(a) Compute the voltage at \(25^{\circ} \mathrm{C}\) of an electrochemical cell consisting of pure lead immersed in a \(5 \times 10^{-2} M\) solution of \(\mathrm{Pb}^{2+}\) ions, and pure tin in a \(0.25 M\) solution of \(\mathrm{Sn}^{2+}\) ions. (b) Write the spontaneous electrochemical reaction.

The Nernst equation is a fundamental tool in electrochemistry, allowing scientists and students alike to calculate the voltage of an electrochemical cell under non-standard conditions. It provides a precise relationship between the cell voltage, the standard cell potential, and the reaction quotient, which takes into account ion concentration and temperature.

Mathematically, it is represented as: \[E_{cell} = E_{cell}^{\text{\textdegree}} - \frac{RT}{nF} \ln(Q)\]Where:\

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• \(E_{cell}\) is the cell potential at non-standard conditions.
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• \(E_{cell}^{\text{\textdegree}}\) is the standard cell potential.
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• \(R\) is the universal gas constant (8.314 J/K·mol).
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• \(T\) is the absolute temperature in Kelvin (K).
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• \(n\) is the number of moles of electrons exchanged.
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• \(F\) is the Faraday constant (96485 C/mol).
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• \(Q\) is the reaction quotient, which is the ratio of the concentrations of the reactants and products.
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Applying the Nernst equation involves careful consideration of ion concentrations and temperature to predict how cell potential will change along with these variables. It is especially useful for understanding how batteries and fuel cells operate under various conditions and is widely used in laboratories to determine the feasibility of redox reactions.

Understanding standard reduction potentials (\(E_{cell}^{\text{\textdegree}}\) is key to grasping the innate tendency of a species to gain electrons, thereby undergoing reduction. These potentials are measured under standard conditions, which is typically 25 degrees Celsius, 1 atm pressure, and 1 M concentration for each ion.

These values serve as a reference to compare different half-cell potential and they are crucial for predicting which direction an electrochemical reaction will proceed. The more positive the reduction potential, the greater is the substance's affinity for electrons and inclination to be reduced.

Connection to Cell Potential

In an electrochemical cell, the half-cell with the higher standard reduction potential acts as the cathode, as in the case with the lead (\(Pb^{2+}\) in the example provided. Conversely, the half-cell with the lower standard reduction potential becomes the anode, like tin (\(Sn^{2+}\). The difference in these potentials gives us the standard cell potential, which is a crucial step in calculating the overall cell voltage.

Spontaneous electrochemical reactions are naturally occurring processes that result in a net release of energy, allowing them to proceed without any external energy source. These reactions are essential for batteries to function as they drive the flow of electrons from the anode to the cathode, creating electric current.

The spontaneity of these reactions depends on the cell potential being positive when the standard reduction potentials are applied and can be affected by the concentrations of the reactants and products, as shown when applying the Nernst equation.

Critical Role in Everyday Life

These spontaneous reactions have applications that extend far beyond the classroom—they are the basis for powering electronic devices, starting vehicles, and storing renewable energy. In the example provided, the reaction between \(Pb^{2+}\) and \(Sn\) is spontaneous, underscored by the resulting positive voltage calculation. By studying these reactions, we gain insights into designing more effective and efficient energy sources for a wide range of technologies.