Question

What is the ratio of uncomplexed to complexed $Z{{n}^{2+}}$ ion in a solution that is 10M in $N{{H}_{3}}$ , if the stability constant of ${{[Zn{{(N{{H}_{3}})}_{4}}]}^{2+}}$ is $3\times {{10}^{9}}$?
(a) $3.3\times {{10}^{-9}}$
(b) $3.3\times {{10}^{-11}}$
(c) $3.3\times {{10}^{-14}}$
(d) $3.3\times {{10}^{-13}}$

Answer 1

Hint: By stability of the complexes we means the formation of the complex at equilibrium and if the interaction is strong, the complex formed is thermodynamically very stable and thermodynamic stability is expressed in terms of the stability constant and the stability constant for n equilibria is given as ${{K}_{n}}=\dfrac{\left[ M{{L}_{n}} \right]}{\left[ M{{L}_{n-1}}\text{ } \right]\left[ L \right]}$ and we know the value of the stability constant and from this, we can easily find the ratio of the uncomplexed to complexed $Z{{n}^{2+}}$ ion. Now solve it.

Complete answer:
The most important aspect of the metal complexes is the stability of the complexes formed and the stability depends upon the nature of the metal and the ligands as well as on reaction conditions. A compound may be unstable with respect to a particular reactant or condition such as heat, light, acid, base etc.
By the term stability of a compound we mean that the compound exists and under suitable conditions may be stored for a long period of time.
The thermodynamic stability of a complex is the measure of the extent of formation of a complex at equilibrium. The interactions between the metal ion and the ligands may be regarded as Lewis-acid base reactions.
Let us consider the formation of the complex ,$M{{L}_{n}}$as;
     \[M+nL\rightleftharpoons M{{L}_{n}}\]
The reaction proceeds by the following steps;
     $M+L\rightleftharpoons ML$
And the equilibrium constant for this reaction will be;
      \[{{K}_{1}}=\dfrac{\left[ ML \right]}{\left[ M \right]\left[ L \right]}\]
Now, ML adds to the molecule of the ligand L as;
$\begin{align}
  & ML+L\rightleftharpoons M{{L}_{2}}\text{ and} \\
 & {{K}_{2}}=\dfrac{\left[ M{{L}_{2}} \right]}{\left[ ML \right]\left[ L \right]} \\
\end{align}$
\[\]
Similarly, $M{{L}_{n}}$is formed by the following reactions:
                                                         $\begin{align}
  & M{{L}_{2}}+L\rightleftharpoons M{{L}_{3}}\text{ and }{{K}_{3}}=\dfrac{\left[ M{{L}_{3}} \right]}{\left[ M{{L}_{2}}\text{ } \right]\left[ L \right]} \\
 & M{{L}_{n-1}}+L\rightleftharpoons M{{L}_{n}}\text{ and }{{K}_{n}}=\dfrac{\left[ M{{L}_{n}} \right]}{\left[ M{{L}_{n-1}}\text{ } \right]\left[ L \right]}
\end{align}$
There will be in such equilibria , where n represents maximum coordination number of the metal ion for the ligand L. The equilibrium constants ${{K}_{1}},{{K}_{2}},{{K}_{3}}............{{K}_{n}}$ are known as stepwise stability constants.
Now considering the statement;
The formation of zinc-ammonia complex occurs as;
\[Z{{n}^{2+}}+4N{{H}_{3}}\rightleftharpoons {{[Zn{{(N{{H}_{3}})}_{4}}]}^{2+}}\]
The stability constant, K is as;
 \[K=\dfrac{{{\left[ Zn{{(N{{H}_{3}})}_{4}} \right]}^{2+}}}{\left[ Z{{n}^{2+}} \right]{{\left[ N{{H}_{3}} \right]}^{4}}}\]
The stability constant of the complex i.e. ${{[Zn{{(N{{H}_{3}})}_{4}}]}^{2+}}$=$3\times {{10}^{9}}$(given)
Then;
\[\begin{align}
  & 3\times {{10}^{9}}=\dfrac{{{\left[ Zn{{(N{{H}_{3}})}_{4}} \right]}^{2+}}}{\left[ Z{{n}^{2+}} \right]{{\left[ N{{H}_{3}} \right]}^{4}}} \\
 & 3\times {{10}^{9}}=\dfrac{{{\left[ Zn{{(N{{H}_{3}})}_{4}} \right]}^{2+}}}{\left[ Z{{n}^{2+}} \right]{{\left[ 10 \right]}^{4}}}\text{ (}N{{H}_{3}}=10\text{ (}given)) \\
 & 3\times {{10}^{9}}=\dfrac{{{\left[ Zn{{(N{{H}_{3}})}_{4}} \right]}^{2+}}}{\left[ Z{{n}^{2+}} \right]10000} \\
 & 3\times {{10}^{13}}=\dfrac{{{\left[ Zn{{(N{{H}_{3}})}_{4}} \right]}^{2+}}}{\left[ Z{{n}^{2+}} \right]} \\
\end{align}\]
In the statement, we have been asked the ratio of uncomplexed to complexed $Z{{n}^{2+}}$ ion in a solution, then;
 \[\begin{align}
  & \dfrac{1}{3\times {{10}^{13}}}=\dfrac{\left[ Z{{n}^{2+}} \right]}{{{\left[ Zn{{(N{{H}_{3}})}_{4}} \right]}^{2+}}} \\
 & \dfrac{1}{3\times {{10}^{13}}}=\dfrac{\left[ Z{{n}^{2+}} \right]}{{{\left[ Zn{{(N{{H}_{3}})}_{4}} \right]}^{2+}}} \\
 & \dfrac{\left[ Z{{n}^{2+}} \right]}{{{\left[ Zn{{(N{{H}_{3}})}_{4}} \right]}^{2+}}}=3.3\times {{10}^{-14}} \\
\end{align}\]
Hence, the ratio of uncomplexed to complexed $Z{{n}^{2+}}$ ion in a solution that is 10M in $N{{H}_{3}}$ is \[\dfrac{\left[ Z{{n}^{2+}} \right]}{{{\left[ Zn{{(N{{H}_{3}})}_{4}} \right]}^{2+}}}=3.3\times {{10}^{-14}}\]

So, option (c) is correct.

Note:
There are two types of stabilities for the metal complexes i.e. thermodynamic stability and kinetic stability. The thermodynamic stability refers to equilibrium constants and deals with the bond energies, stability constants, redox potentials etc. On the other hand, kinetic stability deals with rates and mechanisms of chemical reactions.