Consider the cell \(\mathrm{Ag}(\mathrm{s})\left|\mathrm{Ag}^{+}\left(\mathrm{aq}, 5.0 \mathrm{mmol} \cdot \mathrm{L}^{-1}\right)\right| \mid \mathrm{Ag}^{+}(\mathrm{aq}\), \(\left.0.15 \mathrm{~mol} \cdot \mathrm{L}^{-1}\right) \mid \mathrm{Ag}(\mathrm{s})\). Can this cell do work? If so, what is the maximum work that it can perform (per mole of \(\mathrm{Ag}\) )?

The Nernst Equation is a fundamental formula in electrochemistry that allows us to calculate the cell potential at any given conditions, taking into account the concentration of ions involved in the redox reaction. The complete form of the Nernst equation is:
\[\begin{equation}E = E^\circ - \frac{RT}{nF}\ln\left(\frac{\text{Products}}{\text{Reactants}}\right)\end{equation}\]
where:

• E is the cell potential under non-standard conditions.
• E° is the standard cell potential.
• R is the universal gas constant.
• T is the temperature in Kelvin.
• F is Faraday's constant.
• n is the number of moles of electrons transferred.
• The natural logarithm (ln) of the reaction quotient (ratio of product ions to reactant ions).

However, when dealing with solutions at 25°C (298K), we can use a simplified form, as seen in the textbook solution, which replaces RT/nF with 0.0592 volts when n equals 1. By using the concentrations of ions in the two half-cells, we can substitute into the Nernst equation to find the actual potential of the galvanic cell.

Cell potential, often denoted as E, is the ability of a galvanic or voltaic cell to produce an electric current as a result of a redox reaction. It is the difference in potential between the two electrodes of the cell when no current is flowing. The potential arises due to the chemical energy from the reaction being converted to electrical energy. In the given exercise, we use the Nernst equation to determine the cell potential because the concentrations of the Ag⁺ ions in the two half-cells are different, leading to a potential that can drive the flow of electrons and do work. The sign of the cell potential tells us which direction the electrons will naturally flow; a positive potential means the reaction is spontaneous and can do work, while a negative potential indicates a non-spontaneous reaction.

Electrochemistry is a branch of chemistry that studies the movement of electrons and ions and their interactions with matter, particularly during redox reactions. In these reactions, oxidation and reduction processes occur, often resulting in the generation or use of electrical energy. Our exercise showcases a practical application of electrochemistry, where we examine how a galvanic cell can harness the chemical energy from the movement of silver ions to perform electrical work. The principles of electrochemistry are applied in various technologies such as batteries, fuel cells, and electrolysis processes.

Faraday's constant, symbolized as F, is a fundamental physical constant used in electrochemistry that represents the charge of one mole of electrons. Its value is approximately 96,485 coulombs per mole. This constant becomes particularly important when relating the chemical changes to the flow of electrical current. In the calculation of maximum work, W_max, from our cell potential, we use Faraday's constant in the formula along with the number of moles of electrons transferred (n) and the calculated cell potential (E_cell). The significance of Faraday's constant lies in its ability to link the amount of substance (in moles) with the quantity of electricity involved in an electrochemical process.