A voltaic cell consists of an \(\mathrm{Mn} / \mathrm{Mn}^{2+}\) half-cell and a \(\mathrm{Pb} / \mathrm{Pb}^{2+}\) half-cell. Calculate \(\left[\mathrm{Pb}^{2+}\right]\) when \(\left[\mathrm{Mn}^{2+}\right]\) is \(1.4 M\) and \(E_{\text {cell }}\) is \(0.44 \mathrm{~V}\).

The Nernst equation is fundamental in electrochemistry. It provides a way to calculate the cell potential under non-standard conditions, which is essential for understanding how real-world cells operate. The equation is: \[ E_{\text{cell}} = E^{\text{o}}_{\text{cell}} - \frac{0.0592}{n} \text{log} \frac{[\text{Oxidant}]}{[\text{Reductant}]} \] This formula tells us how to adjust the standard cell potential based on the concentrations of the reactants and products. Here’s a breakdown of the variables:

• E_{\text{cell}}: The cell potential under non-standard conditions.
• E^{\text{o}}_{\text{cell}}: The standard cell potential.
• n: The number of moles of electrons transferred in the reaction.
• [\text{Oxidant}]: Concentration of the oxidized species.
• [\text{Reductant}]: Concentration of the reduced species.

Please note, accurate and precise calculation of the potential relies on correct inputs for each of these values.

Standard reduction potential is a measure of the tendency of a chemical species to be reduced, and it’s measured under standard conditions (1 M concentration, 1 atm pressure, 25°C). Reduction potentials are often tabulated for various half-reactions. For example:

• Mn^{2+}/Mn has a standard reduction potential of -1.18 V.
• Pb^{2+}/Pb has a standard reduction potential of -0.13 V.

This means that lead ions are more likely to accept electrons and be reduced than manganese ions, as a less negative value indicates a greater tendency to gain electrons.In a galvanic (voltaic) cell, the half-reaction with the higher reduction potential occurs at the cathode, and the one with the lower reduction potential takes place at the anode. Understanding these potentials is crucial for predicting and calculating the behavior of different electrochemical cells.

In electrochemistry, concentrations of species play a critical role in determining cell potential, especially under non-standard conditions. Here’s a focused example on how to calculate the concentration. We start from the Nernst equation: \[ E_{\text{cell}} = E^{\text{o}}_{\text{cell}} - \frac{0.0592}{n} \text{log} \frac{[\text{Mn}^{2+}]}{[\text{Pb}^{2+}]} \] Plug the given values into the equation: \[ 0.44 \text{ V} = 1.05 \text{ V} - \frac{0.0592}{2} \text{log} \frac{1.4}{[\text{Pb}^{2+}]} \] Solving for [\text{Pb}^{2+}], we'll isolate the logarithmic term:\[ 0.44 \text{ V} - 1.05 \text{ V} = -\frac{0.0592}{2} \text{log} \frac{1.4}{[\text{Pb}^{2+}]} \]\[ -0.61 \text{ V} = -\frac{0.0592}{2} \text{log} \frac{1.4}{[\text{Pb}^{2+}]} \]Then solve for the log term and exponentiate both sides to find:\[ [\text{Pb}^{2+}] = \frac{1.4}{10^{20.61}} \thickapprox 2.7 \times 10^{-21} \text{ M} \] This shows how significant the influence of concentration is on the cell potential.

The cell potential, \( E_{\text{cell}} \), is the difference in potential energy between the two electrodes of an electrochemical cell. It drives the flow of electrons from the anode to the cathode and is measured in volts (V). The standard cell potential, \( E^{\text{o}}_{\text{cell}} \), is calculated using the standard reduction potentials of the half-reactions. Here it’s computed as: \[ E^{\text{o}}_{\text{cell}} = E^{\text{o}}_{\text{cathode}} - E^{\text{o}}_{\text{anode}} \]For our Mn/Pb cell, this results in:\[ E^{\text{o}}_{\text{cell}} = -0.13 \text{ V} - (-1.18 \text{ V}) = 1.05 \text{ V} \] Under non-standard conditions, the actual cell potential is adjusted using the Nernst equation. Hence, understanding both standard and actual cell potentials are vital for accurately predicting the cell’s behavior in various operating conditions.