Question

${{H}_{3}}P{{O}_{4}}$ is tribasic acid with $p{{K}_{a1}},p{{K}_{a2}}\text{ }and\text{ }p{{K}_{a3}}$ 2.12, 7.21 and 12.32 respectively. It is used in fertilizer productions and its various salts are used in food, detergent, toothpaste, and in mineral treatment.
Small quantities of ${{H}_{3}}P{{O}_{4}}$ are used in imparting the sour or tart taste to soft drinks, such as coca cola, and beers, in which ${{H}_{3}}P{{O}_{4}}$ is present 0.05 % by weight (density = 1.0 g/ml). 10-3 M ${{H}_{3}}P{{O}_{4}}$ (pH=7) is used in fertilizers as an aqueous soil digesting. Plants can absorb zinc in water soluble form only. Zinc phosphate is the source of zinc and $PO_{3}^{2-}$ ions in the soil, $~{{K}_{sp}}$ of zinc phosphate = $9.1\text{ }\times {{10}^{-33}}$ , $\left[ Z{{n}^{2+}} \right]$ ion in the soil is:
A. $2.9\times {{10}^{-11}}M$
B. $4.0\times {{10}^{-10}}M$
C. $3.0\times {{10}^{-6}}M$
D. $9.1\times {{10}^{-5}}M$

Answer 1

Hint:. First we have to calculate the ${{K}_{a1}},{{K}_{a2}}\text{ }and\text{ }{{K}_{a3}}$ from $p{{K}_{a1}},p{{K}_{a2}}\text{ }and\text{ }p{{K}_{a3}}$ by using the following relationship between ${{K}_{a}}$ and $p{{K}_{a}}$ .
\[{{K}_{a}}={{10}^{-p{{K}_{a}}}}\]
Here, ${{K}_{a}}$ = acid dissociation constant.
$p{{K}_{a}}$ = ionization constant.

Complete step by step answer:
- In the question it is given that pH of the solution is 7.
- Then $[{{H}_{3}}{{O}^{+}}]={{10}^{-7}}$
- In the question it is given that the $p{{K}_{a1}},p{{K}_{a2}}\text{ }and\text{ }p{{K}_{a3}}$ values are 2.12, 7.21 and 12.32 respectively.
- The ${{K}_{a1}},{{K}_{a2}}\text{ }and\text{ }{{K}_{a3}}$ values will be $7.59\times {{10}^{-3}},6.17\times {{10}^{-8}}and4.79\times {{10}^{-13}}$ respectively.

- Therefore for tribasic acid
\[=\dfrac{{{K}_{a1}}\text{ }{{K}_{a2}}\text{ }{{K}_{a3}}}{{{K}_{a1}}\text{ }{{K}_{a2}}\text{ }{{K}_{a3}}+{{K}_{a1}}\text{ }{{K}_{a2}}\text{ }\!\![\!\!\text{ }{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{+}}}\text{ }\!\!]\!\!\text{ +}{{K}_{a1}}\text{ }\!\![\!\!\text{ }{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{+}}}{{\text{ }\!\!]\!\!\text{ }}^{\text{2}}}+{{\text{ }\!\![\!\!\text{ }{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{+}}}\text{ }\!\!]\!\!\text{ }}^{3}}}\]

- Substitute all the known values in the above equation to get the value.
     \[\begin{align}
  & f=\dfrac{{{K}_{a1}}\text{ }{{K}_{a2}}\text{ }{{K}_{a3}}}{{{K}_{a1}}\text{ }{{K}_{a2}}\text{ }{{K}_{a3}}+{{K}_{a1}}\text{ }{{K}_{a2}}\text{ }\!\![\!\!\text{ }{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{+}}}\text{ }\!\!]\!\!\text{ +}{{K}_{a1}}\text{ }\!\![\!\!\text{ }{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{+}}}{{\text{ }\!\!]\!\!\text{ }}^{\text{2}}}+{{\text{ }\!\![\!\!\text{ }{{\text{H}}_{\text{3}}}{{\text{O}}^{\text{+}}}\text{ }\!\!]\!\!\text{ }}^{3}}} \\
 & =\dfrac{7.59\times {{10}^{-3}}\times 6.17\times {{10}^{-8}}\times 4.79\times {{10}^{-13}}}{7.59\times {{10}^{-3}}\times 6.17\times {{10}^{-8}}\times 4.79\times {{10}^{-13}}+7.59\times {{10}^{-3}}\times 6.17\times {{10}^{-8}}\times {{10}^{-7}}+7.59\times {{10}^{-3}}\times 6.17\times {{10}^{-8}}\times {{10}^{-7}}+7.59\times {{10}^{-3}}\times {{({{10}^{-7}})}^{2}}} \\
 & =1.83\times {{10}^{-6}} \\
\end{align}\]

- Assume the solubility product of Zinc phosphate ($Z{{n}_{3}}{{(P{{O}_{4}})}_{2}}$ ) is SM.
\[\begin{align}
  & [Z{{n}^{2+}}]=3SM \\
 & [PO_{4}^{3-}]=f\times 2S \\
\end{align}\]

- The formula of solubility product of zinc sulphate is
\[\begin{align}
 & {{K}_{sp}}=[Z{{n}^{2+}}]{{[PO_{4}^{3-}]}^{2}} \\
 & 9.1\times {{10}^{-33}}={{(3S)}^{3}}\times {{(f\times 2S)}^{2}} \\
 & 9.1\times {{10}^{-33}}={{(3S)}^{3}}\times {{(1.83\times {{10}^{-6}}\times 2S)}^{2}} \\
 & S=3.0\times {{10}^{-5}}M \\
\end{align}\]

- Thus the solubility product of the $Z{{n}_{3}}{{(P{{O}_{4}})}_{2}}$ is $3.0\times {{10}^{-5}}M.$
- But we have to calculate the concentration of the zinc.
\[[Z{{n}^{2+}}] = 3S = 3\times 3.0\times {{10}^{-5}}=9\times {{10}^{-5}}M\]
- Therefore the concentration of zinc is $9\times {{10}^{-5}}M$ .
So, the correct answer is “Option D”.

Note: ${{H}_{3}}P{{O}_{4}}$ is called phosphoric acid and it is a tribasic acid because it releases three hydrogen ions when dissolved in water. When phosphoric acid reacts with zinc it forms a zinc phosphate compound as the product.