Question

Calculate the emf of the following concentration cell: $$ \mathrm{Mg}(s)\left|\mathrm{Mg}^{2+}(0.24 M) \| \mathrm{Mg}^{2+}(0.53 M)\right| \mathrm{Mg}(s) $$

Short answer

The emf of the concentration cell is approximately 0.0102 V.

Step-by-step solution

Understand the Situation

We have a concentration cell, which involves the same element, magnesium in this case, at different concentrations in the anode and cathode half-cells.

Write the Nernst Equation

The emf of a concentration cell can be calculated using the Nernst equation: \[ E = E^0 - \frac{RT}{nF} \ln \frac{[\text{Concentration of Mg}^{2+}_{\text{anode}}]}{[\text{Concentration of Mg}^{2+}_{\text{cathode}}]} \] where \( E^0 \) is 0 for the concentration cell and \( n \) (number of moles of electrons) is 2 for Mg.

Simplify the Nernst Equation

Since \( E^0 = 0 \) for a concentration cell of the same element, the equation simplifies to: \[ E = - \frac{RT}{nF} \ln \frac{[\text{Mg}^{2+}]_{\text{anode}}}{[\text{Mg}^{2+}]_{\text{cathode}}} \] Substitute the given concentrations. R is the universal gas constant (8.314 J mol\(^{-1}\) K\(^{-1}\)), T is the temperature in Kelvin (assume 298 K), n is 2 for magnesium, and F is the Faraday constant (96485 C mol\(^{-1}\)).

Plug in Numerical Values

Substitute the values into the simplified equation: \[ E = - \frac{8.314 \times 298}{2 \times 96485} \ln \frac{0.24}{0.53} \]

Calculate the Natural Logarithm

Calculate \( \ln \frac{0.24}{0.53} \). First, divide 0.24 by 0.53 to get approximately 0.4528, then find its natural logarithm to get approximately -0.7923.

Compute the EMF

Multiply the values to find the emf:\[ E = - \frac{2477.972}{192970} \times (-0.7923) \approx 0.0102 \text{ V} \]

Conclusion: Report the EMF

The calculated emf for the concentration cell is approximately 0.0102 V.

Key concepts

Nernst equation

The Nernst equation is a fundamental tool that allows us to calculate the electromotive force (EMF) of a cell under non-standard conditions. It is especially useful for concentration cells, where the concentration of ions in each half-cell matters. The general form of the Nernst equation is: \[ E = E^0 - \frac{RT}{nF} \ln \frac{[\text{anode}]}{[\text{cathode}]} \] In this equation:

• \(E^0\) is the standard electrode potential, which is zero for a concentration cell involving the same electrode material on both sides.
• \(R\) stands for the universal gas constant, valued at 8.314 J mol\(^{-1}\) K\(^{-1}\).
• \(T\) represents the temperature in Kelvin (often taken as 298 K, which is about 25°C).
• \(n\) is the number of moles of electrons involved in the reaction, which for magnesium is 2.
• \(F\) is the Faraday constant, approximately 96485 C mol\(^{-1}\).

For a magnesium concentration cell, calculate the EMF by substituting the concentration of \(\text{Mg}^{2+}\) ions in the anode and cathode.

electromotive force (EMF)

Electromotive force (EMF) is the voltage generated by a battery or cell when there is no current flow. It represents the potential difference between two terminals. In the context of concentration cells, the EMF is generated due to the difference in ion concentrations in the separate half-cells. The formula for calculating EMF in a concentration cell boils down to the difference in concentration of ions within the two compartments. The greater the difference between ion concentrations, the higher the potential or EMF of the cell.
The specific formula as we apply in concentration cells is simplified from the Nernst equation due to \(E^0 = 0\), and expressed as: \[ E = - \frac{RT}{nF} \ln \frac{[\text{ion concentration in anode}]}{[\text{ion concentration in cathode}]} \] This highlights how EMF reflects a system not yet at equilibrium.

magnesium ions

Magnesium ions, \(\text{Mg}^{2+}\), play a crucial role in concentration cells where magnesium metal is used.
In this case, the half-cell reactions involve the oxidation and reduction of \(\text{Mg}\) and \(\text{Mg}^{2+}\), creating a potential difference based on the disparity in ion concentrations.

• At the anode, magnesium metal dissolves into the solution, increasing the concentration of \(\text{Mg}^{2+}\) ions: \( \text{Mg (s)} \rightarrow \text{Mg}^{2+} (aq) + 2e^- \).
• At the cathode, \(\text{Mg}^{2+}\) ions from the solution gain electrons and deposit as metallic \(\text{Mg}\): \( \text{Mg}^{2+} (aq) + 2e^- \rightarrow \text{Mg (s)} \).

The difference in concentrations of \(\text{Mg}^{2+}\) between the two half-cells drives the flow of electrons through the external circuit, providing the EMF.

half-cells

A concentration cell is comprised of two half-cells, each containing an electrode and an electrolyte in which ions are dissolved. In a magnesium concentration cell configuration, each half-cell contains magnesium electrodes and \(\text{Mg}^{2+}\) as the electrolyte at varying concentrations.
The half-cell with the lower concentration of \(\text{Mg}^{2+}\) acts as the anode, where oxidation takes place. Electrons are released here. The other half-cell with the higher \(\text{Mg}^{2+}\) concentration serves as the cathode, where reduction occurs, and electrons are consumed.

• In a voltaic cell, each half-cell is crucial for maintaining the flow and balance of ions and electrons across the system.
• The salt bridge or porous barrier typically connects them, allowing ionic transfer without mixing the different solutions, preserving the overall neutrality.

Understanding half-cells helps in grasping how redox reactions lead to electrical energy generation.

chemical equilibrium

Chemical equilibrium in a concentration cell is a state where the potential energy difference, or EMF, becomes zero. This happens when the concentrations of ions in both the anode and cathode half-cells equalize. Until equilibrium, a concentration gradient exists, providing the driving force for electrons to flow from the anode to the cathode. As this flow occurs, the concentration of ions gradually evens out.

• The Nernst equation helps demonstrate how EMF decreases as equilibrium is approached, following the natural logarithm of the concentration ratio.
• Eventually, no net electron flow happens since the system has achieved equilibrium and the EMF effectively becomes negligible.

The concept of chemical equilibrium is key to understanding why and when a concentration cell stops working.