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Mobile App Chapter 19: Problem 102
Find the solubility of \(\mathrm{AgI}\) in \(2.5 \mathrm{M} \mathrm{NH}_{3}\left[K_{\mathrm{sp}}\right.\) of \(\mathrm{AgI}=\) \(8.3 \times 10^{-17} ; K_{\mathrm{f}}\) of \(\left.\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}=1.7 \times 10^{7}\right]\)
Short Answer
Expert verified
The solubility of \(AgI\text{ in }\text{2.5 M NH}_3\text{ is }\ 9.39 \times 10^{-5} \text{M}.
Step by step solution
01
Write the Dissolution Reaction
First, write the dissolution reaction of \(\text{AgI}\): \[ \text{AgI}_{(s)} \rightarrow \text{Ag}^{+}_{(aq)} + \text{I}^{-}_{(aq)} \]
02
Write the Formation Reaction
Next, write the formation reaction for the complex ion \(\text{Ag(NH}_3)_2^+\): \[ \text{Ag}^{+}_{(aq)} + 2\text{NH}_3 \rightarrow \text{Ag(NH}_3)_2^+ \]
03
Calculate the Overall Equilibrium Constant
The overall equilibrium constant for the combined reaction is the product of the solubility product constant \(K_{sp}\) and the formation constant \(K_f\): \[ K_{\text{overall}} = K_{sp} \times K_f \] Substituting given values: \[ K_{\text{overall}} = (8.3 \times 10^{-17}) \times (1.7 \times 10^{7}) = 1.411 \times 10^{-9} \]
04
Write the Overall Reaction
Combine the dissolution and complex formation reactions: \[ \text{AgI}_{(s)} + 2\text{NH}_3 \rightarrow \text{Ag(NH}_3)_2^+ + \text{I}^{-}_{(aq)} \]
05
Define Solubility
Let \(S\) be the solubility of \(\text{AgI}\) in \(\text{NH}_3\) solution. The concentration of \(\text{I}^{-}\) ions will be the same as \(S\) and the concentration of \(\text{Ag(NH}_3)_2^+\) will be \(S\).
06
Apply the Overall Equilibrium Constant
Using the overall equilibrium constant \(K_{\text{overall}}\), we get: \[ K_{\text{overall}} = [\text{Ag(NH}_3)_2^+][\text{I}^{-}]/[\text{NH}_3]^2 \] Substitute the known values: \[ 1.411 \times 10^{-9} = \frac{S \times S}{(2.5)^2} \]
07
Solve for Solubility
Solve for \(S\): \[ 1.411 \times 10^{-9} = \frac{S^2}{6.25} \] \[ S^2 = 1.411 \times 10^{-9} \times 6.25 \] \[ S^2 = 8.81875 \times 10^{-9} \] \[ S = \text{√}(8.81875 \times 10^{-9}) \] \[ S = 9.39 \times 10^{-5} \text{M}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Dissolution Reaction
Understanding the dissolution reaction is crucial for solving solubility problems. The dissolution reaction refers to the process where a solid compound dissolves in a solvent to form ions. For AgI, the dissolution reaction can be written as:
\[ \text{AgI}_{(s)} \rightarrow \text{Ag}^{+}_{(aq)} + \text{I}^{-}_{(aq)} \] Here, AgI dissociates into its constituent ions, Ag+ and I-. This process is essential as it provides the ions necessary for other reactions like complex formation. By knowing the dissociation reaction, we can set up the framework for calculating how much of the solid can dissolve in a given medium.
To further grasp this concept:
\[ \text{AgI}_{(s)} \rightarrow \text{Ag}^{+}_{(aq)} + \text{I}^{-}_{(aq)} \] Here, AgI dissociates into its constituent ions, Ag+ and I-. This process is essential as it provides the ions necessary for other reactions like complex formation. By knowing the dissociation reaction, we can set up the framework for calculating how much of the solid can dissolve in a given medium.
To further grasp this concept:
- Recognize that dissociation occurs in aqueous solutions.
- Identify the ions produced.
- Determine the balanced equation of the dissolution.
Formation Reaction
The formation reaction is the process where ions in solution form a complex ion. For Ag+, in the presence of NH3, the formation reaction can be written as:
\[ \text{Ag}^{+}_{(aq)} + 2\text{NH}_3 \rightarrow \text{Ag(NH}_3)_2^+ \] This reaction shows how Ag+ combines with two NH3 molecules to form the complex ion Ag(NH3)2+. The formation reaction is key as it influences the overall solubility of compounds in solution.
Points to remember:
\[ \text{Ag}^{+}_{(aq)} + 2\text{NH}_3 \rightarrow \text{Ag(NH}_3)_2^+ \] This reaction shows how Ag+ combines with two NH3 molecules to form the complex ion Ag(NH3)2+. The formation reaction is key as it influences the overall solubility of compounds in solution.
Points to remember:
- Complex ions are typically more stable in solution.
- The number of ligands (NH3) around the central ion (Ag+) depends on the specific reaction.
- Formation reactions move in equilibrium, meaning they can go forward or reverse.
Overall Equilibrium Constant
The overall equilibrium constant (K_overall) for a reaction combines different individual equilibrium constants. For our problem, we multiply the solubility product constant (K_sp) and the formation constant (K_f): \[ K_{\text{overall}} = K_{sp} \times K_f \] Using the provided values:
\[ K_{\text{overall}} = (8.3 \times 10^{-17}) \times (1.7 \times 10^{7}) = 1.411 \times 10^{-9} \]
This new constant reflects the combined stability and solubility properties of both the dissolution and the formation reactions. It helps in determining the system's behavior in a specific solution context.
Why it matters:
\[ K_{\text{overall}} = (8.3 \times 10^{-17}) \times (1.7 \times 10^{7}) = 1.411 \times 10^{-9} \]
This new constant reflects the combined stability and solubility properties of both the dissolution and the formation reactions. It helps in determining the system's behavior in a specific solution context.
Why it matters:
- Shows how equilibrium constants can be combined.
- Provides a way to simplify complex multi-step reactions.
- Helps in calculating overall solubility in complex systems.
Solubility Product Constant
The solubility product constant (K_sp) is a special type of equilibrium constant applicable to the dissolution of sparingly soluble salts. For AgI, the K_sp value is given as: \[ K_{sp} = 8.3 \times 10^{-17} \] This value signifies how much of the solid can dissolve in water at equilibrium. The lower the K_sp, the less soluble the compound.
Crucial pointers:
Crucial pointers:
- K_sp values are specific to temperature.
- Used to predict if a precipitate will form in a solution.
- Helps in calculating the maximum possible ion concentration in a solution.
Formation Constant
The formation constant (K_f) measures the stability of a complex ion in solution. For the complex ion Ag(NH3)2+, K_f is: \[ K_f = 1.7 \times 10^7 \] This large value indicates a highly stable complex, which shifts the equilibrium towards the complex ion formation. It's vital in solving solubility problems where complex ions are involved.
What you need to know:
What you need to know:
- K_f values denote the strength of complex ion formation.
- Larger K_f means more stable and prevalent complex ions in solution.
- Essential for predicting reactions involving complex ions and their solubility.
