A Cu/Cu²⁺ concentration cell has a voltage of 0.22 V at 25 °C. The concentration of Cu²⁺ in one of the half-cells is 1.5×10^–3 M. What is the concentration of Cu²⁺ in the other half-cell? (Assume the concentration in the unknown cell is the lower of the two concentrations.)

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Identify that the problem involves a concentration cell, which is a type of galvanic cell where the electrodes are the same material, but the ion concentrations are different.

Use the Nernst equation to relate the cell potential to the concentrations of the ions: \(E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln \frac{[\text{Cu}^{2+}]_{\text{cathode}}}{[\text{Cu}^{2+}]_{\text{anode}}}\).

Since this is a concentration cell, \(E^\circ_{cell} = 0\) because the electrodes are identical, so the equation simplifies to \(E_{cell} = -\frac{RT}{nF} \ln \frac{[\text{Cu}^{2+}]_{\text{cathode}}}{[\text{Cu}^{2+}]_{\text{anode}}}\).

Substitute the given values into the Nernst equation: \(0.22 = -\frac{(8.314)(298)}{(2)(96485)} \ln \frac{1.5 \times 10^{-3}}{[\text{Cu}^{2+}]_{\text{anode}}}\).

Solve the equation for \([\text{Cu}^{2+}]_{\text{anode}}\), which represents the concentration of Cu\(^{2+}\) in the other half-cell.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Nernst Equation

The Nernst Equation relates the cell potential to the concentrations of the reactants and products in an electrochemical cell. It is expressed as E = E° - (RT/nF) ln(Q), where E is the cell potential, E° is the standard cell potential, R is the gas constant, T is the temperature in Kelvin, n is the number of moles of electrons transferred, F is Faraday's constant, and Q is the reaction quotient. This equation is essential for calculating the voltage of concentration cells.

Concentration Cell

A concentration cell is a type of electrochemical cell where the two half-cells have the same electrodes but different concentrations of the same ion. The potential difference arises from the difference in concentration, driving the spontaneous flow of electrons from the higher concentration to the lower concentration. In this case, the cell generates voltage due to the concentration gradient of Cu2+ ions.

Reaction Quotient (Q)

The reaction quotient (Q) is a measure of the relative concentrations of products and reactants at any point in a reaction. For a concentration cell involving Cu/Cu2+, Q is calculated as the ratio of the concentration of Cu2+ in the anode half-cell to that in the cathode half-cell. Understanding Q is crucial for applying the Nernst Equation to find the unknown concentration in the cell.

A Cu/Cu2+ concentration cell has a voltage of 0.22 V at 25 °C. The concentration of Cu2+ in one of the half-cells is 1.5×10–3 M. What is the concentration of Cu2+ in the other half-cell? (Assume the concentration in the unknown cell is the lower of the two concentrations.)