An aqueous solution that \(2.00 \mathrm{M}\) in \(\mathrm{AgNO}_{3}\) is slowly added from a buret to an aqueous solution that is \(0.0100 \mathrm{M}\) in \(\mathrm{Cl}^{-}\) and \(0.250 \mathrm{M}\) in \(\mathrm{I}^{-}\). (a) Which ion, \(\mathrm{Cl}^{-}\) or \(\mathrm{I}^{-}\), is the first to precipitate? (b) When the second ion begins to precipitate, what is the remaining concentration of the first ion? (c) Is the separation of \(\mathrm{Cl}^{-}\) and \(\mathrm{I}^{-}\) feasible by fractional precipitation in this solution?

Understanding the solubility of compounds is a key part of predicting the outcome of reactions in solution. In chemistry, the solubility product constant (Ksp) is a vital concept that helps us do just that. It quantifies the solubility of a sparingly soluble compound under a given set of conditions. The Ksp is the product of the concentrations of the ionic constituents of a compound, each raised to the power of its coefficient in the balanced equation.

For a generic salt AB that dissociates into A+ and B-, the Ksp expression would be written as: \[ Ksp = [A^+][B^-] \]
Where \[A+\] and \[B-\] are the concentrations of the ions in moles per liter. The value of Ksp is set for a particular compound at a specific temperature, and it becomes a crucial factor in predicting whether a precipitate will form in a solution.

When multiple ions are present in a solution, it's useful to know the order in which they will precipitate. This follows the rule that the ion with the lowest solubility (thus the smallest Ksp) will precipitate first. In a solution containing both chloride (Cl-) and iodide (I-) ions, for instance, we can compare their Ksp values when reacted with a common cation like silver (Ag+).

Determining Precipitation Sequence

For two silver salts, say AgCl and AgI, the respective Ksp values are key indicators. A smaller Ksp implies a salt is less soluble. Thus, AgI with its smaller Ksp compared to AgCl will precipitate out of solution first upon the addition of Ag+ ions. This is because it reaches its solubility limit more quickly than the salt with the higher Ksp.

Calculating ion concentrations is critical when we wish to determine at what point a precipitate will form or to assess the feasibility of a separation process like fractional precipitation.

To find the concentration of ions left in solution after a reaction, we can use stoichiometry and the initial concentrations of the reactants. For the exercise given, when Ag+ is added to a mixture of Cl- and I-, we calculate the remaining Cl- concentration after all I- has precipitated by acknowledging that the AgI reaction has gone to completion. The concentration of Cl- remains unchanged since no AgCl has formed yet. This step is vital in understanding how much more Ag+ needs to be added to the solution to start precipitating the next ion.

The relationship between solubility and Ksp is reciprocal; as one increases, the other decreases. Knowing this helps in fractional precipitation, a technique used to separate ions based on their differing solubilities. The solubility of an ion in a solution can be calculated from its Ksp value if we have the concentration of the other ion present in the solution.

For example, if we want to separate Cl- from I- by precipitation, we add Ag+ slowly and utilize the difference in Ksp values. The lower the Ksp of the compound, the less soluble it is, resulting in its precipitation at a lower concentration of Ag+. With the known initial concentration of Cl- and its Ksp with Ag+, we can precisely calculate the threshold concentration of Ag+ needed to start precipitating Cl-, thus confirming whether fractional precipitation would successfully separate the two ions.