Which graph correctly correlates `E_("cell")` as a function of concentration for the cell (for different values of M nad M' `Zn(s)+Cu^(2+)(M)rarrZn^(2+(M')+Cu(s), E_("cell")^(@)=1.10V`
X - axis : `log_(10).([Zn^(2+)])/([Cu^(2+)]),` Y - axis : `E_("cell")`

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to determine the relationship between the cell potential \( E_{\text{cell}} \) and the concentrations of the reactants and products in the electrochemical cell reaction given by:

\[ \text{Zn}(s) + \text{Cu}^{2+}(M) \rightleftharpoons \text{Zn}^{2+}(M') + \text{Cu}(s) \]

with a standard cell potential \( E^\circ_{\text{cell}} = 1.10 \, \text{V} \).

### Step 1: Write the Nernst Equation

The Nernst equation relates the cell potential to the concentrations of the reactants and products:

\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.059}{n} \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) \]

where:
- \( E_{\text{cell}} \) is the cell potential under non-standard conditions.
- \( E^\circ_{\text{cell}} \) is the standard cell potential.
- \( n \) is the number of electrons transferred in the reaction (which is 2 for this reaction).
- \( [\text{Zn}^{2+}] \) and \( [\text{Cu}^{2+}] \) are the concentrations of zinc and copper ions, respectively.

### Step 2: Substitute Values into the Nernst Equation

Substituting \( n = 2 \) into the Nernst equation gives:

\[ E_{\text{cell}} = 1.10 \, \text{V} - \frac{0.059}{2} \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) \]

This simplifies to:

\[ E_{\text{cell}} = 1.10 \, \text{V} - 0.0295 \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) \]

### Step 3: Define the Variables for the Graph

In the problem, we are asked to plot \( E_{\text{cell}} \) as a function of \( \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) \). We can rewrite the equation as:

\[ E_{\text{cell}} = 1.10 \, \text{V} - 0.0295 \cdot \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) \]

### Step 4: Analyze the Graph

The equation can be rearranged to the form of a straight line \( y = mx + c \):

- \( y \) corresponds to \( E_{\text{cell}} \)
- \( m \) (the slope) is negative (\(-0.0295\))
- \( x \) corresponds to \( \log \left( \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \right) \)
- \( c \) is the y-intercept (\(1.10 \, \text{V}\))

Since the slope is negative, this indicates that as the logarithm of the concentration ratio increases, the cell potential \( E_{\text{cell}} \) decreases.

### Conclusion

The graph that correctly represents this relationship will be a straight line with a negative slope. Therefore, the correct option is **B**, which shows a downward sloping line.