Chapter 12, Problem 12.18P

(a)

Interpretation Introduction

Interpretation:

The equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$ should be given.

Concept Introduction:

Formation Constant:

In the reaction of metal with ligand, the equilibrium constant is called as formation constant or the stability constant.

${\text{MY}}^{\text{n4-}}\underset{}{\rightleftharpoons }{\text{MY}}^{\text{n+}}+{\text{Y}}^{\text{4-}}$

The formation constant for above complex reaction is,

${\text{K}}_{\text{f}}\text{=}\frac{{\text{[MY}}^{\text{n-4}}\text{]}}{{\text{[M}}^{\text{n+}}{\text{][Y}}^{\text{4-}}\text{]}}$

Where,

${\text{K}}_{\text{f}}$ is formation constant

${\text{[M}}^{\text{+}}\text{]}$ is concentration of metal ion

${\text{[Y}}^{\text{4-}}\text{]}$ is concentration of ligand

${\text{[MY}}^{\text{n4-}}\text{]}$ is concentration of the acid

Answer to Problem 12.18P

The equations for formation constants ${\text{K}}_1\text{=}\frac{\text{[FeT]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}$

The equations for formation constants ${\text{K}}_2\text{=}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][FeT]}}$

Explanation of Solution

To give the equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$

Given,

Figure 1

$\begin{array}{l}{\text{Fe}}^{\text{3+}}\text{+T}\stackrel{{\text{K}}_{\text{1}}}{\rightleftharpoons }\text{FeT}\\ \\ {\text{Fe}}^{\text{3+}}\text{+FeT}\stackrel{{\text{K}}_{\text{2}}}{\rightleftharpoons }{\text{Fe}}_{\text{2}}\text{T}\end{array}$

We know the formation constant for forward reaction is,

${\text{K}}_{\text{f}}\text{=}\frac{{\text{[MY}}^{\text{n-4}}\text{]}}{{\text{[M}}^{\text{n+}}{\text{][Y}}^{\text{4-}}\text{]}}$

Apply the respective reactant and product in above equation to gives the equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$.

The equations for formation constants ${\text{K}}_1\text{=}\frac{\text{[FeT]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}$

The equations for formation constants ${\text{K}}_2\text{=}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][FeT]}}$

Conclusion

The equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$ was given.

(b)

Interpretation Introduction

Interpretation:

The given respected equations should be explained.

Concept Introduction:

Formation Constant:

In the reaction of metal with ligand, the equilibrium constant is called as formation constant or the stability constant.

${\text{MY}}^{\text{n4-}}\underset{}{\rightleftharpoons }{\text{MY}}^{\text{n+}}+{\text{Y}}^{\text{4-}}$

The formation constant for above complex reaction is,

${\text{K}}_{\text{f}}\text{=}\frac{{\text{[MY}}^{\text{n-4}}\text{]}}{{\text{[M}}^{\text{n+}}{\text{][Y}}^{\text{4-}}\text{]}}$

Where,

${\text{K}}_{\text{f}}$ is formation constant

${\text{[M}}^{\text{+}}\text{]}$ is concentration of metal ion

${\text{[Y}}^{\text{4-}}\text{]}$ is concentration of ligand

${\text{[MY}}^{\text{n4-}}\text{]}$ is concentration of the acid

Answer to Problem 12.18P

The equations for formation constants ${\text{K}}_1\text{=}\frac{\text{[FeT]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}$

The equations for formation constants ${\text{K}}_2\text{=}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][FeT]}}$

Explanation of Solution

To give the equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$

Given,

Figure 1

$\begin{array}{l}{\text{Fe}}^{\text{3+}}\text{+T}\stackrel{{\text{K}}_{\text{1}}}{\rightleftharpoons }\text{FeT}\\ \\ {\text{Fe}}^{\text{3+}}\text{+FeT}\stackrel{{\text{K}}_{\text{2}}}{\rightleftharpoons }{\text{Fe}}_{\text{2}}\text{T}\end{array}$

We know the formation constant for forward reaction is,

${\text{K}}_{\text{f}}\text{=}\frac{{\text{[MY}}^{\text{n-4}}\text{]}}{{\text{[M}}^{\text{n+}}{\text{][Y}}^{\text{4-}}\text{]}}$

Apply the respective reactant and product in above equation to gives the equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$.

The equations for formation constants ${\text{K}}_1\text{=}\frac{\text{[FeT]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}$

The equations for formation constants ${\text{K}}_2\text{=}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][FeT]}}$

The above equations are combined to give,

$\begin{array}{l}{\text{K}}_{\text{1}}\text{=}\frac{{\text{[Fe}}_{\text{a}}{\text{T][Fe}}_{\text{a}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}\\ \text{=}\frac{{\text{[Fe}}_{\text{a}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}\text{+}\frac{{\text{[Fe}}_{\text{b}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}\\ \text{=}{\text{K}}_{\text{1}}{\text{+K}}_{\text{2}}\end{array}$

The reciprocal of ${\text{K}}_2$ in given equation is,

$\begin{array}{l}\frac{\text{1}}{{\text{K}}_{\text{2}}}\text{=}\frac{{\text{[Fe}}^{\text{3+}}{\text{][Fe}}_{\text{a}}{\text{T][Fe}}_{\text{b}}\text{T]}}{{\text{[Fe}}_{\text{2}}\text{T]}}\\ \text{=}\frac{{\text{[Fe}}^{\text{3+}}{\text{][Fe}}_{\text{a}}\text{T]}}{{\text{[Fe}}_{\text{2}}\text{T]}}\text{+}\frac{{\text{[Fe}}^{\text{3+}}{\text{][Fe}}_{\text{b}}\text{T]}}{{\text{[Fe}}_{\text{2}}\text{T]}}\\ \text{=}\frac{\text{1}}{{\text{K}}_{\text{2b}}}\text{+}\frac{\text{1}}{{\text{K}}_{\text{2a}}}\end{array}$

From the above derivation s the given equations are derived and they are shown correct relationship with corresponding reactions.

Conclusion

The given respected equations should be explained.

(c)

Interpretation Introduction

Interpretation:

Respect with the given reaction, the given expression should be explained.

Concept Introduction:

Formation Constant:

In the reaction of metal with ligand, the equilibrium constant is called as formation constant or the stability constant.

${\text{MY}}^{\text{n4-}}\underset{}{\rightleftharpoons }{\text{MY}}^{\text{n+}}+{\text{Y}}^{\text{4-}}$

The formation constant for above complex reaction is,

${\text{K}}_{\text{f}}\text{=}\frac{{\text{[MY}}^{\text{n-4}}\text{]}}{{\text{[M}}^{\text{n+}}{\text{][Y}}^{\text{4-}}\text{]}}$

Where,

${\text{K}}_{\text{f}}$ is formation constant

${\text{[M}}^{\text{+}}\text{]}$ is concentration of metal ion

${\text{[Y}}^{\text{4-}}\text{]}$ is concentration of ligand

${\text{[MY}}^{\text{n4-}}\text{]}$ is concentration of the acid

Explanation of Solution

To give the equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$

Given,

Figure 1

$\begin{array}{l}{\text{Fe}}^{\text{3+}}\text{+T}\stackrel{{\text{K}}_{\text{1}}}{\rightleftharpoons }\text{FeT}\\ \\ {\text{Fe}}^{\text{3+}}\text{+FeT}\stackrel{{\text{K}}_{\text{2}}}{\rightleftharpoons }{\text{Fe}}_{\text{2}}\text{T}\end{array}$

The given equation is,

${\text{K}}_{\text{1a}}{\text{K}}_{\text{2b}}\text{=}{\text{K}}_{\text{1b}}{\text{K}}_{\text{2a}}$

We know the formation constant for forward reaction is,

${\text{K}}_{\text{f}}\text{=}\frac{{\text{[MY}}^{\text{n-4}}\text{]}}{{\text{[M}}^{\text{n+}}{\text{][Y}}^{\text{4-}}\text{]}}$

Apply the respective reactant and product in above equation to gives the equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$.

The equations for formation constants ${\text{K}}_1\text{=}\frac{\text{[FeT]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}$

The equations for formation constants ${\text{K}}_2\text{=}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][FeT]}}$

The above equations are combined to give,

$\begin{array}{l}{\text{K}}_{\text{1a}}{\text{K}}_{\text{2b}}\text{=}\frac{{\text{[Fe}}_{\text{a}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}{\text{][Fe}}_{\text{a}}\text{T]}}\\ \\ {\text{K}}_{\text{1a}}{\text{K}}_{\text{2b}}\text{=}\frac{{\text{[Fe}}_{\text{b}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}{\text{][Fe}}_{\text{b}}\text{T]}}\end{array}$

To cancel the common the terms, ${\text{[Fe}}_{\text{a}}\text{T]}$ and ${\text{[Fe}}_{\text{b}}\text{T]}$ is above equations to give the relationship of given equation ${\text{K}}_{\text{1a}}{\text{K}}_{\text{2b}}\text{=}{\text{K}}_{\text{1b}}{\text{K}}_{\text{2a}}$.

Conclusion

Respect with the given reaction, the given expression was explained.

(d)

Interpretation Introduction

Interpretation:

From the given equations, the three unknown values are should be determined.

Concept Introduction:

Formation Constant:

In the reaction of metal with ligand, the equilibrium constant is called as formation constant or the stability constant.

${\text{MY}}^{\text{n4-}}\underset{}{\rightleftharpoons }{\text{MY}}^{\text{n+}}+{\text{Y}}^{\text{4-}}$

The formation constant for above complex reaction is,

${\text{K}}_{\text{f}}\text{=}\frac{{\text{[MY}}^{\text{n-4}}\text{]}}{{\text{[M}}^{\text{n+}}{\text{][Y}}^{\text{4-}}\text{]}}$

Where,

${\text{K}}_{\text{f}}$ is formation constant

${\text{[M}}^{\text{+}}\text{]}$ is concentration of metal ion

${\text{[Y}}^{\text{4-}}\text{]}$ is concentration of ligand

${\text{[MY}}^{\text{n4-}}\text{]}$ is concentration of the acid

Answer to Problem 12.18P

Three unknown values are,

$\begin{array}{l}\text{[T] }\text{=0}{\text{.27}}_{\text{7}}\\ {\text{[Fe}}_{\text{a}}\text{T] }\text{=0}{\text{.55}}_{\text{7}}\\ {\text{[Fe}}_{\text{2}}\text{T] }\text{=0}{\text{.7}}_{\text{7}}\end{array}$

Explanation of Solution

To give the equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$

Given,

Figure 1

$\begin{array}{l}{\text{Fe}}^{\text{3+}}\text{+T}\stackrel{{\text{K}}_{\text{1}}}{\rightleftharpoons }\text{FeT}\\ \\ {\text{Fe}}^{\text{3+}}\text{+FeT}\stackrel{{\text{K}}_{\text{2}}}{\rightleftharpoons }{\text{Fe}}_{\text{2}}\text{T}\end{array}$

The given equation is,

${\text{K}}_{\text{1a}}{\text{K}}_{\text{2b}}\text{=}{\text{K}}_{\text{1b}}{\text{K}}_{\text{2a}}$

We know the formation constant for forward reaction is,

${\text{K}}_{\text{f}}\text{=}\frac{{\text{[MY}}^{\text{n-4}}\text{]}}{{\text{[M}}^{\text{n+}}{\text{][Y}}^{\text{4-}}\text{]}}$

Apply the respective reactant and product in above equation to gives the equations for formation constants ${\text{K}}_{\text{1}}$ and ${\text{K}}_2$.

The equations for formation constants ${\text{K}}_1\text{=}\frac{\text{[FeT]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}\mathrm{...1}$

The equations for formation constants ${\text{K}}_2\text{=}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][FeT]}}\mathrm{...2}$

The above equations are combined to give,

$\begin{array}{l}{\text{K}}_{\text{1a}}{\text{K}}_{\text{2b}}\text{=}\frac{{\text{[Fe}}_{\text{a}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}{\text{][Fe}}_{\text{a}}\text{T]}}\mathrm{...3}\\ \\ {\text{K}}_{\text{1a}}{\text{K}}_{\text{2b}}\text{=}\frac{{\text{[Fe}}_{\text{b}}\text{T]}}{{\text{[Fe}}^{\text{3+}}\text{][T]}}\frac{{\text{[Fe}}_{\text{2}}\text{T]}}{{\text{[Fe}}^{\text{3+}}{\text{][Fe}}_{\text{b}}\text{T]}}\mathrm{...}4\end{array}$

Plugged the equation 1 and in 3 and equation 2 in 4,

$\begin{array}{l}\text{=}\frac{{\text{[FeT]}}^{\text{2}}}{{\text{(1-[Fe]-[Fe}}_{\text{2}}\text{T]}{\text{[Fe}}_{\text{a}}\text{T]}}\\ \text{=}\frac{{\text{(0}\text{.8-2}{\text{[Fe}}_{\text{2}}\text{T])}}^{\text{2}}}{\text{{1-(0}{\text{.8-2[Fe}}_{\text{2}}\text{T])-}{\text{[Fe}}_{\text{2}}{\text{T]}[Fe}}_{\text{2}}\text{T]}}\end{array}$

To solve the quadratic equation to give the ${\text{[Fe}}_{\text{2}}\text{T]}$

${\text{[Fe}}_{\text{2}}\text{T]}\text{=0}\text{.0773}$

The given value of $\text{[FeT] is}$ $\text{0}\text{.645}$

Using the calculated ${\text{[Fe}}_{\text{2}}\text{T]}$ and give values $\text{[FeT]}$ , $\text{[T] =}\text{0}\text{.2773}$ and ${\text{K1}}_{\text{a}}{\text{/K1}}_{\text{b}}=6.0$ into our calculations.

$\begin{array}{l}\frac{{\text{K1}}_{\text{a}}}{{\text{K1}}_{\text{b}}}\text{=}\frac{{\text{[Fe}}_{\text{a}}\text{T]}}{{\text{[Fe}}_{\text{b}}\text{T]}}\\ =6.0\end{array}$

$\begin{array}{l}{\text{[Fe}}_{\text{a}}\text{T]}\text{=}\left(\frac{\text{6}\text{.0}}{\text{7}\text{.0}}\right)\text{[FeT]}\\ \text{=}\text{0}\text{.5532}\\ {\text{[Fe}}_{\text{b}}\text{T]}\text{=}\left(\frac{\text{1}\text{.0}}{\text{7}\text{.0}}\right)\text{[FeT]}\\ \text{=}\text{0}\text{.0922}\end{array}$

From the above calculations three unknown values are,

$\begin{array}{l}\text{[T] }\text{=0}{\text{.27}}_{\text{7}}\\ {\text{[Fe}}_{\text{a}}\text{T] }\text{=0}{\text{.55}}_{\text{7}}\\ {\text{[Fe}}_{\text{2}}\text{T] }\text{=0}{\text{.7}}_{\text{7}}\end{array}$

Conclusion

From the given equations, the three unknown values are determined.