Question

Which graph correctly correlates \(E_{\text {cell }}\) as a function of concentrations for the cell $$ \mathrm{Zn}(s)+2 \mathrm{Ag}^{+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+2 \mathrm{Ag}(s), \quad E_{\text {cell }}^{\circ}=1.56 \mathrm{~V} $$ \(Y\) -axis : \(E_{\text {cell }}, X\) -axis : \(\log _{10} \frac{\left[\mathrm{Zn}^{2+}\right]}{\left[\mathrm{Ag}^{+}\right]}\)

Short answer

The correct graph shows a straight line with a negative slope, indicating an inverse relationship between the cell potential, \(E_{\text{cell}}\), and the logarithm of the concentration ratio of zinc ions to silver ions.

Step-by-step solution

Understanding the Nernst Equation

Realize that the potential of an electrochemical cell under non-standard conditions is given by the Nernst equation: \(E_{\text{cell}} = E_{\text{cell}}^{\circ} - \frac{0.05916}{n} \log_{10} \frac{[\text{Products}]}{[\text{Reactants}]}\).In this reaction, \(n=2\), the number of moles of electrons transferred.

Identify the Cell Reaction Components

Identify that \(E_{\text{cell}}^{\circ} = 1.56 \text{V} \), \([\text{Products}] = [\text{Zn}^{2+}]\), and \([\text{Reactants}] = [\text{Ag}^{+}]^{2}\) because two moles of silver ions are involved for every mole of zinc ion formed.

Apply the Logarithm to Ratio of Concentrations

Rewrite the Nernst equation to express \(E_{\text{cell}}\) as a function of the concentrations. Since \(n=2\), the equation simplifies to \(E_{\text{cell}} = E_{\text{cell}}^{\circ} - 0.02958 \log_{10} \frac{[\text{Zn}^{2+}]}{[\text{Ag}^{+}]^{2}}\). For simplicity, consider the reaction quotient \(Q\) in the logarithm: \(\log_{10} \frac{[\text{Zn}^{2+}]}{[\text{Ag}^{+}]^{2}}\) can be expressed as \(\log_{10} \left([\text{Zn}^{2+}]\right) - 2\log_{10} \left([\text{Ag}^{+}]\right)\).

Construct the Graph

Since the relationship is linear with a negative slope due to the minus sign in the Nernst equation, the graph of \(E_{\text{cell}}\) as a function of \(\log_{10} \frac{[\text{Zn}^{2+}]}{[\text{Ag}^{+}]}\) should be a straight line with negative slope. If the concentration ratio increases (meaning more zinc ions or fewer silver ions), the cell potential decreases; and vice versa. Thus, the correct graph will have a negative slope showing an inverse relationship between \(E_{\text{cell}}\) and \(\log_{10} \frac{[\text{Zn}^{2+}]}{[\text{Ag}^{+}]}\).

Key concepts

Electrochemical Cell Potential

The electrochemical cell potential, often represented as E_cell, is a measure of the driving force behind the electrical current in an electrochemical cell. It is determined by the difference in potential energy between the electrons in the anode and cathode.

Under standard conditions, this potential is denoted as E_cell^° and measured when all reactants and products are at 1 M concentration, atmospheric pressure, and room temperature (usually 25°C). For the given zinc-silver cell reaction, E_cell^° = 1.56 V indicates a strong tendency for the reaction to occur, as standard potentials above 0 V suggest a spontaneous reaction.

However, real-world scenarios rarely meet these perfect conditions. Thus, the Nernst equation is employed to calculate the cell potential under non-standard conditions. It considers the actual concentrations of ions involved in the electrochemical reaction to provide a more accurate cell potential value. By understanding and applying the Nernst equation, students can predict how changes in ion concentrations affect E_cell.

Reaction Quotient Q in Electrochemistry

The reaction quotient, Q, is a crucial concept in electrochemistry that represents the ratio of concentrations of reaction products to reactants, raised to the power of their coefficients in the balanced equation. For the electrochemical cell in question,

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

The Q reflects the instantaneous state of the reaction, differing from the equilibrium constant K which only applies when the reaction has reached equilibrium. As the reaction progresses towards equilibrium, Q changes and approaches K.

In the Nernst equation, Q serves as a variable that directly influences E_cell. The equation correlates the cell potential to the reaction quotient, encapsulating the dependency of the cell voltage on the concentrations of the electrolytic solutions. A higher concentration of products or lower concentration of reactants leads to a larger Q, which typically results in a reduction of the cell potential, as per the formula.

Logarithmic Concentration Ratio

The logarithmic concentration ratio is the foundation of the Nernst equation. It provides a way to relate the concentrations of ionic species in an electrochemical reaction to the cell potential.

In the provided example, the equation simplifies to:

\[ \log_{10} \frac{[\text{Zn}^{2+}]}{[\text{Ag}^{+}]^2} \]

which is further dissected into individual logarithmic terms of the concentrations involved. This treatment results from the properties of logarithms, where the log of a ratio is the difference between the logs of the numerator and denominator.

Understanding the Logarithmic Scale

The logarithmic scale compresses the wide range of concentration values into a smaller, more manageable scale. This trait is particularly helpful when dealing with the vast differences in concentration levels that can occur in electrochemistry.

Moreover, since the graph plots E_cell against the logarithm of concentration ratios, a linear relationship is observed. This linearity is a direct consequence of the logarithmic term in the Nernst equation, fundamentally linking the logarithmic concentration ratio to the cell potential in a predictable manner.