From the following information, calculate the solubility product of AgBr: $$ \begin{array}{ll} \mathrm{Ag}^{+}(a q)+e^{-} \longrightarrow \mathrm{Ag}(s) & E^{\circ}=0.80 \mathrm{~V} \\ \operatorname{AgBr}(s)+e^{-} \longrightarrow \mathrm{Ag}(s)+\mathrm{Br}^{-}(a q) & E^{\circ}=0.07 \mathrm{~V} \end{array} $$

Understanding electrochemical cell reactions is crucial when studying the behavior of substances like silver bromide (AgBr) in solution. These reactions involve the exchange of electrons between chemical species, particularly through reduction and oxidation processes—redox reactions for short.

In our example, two half-cell reactions are provided, and they depict how silver ions (Ag⁺) and silver bromide (AgBr) can each accept an electron (e^-) to form solid silver (Ag). By analyzing these reactions, we can determine what happens when AgBr dissolves in water. With one half-reaction involving the reduction of silver ions to solid silver, and the other representing the reduction of silver bromide to produce solid silver and bromide ions, we can intuitively understand a core principle of electrochemistry: the spontaneity of a reaction can be assessed by evaluating the associated standard reduction potentials.

The Nernst equation is a bridge between electrochemistry and thermodynamics. It relates the electrochemical potential of a cell to the concentration of reactants using the formula: \[ E = E^{\circ} - \frac{0.059}{n} \log Q \] where \(E\) is the cell potential, \(E^{\circ}\) is the standard cell potential, \(n\) is the number of moles of electrons transferred, and \(Q\) is the reaction quotient, which reflects the ratio of the concentrations of reactants and products.

For a system at equilibrium, the cell potential \(E\) is zero because no net reaction occurs. The Nernst equation then simplifies to relate the standard potential and the equilibrium constant (in our case, the solubility product, \(K_{sp}\)). This relationship allows chemists to calculate the extent to which a salt like AgBr will dissolve in solution, providing a means to predict solubility behavior from electrochemical data.

The concept of chemical equilibrium is central to understanding solubility products. Equilibrium occurs when the forward and reverse rates of a reaction are equal, resulting in no net change in the concentration of reactants and products over time. In the solubility context, the equilibrium is between the solid ionic compound and its dissolved ions.

In the presented problem, the equilibrium is established between the solid AgBr and its ions Ag⁺ and Br^- in aqueous solution. At this point, the rate at which AgBr dissolves equals the rate at which Ag⁺ and Br^- ions combine to form the solid. The expression for the equilibrium constant for this dissolving reaction is equivalent to the solubility product (\(K_{sp}\)). Determining \(K_{sp}\) gives us a quantitative measure of the solubility of the compound, which is especially useful when predicting the extent of a substance's solubility in various conditions.

Standard reduction potentials (\(E^{\circ}\)) are values that indicate the tendency of a chemical species to acquire electrons and thereby be reduced. Measured in volts, these values are determined under standard conditions, which entail a temperature of 298 K, a 1 M concentration for each ion, and a pressure of 1 atmosphere for any gases involved.

In our exercise, two half-reactions with different standard reduction potentials are given. By comparing these potentials, we can predict the direction in which the overall reaction will proceed. A more positive \(E^{\circ}\) value indicates a greater tendency to become reduced. The subtraction of these values as seen in the step by step solution is a method to determine the overall cell potential under standard conditions. Understanding these potentials is vital as they signal the intrinsic capacity of the species to participate in redox reactions, which directly impacts its solubility behavior in electrochemical cells.