Problem 40 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

An excess of liquid mercury is added to an acidified solution of \(1.0 \times\) \(10^{-3} \mathrm{M} \mathrm{Fe}^{3+}\). It is found that \(5 \%\) of \(\mathrm{Fe}^{3+}\) remains at equilibrium at \(25^{\circ} \mathrm{C}\). Calculate \(E_{\mathrm{Hg}_{2}^{2+} \mid \mathrm{Hg}}\), assuming that the only reaction that occurs is \(2 \mathrm{Hg}+2 \mathrm{Fe}^{3+} \longrightarrow \mathrm{Hg}_{2}^{2+}+2 \mathrm{Fe}^{2+}\) (Given \(\left.E_{\mathrm{Fe}^{3+} \mid \mathrm{Fe}^{2+}}^{\circ}=0.77 \mathrm{~V} .\right)\)

Step-by-Step Solution

Answer

The electrode potential \(E_{\mathrm{Hg}_2^{2+} | \mathrm{Hg}}\) is approximately \(0.6943 \text{ V}\).

Step 1: Calculate concentration of Fe3+ at equilibrium

Initially, we have a concentration of \(1.0 \times 10^{-3} \text{ M} \) for \(\mathrm{Fe^{3+}}\). Since \(5\%\) remains at equilibrium, the concentration of \(\mathrm{Fe^{3+}}\) at equilibrium will be \(0.05 \times 1.0 \times 10^{-3} = 5.0 \times 10^{-5} \text{ M}\).

Step 2: Calculate concentration of Fe2+ at equilibrium

Since \(95\%\) of \(\mathrm{Fe^{3+}}\) converted to \(\mathrm{Fe^{2+}}\), the concentration of \(\mathrm{Fe^{2+}}\) at equilibrium is \(0.95 \times 1.0 \times 10^{-3} = 9.5 \times 10^{-4} \text{ M}\).

Step 3: Use the Nernst Equation for Fe3+/Fe2+

The Nernst equation for the reduction half-cell \(\mathrm{Fe^{3+}/Fe^{2+}}\) is:\[E = E^{\circ} - \frac{RT}{nF} \ln\left(\frac{\text{[Fe}^{2+}\text{]}}{\text{[Fe}^{3+}\text{]}}\right)\]Since we are at \(25^{\circ}C\), the Nernst equation simplifies to:\[E = 0.77 - \frac{0.05916}{1} \log\left(\frac{9.5 \times 10^{-4}}{5.0 \times 10^{-5}}\right)\]

Step 4: Calculate cell potential using Nernst equation

Plug in the values in the simplified Nernst equation:\[E = 0.77 - 0.05916 \cdot \log(19)\]\[\log(19) \approx 1.279\]\[E = 0.77 - 0.0757 \approx 0.6943 \text{ V}\]

Step 5: Calculate E for Hg2+/Hg system

To find \(E_{\mathrm{Hg}_2^{2+} | \mathrm{Hg}}\), we use the overall cell potential where the net potential for the reaction is 0 (since it's at equilibrium between the two half-cells): \[E_{\mathrm{cell}} = E_{\text{cathode}} - E_{\text{anode}} = 0\]Letting \(E_{\text{cathode}} = E_{Hg_2^{2+} | Hg}\) and \(E_{\text{anode}} = E_{\mathrm{Fe}^{3+} | \mathrm{Fe}^{2+}}\), then:\[E_{\mathrm{Hg}_2^{2+} | \mathrm{Hg}} = E_{\mathrm{Fe}^{3+} | \mathrm{Fe}^{2+}}\]So, \(E_{\mathrm{Hg}_2^{2+} | \mathrm{Hg}} \approx 0.6943 \text{ V}\).

Key Concepts

Nernst Equation

In electrochemistry, the Nernst Equation is fundamental for understanding and calculating the potential of an electrode under non-standard conditions. It relates the standard electrode potential to the actual concentration of the ions in a redox reaction.
This equation helps us find the real-world voltage of electrochemical cells, factoring in concentrations.The equation is expressed as:\[E = E^\circ - \frac{RT}{nF} \ln(Q)\]Where:

• \(E\) is the electrode potential at non-standard conditions.
• \(E^\circ\) is the standard electrode potential.
• \(R\) is the universal gas constant (8.314 J/(mol·K)).
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons transferred in the reaction.
• \(F\) is the Faraday constant (approximately 96485 C/mol).
• \(Q\) is the reaction quotient, similar in form to the equilibrium constant, but applicable at any point during the reaction.

At room temperature (25°C), the Nernst Equation simplifies to:\[E = E^\circ - \frac{0.05916}{n} \log(Q)\]This form makes it relatively straightforward to plug in concentrations and determine electrode potentials. Remember, the Nernst Equation is essential for predicting how cells behave outside standard conditions, a frequent scenario in real-life applications.

Electrode Potentials

Electrode potentials, also known as standard electrode potentials when measured under standard conditions (1 M concentration, 1 atm pressure, and 25°C), are values that describe the energy difference between an electrode and its ion in solution. These values are critical in predicting the direction of spontaneous redox reactions.
Each half-reaction in a redox process is characterized by its standard electrode potential \(E^\circ\). An electrode with a high \(E^\circ\) readily gains electrons and functions as a good oxidizing agent, whereas an electrode with a low \(E^\circ\) is more likely to lose electrons, functioning as a reducing agent.The net cell potential, denoted as \(E_{\text{cell}}\), is calculated by:\[E_{\text{cell}} = E_{\text{cathode}} - E_{\text{anode}}\]In electrochemical cells, oxidation occurs at the anode and reduction occurs at the cathode. A positive \(E_{\text{cell}}\) indicates a spontaneous reaction, while a negative \(E_{\text{cell}}\) suggests non-spontaneity. Through this understanding, electrode potentials can be used not only to measure the feasibility of electrochemical reactions but also to determine the direction of electron flow in a cell.

Redox Reactions

Redox reactions, or oxidation-reduction reactions, are chemical processes in which the oxidation states of atoms are changed. These reactions involve the transfer of electrons between two substances.
In a redox reaction:

• The substance that loses electrons is oxidized, while the substance that gains electrons is reduced.
• This concept forms the basis of many processes in both chemistry and biology, such as cellular respiration and photosynthesis.
• Redox reactions can be split into two half-reactions: the oxidation half and the reduction half.

For each half-reaction, we can assign an electrode potential. This potential allows us to predict which species will be oxidized or reduced. The electrode with a higher standard potential will act as the reduction site, whereas the one with a lower potential will be the oxidation site. Balancing redox reactions involves ensuring that the number of electrons lost in oxidation equals the number. These principles form the backbone of electrochemical cells, batteries, and many other technologies we rely on in daily life.