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Chapter 17: Problem 107

The value of \(K_{s p}\) for \(\mathrm{Mg}_{3}\left(\mathrm{AsO}_{4}\right)_{2}\) is \(2.1 \times 10^{-20}\) . The \(\mathrm{AsO}_{4}^{3-}\) ion is derived from the weak acid \(\mathrm{H}_{3} \mathrm{AsO}_{4}\left(\mathrm{pK}_{a 1}=\right.\) \(2.22 ; \mathrm{p} K_{a 2}=6.98 ; \mathrm{p} K_{a 3}=11.50 )\) . (a) Calculate the molar solubility of \(\mathrm{Mg}_{3}\left(\mathrm{AsO}_{4}\right)_{2}\) in water. (b) Calculate the pH of a saturated solution of \(\mathrm{Mg}_{3}\left(\mathrm{AsO}_{4}\right)_{2}\) in water.

Short Answer

Expert verified
(a) The molar solubility of Mg₃(AsO₄)₂ in water is approximately \(4.07 \times 10^{-7} M\). (b) The pH of a saturated solution of Mg₃(AsO₄)₂ in water is approximately 5.41.

Step by step solution

01

Set up the solubility equilibrium expression

Write the balanced dissolution equation for Mg₃(AsO₄)₂ and the corresponding solubility equilibrium constant expression involving the Ksp: Mg₃(AsO₄)₂(s) ⇌ 3Mg²⁺(aq) + 2AsO₄³⁻(aq) Ksp = [Mg²⁺]³[AsO₄³⁻]²
02

Assume molar solubility

Let's assume the molar solubility of Mg₃(AsO₄)₂ is 's': for every mole of the salt dissolves, we obtain 3 moles of Mg²⁺ and 2 moles of AsO₄³⁻ ions. Substitute this into the Ksp equation: Ksp = (3s)³(2s)²
03

Solve for 's'

We have the value of Ksp, which is 2.1 x 10⁻²⁰, so we can solve for 's': 2.1 x 10⁻²⁰ = (3s)³(2s)² Solve for s: s ≈ 4.07 x 10⁻⁷ M This is the molar solubility of Mg₃(AsO₄)₂ in water.
04

Find the concentrations of H₃AsO₄ species

The AsO₄³⁻ ion is derived from the weak acid H₃AsO₄. Hence, we can find the concentrations of the individual species of H₃AsO₄: H₃AsO₄ ⇌ H⁺ + H₂AsO₄⁻ with Ka1 = 10⁻².²² H₂AsO₄⁻ ⇌ H⁺ + HAsO₄²⁻ with Ka2 = 10⁻⁶.⁹⁸ HAsO₄²⁻ ⇌ H⁺ + AsO₄³⁻ with Ka3 = 10⁻¹¹.⁵⁰ At equilibrium, AsO₄³⁻ concentration is s ≈ 2s (from Step 3 result) We can now use the given Ka values and the equilibrium concentrations to set up and solve a system of equations for the H⁺ concentration.
05

Calculate the H⁺ concentration

Use the Ka expressions to find the H⁺ concentration: Ka1 = [H⁺][H₂AsO₄⁻]/[H₃AsO₄] Ka2 = [H⁺][HAsO₄²⁻]/[H₂AsO₄⁻] Ka3 = [H⁺][AsO₄³⁻]/[HAsO₄²⁻] Substitute the equilibrium concentrations s and 2s as well as pKãs (Ka = 10^{-pKa}): [H⁺]²(10^{11.50 -2.22}) ≈ 2s(10^{2.22 - 6.98}[H⁺] + 10^{-6.98}[H₂AsO₄⁻]) Solve for [H⁺]: [H⁺] ≈ 3.87 x 10⁻⁶ M
06

Calculate the pH of the saturated solution

Now we can determine the pH of the saturated solution using the simple formula: pH = -log[H⁺] pH ≈ 5.41 Thus, the pH of a saturated solution of Mg₃(AsO₄)₂ in water is approximately 5.41.

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Most popular questions from this chapter

A buffer is prepared by adding 20.0 g of sodium acetate \(\left(\mathrm{CH}_{3} \mathrm{COONa}\right)\) to 500 \(\mathrm{mL}\) of a 0.150 \(\mathrm{M}\) acetic acid \(\left(\mathrm{CH}_{3} \mathrm{COOH}\right)\) solution. (a) Determine the pH of the buffer. (b) Write the complete ionic equation for the reaction that occurs when a few drops of hydrochloric acid are added to the buffer. (c) Write the complete ionic equation for the reaction that occurs when a few drops of sodium hydroxide solution are added to the buffer.

Fluoridation of drinking water is employed in many places to aid in the prevention of tooth decay. Typically. the F- ion concentration is adjusted to about 1 ppm. Some water supplies are also "hard"; that is, they contain certain cations such as \(\mathrm{Ca}^{2+}\) that interfere with the action of soap. Consider a case where the concentration of \(\mathrm{Ca}^{2+}\) is 8 ppm. Could a precipitate of \(\mathrm{CaF}_{2}\) form under these conditions? (Make any necessary approximations.)

The solubility of \(\mathrm{CaCO}_{3}\) is pH dependent. (a) Calculate the molar solubility of \(\mathrm{CaCO}_{3}\left(K_{s p}=4.5 \times 10^{-9}\right)\) neglecting the acid-base character of the carbonate ion. (b) Use the \(K_{b}\) expression for the \(\mathrm{CO}_{3}^{2-}\) ion to determine the equilibrium constant for the reaction $$\mathrm{CaCO}_{3}(s)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons_{\mathrm{Ca}^{2+}(a q)+\mathrm{HCO}_{3}^{-}(a q)+\mathrm{OH}^{-}(a q)}$$ (c) If we assume that the only sources of \(\mathrm{Ca}^{2+}, \mathrm{HCO}_{3}^{-}\) and \(\mathrm{OH}^{-}\) ions are from the dissolution of \(\mathrm{CaCO}_{3},\) what is the molar solubility of \(\mathrm{CaCO}_{3}\) using the equilibrium expression from part (b)? \((\boldsymbol{d} )\)What is the molar solubility of \(\mathrm{CaCO}_{3}\) at the pH of the ocean \((8.3) ?(\mathbf{e})\) If the \(\mathrm{pH}\) is buffered at \(7.5,\) what is the molar solubility of \(\mathrm{CaCO}_{3} ?\)

Assume that 30.0 \(\mathrm{mL}\) of a 0.10 \(\mathrm{M}\) solution of a weak base \(\mathrm{B}\) that accepts one proton is titrated with a 0.10\(M\) solution of the monoprotic strong acid HA. (a) How many moles of HA have been added at the equivalence point? (b) What is the predominant form of B at the equivalence point? (a) Is the pH \(7,\) less than \(7,\) or more than 7 at the equivalence point?\( (\mathbf{d} )\) Which indicator, phenolphthalein or methyl red, is likely to be the better choice for this titration?

Predict whether the equivalence point of each of the following titrations is below, above, or at \(\mathrm{pH} 7 :\) (a) formic acid titrated with \(\mathrm{NaOH},(\mathbf{b})\) calcium hydroxide titrated with perchloric acid, (c) pyridine titrated with nitric acid.
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