Problem 73 - Chemistry The Molecular Science

Question

Write the chemical equation for the formation of each complex ion and write its formation constant expression. (a) \(\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}\) (b) \(\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}\)

Step-by-Step Solution

Answer

(a) \(K_f = \frac{\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}}{[\mathrm{Ag}^{+}][\mathrm{CN}^{-}]^2}\); (b) \(K_f = \frac{\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}}{[\mathrm{Cd}^{2+}][\mathrm{NH}_{3}]^4}\)."

Step 1: Write the chemical equation for (a)

The formation of the complex ion \(\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}\) involves silver ion (\(\mathrm{Ag}^{+}\)) and cyanide ions \((\mathrm{CN}^{-})\). The balanced chemical equation is: \[ \mathrm{Ag}^{+} + 2\,\mathrm{CN}^{-} \rightarrow \left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-} \]

Step 2: Write the formation constant expression for (a)

The formation constant \(K_f\) for the complex \(\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}\) is given by:\[ K_f = \frac{\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}}{[\mathrm{Ag}^{+}][\mathrm{CN}^{-}]^2} \]

Step 3: Write the chemical equation for (b)

The formation of the complex ion \(\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}\) involves cadmium ion \((\mathrm{Cd}^{2+})\) and ammonia molecules \((\mathrm{NH}_{3})\). The balanced chemical equation is:\[ \mathrm{Cd}^{2+} + 4\,\mathrm{NH}_{3} \rightarrow \left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+} \]

Step 4: Write the formation constant expression for (b)

The formation constant \(K_f\) for \(\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}\) is:\[ K_f = \frac{\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}}{[\mathrm{Cd}^{2+}][\mathrm{NH}_{3}]^4} \]

Key Concepts

Formation Constant

When discussing complex ions like \( [\mathrm{Ag}(\mathrm{CN})_{2}]^{-} \) and \( [\mathrm{Cd}(\mathrm{NH}_{3})_{4}]^{2+} \), understanding the formation constant is key. The formation constant, denoted as \( K_f \), is a measure of the stability of a complex ion in solution. It shows how easily the ions join to form the complex.
The equation for \( K_f \) illustrates the extent to which reactants transform into complex ions. For example, for \( [\mathrm{Ag}(\mathrm{CN})_{2}]^{-} \), the equation is:

• \( K_f = \frac{\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}}{[\mathrm{Ag}^{+}][\mathrm{CN}^{-}]^2} \)

The higher the \( K_f \), the more stable the complex ion.
For \( [\mathrm{Cd}(\mathrm{NH}_{3})_{4}]^{2+} \), the formation constant is expressed as:

• \( K_f = \frac{\left[\mathrm{Cd}(\mathrm{NH}_{3})_{4}\right]^{2+}}{[\mathrm{Cd}^{2+}][\mathrm{NH}_{3}]^4} \)

This highlights how the concentration of complex ions depends on the concentration of free ions and ligands.

Chemical Equation

The chemical equation for forming complex ions represents the process where metal ions react with ligands, like cyanide \((\mathrm{CN}^- )\) or ammonia \((\mathrm{NH}_3)\), to form stable complexes.
In the equation for the formation of \([\mathrm{Ag}(\mathrm{CN})_{2}]^{-}\), the silver ion \((\mathrm{Ag}^{+})\) combines with two cyanide ions to form the complex:

• \( \mathrm{Ag}^{+} + 2\,\mathrm{CN}^{-} \rightarrow [\mathrm{Ag}(\mathrm{CN})_{2}]^{-} \)

This balanced equation indicates the stoichiometric relations within the reaction.
For \([\mathrm{Cd}(\mathrm{NH}_{3})_{4}]^{2+}\), cadmium ions \((\mathrm{Cd}^{2+})\) react with four ammonia molecules:

• \( \mathrm{Cd}^{2+} + 4\,\mathrm{NH}_{3} \rightarrow [\mathrm{Cd}(\mathrm{NH}_{3})_{4}]^{2+} \)

The chemical equation not only shows what forms but also specifies the exact ratio needed for the reaction to proceed.

Coordination Chemistry

Coordination chemistry explores how metal ions interact with ligands to form complexes. A complex is a structure consisting of a central metal atom bonded to surrounding ligands. In \([\mathrm{Ag}(\mathrm{CN})_{2}]^{-}\) and \([\mathrm{Cd}(\mathrm{NH}_{3})_{4}]^{2+}\), silver and cadmium serve as the central metal atoms.
Ligands like \(\mathrm{CN}^{-}\) and \(\mathrm{NH}_{3}\) are molecules or ions that donate pairs of electrons to the metal atom, creating a coordinate covalent bond.

• \( [\mathrm{Ag}(\mathrm{CN})_{2}]^{-} \): Cyanide ions act as ligands.
• \( [\mathrm{Cd}(\mathrm{NH}_{3})_{4}]^{2+} \): Ammonia serves as the ligand.

Coordination chemistry helps in understanding complex ion formation, bonding, and stability.
These concepts are crucial for applications ranging from industrial processes to the development of novel materials. Understanding them allows us to manipulate their behavior for various practical uses.