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

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Answer

To solve the problem, we need to analyze the electrochemical cell reaction and how the cell potential \( E_{cell} \) varies with the concentrations of the reactants and products. The cell reaction given is:

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

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

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

Where:
- \( E_{cell}^\circ \) is the standard cell potential (1.56 V in this case).
- \( R \) is the universal gas constant (8.314 J/(mol·K)).
- \( T \) is the temperature in Kelvin (assume 298 K if not specified).
- \( n \) is the number of moles of electrons transferred (2 for this reaction).
- \( F \) is Faraday's constant (96485 C/mol).
- \( Q \) is the reaction quotient.

### Step 2: Determine the Reaction Quotient \( Q \)
For the given reaction, the reaction quotient \( Q \) is defined as:

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

### Step 3: Substitute \( Q \) into the Nernst Equation
Substituting \( Q \) into the Nernst equation, we get:

\[ E_{cell} = E_{cell}^\circ - \frac{RT}{nF} \ln \left( \frac{[\text{Zn}^{2+}]}{[\text{Ag}^+]^2} \right) \]

### Step 4: Convert the Natural Logarithm to Logarithm Base 10
To express the equation in terms of logarithm base 10, we use the conversion:

\[ \ln x = 2.303 \log_{10} x \]

Thus, the equation becomes:

\[ E_{cell} = E_{cell}^\circ - \frac{2.303RT}{nF} \log_{10} \left( \frac{[\text{Zn}^{2+}]}{[\text{Ag}^+]^2} \right) \]

### Step 5: Rearranging the Equation
Rearranging the equation gives us:

\[ E_{cell} = E_{cell}^\circ - \frac{2.303RT}{nF} \log_{10} [\text{Zn}^{2+}] + \frac{2 \cdot 2.303RT}{nF} \log_{10} [\text{Ag}^+] \]

### Step 6: Identify the Slope and Intercept
From the rearranged equation, we can identify:
- The y-intercept is \( E_{cell}^\circ = 1.56 \, V \).
- The slope is negative because of the negative sign in front of the concentration terms.

### Step 7: Conclusion about the Graph
The graph of \( E_{cell} \) vs. \( \log_{10} \left( \frac{[\text{Zn}^{2+}]}{[\text{Ag}^+]^2} \right) \) will have:
- A negative slope.
- A y-intercept at \( 1.56 \, V \).

Thus, the correct graph will show a downward trend starting from \( 1.56 \, V \) on the y-axis.