Problem 77
Question
A solution contains 0.018 molel each of \(\mathrm{I}^{-}, \mathrm{Br}^{-},\) and \(\mathrm{Cl}^{-}\). When the solution is mixed with \(200 . \mathrm{mL}\) of \(0.24\) \(M\) \(\mathrm{AgNO}_{3}\), what mass of \(\mathrm{AgCl}(s)\) precipitates out, and what is \(\left[\mathrm{Ag}^{+}\right] ?\) Assume no volume change. $$\begin{aligned} \operatorname{AgI}: K_{\mathrm{sp}} &=1.5 \times 10^{-16} \\ \operatorname{AgBr}: K_{\mathrm{sp}} &=5.0 \times 10^{-13} \\ \mathrm{AgCl}: K_{\mathrm{sp}} &=1.6 \times 10^{-10} \end{aligned}$$
Step-by-Step Solution
Verified Answer
The mass of AgCl(s) that precipitates out is 2.5812 g, and the final concentration of Ag⁺ is 0 M.
1Step 1: Determine moles of AgNO₃
200 mL of 0.24 M AgNO₃ solution contains 0.24 mol/L × 0.2 L = 0.048 moles of AgNO₃.
2Step 2: Determine which compound precipitates first
We will determine the solubility quotient Q for all three compounds and compare it with their Ksp.
Q = ([Ag⁺][X⁻]) / Ksp, where X is either I, Br, or Cl.
Using initial moles of 0.018 mol of X⁻ in 0.2 L
Initial [X⁻] = 0.018 mol / 0.2 L = 0.09 M
Initial [Ag⁺] = 0.24 M
AgI: Q = (0.24)(0.09) / 1.5 × 10⁻¹⁶ = 1.44 × 10¹⁵
AgBr: Q = (0.24)(0.09) / 5.0 × 10⁻¹³ = 4.32 × 10¹¹
AgCl: Q = (0.24)(0.09) / 1.6 × 10⁻¹⁰ = 1.44 × 10⁹
Since AgI has the highest Q (compared to its Ksp), it will precipitate first.
3Step 3: Determine remaining moles of Ag⁺ and X⁻ ions
Once AgI precipitates, we are left with 0.048 - 0.018 = 0.030 moles of Ag⁺.
Since AgBr has the next highest Q, it will precipitate next. Once AgBr precipitates, we are left with 0.030 - 0.018 = 0.012 moles of Ag⁺.
4Step 4: Determine moles and mass of AgCl precipitated
Now that there are 0.012 moles of Ag⁺ left, all remaining Cl⁻ ions (0.018 moles) will react with Ag⁺ to form AgCl.
Moles of AgCl precipitated = 0.018 moles
Mass of AgCl precipitated = (moles of AgCl) × (molar mass of AgCl)
= 0.018 moles × (107.9 + 35.5) g/mol
= 0.018 × 143.4 g/mol
= 2.5812 g
5Step 5: Calculate the final concentration of Ag⁺
Moles of Ag⁺ remaining after AgCl precipitates = 0.012 - 0.018 = -0.006 moles
Since this value is negative, it means all Ag⁺ ions have reacted with the Cl⁻ ions.
Final [Ag⁺] = 0 M (since no more Ag⁺ is left in the solution)
The mass of AgCl(s) that precipitates out is 2.5812 g, and the final concentration of Ag⁺ is 0 M.
Key Concepts
Solubility Product Constant (Ksp)Molar ConcentrationSolubility Quotient (Q)
Solubility Product Constant (Ksp)
Understanding the solubility product constant, or Ksp, is crucial for predicting whether a precipitate will form in a solution. The Ksp is an equilibrium constant that applies to the dissolution of sparingly soluble salts.
Let's simplify this concept: Imagine a dance floor where certain dance partners (ions) prefer not to dance together (solid state) and only mix with the crowd (solution) to a limited extent. The Ksp tells us how much of this salt will 'dance' in solution before the floor becomes too crowded, inducing some pairs to leave the floor and form a solid precipitate.
The Ksp value is unique for each salt at a given temperature and is determined by the concentrations of the ions in their saturated solution. For instance, in the exercise:
Let's simplify this concept: Imagine a dance floor where certain dance partners (ions) prefer not to dance together (solid state) and only mix with the crowd (solution) to a limited extent. The Ksp tells us how much of this salt will 'dance' in solution before the floor becomes too crowded, inducing some pairs to leave the floor and form a solid precipitate.
The Ksp value is unique for each salt at a given temperature and is determined by the concentrations of the ions in their saturated solution. For instance, in the exercise:
- AgCl: Ksp = \(1.6 \times 10^{-10}\)
Molar Concentration
Molar concentration, often represented by square brackets [], is a measure of the number of moles of a solute that is present in a unit volume of solution. It is typically expressed in moles per liter (mol/L).
As an analogy, think of molar concentration as the number of people in a room per square meter. The more people there are, the more crowded the space becomes. Similarly, the higher the molar concentration of a solute in a solution, the more particles are available to interact and possibly form a precipitate.
In our precipitation reaction scenario, the initial molar concentration of Ag⁺ ions in the AgNO₃ solution was 0.24 M. This concentration helps determine which silver halide precipitates first, as the ions from the salt meet up in the solution, potentially hitting their solubility limit and starting to precipitate. Proper understanding of molar concentration is also the key to calculating the final concentration of ions in a solution after a reaction takes place.
As an analogy, think of molar concentration as the number of people in a room per square meter. The more people there are, the more crowded the space becomes. Similarly, the higher the molar concentration of a solute in a solution, the more particles are available to interact and possibly form a precipitate.
In our precipitation reaction scenario, the initial molar concentration of Ag⁺ ions in the AgNO₃ solution was 0.24 M. This concentration helps determine which silver halide precipitates first, as the ions from the salt meet up in the solution, potentially hitting their solubility limit and starting to precipitate. Proper understanding of molar concentration is also the key to calculating the final concentration of ions in a solution after a reaction takes place.
Solubility Quotient (Q)
The solubility quotient, or Q, is the calculated 'product' of the molar concentrations of the ions that compose a salt at any given moment, not necessarily at equilibrium. This value is extremely useful when you want to determine if a precipitate will form under certain conditions.
Imagine you are filling a bag with groceries until it reaches its limit. Q represents the amount of groceries in the bag at any point during filling, while Ksp is the maximum capacity of the bag before it starts to tear. Once your calculated Q exceeds the known Ksp for the compound, the bag 'tears' and the compound begins to precipitate.
Returning to our exercise, the calculated Q values for AgI, AgBr, and AgCl were:
Imagine you are filling a bag with groceries until it reaches its limit. Q represents the amount of groceries in the bag at any point during filling, while Ksp is the maximum capacity of the bag before it starts to tear. Once your calculated Q exceeds the known Ksp for the compound, the bag 'tears' and the compound begins to precipitate.
Returning to our exercise, the calculated Q values for AgI, AgBr, and AgCl were:
- AgI: Q = \(1.44 \times 10^{15}\)
- AgBr: Q = \(4.32 \times 10^{11}\)
- AgCl: Q = \(1.44 \times 10^9\)
Other exercises in this chapter
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