A silver wire coated with \(\mathrm{AgCl}\) is sensitive to the presence of chloride ion because of the half-cell reaction \(\mathrm{AgCl}(s)+e^{-} \rightleftharpoons \mathrm{Ag}(s)+\mathrm{Cl}^{-} \quad E_{\mathrm{AgC}}^{\circ}=0.2223 \mathrm{~V}\) A student, wishing to measure the chloride ion concentration in a number of water samples, constructed a galvanic cell using the \(\mathrm{AgCl}\) electrode as one half-cell and a copper wire dipping into \(1.00 \mathrm{M} \mathrm{CuSO}_{4}\) solution as the other half-cell. In one analysis, the potential of the cell was measured to be \(0.0895 \mathrm{~V}\) with the copper half-cell serving as the cathode. What was the chloride ion concentration in the water? (Take \(E_{\mathrm{Cu}^{2+}}^{\circ}=0.3419 \mathrm{~V}\).)

The Nernst equation is a fundamental relation in electrochemistry that bridges the gap between the electric potential of a galvanic cell under standard conditions and the potential measured under non-standard conditions. In essence, it allows us to calculate the cell potential when reactant and product concentrations differ from their standard state values.

The general form of the Nernst equation is: \[E_{\text{cell}} = E_{\text{cell}}^{\circ} - \frac{0.0592}{n}\log Q\]where \(E_{\text{cell}}^{\circ}\) denotes the standard cell potential, \(n\) the number of moles of electrons exchanged in the electrochemical reaction, and \(Q\) the reaction quotient -- a ratio akin to the equilibrium constant but for conditions that may not be at equilibrium.

The equation essentially relates the measured cell potential to the ion concentrations (activities) involved in the reaction, making it an indispensable tool for chemists and electrochemists, particularly when determining unknown concentrations as in the exercise with the chloride ion.

A galvanic cell, also known as a voltaic cell, is a device that generates electrical energy through spontaneous redox reactions. It consists of two half-cells, each containing an electrode immersed in an electrolyte, where oxidation occurs at the anode and reduction at the cathode. A salt bridge or porous membrane often separates the two half-cells to allow ions to flow and maintain electrical neutrality.

In the context of our example, the student used a galvanic cell comprising a silver chloride electrode and a copper wire in a copper sulfate solution, a clever setup to harness the underlying redox chemistry for analytical purposes. Cells like this are at the heart of batteries and are foundational in the study of electrochemistry, providing a practical way to convert chemical energy into electrical energy.

Standard reduction potential (\(E^{\circ}\)) is the voltage associated with a reduction reaction at an electrode when all solutes are at 1 M concentration and all gases are at 1 ATM pressure, at a temperature of 25°C (298 K). This intrinsic property of a substance measures its tendency to gain electrons and be reduced.

Values for standard reduction potential are determined under these standardized conditions and are crucial for predicting the direction of redox reactions and calculating the standard cell potential. The more positive the standard reduction potential, the greater the substance's affinity for electrons. In our exercise, the standard reduction potential for the \(\mathrm{AgCl}\) electrode and the copper electrode were given, providing the necessary information to calculate the standard cell potential.

Cell potential, commonly denoted as \(E_{\text{cell}}\), represents the driving force behind the movement of electrons in an electrochemical cell. This potential difference between the anode and cathode is what propels electrons through the circuit, enabling electrical work to be done.

The standard cell potential \(E_{\text{cell}}^{\circ}\) can be calculated by taking the difference between the cathode's and anode's standard reduction potentials. When not under standard conditions, \(E_{\text{cell}}\) varies based on the actual concentrations of the electrochemical species involved, which is where the Nernst equation comes into play. The calculated \(E_{\text{cell}}\) for the student's setup using the Nernst equation showed us how cell potential can be used to deduce the concentration of chloride ions in the solution.