The solubility product for \(\operatorname{Cul}(s)\) is \(1.1 \times 10^{-12}\). Calculate the value of \(\mathscr{E}^{\circ}\) for the half-reaction $$\mathrm{CuI}+\mathrm{e}^{-} \longrightarrow \mathrm{Cu}+\mathrm{I}^{-}$$

The solubility product constant, or Ksp, is fundamental in predicting the solubility of a compound in water. It quantifies the extent to which a solid dissolves in an aqueous solution to form its constituent ions. The constant is specific to each substance and temperature-dependent.

For a generic salt AB that dissociates into A+ and B- ions, the Ksp is written as: \[ K_{sp} = [A^{+}][B^{-}] \].
In the case of copper(I) iodide (CuI), the Ksp value is provided as \(1.1 \times 10^{-12}\). This is a very low value, indicating that CuI is only slightly soluble in water. In an equilibrium scenario, the concentrations of the ions Cu+ and I- are equal due to the 1:1 stoichiometry of their dissociation. The equilibrium concentrations are critical for calculating the solubility of the salt, as well as in applying the Nernst equation to determine the standard cell potential.

The Nernst equation is a key tool in electrochemistry, enabling us to calculate the electric potential of a half-cell or full electrochemical cell under non-standard conditions. It accounts for the temperature, ion concentration (or partial pressures of gases), and other relevant factors affecting the cell's potential.

The equation is given by: \[ \mathscr{E} = \mathscr{E}^{\circ} - \frac{RT}{nF} \ln Q \],where \(\mathscr{E}\) is the cell potential, \(\mathscr{E}^{\circ}\) is the standard cell potential, R is the universal gas constant, T is the temperature in Kelvin, n is the number of moles of electrons exchanged, F is the Faraday constant, and Q is the reaction quotient, representing the ratio of products over reactants.
Applying the Nernst equation requires knowledge of the equilibrium concentrations of ions. In the example provided, this equation allowed us to connect the physical chemistry of solubility with the electrical chemistry involved in the half-cell potential.

When a salt is in a dynamic equilibrium in an aqueous solution, the concentrations of the ions produced by its dissociation are constant, and these are the equilibrium concentrations. They are crucial for predicting the outcome of reactions, calculating solubility, and applying the Nernst equation.

In the case of the dissolution of CuI in water, the equilibrium concentrations were found by setting the solubility product equal to the product of the individual ions' concentrations, each designated as x. By solving the expression \(K_{sp} = x^2\), we obtained the equilibrium concentration of each ion in solution. Here, the concentration appears squared because of the stoichiometry of the reaction where one mole of CuI produces one mole of Cu+ and one mole of I- ions.

Calculating equilibrium concentrations is often the first step in establishing reaction quotients or determining the direction in which a reaction will proceed, further underlying its significance in the analytical aspects of chemistry.

Half-reactions split an overall redox reaction into its two complementary parts: oxidation and reduction. Each half-reaction involves the transfer of electrons, and understanding the electrochemistry of each half-reaction is essential for calculating cell potentials and balancing redox reactions.

The given half-reaction \(\text{CuI} + \text{e}^{-} \longrightarrow \text{Cu} + \text{I}^{-}\) shows the reduction of copper(I) iodide to copper metal and iodide ion. In this example, calculating the standard cell potential begins with understanding the dissociation of CuI and utilizing the Nernst equation. The steps include balancing the half-reaction, calculating equilibrium concentrations, and finally finding the standard cell potential which indicates the tendency of the reaction to proceed towards the reduction of CuI.

This connection of half-reaction calculations with the solubility product and Nernst equation amplifies their role in a wide array of chemical analyses, from electroplating to energy storage in batteries.