Question

For the voltaic cell, $$\begin{array}{l} \mathrm{Ag}(\mathrm{s}) | \mathrm{Ag}^{+}(0.015 \mathrm{M}) \| \mathrm{Fe}^{3+}(0.055 \mathrm{M}) \\ \quad \mathrm{Fe}^{2+}(0.045 \mathrm{M}) | \mathrm{Pt}(\mathrm{s}) \end{array}$$ (a) what is \(E_{\text {cell initially? }}\) (b) As the cell operates, will \(E_{\text {cell increase }}\) decrease, or remain constant with time? Explain. (c) What will be \(E_{\text {cell }}\) when \(\left[\mathrm{Ag}^{+}\right]\) has increased to \(0.020 \mathrm{M} ?\) (d) What will be \(\left[\mathrm{Ag}^{+}\right]\) when \(E_{\text {cell }}=0.010 \mathrm{V} ?\) (e) What are the ion concentrations when \(E_{\text {cell }}=0 ?\)

Short answer

(a) Use the Nernst equation with the given concentrations to calculate the initial E_cell. (b) E_cell will decrease over time as the system approaches equilibrium. (c) Again use the Nernst equation but with [Ag+] = 0.020M. (d) Rearrange the Nernst equation to find [Ag+] when E_cell = 0.010V. (e) At equilibrium (E_cell = 0), use the Nernst equation to find the concentrations of Ag+, Fe^2+, and Fe^3+.

Step-by-step solution

Define the Half-Reactions and Determine the Standard Cell Potential

The half-reactions in the given voltaic cell are: \[ \mathrm{Ag}^{+} + e^{-} \rightarrow \mathrm{Ag (s)} \text{ and } \mathrm{Fe}^{3+} + e^{-} \rightarrow \mathrm{Fe}^{2+} .\] The standard reduction potential for Ag+/Ag is 0.80V and for Fe^3+/Fe^2+ is 0.77V. The cell potential (\(E_{\text {cell }}\)) under standard conditions can be calculated by subtracting the reduction potential of the anode from that of the cathode \(E_{\text {cell }}^0 = E_{\text {cathode }}^0 - E_{\text {anode }}^0 = 0.80 - 0.77 = 0.03V.\)

Calculate Initial Cell Potential

The cell potential (\(E_{\text {cell }}\)) under non-standard conditions can be calculated using the Nernst equation: \[E_{\text {cell }} = E_{\text {cell }}^0 - \frac{0.0591}{n} \log Q ,\] where n is the number of electrons transferred in the redox reaction and Q is the reaction quotient. Substitute \(E_{\text {cell }}^0 = 0.03V\), n = 1, and Q = \([Fe^{2+}]/[Fe^{3+}][Ag^+]\) = (0.045M)/(0.055M * 0.015M) to find \(E_{\text {cell }}\).

Determine if Cell Potential Changes Over Time

Since the voltaic cell operates under non-equilibrium conditions, the cell potential will decrease over time as the cell reaches equilibrium (Q becomes larger).

Calculate Cell Potential for [Ag+] = 0.020M

Use the Nernst equation with \(E_{\text {cell }}^0 = 0.03V\), n = 1, and an updated Q = \([Fe^{2+}]/[Fe^{3+}][Ag^+]\) = (0.045M)/(0.055M * 0.020M) to find the new \(E_{\text {cell }}\).

Calculate [Ag+] for E_cell = 0.010V

Rearrange the Nernst equation to solve for [Ag+]: \[ [Ag+] = [Fe^{2+}] / [Fe^{3+}] * 10^{(E_{cell}^0 - E_{cell})n/0.0591} .\] Substitute \(E_{\text {cell }}^0 = 0.03V\), \(E_{\text {cell }} = 0.010V\), n = 1, [Fe^3+] = 0.055M, and [Fe^2+] = 0.045M to find [Ag+].

Calculate Ion Concentrations at Equilibrium (E_cell = 0)

At equilibrium, E_cell = 0. Use the Nernst equation set to 0 = \(E_{\text {cell }}^0\) - (0.0591/n) * logQ to solve for Q at equilibrium. Q = \([Fe^{2+}]/[Fe^{3+}][Ag^+]\) at equilibrium can then be used to determine the concentrations of different ions.

Key concepts

Standard Cell Potential

The standard cell potential, represented as \(E_{\text{cell}}^0\), is a crucial parameter in understanding how electrochemical cells, such as voltaic cells, function. It is the voltage difference between two half-cells at standard conditions, which typically means solutes at 1M concentration, gases at 1 atm pressure, and a temperature of 25°C (298K). This potential is determined by the tendency of the reaction to occur and is calculated by subtracting the standard reduction potential of the anode from the cathode. In our exercise, the standard cell potential for a cell involving silver and iron ions was calculated as \(E_{\text{cell}}^0 = 0.80V - 0.77V = 0.03V\).
Understanding this concept is fundamental because it sets the baseline for how we measure and predict the behavior of the cell under non-standard conditions using equations like the Nernst equation.

Nernst Equation

The Nernst equation allows us to calculate the electrode potential of a half-cell or the overall potential of an electrochemical cell under non-standard conditions. It takes into consideration the effect of ion concentration, temperature, and the number of electrons transferred in the reaction on the cell potential. The equation is written as:
\[E_{\text{cell}} = E_{\text{cell}}^0 - \left(\frac{0.0591}{n}\right) \log Q\]
where \(E_{\text{cell}}^0\) is the standard cell potential, \(n\) is the number of moles of electrons exchanged, and \(Q\) is the reaction quotient (ratio of products to reactants). By substituting the appropriate values into the Nernst equation, we can find out the actual cell potential at any moment, such as when the cell operation leads to changes in ion concentrations.

Electrochemical Cell Operations

Understanding electrochemical cell operations involves recognizing that as a voltaic cell operates, redox reactions at the electrodes cause changes in the concentrations of the reacting species. In a typical operation, electrons flow from the anode to the cathode through an external circuit, while ions move within the surrounding electrolyte to maintain the charge balance. Over time, these reactions will shift towards equilibrium, affecting the cell potential. This change happens because, as the concentration of reactants and products changes, so does the reaction quotient \(Q\), which, as dictated by the Nernst equation, alters the cell's voltage.

Concentration Changes in Electrolytic Solutions

In voltaic cells, the concentration of ions in the electrolytic solutions changes as the cell delivers current. For instance, as the cell operates, the concentration of silver ions (\([Ag^+]\)) will increase as silver is deposited onto the electrode. This alteration in ion concentration influences the reaction quotient \(Q\) and, consequently, the cell potential as described by the Nernst equation. If left unchecked, these changes will continue until the cell reaches equilibrium, at which point the cell potential drops to zero, the flow of electrons stops, and the reaction ceases to progress.

Redox Reactions

Redox reactions are chemical reactions where electrons are transferred between substances. They are the heart of any electrochemical cell's operation, including voltaic cells. Each half-cell in a voltaic cell contains a half-reaction; one is oxidation (loss of electrons), and the other is reduction (gain of electrons). In the exercise provided, we analyzed the redox reaction between silver and iron ions. As these half-reactions proceed, they drive the flow of electrons through the external circuit and ions through the solution, which in turn affects the cell potential. Understanding how these reactions contribute to the cell's overall behavior is essential for explaining changes in cell potential over time and for calculating the equilibrium conditions.