Question

\[M{{(OH)}_{x}}\] has \[{{K}_{sp}}\] = \[27\times {{10}^{-12}}\] and solubility in water is \[{{10}^{-3}}\] M. Calculate the value of x.

Answer 1

Hint: The solubility product is a type of equilibrium constant and its value depends on temperature of the solution. It is denoted with a symbol \[{{K}_{sp}}\]. \[{{K}_{sp}}\] usually increases with an increase in temperature due to increased solubility.

Complete Solution :
Whenever we are going to add \[M{{(OH)}_{x}}\] into water. It is going to convert into ions as follows.
\[M{{(OH)}_{x}}\xrightarrow{{}}{{M}^{x+}}+X\text{ }O{{H}^{-}}\]

- The Solubility product of the above equation is as follows.
\[{{K}_{sp}}=\text{ }\!\![\!\!\text{ }{{M}^{x+}}]\text{ }{{[O{{H}^{-}}]}^{x}}\]
We know that the solubility product for any reaction is equal to the product of concentration of the products.
- In the question it is given that \[{{K}_{sp}}\] = \[27\times {{10}^{-12}}\],\[\text{ }\!\![\!\!\text{ }{{M}^{x+}}]\]= \[{{10}^{-3}}\] M . Substitute all these values in the above formula to get the value of x.
\[\begin{align}
  & 27\times {{10}^{-12}}=({{10}^{-3}}){{(X{{10}^{-3}})}^{x}} \\
 & {{X}^{x}}{{({{10}^{-3}})}^{x+1}}=27\times {{10}^{-12}} \\
 & {{X}^{x}}{{({{10}^{-3}})}^{x+1}}={{(3)}^{3}}\times {{({{10}^{-3}})}^{3+1}} \\
 & X=3 \\
\end{align}\]
- By comparing the both sides in the above expression we will get the value of X = 3.
So, the molecular formula of the compound is \[M{{(OH)}_{3}}\].

Additional Information:
- If the value of a solubility product is small, the lower the solubility. Solubility products are useful in predicting whether a precipitate will form under specified conditions or not.
- Solubility products are also helpful in choosing conditions for the reaction. Under which two chemical substances in solution can be separated by fractional precipitation.
- The solubility product of a number of compounds has been experimentally determined and can be used to calculate solubility at a specified temperature.

Note: Don’t be confused with the terms solubility product and Solubility. Both are not the same.
- Solubility product: The solubility product is a type of equilibrium constant and its value depends on the temperature of the solution.
- Solubility: “The amount of a substance that will dissolve in a given amount of solvent to give a saturated solution under specified conditions”.