Problem 9 - Chemistry: The Central Science in SI Units, Global Edition

Question

Consider the half-reaction \(\mathrm{Ag}^{+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{Ag}(s)\) (a) Which of the lines in the following diagram indicates how the reduction potential varies with the concentration of \(\mathrm{Ag}^{+}(a q) ?(\mathbf{b})\) What is the value of \(E_{\text {red }}\) when \(\log \left[\mathrm{Ag}^{+}\right]=0 ?\)

Step-by-Step Solution

Answer

The reduction potential varies linearly with the log of the Ag+ concentration, as shown by the equation \(E = E^0 - \frac{2.303 RT}{F} \log{[\mathrm{Ag}^+]}\). When \(\log{[\mathrm{Ag}^+]} = 0\), the reduction potential is equal to the standard reduction potential (\(E^0\)).

Step 1: The Nernst equation allows us to determine the reduction potential of half-reactions at various ion concentrations. It is expressed as: \[E = E^0 - \frac{2.303 RT}{nF} \log{Q}\] where E is the potential, \(E^0\) is the standard reduction potential, R is the gas constant, T is the temperature, n is the number of electrons transferred, F is the Faraday constant, and Q is the reaction quotient. #Step 2: Applying the Nernst Equation to the Given Half-Reaction#

The given half-reaction is: \(\mathrm{Ag}^{+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{Ag}(s)\). In this half-reaction, only one electron is transferred (n = 1). The reaction quotient (Q) is equal to the concentration of Ag+ ions. We can rewrite the Nernst equation for this specific reaction as: \[E = E^0 - \frac{2.303 RT}{F} \log{[\mathrm{Ag}^+]}\] #Step 3: Analyzing the Relationship between Reduction Potential and Ag+ Concentration#

Step 2: As seen in the equation from step 2, \(E = E^0 - \frac{2.303 RT}{F} \log{[\mathrm{Ag}^+]}\), the reduction potential (E) is linearly dependent on the log of the Ag+ concentration. #Step 4: Calculating the Reduction Potential When log[Ag+] = 0#

When log[Ag+] = 0, the Nernst equation becomes: \[E = E^0 - \frac{2.303 RT}{F} \times 0\] Since multiplying by zero eliminates the second term, this simplifies to: \[E = E^0\] So when log[Ag+] = 0, the reduction potential is equal to the standard reduction potential (\(E^0\)).

Key Concepts

Reduction Potential

Reduction potential is a measure of the tendency of a chemical species to gain electrons and thus be reduced. It is quantified in volts and often represented as \(E\).
In electrochemistry, understanding reduction potential helps predict whether a redox reaction will occur and in which direction. The standard reduction potential, \(E^0\), is measured under standard conditions: 1 M concentrations, 1 atm pressure, and a specific temperature (usually 298 K).
The Nernst equation allows us to calculate the reduction potential under non-standard conditions by considering ion concentrations. This adjusted potential reflects real-world scenarios, enhancing our comprehension of various chemical processes.

Half-Reaction

A half-reaction showcases either the oxidation or the reduction process in a redox reaction. It separately highlights how electrons are transferred in reactions.
For example, consider the half-reaction:

• \(\mathrm{Ag}^{+}(aq) + \mathrm{e}^{-} \rightarrow \mathrm{Ag}(s)\)

This indicates the reduction of silver ions (\(\mathrm{Ag}^{+}\)) to metallic silver (\(\mathrm{Ag}\)).
Half-reactions help in balancing redox equations and in applying the Nernst equation, which focuses on these specific segments to determine changes in potential with varying conditions. Understanding half-reactions is crucial for analyzing and solving electrochemical problems effectively.

Concentration Dependence

The concentration of ions in a solution significantly influences the reduction potential, as emphasized by the Nernst equation. This equation illustrates that reduction potential decreases with an increase in ion concentration and vice versa.
Here's the modified Nernst equation for reference:

• \(E = E^0 - \frac{2.303 RT}{F} \log{[\mathrm{Ag}^+]}\)

When the concentration of \(\mathrm{Ag}^+\) ions changes, the \(\log{[\mathrm{Ag}^+]}\) term also changes, thus altering the overall potential \(E\).
This dependency is crucial for real-world applications where ion concentrations are rarely at standard values. Understanding this concept helps in predicting and controlling the behavior of electrochemical reactions in various concentrations.