x mg of Mg(OH)$_2$ (molar mass = 58) is required to be dissolved in 1.0 L of water to produce a pH of 10.0 at 298 K. The value of x is ____ mg. (Nearest integer) (Given: Mg(OH)$_2$ is assumed to dissociate completely in H$_2$O)
x mg of Mg(OH)$_2$ (molar mass = 58) is required to be dissolved in 1.0 L of water to produce a pH of 10.0 at 298 K. The value of x is ____ mg. (Nearest integer) (Given: Mg(OH)$_2$ is assumed to dissociate completely in H$_2$O)
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Correct Answer: 3
Solution and Explanation
Solution:
To determine the mass of Mg(OH)2 required for a pH of 10.0, we first ascertain the hydroxide ion concentration [OH−].
1. Calculate pOH from the given pH: pOH = 14.0 - 10.0 = 4.0.
2. Calculate [OH−]: [OH−] = 10−pOH = 10−4.0 = 1.0 × 10−4 M.
3. Given the complete dissociation of Mg(OH)2: Mg(OH)2(s) → Mg2+(aq) + 2OH−(aq), the concentration of OH− is double the molarity of Mg(OH)2. Therefore, 2[Mg(OH)2] = [OH−], which implies [Mg(OH)2] = [OH−] / 2 = 0.5 × 10−4 M.
4. Calculate the moles of Mg(OH)2 needed for a 1.0 L solution: Moles of Mg(OH)2 = [Mg(OH)2] × Volume (L) = 0.5 × 10−4 mol/L × 1.0 L = 0.5 × 10−4 mol.
5. Convert moles to grams using the molar mass of Mg(OH)2 (58 g/mol): Mass = moles × molar mass = 0.5 × 10−4 mol × 58 g/mol = 2.90 × 10−3 g.
6. Convert grams to milligrams: Mass = 2.90 × 10−3 g × 1000 mg/g = 2.90 mg.
Rounded to the nearest integer, x = 3 mg. This value falls within the specified range (3,3).
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