Question

A solution at \(25^{\circ} \mathrm{C}\) contains 1.0\(M \mathrm{Cu}^{2+}\) and \(1.0 \times 10^{-4} M\) \(\mathrm{Ag}^{+}\). Which metal will plate out first as the voltage is gradually increased when this solution is electrolyzed? (Hint: Use the Nernst equation to calculate \(\mathscr{E}\) for each half-reaction.)

Short answer

Copper will plate out first as the voltage is gradually increased when the given solution is electrolyzed. This is because its cell potential (\(E_{\mathrm{Cu^{2+}/Cu}} = +0.337V\)) is lower than the cell potential of silver (\(E_{\mathrm{Ag^{+}/Ag}} = +1.0358V\)).

Step-by-step solution

Write the half-cell reactions and look up their standard reduction potentials

Before we can use the Nernst equation, we need to know the half-cell reactions and their corresponding standard reduction potentials. The half-cell reactions are: Copper: \[\mathrm{Cu}^{2+} + 2e^- \rightarrow \mathrm{Cu}\] Silver: \[\mathrm{Ag}^{+} + e^- \rightarrow \mathrm{Ag}\] Next, check a table of standard reduction potentials and find the values for these reactions: Copper: \(E_{\mathrm{Cu^{2+}/Cu}}^{\circ} = +0.337\, \mathrm{V}\) Silver: \(E_{\mathrm{Ag^{+}/Ag}}^{\circ} = +0.799\, \mathrm{V}\)

Write the Nernst equation and plug in the known values

The Nernst equation relates the cell potential to the standard cell potential and the concentration of species present in the half-reactions: \[E = E^{\circ} - \frac{0.0592}{n}\log{Q}\] Here, - \(E\) is the cell potential, - \(E^{\circ}\) is the standard cell potential, - \(n\) is the number of electrons transferred in the half-reaction, - \(Q\) is the reaction quotient, which in this case is the concentration of the metal ions. Next, plug in the known values for the copper and silver half-cell reactions: Copper: \[E_{\mathrm{Cu^{2+}/Cu}} = +0.337 - \frac{0.0592}{2}\log{\frac{[\mathrm{Cu^{2+}]} }{[\mathrm{Cu}]}}\] Since the concentration of solid copper is 1, we can simplify this equation. Given that the concentration of \(\mathrm{Cu^{2+}}\) ions is \(1.0M\),, \[E_{\mathrm{Cu^{2+}/Cu}} = +0.337 - \frac{0.0592}{2}\log{1.0}\] Silver: \[E_{\mathrm{Ag^{+}/Ag}} = +0.799 - \frac{0.0592}{1}\log{\frac{[\mathrm{Ag^{+}]} }{[\mathrm{Ag}]}}\] And, since the concentration of solid silver is 1, we can simplify this equation. Given that the concentration of \(\mathrm{Ag^{+}}\) ions is \(1.0 \times 10^{-4} M\), \[E_{\mathrm{Ag^{+}/Ag}} = +0.799 - \frac{0.0592}{1}\log{(1.0 \times 10^{-4})}\]

Calculate the cell potentials for copper and silver half-cells

Determine the cell potentials for both the copper and silver half-cell reactions by solving the above equations: Copper: \(E_{\mathrm{Cu^{2+}/Cu}} = +0.337V\) (since the logarithm of 1 is 0) Silver: \(E_{\mathrm{Ag^{+}/Ag}} = +0.799 - 0.0592 \times (-4) = +0.799 + 0.2368 = +1.0358V\)

Compare the cell potentials to identify the metal which will plate out first

Now that we know the cell potentials for both half-cells, we can determine which metal will plate out first by comparing the values. The metal with a lower cell potential will plate out first: Copper: \(E_{\mathrm{Cu^{2+}/Cu}} = +0.337V\) Silver: \(E_{\mathrm{Ag^{+}/Ag}} = +1.0358V\) Since the cell potential for the copper half-cell reaction is lower, copper will plate out first as the voltage is gradually increased when the given solution is electrolyzed.

Key concepts

Nernst Equation

The Nernst Equation is a crucial tool in electrochemistry that helps calculate the cell potential under non-standard conditions. It essentially adjusts the standard cell potential, considering the concentrations of the ions involved in the reactions. Here's how it is structured:

\[E = E^{\circ} - \frac{0.0592}{n}\log{Q}\]

- **\(E\)**: Cell potential under non-standard conditions.
- **\(E^{\circ}\)**: Standard cell potential.
- **\(n\)**: Number of electrons transferred in the half-reaction.
- **\(Q\)**: Reaction quotient, calculated as the product of concentrations of the products divided by the product of concentrations of the reactants.

The Nernst Equation becomes especially useful when the concentrations are not at the standard 1M. For instance, in the original problem, it helps predict whether copper or silver will plate out first by plugging in the differing ion concentrations into the formula. This outcome is based on comparing the calculated cell potentials for both metals.

Standard Reduction Potential

The term "Standard Reduction Potential" refers to the ability of a species to be reduced, measured at standard conditions (1M concentration, 1 atm pressure, and 25°C). This is crucial in understanding which metal will plate out first, as it tells us how likely it is for a given ion to gain electrons and convert to a solid metal.

- **Copper**: The equation is \(\mathrm{Cu}^{2+} + 2e^- \rightarrow \mathrm{Cu}\), with a standard reduction potential of \(E_{\mathrm{Cu^{2+}/Cu}}^{\circ} = +0.337 \, \mathrm{V}\).
- **Silver**: The equation is \(\mathrm{Ag}^{+} + e^- \rightarrow \mathrm{Ag}\), with a standard reduction potential of \(E_{\mathrm{Ag^{+}/Ag}}^{\circ} = +0.799 \, \mathrm{V}\).

This information is vital for using the Nernst Equation, as it provides the baseline potential that is adjusted according to the reaction's conditions. Knowing these values, we enhance our understanding of which reactions are more favorable under given conditions.

Half-Cell Reactions

Half-cell reactions are individual oxidation or reduction processes that occur in separate compartments of an electrochemical cell. Each half-cell involves a specific metal ion gaining or losing electrons. This separation is fundamental to calculating the overall cell potential and essential for using the Nernst Equation.

**Understanding Half-Cell Reactions:**
- In any electrochemical setup, each electrode undergoes its own half-reaction.
- An oxidation reaction occurs at the anode, while a reduction reaction takes place at the cathode.

For the reactions considered in this task:
- **Copper Half-Cell:** \(\mathrm{Cu}^{2+} + 2e^- \rightarrow \mathrm{Cu}\) - Here, copper ions gain electrons to become solid copper.
- **Silver Half-Cell:** \(\mathrm{Ag}^{+} + e^- \rightarrow \mathrm{Ag}\) - Here, silver ions gain electrons to become solid silver.

Recognizing and writing out these reactions is the first step in applying the Nernst Equation and determining the sequence of electrodeposition, which helps in experiments like copper and silver plating.