Exercises: Solubility, Ksp and precipitation

At $25\,^{\circ}\mathrm{C}$, the solubility product of barium sulfate, $\mathrm{BaSO_4}$, is $K_{\mathrm{ps}} = 1.1 \times 10^{-10}$. Assuming ideal behaviour in aqueous solution and no other side reactions: a) Write the equation for the dissolution equilibrium of the solid and deduce the algebraic relationship between $K_{\mathrm{ps}}$ and the molar solubility, $s$. b) Calculate the molar solubility of the salt in pure water at that temperature and express this solubility in grams per litre. Data: atomic masses: $\mathrm{Ba} = 137.3\,\text{u}$; $\mathrm{S} = 32.1\,\text{u}$; $\mathrm{O} = 16.0\,\text{u}$. Treat the solutions as ideal and ignore any side equilibria not stated.

Calcium fluoride, $\mathrm{CaF_2}$, is a sparingly soluble salt of type $\mathrm{AB_2}$ whose solubility product at $25\,^{\circ}\mathrm{C}$ is $K_{\mathrm{ps}} = 3.9 \times 10^{-11}$. a) Write the heterogeneous dissolution equilibrium and obtain the expression relating $K_{\mathrm{ps}}$ to the molar solubility $s$ in pure water. b) Determine, with justification, the concentration of fluoride ions, $\mathrm{F^-}$, in a saturated solution at $25\,^{\circ}\mathrm{C}$ and calculate the maximum mass of $\mathrm{CaF_2}$ that can dissolve in $500\,\text{mL}$ of distilled water. Data: atomic masses: $\mathrm{Ca} = 40.1\,\text{u}$; $\mathrm{F} = 19.0\,\text{u}$. Assume the solutions are ideal and ignore any secondary equilibria not stated.

You have a saturated solution of lead(II) chloride, $\mathrm{PbCl_2}$, at $25\,^{\circ}\mathrm{C}$, a salt whose solubility product is $K_{\mathrm{ps}} = 1.7 \times 10^{-5}$. a) Calculate the molar solubility of $\mathrm{PbCl_2}$ in pure water at that temperature. b) Calculate the molar solubility of this salt in a $0.10\,\text{mol}\times\text{L}^{-1}$ aqueous solution of sodium chloride, $\mathrm{NaCl}$, assuming complete dissociation of the latter salt. Compare both results, justifying the observed change using Le Chatelier's principle. If you use a common-ion approximation, check that the contribution of the dissolved salt to the initial chloride concentration is less than 5 %. Assume the solutions are ideal and ignore any secondary equilibria not stated.

In the laboratory, a precipitation test is planned by mixing two homogeneous aqueous solutions according to the experimental scheme shown below:

Exactly $50.0\,\text{mL}$ of an aqueous solution of silver nitrate, $\mathrm{AgNO_3}$, of concentration $0.010\,\text{mol}\times\text{L}^{-1}$ are mixed with $50.0\,\text{mL}$ of an aqueous solution of sodium chloride, $\mathrm{NaCl}$, of concentration $0.020\,\text{mol}\times\text{L}^{-1}$, at $25\,^{\circ}\mathrm{C}$. Assuming additive volumes: a) Calculate the ionic reaction quotient, $Q_{\mathrm{ps}}$, immediately after mixing and before any precipitation takes place, and justify whether silver chloride, $\mathrm{AgCl}$, will precipitate. b) Calculate the residual concentration of silver ions, $\mathrm{Ag^+}$, remaining in the solution once the final precipitation equilibrium has been reached. Datum at $25\,^{\circ}\mathrm{C}$: $K_{\mathrm{ps}}(\mathrm{AgCl}) = 1.8 \times 10^{-10}$. Assume the solutions are ideal and ignore any secondary equilibria not stated.

An aqueous solution simultaneously contains iodide ions, $\mathrm{I^-}$, and chloride ions, $\mathrm{Cl^-}$, both at a concentration of $0.010\,\text{mol}\times\text{L}^{-1}$. A concentrated solution of silver nitrate, $\mathrm{AgNO_3}$, is added slowly, drop by drop, with no appreciable change in total volume. a) State, with the appropriate calculation, which of the two silver halides will begin to precipitate first. b) Calculate the concentration of $\mathrm{I^-}$ ions remaining in solution at the exact moment the second halide begins to precipitate. Data at $25\,^{\circ}\mathrm{C}$: $K_{\mathrm{ps}}(\mathrm{AgI}) = 8.3 \times 10^{-17}$; $K_{\mathrm{ps}}(\mathrm{AgCl}) = 1.8 \times 10^{-10}$. Assume the solutions are ideal and ignore any secondary equilibria not stated.

There is a common misconception that the solubility of all salts always increases when the temperature is raised. To analyse this rigorously from a physical-chemistry viewpoint: a) Apply Le Chatelier's principle to the heterogeneous equilibrium $\mathrm{M_m X_n (s)} \rightleftharpoons m\,\mathrm{M^{n+}(aq)} + n\,\mathrm{X^{m-}(aq)}$, distinguishing the cases in which the standard enthalpy of dissolution is endothermic ($\Delta H^{\circ}_{\mathrm{dis}} > 0$) or exothermic ($\Delta H^{\circ}_{\mathrm{dis}} < 0$). b) It is known experimentally that for cerium(III) sulfate, $\mathrm{Ce_2(SO_4)_3}$, dissolution in water is strongly exothermic. Deduce how the value of its constant $K_{\mathrm{ps}}$ and its solubility will change on heating the mixture, and state a physical method to increase the amount of precipitate obtained from a saturated solution of this salt. Assume the solid phase is retained and that the enthalpy of dissolution keeps its sign over the range studied.

Magnesium hydroxide, $\mathrm{Mg(OH)_2}$, is a sparingly soluble compound commonly used as a stomach antacid. At $25\,^{\circ}\mathrm{C}$, its solubility product is $K_{\mathrm{ps}} = 5.6 \times 10^{-12}$. In an aqueous solution containing an initial magnesium ion concentration $[\mathrm{Mg^{2+}}] = 0.020\,\text{mol}\times\text{L}^{-1}$: a) Determine the value of $\mathrm{pH}$ above which the formation of a white precipitate of $\mathrm{Mg(OH)_2}$ will begin to be observed as a strong base is gradually added. b) Calculate the value of $\mathrm{pH}$ needed for the residual concentration of $\mathrm{Mg^{2+}}$ ions in solution to be reduced to $0.10\,\%$ of its initial value (quantitative precipitation of the cation). Datum at $25\,^{\circ}\mathrm{C}$: $K_w = 1.0 \times 10^{-14}$. Assume the change in volume during the pH adjustment is negligible. Assume the solutions are ideal and ignore any secondary equilibria not stated.

An aqueous waste solution from a galvanic process contains a mixture of $\mathrm{Fe^{3+}}$ and $\mathrm{Mg^{2+}}$ cations, both at a concentration of $0.050\,\text{mol}\times\text{L}^{-1}$ at $25\,^{\circ}\mathrm{C}$. The aim is to separate both ions selectively by adjusting the $\mathrm{pH}$ of the medium in a controlled way through the addition of a strong base. a) Calculate the $\mathrm{pH}$ range in which practically all the iron(III) precipitates as iron(III) hydroxide, $\mathrm{Fe(OH)_3}$, without any magnesium hydroxide, $\mathrm{Mg(OH)_2}$, beginning to precipitate. Consider the separation quantitative when $[\mathrm{Fe^{3+}}] \le 1.0 \times 10^{-5}\,\text{mol}\times\text{L}^{-1}$. b) Justify whether adding a buffer solution of $\mathrm{NH_3}/\mathrm{NH_4^+}$ that fixes the $\mathrm{pH}$ at $9.00$ is suitable for carrying out this clean separation. Data at $25\,^{\circ}\mathrm{C}$: $K_{\mathrm{ps}}[\mathrm{Fe(OH)_3}] = 2.8 \times 10^{-39}$; $K_{\mathrm{ps}}[\mathrm{Mg(OH)_2}] = 5.6 \times 10^{-12}$; $K_w = 1.0 \times 10^{-14}$. Assume the change in volume during the pH adjustment is negligible. Assume the solutions are ideal and ignore any secondary equilibria not stated.

$200\,\text{mL}$ of a $0.050\,\text{mol}\times\text{L}^{-1}$ aqueous solution of lead(II) nitrate, $\mathrm{Pb(NO_3)_2}$, are mixed with $300\,\text{mL}$ of a $0.10\,\text{mol}\times\text{L}^{-1}$ aqueous solution of sodium sulfate, $\mathrm{Na_2SO_4}$, at $25\,^{\circ}\mathrm{C}$. Assuming additive volumes: a) Show formally that a precipitate of lead(II) sulfate, $\mathrm{PbSO_4}$, forms, and calculate the mass in grams of the dry precipitate obtained after filtering. b) Calculate the final molar concentration of all the ions present in the resulting solution at equilibrium ($\mathrm{Pb^{2+}}$, $\mathrm{SO_4^{2-}}$, $\mathrm{Na^+}$ and $\mathrm{NO_3^-}$). Data: $K_{\mathrm{ps}}(\mathrm{PbSO_4}) = 1.6 \times 10^{-8}$ at $25\,^{\circ}\mathrm{C}$; atomic masses: $\mathrm{Pb} = 207.2\,\text{u}$; $\mathrm{S} = 32.1\,\text{u}$; $\mathrm{O} = 16.0\,\text{u}$. Assume the solutions are ideal and ignore any secondary equilibria not stated.

Topic challenge. Why do boilers in your area become blocked? Act as a technical adviser and investigate the impact of domestic and industrial water hardness.

1. Real data: Look up in the EPA (Environmental Protection Agency) drinking water reports, on the Uisce Éireann (Irish Water) website or in your local authority's water supply data the concentration of $Ca^{2+}$ (in $mg/L$) and estimate an average concentration of $[CO_3^{2-}] = 1.2 \times 10^{-4}\text{ M}$.

2. Equilibrium and precipitation: Set out the dynamic dissolution equilibrium of calcium carbonate ($K_{ps} = 3.4 \times 10^{-9}$). Convert your data to molarity, calculate the reaction quotient $Q$ at $25\text{ }^\circ\text{C}$ and predict whether the system precipitates spontaneously.

3. Le Chatelier at industrial level: Analyse the equilibrium $Ca^{2+}(aq) + 2HCO_3^-(aq) \rightleftharpoons CaCO_3(s) + CO_2(g) + H_2O(l)$ (endothermic process). Justify, using Le Chatelier's principle, how the rise in temperature in industrial heat exchangers and the degassing of $CO_2$ affect it. Suggest a domestic acidic reagent to shift it in the reverse direction and dissolve the limescale.

4. Final product: Produce a one-page executive report with the detailed calculations ($Q$ versus $K_{ps}$), the thermodynamic justification and your maintenance protocol.

You will be assessed on:

• Rigour in the molar calculation of concentrations, determination of $Q$ and critical comparison with $K_{ps}$.

• Correct application of Le Chatelier's principle to predict shifts in heterogeneous dynamic equilibria.

• Coherence and chemical feasibility of the technical solution proposed to mitigate scaling.