Which graph correctly correlates `E_(cell)` as a function of concentration for the cell
`Zn(s)+2Ag^+(aq)toZn^(2+)(aq)+2Ag(s), E_(cell)^(@)=1.56V`
y-axis: `E_(cell)`, X-axis: `log_(10)"([Zn^(2+)])/([Ag^+]^2)`

Options

A. (a)
B. (b)
C. (c)
D. (d)

Answer

To solve the problem, we will follow these steps:

### Step 1: Write the Nernst Equation
The Nernst equation relates the cell potential \( E_{cell} \) to the standard cell potential \( E^{\circ}_{cell} \) and the concentrations of the reactants and products. For the given reaction:

\[ \text{Zn(s)} + 2\text{Ag}^+(aq) \rightarrow \text{Zn}^{2+}(aq) + 2\text{Ag(s)} \]

The Nernst equation can be written as:

\[ E_{cell} = E^{\circ}_{cell} - \frac{RT}{nF} \ln Q \]

where \( Q \) is the reaction quotient given by:

\[ Q = \frac{[\text{Zn}^{2+}]}{[\text{Ag}^+]^2} \]

### Step 2: Substitute Values into the Nernst Equation
Given \( E^{\circ}_{cell} = 1.56 \, V \) and \( n = 2 \) (since 2 electrons are transferred in the reaction), we can rewrite the Nernst equation as:

\[ E_{cell} = 1.56 - \frac{2.303 \cdot RT}{2F} \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Ag}^+]^2} \right) \]

### Step 3: Rearranging the Equation
To correlate \( E_{cell} \) with the logarithmic term, we can express it as:

\[ E_{cell} = 1.56 - \frac{2.303RT}{2F} \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Ag}^+]^2} \right) \]

### Step 4: Identify the Variables for the Graph
In the context of the graph, we have:

- **Y-axis**: \( E_{cell} \)
- **X-axis**: \( \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Ag}^+]^2} \right) \)

### Step 5: Determine the Slope and Intercept
From the equation, we can see that:

- The intercept (when \( \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Ag}^+]^2} \right) = 0 \)) is \( 1.56 \, V \).
- The slope is negative because of the negative sign in front of the logarithmic term.

### Step 6: Analyze the Graphs
Now we need to analyze the provided graphs based on the slope and intercept:

- The correct graph should have:
- A negative slope.
- An intercept at \( 1.56 \, V \) on the Y-axis.

### Conclusion
After analyzing the graphs, the one that matches these criteria is Graph B.