Question

The value of \(E_{\text {cell }}^{\circ}\) for the reaction \(\mathrm{Zn}(\mathrm{s})+\) \(\mathrm{Pb}^{2+}(\mathrm{aq}) \longrightarrow \mathrm{Zn}^{2+}(\mathrm{aq})+\mathrm{Pb}(\mathrm{s})\) is \(0.66 \mathrm{V} .\) This means that for the reaction \(\mathrm{Zn}(\mathrm{s})+\mathrm{Pb}^{2+}(0.01 \mathrm{M})\) \(\rightarrow \mathrm{Zn}^{2+}(0.10 \mathrm{M})+\mathrm{Pb}(\mathrm{s}), E_{\text {cell }}\) equals \((\mathrm{a}) 0.72 \mathrm{V}\) (b) \(0.69 \mathrm{V} ;\) (c) \(0.66 \mathrm{V} ;\) (d) \(0.63 \mathrm{V}\)

Short answer

The correct answer is (d) 0.63 V.

Step-by-step solution

Understanding the Nernst Equation

The Nernst equation allows us to calculate the potential of a cell under non-standard conditions (not at equilibrium). It relates the cell potential at any point in time to the standard cell potential, temperature, reaction quotient, and the number of electrons transferred in the redox reaction. The formula usually takes two forms: \n1. \(E_{cell}=E_{cell}^{o}-\frac{0.0592}{n}logQ\) at 25°C \n2. \(E_{cell}=E_{cell}^{o}-\frac{(RT)}{nF}lnQ\) at other temperatures. Here, \(R\) is the gas constant (8.314 J/K mol), \(F\) is Faraday's constant (96485 C/mol), \(T\) is the temperature (298 K), and \(n\) is the moles of electrons transferred in the redox reaction. Hence, for first part, we'll use the equation at 25°C.

Identifying the Values

We have \(E_{cell}^{o}=0.66 V\), \(Q=\frac{[Zn^{2+}]}{[Pb^{2+}]}=\frac{0.10 M}{0.01 M}=10\), and \(n=2\) electrons are exchanged in the reaction. In the Nernst equation, we plug these values in.

Solving the Equation

Insert the given values into the formula: \(E_{cell}=0.66 V- \frac{0.0592}{2}log10 = 0.66 V - 0.03 V = 0.63 V \) Thus, we've calculated the cell potential under the given conditions.

Key concepts

Electrochemistry

Electrochemistry is an essential branch of chemistry that explores the relationship between chemical changes and electrical energy. It involves the study of cells that generate electricity (galvanic cells) and cells that consume electricity to cause chemical changes (electrolytic cells). A critical concept in electrochemistry is the **electrode potential**, which describes the ability of a given cell to produce an electric current.

In electrochemical reactions, electrons are transferred between species. This transfer of electrons results in either a generation of electrical energy, like in batteries, or consumption of electrical energy, as seen in electroplating. Understanding how to calculate the potential of electrochemical cells is fundamental to many industries, particularly those involved in energy storage and metal purification.

In practical applications, electrochemistry is used in:

• Batteries – Which transform chemical energy into electrical energy.
• Fuel cells – Which generate electricity through chemical reactions.
• Corrosion study – Which involves understanding and preventing material degradation.

Redox Reactions

Redox reactions, or reduction-oxidation reactions, are chemical processes involving the transfer of electrons between two substances. They are integral to electrochemical reactions. ‘Redox’ is a combination of the terms ‘reduction’ and ‘oxidation.’

- **Oxidation** is the process where an atom or molecule loses electrons, increasing its oxidation state. - **Reduction** is the gain of electrons, leading to a decrease in oxidation state.

In such reactions, the substance that gives away electrons is the **reducing agent**, while the one that gains electrons is the **oxidizing agent**. For any redox reaction, there is always a corresponding oxidation and reduction event that occurs.

These reactions are foundational for electrochemistry, as they facilitate the flow of electrons, which is the basis for electrical current generation. In the context of our exercise, zinc ( Zn ) acts as a reducing agent by releasing electrons, whereas lead ions ( Pb^{2+} ) act as oxidizing agents by accepting electrons. Balancing these electron transfers is crucial to understanding and correctly predicting the outcomes of redox reactions.

Cell Potential Calculation

In electrochemistry, calculating the cell potential is essential for predicting whether a redox reaction can occur spontaneously under given conditions. This is where the **Nernst Equation** comes in.

The Nernst Equation allows us to calculate the cell potential (E_{cell}) at non-standard conditions by considering the reaction quotient (Q) and the number of moles of electrons transferred (n). It takes into account the concentration of reactants and products, which affects the overall electromotive force (EMF) of the cell: \[E_{cell} = E_{cell}^{o} - \frac{0.0592}{n} \log{Q} \quad \text{at 25°C}.\] This modification from the standard cell potential (E_{cell}^{o}) to the actual cell potential helps us understand how changes in conditions, like concentration and temperature, affect the reaction.

• **E_{cell}^{o}**: Standard cell potential, measured under standard conditions (1M concentration, 1 atm pressure, 25°C).
• **Q**: Reaction quotient, given by the ratio of concentrations of products to reactants.
• **n**: Number of electrons exchanged per molecule of reaction.

For our exercise, with given values of **E_{cell}^{o}** at 0.66 V, **Q** at 10, and **n** at 2, plugging these into the Nernst Equation helps determine the actual cell potential, resulting in a calculated value of 0.63 V, indicating potential shift influenced by concentration changes.