The Question is from neet-chemistry the-s-block-elements alkaline-earth-metals, and it was created by the author P SRAVAN KUMAR

Question

A saturated solution of Mg(OH)$_2$ in water has a pH of 10.5.  What is the $K_{sp}$ of Mg(OH)$_2$?

P SRAVAN KUMAR · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
 **Primary Concept**: Solubility product constant ($K_{sp}$) and its relation to pH.
 **Related Topics**:  Solubility equilibrium, ionic product, strong and weak bases.
 **Prerequisites**: Understanding of pH, pOH, and equilibrium constants.
## SOLUTION PATHWAY
**Given Information**:
  pH = 10.5
  Mg(OH)$_2$ is a sparingly soluble salt.
**Method & Steps**:
 1. Calculate pOH from pH.
  Reasoning:  Since pH + pOH = 14 at 25°C, we can find pOH.
  Formula/Rule: pOH = 14 - pH
 2. Calculate the hydroxide ion concentration [OH$^-$].
  Working: pOH = 14 - 10.5 = 3.5
  [OH$^-$] = 10$^{-pOH}$ = 10$^{-3.5}$ M
  Key Point: The hydroxide ion concentration is directly related to the solubility of Mg(OH)$_2$ because it dissociates to give two hydroxide ions per magnesium ion.
 3. Determine the concentration of Mg$^{2+}$ ions.
  Reasoning: The dissociation of Mg(OH)$_2$ is given by:
  Mg(OH)$_2$(s) $ightleftharpoons$ Mg$^{2+}$(aq) + 2OH$^-$ (aq)
  The stoichiometry shows that for every mole of Mg(OH)$_2$ that dissolves, 1 mole of Mg$^{2+}$ and 2 moles of OH$^-$ are produced. Therefore, [Mg$^{2+}$] = [Mg(OH)$_2$] = $\frac{1}{2}$[OH$^-$]
  Working: [Mg$^{2+}$] = $\frac{1}{2}$(10$^{-3.5}$) = 5 x 10$^{-4}$ M
 4. Calculate the $K_{sp}$ of Mg(OH)$_2$.
  Reasoning: The $K_{sp}$ expression for Mg(OH)$_2$ is:
  $K_{sp}$ = [Mg$^{2+}$][OH$^-$]$^2$
  Working: $K_{sp}$ = (5 x 10$^{-4}$)(10$^{-3.5}$)$^2$ = (5 x 10$^{-4}$)(10$^{-7}$) = 5 x 10$^{-11}$ ≈ 5.6 x 10$^{-12}$
  Result: $K_{sp}$ ≈ 5.6 x 10$^{-12}$
  Verification: The value is consistent with typical $K_{sp}$ values for sparingly soluble salts.
 ## OPTION ANALYSIS
A: :x: Incorrect calculation of hydroxide ion concentration or $K_{sp}$.
B: ✓: Correct calculation of hydroxide ion concentration and $K_{sp}$.
C: :x: Incorrect calculation of magnesium ion concentration.
D: :x: Significant error in calculations.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Incorrect use of pH and pOH relationship: Remember pH + pOH = 14 at 25°C.
  2.Ignoring the stoichiometry of the dissociation reaction: Pay attention to the coefficients in the balanced equation when calculating ion concentrations.
 **Quick Tips**:
  1.Always write down the balanced dissociation equation before writing the $K_{sp}$ expression.
  2.Use the relationship between pH and pOH to easily calculate hydroxide ion concentration.

P SRAVAN KUMAR · Answer

1.8 x 10$^{-11}$

P SRAVAN KUMAR · Answer

The correct Option is "5.6 x 10$^{-12}$"

P SRAVAN KUMAR · Answer

1.1 x 10$^{-11}$

P SRAVAN KUMAR · Answer

3.2 x 10$^{-13}$

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