Problem 80 - Chemistry: An Atoms-Focused Approach

Question

What is the minimum (least negative) cathode potential (versus SHE) needed to clectroplate silver onto cutlery in a solution of \(\mathrm{Ag}^{+}\) and \(\mathrm{NH}_{3}\) in which most of the silver ions are present as the complex \(\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\) and the concentration of \(\mathrm{Ag}^{+}(a q)\) is only \(3.50 \times 10^{-5} \mathrm{M} ?\)

Step-by-Step Solution

Answer

The minimum cathode potential needed to electroplate silver onto cutlery in the given solution is approximately 0.7950 V (versus SHE).

Step 1: Determine the redox reaction

The redox reaction taking place during the electroplating process is the reduction of silver ions, \(\mathrm{Ag}^{+}\), and the formation of metallic silver, \(\mathrm{Ag}(s)\). The reaction can be written as: $$ \mathrm{Ag}^+ + e^- \rightarrow \mathrm{Ag}(s) $$ This is the half-cell reaction we will consider when calculating the minimum cathode potential.

Step 2: Find the standard cell potential

The standard cell potential (\(E^{\circ}\)) for the reduction of silver ions to metallic silver can be found in a reference table (reference electrode). The standard reduction potential for \(\mathrm{Ag}^{+}/\mathrm{Ag}\) couple is \(+0.7996 \,\mathrm{V}\) (versus the Standard Hydrogen Electrode or SHE).

Step 3: Apply the Nernst equation

We can now use the Nernst equation to find the minimum cathode potential needed to electroplate silver onto cutlery. The Nernst equation is given by: $$ E = E^{\circ} - \frac{RT}{nF} \ln{Q} $$ Where \(E\) is the cell potential, \(E^{\circ}\) is the standard cell potential, \(R\) is the gas constant (\(8.314 \,\mathrm{J\, K^{-1}\, mol^{-1}}\)), \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons exchanged in the redox reaction, \(F\) is the Faraday constant (\(96485 \,\mathrm{C\, mol^{-1}}\)), and \(Q\) is the reaction quotient. We are asked to find the minimum cathode potential, so we will plug in the given concentration of \(\mathrm{Ag}^{+}(a q)\), which is \(3.50 \times 10^{-5} \mathrm{M}\). In this case, the reaction quotient \(Q\) can be represented by: $$ Q = \frac{[\mathrm{Ag}^+]_{eq}}{[\mathrm{Ag}^+]_{initial}} $$ Since only the silver ion concentration is involved in the electroplating process, \(Q\) is the concentration of \(\mathrm{Ag}^{+}(a q)\), which is \(3.50 \times 10^{-5}\,\mathrm{M}\). Assume the temperature is \(25^{\circ}\mathrm{C}\), or \(298.15\,\mathrm{K}\). Also, the number of moles of electrons exchanged, \(n\), is equal to 1 since one electron is transferred in the half-cell reaction given. Now we plug all the values into the Nernst equation and solve for \(E\): $$ E = 0.7996 \,\mathrm{V} - \frac{8.314 \,\mathrm{J\, K^{-1}\, mol^{-1}} \cdot 298.15 \,\mathrm{K}}{1 \cdot 96485 \,\mathrm{C\, mol^{-1}}} \ln(3.50 \times 10^{-5}\,\mathrm{M}) $$ After calculating, we get: $$ E \approx 0.7950 \,\mathrm{V} $$ So, the minimum (least negative) cathode potential needed to electroplate silver onto cutlery in the given solution is approximately \(0.7950\,\mathrm{V}\) (versus SHE).

Key Concepts

Nernst Equation

The Nernst equation is a fundamental tool in electrochemistry that helps predict the potential of an electrochemical cell under non-standard conditions. It is an essential formula for understanding how variations in ion concentration affect electrode potential. The equation takes into account several factors:

• **Standard Reduction Potential (\(E^{\circ}\), **): This is the potential difference measured under standard conditions (1 M concentration, 1 atm pressure, and 25°C) for a given redox reaction.
• **Temperature (\(T\), **): It is usually considered to be 298.15 K (25°C) unless specified otherwise.
• **Number of Electrons Exchanged (\(n\), **): This refers to the number of electrons transferred in the redox reaction. For the silver reaction in our exercise, \(n = 1\).
• **Reaction Quotient (\(Q\), **): This is a measure of the relative concentrations of products to reactants. For silver, it equals the concentration of \(\mathrm{Ag}^{+}\) ions.

The Nernst equation is written as:\[ E = E^{\circ} - \frac{RT}{nF} \ln Q \]where \(R\) is the gas constant (8.314 J K\(^{-1}\) mol\(^{-1}\)) and \(F\) is the Faraday constant (96485 C mol\(^{-1}\)).
In the context of the electroplating exercise, the Nernst equation allows for calculating the minimum potential needed to start and sustain the deposition of metallic silver from the \(\mathrm{Ag}^{+}\) solution. By substituting the known values into this equation, we predict the cathode potential necessary for the silver ions to reduce and form solid silver on the cutlery.

Redox Reaction

Redox reactions are chemical processes involving the transfer of electrons between two substances. The term 'redox' stands for reduction-oxidation. This is because, in every redox reaction, one substance is reduced (gains electrons), while the other is oxidized (loses electrons).
For our case, the redox reaction deals with the reduction of silver ions to solid silver:\[ \mathrm{Ag}^+ + e^- \rightarrow \mathrm{Ag}(s) \]Silver ions (\(\mathrm{Ag}^+\)) are reduced by gaining electrons to form metallic silver (\(\mathrm{Ag}(s)\)). This reaction is a half-reaction essential to the process of electroplating which transfers silver onto cutlery.
Key points to understand:

• **Reduction**: Gain of electrons. Here, each \(\mathrm{Ag}^+\) ion gains one electron.
• **Oxidation**: Loss of electrons. However, in this exercise, oxidation isn't explicitly shown because we're focusing on the reduction half-cell reaction.
• **Electroplating**: This utilizes redox reactions to coat the surface of an object with a thin layer of metal, in this case, silver.

Understanding the precise electron movement is crucial in designing the electrochemical cell for effective silver deposition. The electrons delivered from the power source reduce \(\mathrm{Ag}^+\) ions to transfer silver onto the surface of the cutlery, demonstrating a useful application of redox reactions.

Standard Hydrogen Electrode (SHE)

The Standard Hydrogen Electrode (SHE) is a reference electrode standard used to measure electrode potentials in electrochemistry. It serves as the universal reference point from which all half-cell potential measurements are made.
**Key Characteristics of the SHE include:**

• **Hydrogen gas** at 1 atm pressure bubbled over in a solution with a high hydrogen ion concentration (1 M), usually facilitated by strong acids.
• **Inert platinum electrode**: Used to facilitate the redox reactions involving hydrogen ions and electrons.
• **Standard Potential assignment**: The SHE is assigned a potential of 0.00 V at all temperatures, providing a consistent baseline to compare other electrode reactions.

In our electroplating problem, the reduction potential of silver ion is determined relative to the SHE. The silver standard electrode potential is \(+0.7996 \,\mathrm{V}\), indicating this is the voltage difference when compared to the zero potential of the SHE under standard conditions.
**Application in Electroplating:**Using SHE as a baseline ensures our potential calculations for plating processes are consistent and accurate. It highlights how much voltage is required to achieve reduction necessary for plating silver onto the cutlery. Thus, SHE helps in setting the proper electrochemical cell conditions and thereby controlling the quality of the silver deposition effectively.