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Chapter 16: Problem 99

Calculate the solubility of \(\mathrm{AgCN}(s)\left(K_{\mathrm{sp}}=2.2 \times 10^{-12}\right)\) in a solution containing \(1.0 M \mathrm{H}^{+} .\left(K_{\mathrm{a}}\right.\) for \(\mathrm{HCN}\) is \(6.2 \times 10^{-10}\).)

Short Answer

Expert verified
The solubility of silver cyanide (AgCN) in a solution containing 1.0 M hydrogen ions (H₊) is approximately \( s \approx 1.48 \times 10^{-6} \,\mathrm{M} \).

Step by step solution

01

Write the solubility equilibrium reaction for AgCN

The solubility equilibrium reaction for silver cyanide can be represented as follows: \[ \mathrm{AgCN}(s) \rightleftharpoons \mathrm{Ag^{+}}(aq) + \mathrm{CN^{-}}(aq) \]
02

Write the dissociation reaction for HCN

The dissociation reaction for hydrogen cyanide can be represented as: \[ \mathrm{HCN}(aq) + \mathrm{H_2O}(l) \rightleftharpoons \mathrm{H_3O^{+}}(aq) + \mathrm{CN^{-}}(aq) \]
03

Write the expressions for solubility product constant Kₛₚ and acid dissociation constant Kₐ

For the solubility equilibrium reaction of AgCN: \[ K_{\mathrm{sp}} = [\mathrm{Ag^{+}}][\mathrm{CN^{-}}]\] For the dissociation reaction of HCN: \[ K_{\mathrm{a}} = \frac {[\mathrm{H_3O^{+}}][\mathrm{CN^{-}}]}{[\mathrm{HCN}]} \]
04

Relate the concentration of H₊ with the concentration of Ag₊ in the solution

Since the solution contains 1.0 M H₊ ions and Kₐ of HCN is given, we can use the following relationship: \[ [\mathrm{Ag^{+}}] = [\mathrm{CN^{-}}] \]
05

Substitute the expressions in the Kₛₚ equation and solve for the solubility of AgCN

Let the solubility of AgCN be denoted by 's'. Based on Step 4, we can rewrite the Kₛₚ equation as: \[ K_{\mathrm{sp}} = (s)(s) \] Now, substitute the given value of Kₛₚ and solve for 's': \[ 2.2 \times 10^{-12} = (s)(s) \] \[ s^2 = 2.2 \times 10^{-12} \] \[ s = \sqrt{2.2 \times 10^{-12}} \approx 1.48 \times 10^{-6} \]
06

Present the final answer

The solubility of silver cyanide (AgCN) in a solution containing 1.0 M hydrogen ions (H₊) is approximately: \[ s \approx 1.48 \times 10^{-6} \,\mathrm{M} \]

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Most popular questions from this chapter

Write balanced equations for the dissolution reactions and the corresponding solubility product expressions for each of the following solids. a. \(\mathrm{Ag}_{2} \mathrm{CO}_{3}\) b. \(\mathrm{Ce}\left(\mathrm{IO}_{3}\right)_{3}\) c. \(\mathrm{BaF}_{2}\)

Aluminum ions react with the hydroxide ion to form the precipitate \(\mathrm{Al}(\mathrm{OH})_{3}(s)\), but can also react to form the soluble complex ion \(\mathrm{Al}(\mathrm{OH})_{4}^{-} .\) In terms of solubility, \(\mathrm{Al}(\mathrm{OH})_{3}(s)\) will be more soluble in very acidic solutions as well as more soluble in very basic solutions. a. Write equations for the reactions that occur to increase the solubility of \(\mathrm{Al}(\mathrm{OH})_{3}(s)\) in very acidic solutions and in very basic solutions. b. Let's study the \(\mathrm{pH}\) dependence of the solubility of \(\mathrm{Al}(\mathrm{OH})_{3}(s)\) in more detail. Show that the solubility of \(\mathrm{Al}(\mathrm{OH})_{3}\), as a function of \(\left[\mathrm{H}^{+}\right]\), obeys the equation $$S=\left[\mathrm{H}^{+}\right]^{3} K_{\mathrm{sp}} / K_{\mathrm{w}}^{3}+K K_{\mathrm{w}} /\left[\mathrm{H}^{+}\right]$$ where \(S=\) solubility \(=\left[\mathrm{Al}^{3+}\right]+\left[\mathrm{Al}(\mathrm{OH})_{4}^{-}\right]\) and \(K\) is the equilibrium constant for $$\mathrm{Al}(\mathrm{OH})_{3}(s)+\mathrm{OH}^{-}(a q) \rightleftharpoons \mathrm{Al}(\mathrm{OH})_{4}^{-}(a q)$$ c. The value of \(K\) is \(40.0\) and \(K_{\mathrm{se}}\) for \(\mathrm{Al}(\mathrm{OH})_{3}\) is \(2 \times 10^{-32}\). Plot the solubility of \(\mathrm{Al}(\mathrm{OH})_{3}\) in the \(\mathrm{pH}\) range \(4-12\).

Use the following data to calculate the \(K_{\mathrm{sp}}\) value for each solid. a. The solubility of \(\mathrm{Pb}_{3}\left(\mathrm{PO}_{4}\right)_{2}\) is \(6.2 \times 10^{-12} \mathrm{~mol} / \mathrm{L}\). b. The solubility of \(\mathrm{Li}_{2} \mathrm{CO}_{3}\) is \(7.4 \times 10^{-2} \mathrm{~mol} / \mathrm{L}\).

Calculate the molar solubility of \(\mathrm{Co}(\mathrm{OH})_{3}, K_{\mathrm{sp}}=2.5 \times 10^{-43}\).

Which of the following will affect the total amount of solute that can dissolve in a given amount of solvent? a. The solution is stirred. b. The solute is ground to fine particles before dissolving. c. The temperature changes.
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