Chapter 17, Problem 76PS

(a)

Interpretation Introduction

Interpretation:

The solubility of $\text{AgCN}$ in water has to be calculated using $K_{\text{sp}}$ value for $\text{AgCN}$.

Concept introduction:

The solubility of a salt is defined as the maximum amount of salt that can be dissolved in definite amount of solvent. It is expressed in moles per liter or grams per liter. Solubility in terms of moles per liter is called molar solubility and is defined as the number of moles of solute (salt) dissolved in per liter of solution.

Solubility product constant $K_{\text{sp}}$ is an equilibrium constant and is defined as the product of the equilibrium concentration of the ions of the salt raised to the power of their coefficients in the balanced chemical equation.

The expression for $K_{\text{sp}}$ of a salt is given as,

$\begin{array}{l}{\text{A}}_x{\text{B}}_y\left(\text{s}\right)\rightleftharpoons x{\text{A}}^{y+}\left(\text{aq}\right)+y{\text{B}}^{-x}\left(\text{aq}\right)\\ K_{\text{sp}}={\left[{\text{A}}^{y+}\right]}^x{\left[{\text{B}}^{-x}\right]}^y\end{array}$

The solubility of $\text{AgCN}$ in water is $7.75\times 10^{-9}$.

Explanation of Solution

Solubility product constant $K_{\text{sp}}$ for $\text{AgCN}$ is $6.0\times {10}^{-17}$.

$\text{AgCN}$ dissociates as follows in water,

  $\text{AgCN}\left(\text{s}\right)\rightleftharpoons {\text{Ag}}^{+}\left(\text{aq}\right)+{\text{CN}}^{-1}\left(\text{aq}\right)$

The expression for $K_{\text{sp}}$,

$K_{\text{sp}}=\left[{\text{Ag}}^{+}\right]\left[{\text{CN}}^{-1}\right]$ (1)

The ICE table (1) is as follows,

$\begin{array}{cccccc}\text{Equation}& \text{AgCN}\left(\text{s}\right)& \rightleftharpoons & {\text{Ag}}^{+}\left(\text{aq}\right)& +& {\text{CN}}^{-1}\left(\text{aq}\right)\\ \text{Initial }\left(\text{M}\right)& & & 0& & 0\\ \text{Change }\left(\text{M}\right)& & & +s& & +s\\ \text{Equilibrium }\left(\text{M}\right)& & & \text{ s}& & s\end{array}$

Here, $s$ is the molar solubility of $\text{AgCN}$.

Substitute value of the concentration of radium and sulphate ions in equation (1) from the table.

$\begin{array}{c}{K}_{\text{sp}}=\left(s\right)\left(s\right)\\ =s^2\end{array}$

Substitute $6.0\times {10}^{-17}$ for $K_{\text{sp}}$.

$\begin{array}{c}s=\sqrt{6.0\times {10}^{-17}}\\ =7.75\times 10^{-9}mol\cdot L^{-1}\end{array}$

The solubility of $\text{AgCN}$ in water is $7.75\times 10^{-9}$.

 (b)

Interpretation Introduction

Interpretation:

The value of the net equilibrium constant $K_{\text{net}}$ has to be calculated for the given reaction.

$\text{AgCN}\left(\text{s}\right)+{\text{CN}}^{-1}\left(\text{aq}\right)\stackrel{K_{\text{net}}}{\rightleftharpoons }{\left[\text{Ag}{\left(\text{CN}\right)}_2\right]}^{-}\left(\text{aq}\right)$

Concept introduction:

Some metal ions when present in an aqueous solution containing anions or neutral species called Lewis base or ligands having a tendency to donate electron pairs to metal ions then complex ion formation will take place.

Example of metal ions that form complex ions includes,${\text{Cd}}^{2+}$,${\text{Fe}}^{2+}$,${\text{Zn}}^{2+}$,${\text{Ni}}^{2+}$ etc.

Example of Lewis bases includes,${\text{NH}}_3$,${\text{OH}}^{-}$ etc.

The complex ion remains in equilibrium with the metal ion and the ligand called complex ion formation equilibrium and the equilibrium constant is called as formation constant $K_{\text{f}}$.

A larger value of $K_{\text{f}}$ implies that the complex ion formed is more stable. $K_{\text{f}}$ is the measure of the strength of the interaction between the metal ions and the Lewis base to form the complex ion.

For example for general complex ion formation reaction,

$\text{xM}+\text{yL}\rightleftharpoons \left[{\text{M}}_{\text{x}}{\text{L}}_{\text{y}}\right]$

$K_{\text{f}}$ can be given as

$K_{\text{f}}=\frac{\left[{\text{M}}_{\text{x}}{\text{L}}_{\text{y}}\right]}{{\left[\text{M}\right]}^{\text{x}}{\left[\text{L}\right]}^{\text{y}}}$

 Here,

• $\left[{\text{M}}_{\text{x}}{\text{L}}_{\text{y}}\right]$ is the equilibrium concentration of complex ion.
• $\left[\text{M}\right]$ is the equilibrium concentration of metal ion.
• $\left[\text{L}\right]$ is the equilibrium concentration of the ligand.
• $\text{x}$ and $\text{y}$ are the coefficients of metal ion and ligand respectively.

Complex ions are stable and thus formation of these increase the solubility of the salt containing the metal ions same as in complex ions. Effect of complex ion formation on the solubility of salt can be explained as below,

$\text{AgBr}$ when dissolved in water does not dissolve completely and dissociates as follows,

$\text{AgBr}(s)\stackrel{K_{\text{sp}}}{\rightleftharpoons }A{g}^{+}(aq)+{\text{Br}}^{-}(aq)$ (1)

For this reaction $K_{\text{sp}}$ expression is,

$\text{Ksp}=\left[{\text{Ag}}^{+}\right]\left[{\text{Br}}^{-}\right]$

$A{g}^{+}$ ions are capable of forming complex with ligand so when aqueous ammonia solution (strong ligand) is added to the saturated solution of $\text{AgBr}$, $A{g}^{+}$ ions present in the solution form complex with ${\text{NH}}_3$.

${\text{Ag}}^{+}\left(\text{aq}\right)+2{\text{NH}}_3\left(\text{aq}\right)\stackrel{K_{\text{f}}}{\rightleftharpoons }{\left[\text{Ag}{\left({\text{NH}}_3\right)}_2\right]}^{+}\left(\text{aq}\right)$ (2)

The expression for formation constant $K_{\text{f}}$ is given as,

$K_{\text{f}}=\frac{\left[{\left[\text{Ag}{\left({\text{NH}}_3\right)}_2\right]}^{+}\right]}{\left[{\text{Ag}}^{+}\right]{\left[{\text{NH}}_3\right]}^2}$

Complex formation leads to the decrease in the concentration of the $A{g}^{+}$ ions in the solution, as a result, according to Le Chatelier’s principle the equilibrium in equation (1) move in the forward direction producing more of the $A{g}^{+}$ ions and the solubility of the slightly soluble salt $\text{AgBr}$ increases.

Adding two equilibrium equation (1) and (2) new overall equilibrium constant can be defined.

Net chemical equation:

${\text{Ag}}^{+}\left(\text{aq}\right)+2{\text{NH}}_3\left(\text{aq}\right)\stackrel{K_{\text{net}}}{\rightleftharpoons }{\left[\text{Ag}{\left({\text{NH}}_3\right)}_2\right]}^{+}\left(\text{aq}\right)+{\text{Br}}^{-}(aq)$ (3)

Net equilibrium constant can be given as,

$K_{\text{net}}=\left(K_{\text{sp}}\right)\left(K_{\text{f}}\right)$ (4)

Explanation of Solution

The value of net equilibrium constant, $K_{\text{net}}$ for the given reaction is calculated below.

Given:

Refer to the Appendix J and K in the textbook for the value of $K_{\text{sp}}$ and $K_{\text{f}}$ respectively.

The value of solubility product constant, $K_{\text{sp}}$ for $\text{AgCN}$ is $6.0\times {10}^{-17}$ .

The value of formation constant, $K_{\text{f}}$ for ${\left[\text{Ag}{\left(\text{CN}\right)}_2\right]}^{-}$ is $1.3\times {10}^{21}$.

$\text{AgCN}$ dissociates as follows in water,

  $\text{AgCN}\left(\text{s}\right)\stackrel{K_{\text{sp}}}{\rightleftharpoons }{\text{Ag}}^{+}\left(\text{aq}\right)+{\text{CN}}^{-}\left(\text{aq}\right)$ (5)

${\text{Ag}}^{+}$ ions present in solution undergo complex formation with cyanide ions from $\text{KCN}$ to form ${\left[\text{Ag}{\left(\text{CN}\right)}_2\right]}^{-}$ complex. The reaction for complex formation is given as,

  ${\text{Ag}}^{+}\left(\text{aq}\right)+2{\text{CN}}^{-1}\left(\text{aq}\right)\stackrel{K_{\text{f}}}{\rightleftharpoons }{\left[\text{Ag}{\left(\text{CN}\right)}_2\right]}^{-}\left(\text{aq}\right)$ (6)

Add equation (5) and (6) to find the net chemical equation.

  $\text{AgCN}\left(\text{s}\right)+{\text{CN}}^{-1}\left(\text{aq}\right)\stackrel{K_{\text{net}}}{\rightleftharpoons }{\left[\text{Ag}{\left(\text{CN}\right)}_2\right]}^{-}\left(\text{aq}\right)$                                          (7)

The net equilibrium constant, $K_{\text{net}}$ for equation (7) is given as,

$K_{\text{net}}=K_{\text{sp}}\times K_{\text{f}}$

Substitute $6.0\times {10}^{-17}$ for $K_{\text{sp}}$ and $1.3\times {10}^{21}$ for $K_{\text{f}}$.

$\begin{array}{c}{K}_{\text{net}}=\left(6.0\times {10}^{-17}\right)\left(1.3\times {10}^{21}\right)\\ =7.8\times 10^4\end{array}$

The value of the net equilibrium constant, $K_{\text{net}}$ for the reaction of dissolution of  $\text{AgCN}$ in aqueous potassium cyanide solution is $7.8\times 10^4$. The value of $K_{\text{net}}$ is greater than the value of $K_{\text{sp}}$ for $\text{AgCN}$ this implies that the forward reaction is favored increasing the solubility of $\text{AgCN}$.  Therefore solid $\text{AgCN}$ will dissolve in aqueous solution of $\text{KCN}$.

(c)

Interpretation Introduction

Interpretation:

The value of the net equilibrium constant $K_{\text{net}}$ has to be calculated for the given reaction.

$\text{AgCN}\left(\text{s}\right)+2{\text{S}}_2{\text{O}}_3{}^{2-}\left(\text{aq}\right)\stackrel{K_{\text{net}}}{\rightleftharpoons }{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2\right]}^{3-}\left(\text{aq}\right)+{\text{CN}}^{-}\left(\text{aq}\right)$

Concept introduction:

The net equilibrium constant, $K_{\text{net}}$ for the equation is given as,

$K_{\text{net}}=K_{\text{sp}}\times K_{\text{f}}$

Explanation of Solution

The value of net equilibrium constant, $K_{\text{net}}$ for the given reaction is calculated below.

Given:

Refer to the Appendix J and K in the textbook for the value of $K_{\text{sp}}$ and $K_{\text{f}}$ respectively.

The value of solubility product constant, $K_{\text{sp}}$ for $\text{AgCN}$ is $6.0\times {10}^{-17}$ .

The value of formation constant, $K_{\text{f}}$ for ${\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2\right]}^{3-}$ is $2.9\times {10}^{13}$.

$\text{AgCN}$ dissociates as follows in water,

  $\text{AgCN}\left(\text{s}\right)\stackrel{K_{\text{sp}}}{\rightleftharpoons }{\text{Ag}}^{+}\left(\text{aq}\right)+{\text{CN}}^{-}\left(\text{aq}\right)$ (5)

${\text{Ag}}^{+}$ ions present in solution undergo complex formation with thio ions to form ${\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2\right]}^{3-}$ complex. The reaction for complex formation is given as,

  ${\text{Ag}}^{+}\left(\text{aq}\right)+2{\text{S}}_2{\text{O}}_3{}^{2-}\left(\text{aq}\right)\stackrel{K_{\text{f}}}{\rightleftharpoons }{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2\right]}^{3-}\left(\text{aq}\right)$ (6)

Add equation (5) and (6) to find the net chemical equation.

  $\text{AgCN}\left(\text{s}\right)+2{\text{S}}_2{\text{O}}_3{}^{2-}\left(\text{aq}\right)\stackrel{K_{\text{net}}}{\rightleftharpoons }{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2\right]}^{3-}\left(\text{aq}\right)+{\text{CN}}^{-}\left(\text{aq}\right)$                       (7)

The net equilibrium constant, $K_{\text{net}}$ is,

$K_{\text{net}}=K_{\text{sp}}\times K_{\text{f}}$

Substitute $6.0\times {10}^{-17}$ for $K_{\text{sp}}$ and $2.9\times {10}^{13}$ for $K_{\text{f}}$.

$\begin{array}{c}{K}_{\text{net}}=\left(6.0\times {10}^{-17}\right)\left(2.9\times {10}^{13}\right)\\ =1.74\times 10^{-3}\end{array}$

The ICE table (2) is as follows,

$\begin{array}{cccccccc}\text{Equation}& \text{AgCN}\left(\text{s}\right)& +& 2{\text{S}}_2{\text{O}}_3{}^{2-}\left(\text{aq}\right)& \rightleftharpoons & {\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2\right]}^{3-}\left(\text{aq}\right)& +& {\text{CN}}^{-}\left(\text{aq}\right)\\ \text{Initial }\left(\text{M}\right)& & & 0.1& & 0& & 0\\ \text{Change }\left(\text{M}\right)& & & -2s& & +s& & +s\\ \text{Equilibrium }\left(\text{M}\right)& & & 0.1-2s& & s& & s\end{array}$

Here, $s$ is the molar solubility of $\text{AgCN}$.

$K_{\text{net}}=\frac{\left[{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2\right]}^{3-}\right]\left[{\text{CN}}^{-}\right]}{{\left[{\text{S}}_2{\text{O}}_3{}^{2-}\right]}^2}$

Substitute values from the table.

$K_{\text{net}}=\frac{s^2}{{\left(0.1-2s\right)}^2}$

Substitute value of $K_{\text{net}}$.

$\begin{array}{c}\left(1.74\times {10}^{-3}\right)=\frac{s^2}{{\left(0.1-2s\right)}^2}\\ \frac{s}{\left(0.1-2s\right)}=\sqrt{1.74\times {10}^{-3}}\\ =0.042\\ s=3.87\times 10^{-3}\end{array}$

The value of net equilibrium constant, $K_{\text{net}}$ for the reaction of dissolution of $\text{AgCN}$ in aqueous solution containing thio ions is $1.74\times 10^{-3}$. The solubility of $\text{AgCN}$ in the solution is $3.87\times 10^{-3}mol\cdot L^{-1}$. Due to the complex formation, the solubility of $\text{AgCN}$ increases in solution containing $0.1\text{ M}$ of ${\text{S}}_2{\text{O}}_3{}^{2-}$ ions than in pure water