Let's find the cell potential of an electrochemical cell based on the half-reactions provided. We'll outline the process step-by-step.
### Step 1: Identify the Half-Reactions and Their Reduction Potentials
The given half-reactions are:
1. [tex]\( Ag^+ + e^- \rightarrow Ag \)[/tex]
2. [tex]\( Fe \rightarrow Fe^{3+} + 3e^- \)[/tex]
From a standard reduction potential chart (not provided in the problem, but we assume commonly known values in electrochemistry):
- The reduction potential for [tex]\( Ag^+ + e^- \rightarrow Ag \)[/tex] is [tex]\( 0.80 \)[/tex] V.
- The reduction potential for [tex]\( Fe^{3+} + 3e^- \rightarrow Fe \)[/tex] is [tex]\( -0.04 \)[/tex] V.
### Step 2: Identify the Cathode and the Anode
In this cell:
- The [tex]\( Ag^+ \)[/tex] is being reduced to [tex]\( Ag \)[/tex], so this reaction occurs at the cathode.
- The [tex]\( Fe \)[/tex] is being oxidized to [tex]\( Fe^{3+} \)[/tex], so this reaction occurs at the anode.
### Step 3: Determine the Potential of the Cathode and Anode
For the cathode ([tex]\( Ag^+ + e^- \rightarrow Ag \)[/tex]):
- The reduction potential is [tex]\( 0.80 \)[/tex] V.
For the anode ([tex]\( Fe \rightarrow Fe^{3+} + 3e^- \)[/tex]):
- The given reduction potential for [tex]\( Fe^{3+} + 3e^- \rightarrow Fe \)[/tex] is [tex]\( -0.04 \)[/tex] V.
- However, since this reaction is occurring at the anode and involves oxidation, we need to reverse this reaction, therefore reversing the sign of the potential:
[tex]\[ E_{\text{anode}} = -(-0.04 \, \text{V}) = 0.04 \, \text{V} \][/tex]
### Step 4: Calculate the Cell Potential
The cell potential ([tex]\( E_{\text{cell}} \)[/tex]) is calculated using the formula:
[tex]\[ E_{\text{cell}} = E_{\text{cathode}} - E_{\text{anode}} \][/tex]
Plugging in the values:
[tex]\[ E_{\text{cathode}} = 0.80 \, \text{V} \][/tex]
[tex]\[ E_{\text{anode}} = 0.04 \, \text{V} \][/tex]
Therefore:
[tex]\[ E_{\text{cell}} = 0.80 \, \text{V} - 0.04 \, \text{V} = 0.76 \, \text{V} \][/tex]
After verifying the given options:
A. [tex]\( -0.44 \, V \)[/tex]
B. [tex]\( -1.24 \, V \)[/tex]
C. [tex]\( 0.44 \, V \)[/tex]
D. [tex]\( 1.24 \, V \)[/tex]
We see that none of these match our calculated potential of [tex]\( 0.76 \, V \)[/tex]. Thus, given our calculations and standard electrochemical principles, the correct calculated cell potential is [tex]\( 0.76 \, V \)[/tex].
It appears there is a mistake in the options provided as they do not include the correct value. The correct potential of the cell should indeed be [tex]\( 0.76 \, V \)[/tex].
