Key Concepts and Summary
The equilibrium constant for an equilibrium involving the precipitation or dissolution of a slightly soluble ionic solid is called the solubility product, $K_{sp}$, of the solid. For a heterogeneous equilibrium involving the slightly soluble solid $M_pX_q$ and its ions $M^{m+}$ and $X^{n-}$:
$$M_pX_q\;(s)⇌pM^{m+}\;(aq)+qX^{n-}\;(aq)$$the solubility product expression is:
$$K_{sp}=[M^{m+}]^p[X^{n-}]^q$$The solubility product of a slightly soluble electrolyte can be calculated from its solubility; conversely, its solubility can be calculated from its $K_{sp}$, provided the only significant reaction that occurs when the solid dissolves is the formation of its ions.
A slightly soluble electrolyte begins to precipitate when the magnitude of the reaction quotient for the dissolution reaction exceeds the magnitude of the solubility product. Precipitation continues until the reaction quotient equals the solubility product.
Practice Questions
Complete the changes in concentrations for each of the following reactions:
| (a) | $AgI(s)$ | ⇌ | $Ag^+(aq)$ | $+$ | $I^-(aq)$ |
| $x$ | ________ |
| (b) | $CaCO_3(s)$ | ⇌ | $Ca^{2+}(aq)$ | $+$ | $CO_3^{2-}(aq)$ |
| ________ | $x$ |
| (c) | $Mg(OH)_2(s)$ | ⇌ | $Mg^{2+}(aq)$ | $+$ | $2OH^-(aq)$ |
| $x$ | ________ |
| (d) | $Mg_3(PO_4)_2(s)$ | ⇌ | $3Mg^{2+}(aq)$ | $+$ | $2PO_4^{3-}(aq)$ |
| $x$ | ________ |
| (e) | $Ca_5(PO_4)_3OH(s)$ | ⇌ | $5Ca^{2+}(aq)$ | $+$ | $3PO_4^{3-}(aq)$ | $+$ | $OH^-(aq)$ |
| ________ | ________ | $x$ |
Solution
| (a) | $AgI(s)$ | ⇌ | $Ag^+(aq)$ | $+$ | $I^-(aq)$ |
| $x$ | ___x___ |
| (b) | $CaCO_3(s)$ | ⇌ | $Ca^{2+}(aq)$ | $+$ | $CO_3^{2-}(aq)$ |
| ___$x$___ | $x$ |
| (c) | $Mg(OH)_2(s)$ | ⇌ | $Mg^{2+}(aq)$ | $+$ | $2OH^-(aq)$ |
| $x$ | ___$2x$___ |
| (d) | $Mg_3(PO_4)_2(s)$ | ⇌ | $3Mg^{2+}(aq)$ | $+$ | $2PO_4^{3-}(aq)$ |
| $x$ | ___$\frac{2}{3}x$___ |
| (e) | $Ca_5(PO_4)_3OH(s)$ | ⇌ | $5Ca^{2+}(aq)$ | $+$ | $3PO_4^{3-}(aq)$ | $+$ | $OH^-(aq)$ |
| __$5x$___ | ___$3x$____ | $x$ |
How do the concentrations of Ag+ and $CrO_4^{2-}$ in a saturated solution above 1.0 g of solid Ag2CrO4 change when 100 g of solid Ag2CrO4 is added to the system? Explain.
Solution
There is no change. A solid has an activity of 1 whether there is a little or a lot.
How do the concentrations of Pb2+ and S2– change when K2S is added to a saturated solution of PbS?
What additional information do we need to answer the following question: How is the equilibrium of solid silver bromide with a saturated solution of its ions affected when the temperature is raised?
Solution
The solubility of silver bromide at the new temperature must be known. Normally the solubility increases and some of the solid silver bromide will dissolve.
Which of the following slightly soluble compounds has a solubility greater than that calculated from its solubility product because of hydrolysis of the anion present: CoSO3, CuI, PbCO3, PbCl2, Tl2S, KClO4?
Which of the following slightly soluble compounds has a solubility greater than that calculated from its solubility product because of hydrolysis of the anion present: AgCl, BaSO4, CaF2, Hg2I2, MnCO3, ZnS, PbS?
Solution
CaF2, MnCO3, and ZnS
Write the ionic equation for dissolution and the solubility product (Ksp) expression for each of the following slightly soluble ionic compounds:
(a) PbCl2
(b) Ag2S
(c) Sr3(PO4)2
(d) SrSO4
Write the ionic equation for the dissolution and the Ksp expression for each of the following slightly soluble ionic compounds:
(a) LaF3
(b) CaCO3
(c) Ag2SO4
(d) Pb(OH)2
Solution
(a) $LaF_3(s)⇌La^{3+}(aq)+3F^-(aq)\qquad K_{sp}=[La^{3+}]^3$;
(b) $CaCO_3(s)⇌Ca^{2+}(aq)+CO_3^{2-}(aq)\qquad K_{sp}=[Ca^{2+}][CO_3^{2-}]$;
(c) $Ag_2SO_4(s)⇌2Ag^+(aq)+SO_4^{2-}(aq)\qquad K_{sp}=[Ag^+]^2[SO_4^{2-}]$;
(d) $Pb(OH)_2(s)⇌Pb^{2+}(aq)+2OH^-(aq)\qquad K_{sp}=[Pb^{2+}][OH^-]^2$
The Handbook of Chemistry and Physics gives solubilities of the following compounds in grams per 100 mL of water. Because these compounds are only slightly soluble, assume that the volume does not change on dissolution and calculate the solubility product for each.
(a) BaSiF6, 0.026 g/100 mL (contains $SiF_6^{2-}$ ions)
(b) Ce(IO3)4, $1.5×10^{-2}$ g/100 mL
(c) Gd2(SO4)3, 3.98 g/100 mL
(d) (NH4)2PtBr6, 0.59 g/100 mL (contains $PtBr_6^{2-}$ ions)
The Handbook of Chemistry and Physics gives solubilities of the following compounds in grams per 100 mL of water. Because these compounds are only slightly soluble, assume that the volume does not change on dissolution and calculate the solubility product for each.
(a) BaSeO4, 0.0118 g/100 mL
(b) Ba(BrO3)2·H2O, 0.30 g/100 mL
(c) NH4MgAsO4·6H2O, 0.038 g/100 mL
(d) La2(MoO4)3, 0.00179 g/100 mL
Solution
(a)$1.77×10^{-7}$;
(b) $1.6×10^{-6}$;
(c) $2.2×10^{-9}$;
(d) $7.91×10^{-22}$
Use solubility products and predict which of the following salts is the most soluble, in terms of moles per liter, in pure water: CaF2, Hg2Cl2, PbI2, or Sn(OH)2.
Assuming that no equilibria other than dissolution are involved, calculate the molar solubility of each of the following from its solubility product:
(a) KHC4H4O6
(b) PbI2
(c) Ag4[Fe(CN)6], a salt containing the $Fe(CN)_6^{4-}$ ion
(d) Hg2I2
Solution
(a) $2×10^{-2}\;M$;
(b) $1.5×10^{-3}\;M$;
(c) $2.27×10^{-9}\;M$;
(d) $2.2×10^{-10}\;M$;
Assuming that no equilibria other than dissolution are involved, calculate the molar solubility of each of the following from its solubility product:
(a) Ag2SO4
(b) PbBr2
(c) AgI
(d) CaC2O4·H2O
Assuming that no equilibria other than dissolution are involved, calculate the concentration of all solute species in each of the following solutions of salts in contact with a solution containing a common ion. Show that changes in the initial concentrations of the common ions can be neglected.
(a) AgCl(s) in 0.025 M NaCl
(b) CaF2(s) in 0.00133 M KF
(c) Ag2SO4(s) in 0.500 L of a solution containing 19.50 g of K2SO4
(d) Zn(OH)2(s) in a solution buffered at a pH of 11.45
Solution
(a) $6.4×10^{-9}\;M=[Ag^+]$, $[Cl^-]=0.025\;M$.
Check: $\frac{6.4×10^{-9}\;M}{0.025\;M}\times 100\text{%}=2.6×10^{-5}$ %, an insignificant change;
(b) $2.2×10^{-5}\;M=[Ca^{2+}]$, $[F^-]=0.0013\;M$.
Check: $\frac{2.26×10^{-5}\;M}{0.00133\;M}\times 100\text{%}=1.70\text{%}$. This value is less than 5% and can be ignored.
(c) $0.2238\;M=[SO_4^{2-}]$; $[Ag^+]=7.4×10^{-3}\;M$.
Check: $\frac{3.7×10^{-3}\;M}{0.2238\;M}\times 100\text{%}=1.65$; the condition is satisfied.
(d) [OH–] = $2.8×10^{-3}\;M$; $5.7×10^{-12}\;M=[Zn^{2+}]$.
Check:$\frac{5.7×10^{-12}\;M}{2.8×10^{-3}\;M}\times 100\text{%}=2.0×10^{-7}\text{%}$; $x$ is less than 5% of $[OH^-]$ and is therefore negligible.
Assuming that no equilibria other than dissolution are involved, calculate the concentration of all solute species in each of the following solutions of salts in contact with a solution containing a common ion. Show that changes in the initial concentrations of the common ions can be neglected.
(a) TlCl(s) in 1.250 M HCl
(b) PbI2(s) in 0.0355 M CaI2
(c) Ag2CrO4(s) in 0.225 L of a solution containing 0.856 g of K2CrO4
(d) Cd(OH)2(s) in a solution buffered at a pH of 10.995
Assuming that no equilibria other than dissolution are involved, calculate the concentration of all solute species in each of the following solutions of salts in contact with a solution containing a common ion. Show that it is not appropriate to neglect the changes in the initial concentrations of the common ions.
(a) TlCl(s) in 0.025 M TlNO3
(b) BaF2(s) in 0.0313 M KF
(c) MgC2O4 in 2.250 L of a solution containing 8.156 g of Mg(NO3)2
(d) Ca(OH)2(s) in an unbuffered solution initially with a pH of 12.700
Solution
(a) [Cl–] = $7.6×10^{-3}\;M$.
Check:$\frac{7.6×10^{-3}}{0.025}\times 100\text{%}=30\text{%}$. This value is too large to drop x. Therefore solve by using the quadratic equation:
[Ti+] = $3.1×10^{-2}\;M$
[Cl–] = $6.1×10^{-3}\;M$
(b) [Ba2+] = $7.7×10^{-4}\;M$
Check: $\frac{7.7×10^{-4}\;M}{0.0313}\times 100\text{%}=2.4\text{%}$
Therefore, the condition is satisfied.
[Ba2+] = $7.7×10^{-4}\;M$;
[F–] = 0.0321 M;
(c) Mg(NO3)2 = 0.02444 M
$[C_2O_4^{2-}]=2.9×10^{-5}\;M$
Check: $\frac{2.9×10^{-5}\;M}{0.02444\;M}\times 100\text{%}=0.12\text{%}$
The condition is satisfied; the above value is less than 5%.
$[C_2O_4^{2-}]=2.9×10^{-5}\;M$
$[Mg^{2+}]=0.0244\;M$
(d) [OH–] = 0.0501 M
[Ca2+] = $3.15×10^{-3}\;M$
Check: $\frac{3.15×10^{-3}}{0.050}\times 100\text{%}=6.28\text{%}$
This value is greater than 5%, so a more exact method, such as successive approximations, must be used.
[Ca2+] = $2.8×10^{-3}\;M$
[OH–] = $0.053×10^{-2}\;M$
Explain why some of the changes in concentrations of the common ions in the previous cannot be neglected, while all of them could be in the question before that.
Solution
The changes in concentration are greater than 5% and thus exceed the maximum value for disregarding the change in the majority of the options in the previous question (except c). In the question before that all the changes in concentration are less than 5%.
Calculate the solubility of aluminum hydroxide, Al(OH)3, in a solution buffered at pH 11.00.
Refer to this Appendix for solubility products for calcium salts. Determine which of the calcium salts listed is most soluble in moles per liter and which is most soluble in grams per liter.
Solution
CaSO4∙2H2O is the most soluble Ca salt in mol/L, and it is also the most soluble Ca salt in g/L.
Most barium compounds are very poisonous; however, barium sulfate is often administered internally as an aid in the X-ray examination of the lower intestinal tract. This use of BaSO4 is possible because of its low solubility. Calculate the molar solubility of BaSO4 and the mass of barium present in 1.00 L of water saturated with BaSO4.
Public Health Service standards for drinking water set a maximum of 250 mg/L $(2.60×10^{-3}\;M)$ of $SO_4^{2-}$ because of its cathartic action (it is a laxative). Does natural water that is saturated with CaSO4 (“gyp” water) as a result or passing through soil containing gypsum, CaSO4·2H2O, meet these standards? What is the concentration of $SO_4^{2-}$ in such water?
Solution
$4.8×10^{-3}\;M=[SO_4^{2-}]=[Ca^{2+}]$; Since this concentration is higher than $2.60×10^{-3}\;M$, “gyp” water does not meet the standards.
Perform the following calculations:
(a) Calculate [Ag+] in a saturated aqueous solution of AgBr.
(b) What will [Ag+] be when enough KBr has been added to make [Br–] = 0.050 M?
(c) What will [Br–] be when enough AgNO3 has been added to make [Ag+] = 0.020 M?
The solubility product of CaSO4·2H2O is $2.4×10^{-5}$. What mass of this salt will dissolve in 1.0 L of 0.010 M $SO_4^{2-}$?
Solution
Mass (CaSO4·2H2O) = 0.72 g/L
Assuming that no equilibria other than dissolution are involved, calculate the concentrations of ions in a saturated solution of each of the following (see this Appendix for solubility products).
(a) TlCl
(b) BaF2
(c) Ag2CrO4
(d) CaC2O4·H2O
(e) the mineral anglesite, PbSO4
Assuming that no equilibria other than dissolution are involved, calculate the concentrations of ions in a saturated solution of each of the following (see this appendix for solubility products):
(a) AgI
(b) Ag2SO4
(c) Mn(OH)2
(d) Sr(OH)2·8H2O
(e) the mineral brucite, Mg(OH)2
Solution
(a) [Ag+] = [I–] = $1.2×10^{-8}\;M$;
(b) [Ag+] = $2.88×10^{-2}\;M$, $[SO_4^{2-}]=1.44×10^{-2}\;M$;
(c) [Mn2+] = $3.7×10^{-5}\;M$, $[OH^-]=7.4×10^{-5}\;M$;
(d) [Sr2+] = $4.3×10^{-2}\;M$, $[OH^-]=8.6×10^{-2}\;M$;
(e) [Mg2+] = $1.3×10^{-4}\;M$, $[OH^-]=2.6×10^{-4}\;M$;
The following concentrations are found in mixtures of ions in equilibrium with slightly soluble solids. From the concentrations given, calculate Ksp for each of the slightly soluble solids indicated:
(a) AgBr: [Ag+] = $5.7×10^{-7}\;M$, [Br–] = $5.7×10^{-7}\;M$
(b) CaCO3: [Ca2+] = $5.3×10^{-3}\;M$, $[CO_3^{2-}]=9.0×10^{-7}\;M$
(c) PbF2: [Pb2+] = $2.1×10^{-3}\;M$, $[F^-]=4.2×10^{-3}\;M$
(d) Ag2CrO4: [Ag+] = $5.3×10^{-5}\;M$, $[CrO_4^{2-}]=3.2×10^{-3}\;M$
(e) InF3: [In3+] = $2.3×10^{-3}\;M$, $[F^-]=7.0×10^{-3}\;M$
The following concentrations are found in mixtures of ions in equilibrium with slightly soluble solids. From the concentrations given, calculate Ksp for each of the slightly soluble solids indicated:
(a) TlCl: [Tl+] = $1.21×10^{-2}\;M$, $[Cl^-]=1.2×10^{-2}\;M$
(b) Ce(IO3)4: [Ce4+] = $1.8×10^{-4}\;M$, $[IO_3^-]=2.6×10^{-13}\;M$
(c) Gd2(SO4)3: [Gd3+] = 0.132 M, $[SO_4^{2-}]=0.198\;M$
(d) Ag2SO4: [Ag+] = $2.40×10^{-2}\;M$, $[SO_4^{2-}]=2.05×10^{-2}\;M$
(e) BaSO4: [Ba2+] = 0.500 M, $[SO_4^{2-}]=4.6×10^{-8}\;M$
Solution
(a) $1.45×10^{-4}$;
(b) $8.2×10^{-55}$;
(c) $1.35×10^{-4}$;
(d) $1.18×10^{-5}$;
(e) $2.30×10^{-8}$;
Which of the following compounds precipitates from a solution that has the concentrations indicated? (See this Appendix for Ksp values.)
(a) KClO4: [K+] = 0.01 M, $[ClO_4^-]=0.01\;M$
(b) K2PtCl6: [K+] = 0.01 M, $[PtCl_6^{2-}]=0.01\;M$
(c) PbI2: [Pb2+] = 0.003 M, [I–] = $1.3×10^{-3}\;M$
(d) Ag2S: [Ag+] = $1×10^{-10}\;M$, [S2–] = $1×10^{-13}\;M$
Which of the following compounds precipitates from a solution that has the concentrations indicated? (See this Appendix for Ksp values.)
(a) CaCO3: [Ca2+] = 0.003 M, $[CO_3^{2-}]=0.003\;M$
(b) Co(OH)2: [Co2+] = 0.01 M, [OH–] = $1×10^{-7}\;M$
(c) CaHPO4: [Ca2+] = 0.01 M, $[HPO_4^{2-}]=2×10^{-6}\;M$
(d) Pb3(PO4)2: [Pb2+] = 0.01 M, $[PO_4^{3-}]=1×10^{-13}\;M$
Solution
(a) CaCO3 does precipitate.
(b) The compound does not precipitate.
(c) The compound does not precipitate.
(d) The compound precipitates.
Calculate the concentration of sulfate ion when BaSO4 just begins to precipitate from a solution that is 0.0758 M in Ba2+.
Solution
$3.03×10^{-7}$
Calculate the concentration of Sr2+ when SrCrO4 starts to precipitate from a solution that is 0.0025 M in $CrO_4^{2-}$.
Calculate the concentration of $PO_4^{3-}$ when Ag3PO4 starts to precipitate from a solution that is 0.0125 M in Ag+.
Solution
$9.2×10^{-13}\;M$
Calculate the concentration of F– required to begin precipitation of CaF2 in a solution that is 0.010 M in Ca2+.
Calculate the concentration of Ag+ required to begin precipitation of Ag2CO3 in a solution that is $2.50×10^{-6}\;M$ in $CO_3^{2-}$.
Solution
[Ag+] = $1.8×10^{-3}\;M$
What [Ag+] is required to reduce $[CO_3^{2-}]$ to $8.2×10^{-4}\;M$ by precipitation of Ag2CO3?
A volume of 0.800 L of a $2×10^{-4}\;M$ Ba(NO3)2 solution is added to 0.200 L of $5×10^{-4}$ M Li2SO4. Does BaSO4 precipitate? Explain your answer.
Perform these calculations for nickel(II) carbonate. (a) With what volume of water must a precipitate containing NiCO3 be washed to dissolve 0.100 g of this compound? Assume that the wash water becomes saturated with NiCO3 (Ksp = $1.36×10^{-7}$).
(b) If the NiCO3 were a contaminant in a sample of CoCO3 (Ksp = $1.0×10^{-12}$), what mass of CoCO3 would have been lost? Keep in mind that both NiCO3 and CoCO3 dissolve in the same solution.
Solution
(a) 2.28 L; (b) $7.4×10^{-7}$ g