Consider three complexes of \(\mathrm{Ag}^{+}\) and their formation constants, \(K_{\mathrm{f}}\) $$\begin{array}{ll}\hline \text { Complex lon } & K_{\mathrm{f}} \\\\\hline \mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}+ & 1.6 \times 10^{7} \\ \mathrm{Ag}(\mathrm{CN})_{2}^{-} & 5.6 \times 10^{18} \\\\\mathrm{AgBr}_{2}^{-} & 1.3 \times 10^{7} \\ \hline\end{array}$$ Which statements are true? (a) \(\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}{ }^{+}\) is more stable than \(\mathrm{Ag}(\mathrm{CN})_{2}^{-}\). (b) Adding a strong acid \(\left(\mathrm{HNO}_{3}\right)\) to a solution that is \(0.010 \mathrm{M}\) in \(\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\) will tend to dissociate the complex ion into \(\mathrm{Ag}^{+}\) and \(\mathrm{NH}_{4}^{+} .\) (c) Adding a strong acid \(\left(\mathrm{HNO}_{3}\right)\) to a solution that is \(0.010 \mathrm{M}\) in \(\mathrm{AgBr}_{2}^{-}\) will tend to dissociate the complex ion into \(\mathrm{Ag}^{+}\) and \(\mathrm{Br}^{-} .\) (d) To dissolve AgI, one can add either \(\mathrm{NaCN}\) or \(\mathrm{HCN}\) as a source of the cyanide-complexing ligand. Fewer moles of NaCN would be required. (e) Solution \(A\) is \(0.10 M\) in \(B r^{-}\) and contains the complex ion \(\mathrm{AgBr}_{2}^{-}\). Solution B is \(0.10 M\) in \(\mathrm{CN}^{-}\) and contains the complex ion \(\mathrm{Ag}(\mathrm{CN})_{2}-\). Solution B will have more particles of complex ion per particle of \(\mathrm{Ag}^{+}\) than solution \(\mathrm{A}\).

Step by step solution

01

Compare the stability of the complex ions

In this exercise, we are given the formation constants \(K_\mathrm{f}\) for three complexes: 1. \(\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\) has \(K_\mathrm{f} = 1.6 \times 10^{7}\) 2. \(\mathrm{Ag}(\mathrm{CN})_{2}^{-}\) has \(K_\mathrm{f} = 5.6 \times 10^{18}\) 3. \(\mathrm{AgBr}_{2}^{-}\) has \(K_\mathrm{f} = 1.3 \times 10^{7}\) The \(\mathrm{Ag}(\mathrm{CN})_{2}^{-}\) complex ion has the largest \(K_\mathrm{f}\) value, which means it is the most stable out of the three complex ions. So, statement (a) is false.

02

Effect of strong acid on the Ag(NH3)2+ complex ion

Statement (b) asks us to consider the addition of a strong acid (\(\mathrm{HNO}_3\)) to a solution containing the \(\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\) complex ion. A strong acid will provide a high concentration of H+ ions. These H+ ions will react with the \(\mathrm{NH}_3\) ligands around the \(\mathrm{Ag}^{+}\) ion, forming \(\mathrm{NH}_{4}^+\) ions. This will cause the complex ion to dissociate due to the loss of ligands. So, statement (b) is true.

03

Effect of strong acid on the AgBr2- complex ion

Statement (c) asks us to consider the addition of a strong acid (\(\mathrm{HNO}_3\)) to a solution containing the \(\mathrm{AgBr}_{2}^{-}\) complex ion. A strong acid will not react with the \(\mathrm{Br}^{-}\) ions to form a new species; therefore, the complex ion will not dissociate due to the addition of \(\mathrm{HNO}_3\). So, statement (c) is false.

04

Dissolving AgI using NaCN or HCN

Statement (d) considers the use of NaCN or HCN to dissolve AgI by forming a complex ion. \(\mathrm{AgI}\) will react with \(\mathrm{CN}^-\) ions to form the complex ion \(\mathrm{Ag}(\mathrm{CN})_{2}^{-}\). The NaCN compound dissociates more completely in water to provide a higher concentration of \(\mathrm{CN}^{-}\) ions than HCN. Therefore, fewer moles of NaCN are required to dissolve AgI compared to HCN. So, statement (d) is true.

05

Comparing the complex ions in solution A and solution B

Statement (e) is comparing the number of complex ions per particle of \(\mathrm{Ag}^+\) ion in two different solutions. Solution A has \(\mathrm{AgBr}_{2}^{-}\) and 0.10 M \(\mathrm{Br}^-\) ions, while solution B has \(\mathrm{Ag}(\mathrm{CN})_{2}^-\) and is 0.10 M \(\mathrm{CN}^-\). The complex ion \(\mathrm{Ag}(\mathrm{CN})_{2}^-\) has a higher \(K_\mathrm{f}\) value than \(\mathrm{AgBr}_{2}^-\) (as determined earlier), which means it is more stable and less likely to dissociate. So, solution B will have more particles of complex ion per particle of \(\mathrm{Ag}^+\) than solution A. Statement (e) is true. Therefore, the true statements are (b), (d), and (e).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Formation Constants

Understanding the stability of complex ions is crucial when studying coordination chemistry, and it's here where formation constants (also called stability constants) come into play. These constants, commonly represented as 'Kf', provide a quantitative measure of the complex ion's stability in solution. Essentially, the higher the Kf value, the more stable the complex ion.

The formation constant is derived from the equilibrium constant for the reaction where a metal ion binds with ligands to form a complex ion. Mathematically, if we have a metal ion M and ligands L forming a complex ML_n, the formation constant Kf is given by the expression:
$$K_{f} = \frac{[ML_{n}]}{[M][L]^{n}}$$
where [ML_n], [M], and [L] represent the molar concentrations of the complex ion, the metal ion, and the ligands, respectively.

Within the context of the provided exercise, a higher formation constant indicates that a particular complex ion, like \(\mathrm{Ag}(\mathrm{CN})_{2}^{-}\), is significantly more stable than others with lower formation constants, such as \(\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\) and \(\mathrm{AgBr}_{2}^{-}\).

Chemical Equilibrium

Chemical equilibrium is the state in which the concentrations of all reactants and products remain constant over time because the rate of the forward reaction equals the rate of the reverse reaction. Even though the concentrations are stable, the molecular interactions don't cease; instead, they continue at equal and opposite rates.

In the context of complex ion formation, chemical equilibrium involves the dynamic process where the metal ions and ligands assemble and disband at equivalent rates, resulting in a stable formation of complex ions. The equilibrium condition is expressed through the equilibrium constant, which for complex ion formation is the aforementioned formation constant Kf.

It's important for students to internalize that although a system in chemical equilibrium might appear static from a macroscopic perspective, there is a constant microscopic activity where reactants form products, which then revert back to reactants at the same rate. This concept helps to understand why adding or removing substances, such as introducing a strong acid as described in the exercise, affects the equilibrium and shifts the concentration of complex ions.

Ligand Exchange Reactions

Ligand exchange reactions are the processes in which one ligand in a complex ion is replaced by another ligand. These reactions are governed by the same principles as chemical equilibrium, and they play a pivotal role in determining the reactivity and stability of complex ions. Based on the exercise, a strong acid like \(\mathrm{HNO}_3\) can initiate a ligand exchange reaction, provoking the dissociation of certain ligands from the metal center.

In the step by step solution, we've seen that the acid reacts with the \(\mathrm{NH}_{3}\) ligands to form \(\mathrm{NH}_{4}^+\), due to the ligand exchange where \(\mathrm{H}^+\) replaces \(\mathrm{NH}_{3}\) surrounding the \(\mathrm{Ag}^+\) ion, and in doing so, disrupts the stability of the complex ion. This reflection demonstrates the practicality of ligand exchange reactions, which explains why the presence of \(\mathrm{H}^+\) ions would not lead to the dissociation of \(\mathrm{AgBr}_{2}^{-}\), as the \(\mathrm{Br}^-\) ligands don't react with \(\mathrm{H}^+\).

Students should also remember that the feasibility of ligand exchange reactions depends on several factors, including the charge and electron configuration of the metal ion, the donor atoms of the ligands, and the overall thermodynamic stability of the resulting complex.