Problem 72 - General Chemistry

Question

Find the concentrations of \(\mathrm{Ag}^{+}(a q), \mathrm{NH}_{3}(a q)\), and \(\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}(a q)\) at equilibrium when \(0.10 \mathrm{~mol} \mathrm{Ag}^{+}(a q)\) and \(0.10 \mathrm{~mol} \mathrm{NH}_{3}(a q)\) are made up to \(1.00 \mathrm{~L}\) of solution. The dissociation constant, \(K_{d}\), for the complex \(\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}\) is \(5.9 \times 10^{-8}\).

Step-by-Step Solution

Answer

Concentrations are determined using the equilibrium constant and solved for equilibrium concentrations based on initial values.

Step 1: Write the Equilibrium Expression

First, we need to consider the equilibrium of the system: \[ \mathrm{Ag}^+ + 2\mathrm{NH}_3 \leftrightarrow \left[\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right]^+ \]The dissociation constant \(K_d\) is given by:\[ K_d = \frac{[\mathrm{Ag}^+][\mathrm{NH}_3]^2}{\left[\left(\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right)^+\right]} = 5.9 \times 10^{-8} \]

Step 2: Establish Initial Concentrations

Initially, the concentrations in the solution before any reaction occurs are:\[ [\mathrm{Ag}^+] = 0.10 \,\text{M}, [\mathrm{NH}_3] = 0.10 \,\text{M}, \text{and} \, \left[\left(\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right)^+\right] = 0 \,\text{M} \]

Step 3: Define Change in Concentrations

As the reaction reaches equilibrium, let \(x\) be the concentration of \(\left[\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right]^+\) that forms. The changes in concentrations will be:- \([\mathrm{Ag}^+] = 0.10 - x\)- \([\mathrm{NH}_3] = 0.10 - 2x\)- \(\left[\left(\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right)^+\right] = x\)

Step 4: Apply Equilibrium Expression

Substitute the equilibrium concentrations into the equilibrium expression:\[ K_d = \frac{(0.10 - x)(0.10 - 2x)^2}{x} = 5.9 \times 10^{-8} \]

Step 5: Solve for x

To find \(x\), we solve the equation:\[(0.10 - x)(0.10 - 2x)^2 = 5.9 \times 10^{-8} x\]This involves approximating or using numerical methods to find \(x\). Given the small \(K_d\), you may assume \(x\) is small, simplifying to say \((0.10)^3 = 5.9 \times 10^{-8} x\), then solve for \(x\).

Step 6: Calculate Equilibrium Concentrations

Once \(x\) is found, calculate each concentration at equilibrium:- \([\mathrm{Ag}^+] = 0.10 - x\)- \([\mathrm{NH}_3] = 0.10 - 2x\)- \(\left[\left(\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right)^+\right] = x\)

Key Concepts

Dissociation Constant

In chemistry, the dissociation constant is used to describe how well a complex species breaks apart into its components in a reversible reaction at equilibrium. This is often denoted by \( K_d \), and is critical in quantifying the stability of complex ions in solution.
The equilibrium expression for the dissociation of a complex ion typically has the complex on one side, and its individual ions on the other. For the silver-ammonia complex \[\left[\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right]^+ \leftrightarrow \mathrm{Ag}^+ + 2\mathrm{NH}_3\]the dissociation constant is calculated using the formula:\[K_d = \frac{[\mathrm{Ag}^+][\mathrm{NH}_3]^2}{[\mathrm{Ag}\left(\mathrm{NH}_3\right)_2^+]}\]This implies sensitivity: smaller \(K_d\) values mean the complex remains largely intact, suggesting greater stability. In our example, \(K_d\) is quite small (\(5.9 \times 10^{-8}\)), hinting at a very stable silver-ammonia complex that doesn't readily dissociate in solution.

Complex Ions

Complex ions consist of a central metal ion bonded to one or more molecules or ions through coordinate covalent bonds. This bonding results in a charged, multi-atom structure called a complex ion, which is a key concept in transition metal chemistry.
When silver \(\mathrm{Ag}^+\) ions react with ammonia \(\mathrm{NH}_3\) molecules, they form the complex ion: \[\left[\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right]^+\]The ammonia molecules act as ligands, donating their lone pairs to bond with the silver ion. Complex ions are vital in understanding reactions involving transition metals, as they exhibit unique properties such as varied colors and enhanced solubility in water.
The formation and dissociation of complex ions are governed by equilibrium chemistry principles, and understanding their stability or reactivity is essential for predicting behaviors of solutions in which they are present.

Equilibrium Concentrations

Equilibrium concentrations refer to the concentrations of reactants and products that remain constant over time in a reversible reaction, once equilibrium is achieved. For a reaction such as the formation of the silver-ammonia complex:\[\mathrm{Ag}^+ + 2\mathrm{NH}_3 \leftrightarrow \left[\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right]^+\]the system will adjust to reach a state where the formation of the complex balances with its dissociation.
To calculate equilibrium concentrations, you start with known initial concentrations and define how these change as the reaction proceeds to equilibrium. For example:- \([\mathrm{Ag}^+] = 0.10 - x\)- \([\mathrm{NH}_3] = 0.10 - 2x\)- \([\mathrm{Ag}\left(\mathrm{NH}_3\right)_2^+] = x\)Evaluating these expressions alongside the equilibrium expression allows you to solve for \(x\), revealing concentration values of all species at equilibrium. This approach provides insights into the extent of the reaction at any given time.

Silver-Ammonia Complex

The silver-ammonia complex, \(\left[\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right]^+\), is a classic example of a coordination compound, where a silver ion is surrounded by ammonia molecules acting as ligands.
This complex is notable because it exemplifies the unique ability of transition metals to form stable, structured ions with distinct characteristics. - **Ligand behavior:** Ammonia molecules have lone electron pairs allowing them to coordinate covalently with the metal ion. - **Stability:** With a low dissociation constant \(K_d = 5.9 \times 10^{-8}\), the complex is highly stable indicating ammonia surrounds silver very effectively, thus minimizing dissociation. In practice, the presence and behavior of silver-ammonia complex ions influence reactions in analytical chemistry, such as when determining silver ion concentrations in solutions. These complexes can also impact silver compound solubility, a concept useful in various chemical separation processes.