5.3 Gravimetric Analysis and Precipitation Equilibria ($K_{sp}$)

Gravimetric Analysis and Precipitation Equilibria ($K_{sp}$)

Gravimetric analysis encompasses quantitative analytical procedures based on measuring the mass of a pure, stoichiometric chemical substance derived from an analyte. Because mass can be measured with an analytical balance to within $\pm 0.1\text{ mg}$ or better without instrument calibration curves, gravimetry is an absolute primary analytical method. On the PRC Chemist Licensure Examination, gravimetric questions focus on stoichiometric factor calculations, precipitation kinetics, purification mechanisms, and chemical equilibria governing solubility.

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1. Principles of Gravimetric Analysis and Gravimetric Factors

In precipitation gravimetry, the analyte in solution is converted into an insoluble precipitate of known or convertible chemical composition, filtered, washed, dried or ignited, and weighed.

Essential Requirements for Successful Gravimetric Precipitates

1. High Insolubility: The precipitate must be sufficiently insoluble that unprecipitated analyte mass remains below the limit of analytical balance detection (typically $< 0.1\text{ mg}$ loss, or recovery $> 99.99\%$).
2. High Purity: The precipitate must be free from coprecipitated matrix impurities and easily purified by washing.
3. Filterability: Particles must be sufficiently large to be completely retained by standard filter paper (e.g., Whatman No. 42 ashless paper) or sintered-glass/porcelain filtering crucibles without clogging.
4. Definite Stoichiometric Composition: The weighed form must possess a known, stable chemical formula that is non-hygroscopic and chemically unreactive with atmospheric gases ($H_2O, CO_2, O_2$).

The Gravimetric Factor ($GF$)

The Gravimetric Factor ($GF$) is the stoichiometric mass ratio that converts the mass of a weighed precipitate into the equivalent mass of the target analyte: $GF = \frac{a \cdot FW(\text{analyte})}{b \cdot FW(\text{precipitate})}$ where $a$ and $b$ are integers chosen to balance the number of moles of the common element (or chemical equivalent) in the numerator and denominator.

$\text{Mass of Analyte} = \text{Mass of Weighed Precipitate} \times GF$ $\%\text{ Analyte} = \frac{\text{Mass of Precipitate} \times GF}{\text{Mass of Sample}} \times 100\%$

Standard Gravimetric Factors in Analytical Chemistry

Worked Example: Gravimetric Iron Determination

A $0.5432\text{ g}$ sample of iron ore is dissolved in hydrochloric acid, oxidized with nitric acid to convert all iron to $\text{Fe}^{3+}$, and precipitated as hydrous ferric oxide ($\text{Fe(OH)}_3 \cdot x\text{H}_2\text{O}$). After filtration and washing, the precipitate is ignited at $900^\circ\text{C}$ to constant mass, yielding $0.2456\text{ g}$ of pure anhydrous ferric oxide ($\text{Fe}_2\text{O}_3$, $FW = 159.69\text{ g/mol}$).

1. Calculate the percentage of elemental $\text{Fe}$ ($FW = 55.845\text{ g/mol}$): $GF_{\text{Fe}} = \frac{2 \times 55.845}{159.69} = \frac{111.69}{159.69} = 0.69942$ $\text{Mass of Fe} = 0.2456\text{ g} \times 0.69942 = 0.17178\text{ g}$ $\%\text{Fe} = \frac{0.17178\text{ g}}{0.5432\text{ g}} \times 100\% = 31.62\%$
2. Calculate the percentage expressed as Magnetite ($\text{Fe}_3\text{O}_4$, $FW = 231.53\text{ g/mol}$): $GF_{\text{Fe}_3\text{O}_4} = \frac{2 \times 231.53}{3 \times 159.69} = \frac{463.06}{479.07} = 0.96658$ $\text{Mass of } \text{Fe}_3\text{O}_4 = 0.2456\text{ g} \times 0.96658 = 0.23739\text{ g}$ $\%\text{Fe}_3\text{O}_4 = \frac{0.23739\text{ g}}{0.5432\text{ g}} \times 100\% = 43.70\%$

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2. Mechanisms of Precipitate Formation: Nucleation vs. Particle Growth

The physical characteristics (crystal size, surface area, filterability) of a precipitate are governed by the kinetics of particle formation.

Von Weimarn's Ratio and Relative Supersaturation ($RSS$)

P.P. von Weimarn discovered that the initial particle size of a precipitate is inversely proportional to the Relative Supersaturation ($RSS$) of the solution during precipitation: $RSS = \frac{Q - S}{S}$ where:

• $Q$ = instantaneous solute concentration when reagents are mixed.
• $S$ = equilibrium molar solubility of the precipitate under experimental conditions.
• $(Q - S)$ = excess supersaturation driving precipitation.

Two Competing Precipitation Mechanisms

1. Nucleation: A process where a small number of ions or molecules (typically 4–20 units) spontaneously coalesce to create stable solid micro-clusters (nuclei).
   • Rate of nucleation: $R_{nucleation} = k_1 \left(\frac{Q - S}{S}\right)^n$ (where $n > 1$).
   • Dominates when $RSS$ is extremely large $\rightarrow$ produces trillions of tiny nuclei, resulting in colloidal suspensions ($1\text{ nm} - 1\ \mu\text{m}$), which pass through filters and are difficult to wash.
2. Particle Growth: Deposition of dissolved solute ions onto pre-existing nuclei or seed crystals.
   • Rate of particle growth: $R_{growth} = k_2 (Q - S)$.
   • Dominates when $RSS$ is low $\rightarrow$ existing nuclei grow into large, dense, easily filterable crystalline precipitates ($> 10\ \mu\text{m}$).

Practical Experimental Conditions for Obtaining Crystalline Precipitates

To maximize crystal size and purity, analysts deliberately maintain $RSS$ as small as possible:

• Minimize $Q$: Add precipitating reagent slowly dropwise with vigorous mechanical stirring, and use dilute solutions.
• Maximize $S$: Carry out precipitation in hot solutions (elevated temperature increases $S$), and adjust $pH$ to maintain moderate solubility during initial mixing.
• Digestion (Ostwald Ripening): Allow the precipitate to stand in contact with the hot mother liquor for 1–2 hours. According to the Kelvin equation, small particles have higher surface curvature, higher chemical potential, and greater solubility than large particles. Small particles dissolve, and their solute recrystallizes onto the surfaces of larger crystals, increasing average grain size and expelling trapped impurities.
• Homogeneous Precipitation: Precipitating agent is generated in situ throughout the entire solution via a slow chemical reaction, maintaining $Q$ barely above $S$ at all times:
  • Generating Hydroxide ($OH^-$): Urea hydrolysis at $90-100^\circ\text{C}$: $(\text{NH}_2)_2\text{CO} + 3\text{H}_2\text{O} \xrightarrow{\Delta} 2\text{NH}_4^+ + \text{CO}_2 + 2\text{OH}^-$ This slowly raises $pH$ uniformly, yielding dense, crystalline basic salts (e.g., for $\text{Al}^{3+}, \text{Fe}^{3+}$) rather than gelatinous hydroxides.
  • Generating Sulfate ($\text{SO}_4^{2-}$): Hydrolysis of dimethyl sulfate or sulfamic acid.
  • Generating Oxalate ($\text{C}_2\text{O}_4^{2-}$): Hydrolysis of dimethyl oxalate.

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3. Coprecipitation Phenomena and Purification

Coprecipitation is the phenomenon where normally soluble chemical species precipitate alongside the analyte precipitate from solution.

Four Primary Mechanisms of Coprecipitation

Peptization and Washing

• Peptization: The process where a coagulated colloidal precipitate reverts back to a dispersed colloidal sol when washed with pure deionized water.
• Mechanism: Washing with pure water dilutes the electrolyte concentration in the electrical double layer. The repulsive counter-ion barrier expands, overcoming attractive van der Waals forces and disintegrating the aggregates into unfilterable particles.
• Prevention: Always wash colloidal precipitates with an aqueous solution of a volatile electrolyte (e.g., dilute $\text{HNO}_3$ for $\text{AgCl}$; $\text{NH}_4\text{NO}_3$ for $\text{Fe(OH)}_3$). The electrolyte preserves ionic strength to prevent peptization, and volatilizes completely during crucible drying or ignition ($> 100^\circ\text{C}$), leaving no non-volatile residue.

Thermal Decomposition and Volatilization Gravimetry

Precipitates are dried or ignited to remove solvent and convert the precipitate into a stoichiometric weighing form:

• Calcium Oxalate Ignition Stages: $\text{CaC}_2\text{O}_4 \cdot \text{H}_2\text{O}_{(s)} \xrightarrow{100-250^\circ\text{C}} \text{CaC}_2\text{O}_{4(s)} + \text{H}_2\text{O}_{(g)}$ $\text{CaC}_2\text{O}_{4(s)} \xrightarrow{400-500^\circ\text{C}} \text{CaCO}_{3(s)} + \text{CO}_{(g)}$ $\text{CaCO}_{3(s)} \xrightarrow{700-900^\circ\text{C}} \text{CaO}_{(s)} + \text{CO}_{2(g)}$
• Combustion Volatilization Analysis (Pregl-Liebig Method): Carbon and hydrogen in organic samples are combusted in pure $\text{O}_2$. Formed $\text{H}_2\text{O}$ is absorbed in anhydrous magnesium perchlorate (Anhydrone), and $\text{CO}_2$ is absorbed in sodium hydroxide on silicate (Ascarite). The mass increase of each absorption tube yields %H and %C.

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4. Precipitation Equilibria: $K_{sp}$, Ionic Strength, and Chemical Effects

Precipitation reactions in aqueous solution are governed by chemical equilibrium.

Solubility Product ($K_{sp}$) and Molar Solubility ($S$)

For a general sparingly soluble salt $\text{M}_a\text{X}_b$: $\text{M}_a\text{X}_{b(s)} \rightleftharpoons a\ \text{M}^{m+}_{(aq)} + b\ \text{X}^{n-}_{(aq)}$ $K_{sp} = [\text{M}^{m+}]^a [\text{X}^{n-}]^b = (aS)^a (bS)^b = a^a b^b S^{a+b}$ $S = \sqrt[a+b]{\frac{K_{sp}}{a^a b^b}}$

• $1:1$ Salt ($\text{AgCl}, \text{BaSO}_4$): $K_{sp} = S^2 \implies S = \sqrt{K_{sp}}$
• $1:2$ Salt ($\text{PbCl}_2, \text{CaF}_2$): $K_{sp} = 4S^3 \implies S = \sqrt[3]{\frac{K_{sp}}{4}}$
• $1:3$ Salt ($\text{BiI}_3, \text{Fe(OH)}_3$): $K_{sp} = 27S^4 \implies S = \sqrt[4]{\frac{K_{sp}}{27}}$
• $2:3$ Salt ($\text{Ca}_3(\text{PO}_4)_2$): $K_{sp} = 108S^5 \implies S = \sqrt[5]{\frac{K_{sp}}{108}}$

Common Ion Effect vs. Diverse Ion (Salt) Effect

• Common Ion Effect: Adding a soluble salt that shares an ion with the precipitate shifts the equilibrium to the left (Le Chatelier's principle), significantly reducing molar solubility ($S$). For example, the solubility of $\text{BaSO}_4$ ($K_{sp} = 1.1 \times 10^{-10}$) in pure water is $1.05 \times 10^{-5}\text{ M}$. In $0.010\text{ M } \text{Na}_2\text{SO}_4$, the solubility drops to $S = \frac{1.1 \times 10^{-10}}{0.010} = 1.1 \times 10^{-8}\text{ M}$ (a 1,000-fold decrease).
• Ionic Strength ($\mu$) and Diverse Ion (Salt) Effect: Adding an "inert" non-common electrolyte (e.g., $\text{KNO}_3$) increases the ionic strength of the solution: $\mu = \frac{1}{2} \sum_{i=1}^n c_i z_i^2$ According to the Debye-Hückel limiting law, increased ionic strength compresses the ionic atmosphere around ions, lowering their activity coefficients ($\gamma_i < 1$): $\log \gamma_i = -0.51 z_i^2 \sqrt{\mu}$ Because the thermodynamic equilibrium constant is fixed ($K_{sp}^\circ = [\text{M}^{m+}][\text{X}^{n-}]\gamma_M \gamma_X$), a decrease in $\gamma$ forces the molar concentrations $[\text{M}^{m+}]$ and $[\text{X}^{n-}]$ to rise. Therefore, inert salts increase precipitate solubility.

Effect of $pH$ and Complexation on Solubility

• $pH$ Effect: Salts containing basic anions (e.g., carbonates, oxalates, sulfides, fluorides) dissolve in acidic media because protons bind the anion, shifting the dissolution equilibrium to the right: $\text{CaCO}_{3(s)} + \text{H}^+ \rightleftharpoons \text{Ca}^{2+} + \text{HCO}_3^-$
• Complexation Effect: Precipitates dissolve in the presence of ligands that form stable coordination complexes with the cation. For example, $\text{AgCl}$ dissolves in aqueous ammonia: $\text{AgCl}_{(s)} + 2\text{NH}_3 \rightleftharpoons [\text{Ag}(\text{NH}_3)_2]^+ + \text{Cl}^- \quad (K = K_{sp} \cdot \beta_2)$

Fractional Precipitation

When a titrant can precipitate multiple ions present in the same solution, the less soluble compound (the one requiring the lowest titrant concentration to exceed its $K_{sp}$) precipitates first.

• Quantitative separation of ion $A$ from ion $B$ is achievable if $> 99.9\%$ of $A$ precipitates before $B$ begins to precipitate.