem 91 - Chemistry: An Atoms-Focused Approach

Question

A concentration cell can be constructed by using the same half-reaction for both the cathode and anode. What is the value of \(E_{\text {cell }}\) of a concentration cell that combines copper electrodes in contact with \(0.25 M\) copper (11) nitrate and \(0.00075 M\) copper (11) nitrate solutions?

Step-by-Step Solution

Answer

Answer: The value of the cell potential, \(E_{\text{cell}}\), for the concentration cell is approximately 0.104 V.

Step 1: Identify the half-reaction and write the cell notation

The half-reaction happening in the cell is the redox reaction between copper and copper (II) ions: Cu\(^{2+}(aq)\) + 2e\(^-\) → Cu\((s)\) Now we can write the cell notation for the concentration cell. The cell notation for this cell is: Cu\((s)\) | Cu\(^{2+}(0.25 M)\) || Cu\(^{2+}(0.00075 M)\) | Cu\((s)\)

Step 2: Calculate the standard cell potential \(E^0_{\text{cell}}\)

In a concentration cell, since the both half-reactions are the same, and the difference in potential is due to the difference in concentrations of the same species, the standard cell potential \(E^0_{\text{cell}}\) is zero.

Step 3: Use the Nernst equation to find \(E_{\text{cell}}\)

The Nernst equation relates the cell potential, \(E_{\text{cell}}\), to the standard cell potential \(E^0_{\text{cell}}\), the concentration of products, and the concentration of reactants. In this case, the Nernst equation is given by: \(E_{\text{cell}} = E^0_{\text{cell}} - \dfrac{0.0592}{n} \log \dfrac{[\text{Cu}^{2+}]_{\text{cathode}}}{[\text{Cu}^{2+}]_{\text{anode}}}\) Using the information provided, we can plug the values into the equation: \(E_{\text{cell}} = 0 - \dfrac{0.0592}{2} \log \dfrac{0.00075}{0.25}\)

Step 4: Solve for \(E_{\text{cell}}\)

Carrying out the calculation, we obtain the value of \(E_{\text{cell}}\): \(E_{\text{cell}} = -\dfrac{0.0592}{2} \times (-3.52) \approx 0.104 \,\text{V}\) Thus, the value of \(E_{\text{cell}}\) for the concentration cell combining copper electrodes in contact with \(0.25 \,\text{M}\) copper (II) nitrate and \(0.00075 \,\text{M}\) copper (II) nitrate solutions is approximately \(0.104 \,\text{V}\).

Key Concepts

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationship between electrical energy and chemical reactions. It involves the study of how chemical processes can produce electric currents and how electric currents can induce chemical reactions. A fundamental component of electrochemistry is the electrochemical cell, which is a device that can generate electrical energy from chemical reactions or facilitate chemical reactions through the introduction of electrical energy.
The two main types of electrochemical cells are galvanic (or voltaic) cells and electrolytic cells. In a galvanic cell, chemical energy is converted into electric energy through spontaneous redox reactions. These cells are often used in batteries. On the other hand, electrolytic cells use electrical energy to drive non-spontaneous chemical reactions.

Concentration cells are a unique type of galvanic cell where the electrodes are identical but are immersed in solutions of different concentrations. The potential difference in these cells arises from the difference in concentration rather than differences in substances. This feature is highlighted in the exercise involving copper electrodes, which rely on differing copper ion concentrations to drive the electrical generation process.

Nernst Equation

The Nernst Equation provides a crucial link between the concentration of ions in solution and the electromotive force (EMF) of an electrochemical cell. Named after the German physical chemist Walther Nernst, this equation allows us to calculate the actual cell potential under non-standard conditions.

In its general form, the Nernst Equation is:
\[ E_{cell} = E^0_{cell} - \frac{0.0592}{n} \log \frac{[ ext{products}]}{[ ext{reactants}]} \]
where:

• \(E_{ ext{cell}}\) is the cell potential at specific conditions,
• \(E^0_{ ext{cell}}\) is the standard cell potential,
• \(n\) is the number of moles of electrons transferred in the reaction,
• \([ ext{products}]\) and \([ ext{reactants}]\) are the activities (often approximated as concentrations) of products and reactants.

In the context of a concentration cell, the standard cell potential \(E^0_{cell}\) is zero because the half-reactions at the anode and cathode are the same. The equation helps us compute the actual cell potential by considering the concentration gradient, as illustrated in the copper concentration cell.

Cell Notation

Cell notation is a shorthand used to describe the components and direction of reactions in electrochemical cells. This notation is quite useful for quickly conveying complex information about the cell's setup.

In the cell notation, a typical galvanic cell is depicted as:\[ ext{Anode} \,|\, ext{Anode Solution (Concentration)} \,||\, ext{Cathode Solution (Concentration)} \,|\, ext{Cathode}\]Here are some pointers on understanding and writing cell notation:

• The anode (where oxidation occurs) is written on the left, and the cathode (where reduction occurs) on the right.
• A single vertical line (|) represents a phase boundary, such as between solid and aqueous phases.
• A double vertical line (||) represents the salt bridge or membrane that separates the anodic and cathodic compartments.

In the exercise with copper electrodes, the cell notation is written as:\[ ext{Cu}(s) \,|\, ext{Cu}^{2+}(0.25 \text{ M}) \,||\, ext{Cu}^{2+}(0.00075 \text{ M}) \,|\, ext{Cu}(s)\]This provides a clear depiction of a copper concentration cell with significant differences in ion concentration at the anode and cathode.