Problem 46 - Chemistry Matter and Change

Question

Biofuel Cell At the cathode of a biofuel cell, \(\mathrm{Fe}^{3+}\) in potassium hexacyanoiron (III) \(\left(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\right)\) is reduced to \(\mathrm{Fe}^{2+}\) in potassium hexacyanoiron (II) \(\left(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\right) .\) At the anode, reduced nicotinamide-adenine-dinucleotide (NADH) is oxidized to NAD \(^{+} .\) Use the following standard reduction potential to determine the potential of the cell. $$ \begin{array}{ll}{\mathrm{NAD}^{+}+\mathrm{H}^{+}+2 \mathrm{e}^{-} \rightarrow \mathrm{NADH}} & {E^{0}=-0.320 \mathrm{V}} \\\ {\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-}+1 \mathrm{e}^{-} \rightarrow\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}} & {E^{0}=+0.36 \mathrm{V}}\end{array} $$

Step-by-Step Solution

Answer

The standard potential of the cell is 0.04 V.

Step 1: Identifying Half-Reactions

First, identify the two half-reactions occurring at the electrodes. At the cathode, the reduction reaction is \( ext{Fe}^{3+} + e^- \rightarrow ext{Fe}^{2+}\) with \(E^0 = +0.36 \, \text{V}\). At the anode, the oxidation reaction is \(\text{NADH} \rightarrow \text{NAD}^+ + \text{H}^+ + 2e^-\) which is the reverse of the reduction reaction \(\text{NAD}^+ + \text{H}^+ + 2e^- \rightarrow \text{NADH}\) with \(E^0 = -0.320 \, \text{V}\).

Step 2: Determining Oxidation Potential

For the oxidation reaction at the anode, reverse the sign of the standard reduction potential. Therefore, the oxidation of NADH has a standard potential of \(E^0 = +0.320 \text{ V}\).

Step 3: Calculate Cell Potential

The standard cell potential \(E_{ ext{cell}}^0\) is calculated using the formula \(E_{ ext{cell}}^0 = E_{ ext{cathode}}^0 - E_{ ext{anode}}^0\). Substitute the values: \(E_{ ext{cell}}^0 = +0.36 \, \text{V} - (+0.320 \, \text{V})\).

Step 4: Perform the Calculation

Subtract the anode potential from the cathode potential: \(E_{ ext{cell}}^0 = 0.36 \, \text{V} - 0.32 \, \text{V} = 0.04 \, \text{V}\).

Key Concepts

Half-Reactions

In electrochemistry, a half-reaction is simply a part of the overall chemical reaction that happens at each electrode of a biofuel cell. For a clearer understanding, imagine splitting the cell reactions into two parts: oxidation and reduction. At the cathode, reduction takes place, while at the anode, oxidation occurs.
To identify the half-reactions:

• The reduction half-reaction at the cathode is: \[\mathrm{Fe}^{3+} + \mathrm{e}^- \rightarrow \mathrm{Fe}^{2+}\]
• The oxidation half-reaction at the anode is the reverse of the given reduction reaction, and it is:\[\mathrm{NADH} \rightarrow \mathrm{NAD}^+ + \mathrm{H}^+ + 2\mathrm{e}^-\]

For students, understanding half-reactions is key because it breaks down complex reactions into manageable pieces. Each half focuses on either electron loss or gain.

Standard Reduction Potential

Standard reduction potential, denoted as \(E^0\), informs us about the tendency of a species to gain electrons, hence getting reduced. The potential is measured under standard conditions, which typically involve a concentration of 1 M for solutions, a pressure of 1 atm for gases, and a temperature of 25°C (298 K).
In our specific biofuel cell case, the given standard potentials are:

• For the cathode: \[E^0 = +0.36 \, \text{V}\]This value shows the ability of \[\left[\mathrm{Fe} (\mathrm{CN})_{6}\right]^{3-}\]to be reduced to \[\left[\mathrm{Fe} (\mathrm{CN})_{6}\right]^{4-}.\]
• For the anode: The reverse reaction has \[E^0 = -0.320 \, \text{V}\].Thus, \[\mathrm{NAD}^{+}\]can gain electrons to form \[\mathrm{NADH}\].But for oxidation, the sign changes to \[+0.320 \, \text{V}\].

The larger the positive potential, the more favorable the reduction process. These values help in calculating the overall cell potential.

Cell Potential Calculation

The overall cell potential, \(E_{\text{cell}}^0\), gives a measure of the voltage a cell can deliver. Calculating it requires knowing the potentials of the individual half-reactions.
For the biofuel cell:

• The cathode reduction potential is \(+0.36 \, \text{V}\).
• After reversing the anode reaction, the oxidation potential becomes \(+0.320 \, \text{V}\).

The formula to find the standard cell potential is:\[E_{\text{cell}}^0 = E_{\text{cathode}}^0 - E_{\text{anode}}^0\]By substituting the values:\[E_{\text{cell}}^0 = +0.36 \, \text{V} - (+0.320 \, \text{V})\]This results in:\[E_{\text{cell}}^0 = 0.04 \, \text{V}\]This value tells us the cell is capable of generating a potential of 0.04 V when operating under standard conditions.

NADH Oxidation

NADH, or reduced nicotinamide adenine dinucleotide, plays a crucial role in biological processes. It is often involved in transferring electrons as it serves as an electron donor. During biofuel cell operation, NADH is oxidized at the anode, which means it loses electrons.
The oxidation reaction is:\[\mathrm{NADH} \rightarrow \mathrm{NAD}^+ + \mathrm{H}^+ + 2\mathrm{e}^-\]To determine how favorable this oxidation is, we rely on the standard reduction potential, which for the reverse reduction reaction is \(-0.320 \, \text{V}\). Flipping this for oxidation yields \(+0.320 \, \text{V}\). By losing electrons, NADH contributes to the overall potential generated by the cell and supports energy production.

Fe(III) Reduction

The reduction of Fe(III) occurs at the cathode of the biofuel cell and is critical to its operation. This process involves transforming \[\mathrm{Fe}^{3+}\]ions to \[\mathrm{Fe}^{2+}\]ions, signifying electron acceptance.
The half-reaction for this reduction is:\[\mathrm{Fe}^{3+} + \mathrm{e}^- \rightarrow \mathrm{Fe}^{2+}\]The standard reduction potential involved here is \(+0.36 \, \text{V}\), a positive value indicating that the reduction of Fe(III) to Fe(II) is favorable.
The presence of the hexacyanoferrate complexes \(\mathrm{K}_3\left[\mathrm{Fe} (\mathrm{CN})_6\right]\)and \(\mathrm{K}_4\left[\mathrm{Fe} (\mathrm{CN})_6\right]\)gives stability to the Fe ions during the reaction, enabling efficient electron transfer and ultimately contributing to the cell potential of the biofuel cell.