A \(50.0-\mathrm{mL}\) volume of \(0.50 M \mathrm{Fe}\left(\mathrm{NO}_{3}\right)_{3}\) is mixed with \(125 \mathrm{~mL}\) of \(0.25 M \mathrm{Cd}\left(\mathrm{NO}_{3}\right)_{2}\) (a) If aqueous \(\mathrm{NaOH}\) is added, which ion precipitates first? (See Appendix C.) (b) Describe how the metal ions can be separated using \(\mathrm{NaOH}\). (c) Calculate the \(\left[\mathrm{OH}^{-}\right]\) that will accomplish the separation.

The solubility product constant (\text{Ksp}) is a key concept in understanding precipitation reactions. It helps us determine the solubility of a compound in water. Each compound has a unique \text{Ksp} value, which is essentially the product of the concentrations of the ions in a saturated solution of that compound, each raised to the power of their stoichiometric coefficients.
For example, for a general salt \text{AB} that dissociates into \text{A}\(^n$$^+\) and \text{B}\(^n$$^-\) in solution, the \text{Ksp} is given by:
\
\[ \mathrm{K_{sp}} = [\mathrm{A}^{n+}][\mathrm{B}^{n-}] \]
Understanding the \text{Ksp} value for different compounds allows us to predict whether a precipitate will form when two solutions are mixed. It's often used to determine the point at which ions in solution will form a solid precipitate.
For instance, in our exercise, \text{Ksp} of \text{Fe(OH)}\(_3\) is \(6.3 \times 10^{-38}\), and for \text{Cd(OH)}\(_2\) it's \(2.5 \times 10^{-14}\). These values tell us how soluble or insoluble these hydroxides are in water. A smaller \text{Ksp} value indicates a less soluble compound, meaning it will precipitate at lower ion concentrations.

The concentration of hydroxide ions (\text{OH}^-) in a solution plays a crucial role in determining whether a metal ion will precipitate out of solution. The \text{Ksp} expression involves \text{OH}^-, so its concentration directly influences whether the solubility limit has been reached or exceeded.
Using the \text{Ksp} expression, we can solve for \text{OH}^- concentration required for precipitation. For \text{Fe(OH)}\(_3\), the calculation is:
\
\[ (6.3 \times 10^{-38}) = (0.143 \mathrm{M}) (\mathrm{[OH]^{-}})^3 \] \[ \mathrm{[OH]^{-}} = \left(\frac{6.3 \times 10^{-38}}{0.143}\right)^{1/3} \approx 2.52 \times 10^{-13} \mathrm{M} \]
Similarly, for \text{Cd(OH)}\(_2\), the calculation is:
\
\[ (2.5 \times 10^{-14}) = (0.179 \mathrm{M}) (\mathrm{[OH]^{-}})^2 \] \[ \mathrm{[OH]^{-}} = \left(\frac{2.5 \times 10^{-14}}{0.179}\right)^{1/2} \approx 9.91 \times 10^{-7} \mathrm{M} \]
From these calculations, we can see that \text{Fe(OH)}\(_3\) will precipitate at a much lower \text{OH}^- concentration compared to \text{Cd(OH)}\(_2\). This is helpful for separation processes, as you'll want to carefully control the \text{OH}^- levels to selectively precipitate one ion over the other.

The separation of metal ions in solution is a vital technique in analytical and inorganic chemistry. It involves precipitating one ion while keeping another in solution, using carefully controlled conditions.
In our example, we aim to separate \text{Fe}\(^{3+}\) and \text{Cd}\(^{2+}\) ions using \text{NaOH} (sodium hydroxide). By gradually adding \text{NaOH}, we increase the \text{OH}^- concentration without exceeding the point where both ions would precipitate.
The key is to find the \text{OH}^- concentration that allows \text{Fe(OH)}\(_3\) to precipitate while keeping \text{Cd(OH)}\(_2\) soluble. This is done by first calculating the specific \text{OH}^- concentration for the onset of precipitation of each hydroxide:

• For \text{Fe(OH)}\(_3\), \text{[OH]^-} = \(2.52 \times 10^{-13}\) \text{M}
• For \text{Cd(OH)}\(_2\), \text{[OH]^-} = \(9.91 \times 10^{-7}\) \text{M}

Start adding \text{NaOH} until the \text{OH}^- concentration reaches \(2.52 \times 10^{-13}\) \text{M}. At this point, \text{Fe(OH)}\(_3\) will start to precipitate.
Continue to add \text{NaOH} while filtering out the precipitate. Stop before \text{[OH]^-} reaches \(9.91 \times 10^{-7}\) \text{M} to ensure all \text{Fe(OH)}\(_3\) is removed but \text{Cd(OH)}\(_2\) remains in solution.
This method is efficient for separating metal ions in a mixture, leveraging differences in solubility products to selectively precipitate compounds.