Write the half-reactions and devise a galvanic cell (write a cell diagram) to study each of the following reactions: (a) \(\mathrm{AgBr}(\mathrm{s})=\mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{Br}^{-}(\mathrm{aq})\), a solubility equilibrium (b) \(\mathrm{H}^{+}(\mathrm{aq})+\mathrm{OH}^{-}\)(aq) \(\longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\), the Bronsted neutralization reaction (c) \(\mathrm{Cd}(\mathrm{s})+2 \mathrm{Ni}(\mathrm{OH})_{3}(\mathrm{~s}) \rightarrow \mathrm{Cd}(\mathrm{OH})_{2}(\mathrm{~s})+2 \mathrm{Ni}(\mathrm{OH})_{2}(\mathrm{~s})\), the reaction in the nickel-cadmium cell

Solubility equilibrium is a type of dynamic equilibrium that exists when a chemical compound in the solid state is in chemical equilibrium with a solution of that compound. The solubility product (\(K_{sp}\)) represents the level at which a solute dissolves in solution. If more solute is added, it will not dissolve until some of the added solute returns to the solid state; likewise, if some solute is removed from the solution, more solute will dissolve until equilibrium is reached again.

To understand this better, consider the solubility equilibrium in the exercise, \(\mathrm{AgBr}(\mathrm{s}) = \mathrm{Ag}^{+}(\mathrm{aq}) + \mathrm{Br}^{-}(\mathrm{aq})\), where solid silver bromide dissolves to form silver and bromide ions in solution. Over time, the rate at which the solid dissolves to form ions and the rate at which ions come together to form the solid become equal, resulting in a saturated solution.

This equilibrium is crucial in understanding the behavior of ions in solution, particularly in the prediction of precipitate formation and the concentration of ions at saturation.

The Brønsted neutralization reaction involves the reaction of hydronium ions (\(\mathrm{H}^{+}(\mathrm{aq})\)) with hydroxide ions (\(\mathrm{OH}^{-}(\mathrm{aq})\)) to form water. In a Brønsted-Lowry framework, an acid is a proton donor, while a base is a proton acceptor. This neutralization reaction is a key concept in acid-base chemistry and illustrates the Brønsted-Lowry acid-base theory.

In the provided exercise, the reaction \(\mathrm{H}^{+}(\mathrm{aq}) + \mathrm{OH}^{-}(\mathrm{aq}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\) represents a neutralization reaction. However, creating a conventional galvanic cell to study this reaction isn't practical since both substances neutralize each other without producing an electrical potential difference that could drive electron flow through an external circuit. This highlights the importance of electron transfer in generating an electromotive force (EMF) for galvanic cells.

The nickel-cadmium cell is a type of rechargeable battery, which was widely used in portable electronics and tools before the advent of lithium-ion batteries. Its operation is based on electrochemical oxidation and reduction (redox) reactions, occurring at two different electrodes during discharge or charge cycles. The overall cell reaction for a nickel-cadmium battery during discharge can be expressed as \(\mathrm{Cd}(\mathrm{s}) + 2 \mathrm{Ni}({\mathrm{OH})_{3}(\mathrm{s}) \rightarrow \mathrm{Cd}(\mathrm{OH})_{2}(\mathrm{s}) + 2 \mathrm{Ni}(\mathrm{OH})_{2}(\mathrm{s})\).

This redox reaction is driven by the movement of electrons from the cadmium electrode to the nickel electrode through an external circuit, providing electric power. The simultaneous hydroxide ion movement within the electrolyte maintains charge balance. These cells are known for their durability and ability to provide a stable electrical current over a wide range of temperatures and charging states, making them suitable for a variety of applications.

Electrochemical reactions are the foundation of galvanic cells, where chemical energy is converted into electrical energy through redox reactions. These reactions involve the transfer of electrons from one substance to another. In a galvanic cell, the anode is where oxidation occurs, releasing electrons, while the cathode is where reduction takes place, accepting electrons.

Understanding electrochemical reactions requires knowledge of oxidation states, electrode potentials, and the Nernst equation, which describes the relationship between the EMF of the cell and the concentrations of the reacting species. The exercise provided explores these reactions by breaking down the overall chemical reactions into half-reactions, showcasing the individual processes of oxidation and reduction, leading to a better grasp of how galvanic cells operate.

A cell diagram is a shorthand notation that represents the components of a galvanic cell, which are the anode and cathode, as well as their respective electrolytes. It provides a clear and succinct way to visualize the flow of electrons and the direction of chemical reactions within the cell.

For example, the cell diagram for the nickel-cadmium cell from the exercise can be seen as \(\mathrm{Cd}|\mathrm{Cd}^{2+}(\mathrm{aq})||\mathrm{OH}^{-}(\mathrm{aq})|\mathrm{Ni}({\mathrm{OH})_{3}, \mathrm{Ni}(\mathrm{OH})_{2}(\mathrm{s})\). This illustrates how the cadmium electrode (anode) is separated by a salt bridge or porous membrane from the nickel electrode (cathode), allowing ionic conduction without direct mixing of the different electrolytes. Cell diagrams are essential tools for understanding and designing electrochemical cells, such as batteries and electrolytic cells.