Limiting Reactant Precipitation lab report

1. Title and aim

Limiting Reactants, Gravimetric Stoichiometry, and Percentage Yield: Precipitation of Barium Sulfate. Quantitatively verify the law of limiting reactants and stoichiometric mole ratios by precipitating insoluble barium sulfate from standardized aqueous solutions of barium chloride and sodium sulfate, determining the experimental molar mass of BaSO₄ via ordinary least-squares linear regression, and evaluating gravimetric percentage yields and coprecipitation equilibria.

2. Abstract

Six gravimetric precipitation trials were executed across added volumes of 0.1000 M Na₂SO₄ from V = 4.0 mL to V = 24.0 mL reacted with a constant excess of 30.00 mL of 0.1000 M BaCl₂ (yielding isolated dry BaSO₄ masses from 0.0934 g to 0.5601 g). An ordinary least-squares linear regression of precipitate mass m against added sulfate volume V yielded an empirical slope of 0.023336 g/mL, extracting an experimental molar mass for barium sulfate of M = 233.36 g/mol and a baseline intercept of 4.0000e-05 g, demonstrating remarkable agreement with the theoretical formula weight of 233.390 g/mol (difference = 0.0110%). Spot tests on the supernatant confirmed the presence of unreacted excess Ba²⁺ ions, rigorously validating that sulfate acted strictly as the limiting reagent across all trials.

3. Apparatus and fixed settings

• Class A Volumetric Burettes & Pipettes: 50.00 mL Class A calibrated volumetric burettes (tolerance ±0.05 mL) with PTFE stopcocks and 25.00/30.00 mL volumetric transfer pipettes (tolerance ±0.02 mL) for precision dispensing of standardized 0.1000 M BaCl₂ and Na₂SO₄ solutions without transfer parallax error.
• Thermostated Magnetic Stirrer & Digestion Beakers: 250 mL heavy-duty Pyrex borosilicate digestion beakers positioned on digital ceramic hotplate magnetic stirrers equipped with external PT100 temperature sensors, maintaining constant solution temperature at 85.0 ± 1.0 °C and providing controllable vortex agitation.
• Grade 4 Sintered Glass Gooch Crucibles: Porous borosilicate filtering crucibles with fused Grade 4 sintered glass fritted discs (nominal pore diameter 1.0–1.6 µm) providing quantitative retention of microcrystalline barium sulfate without the ash-reduction artifacts associated with burning filter paper.
• Vacuum Filtration Manifold & Neoprene Adapters: Heavy-walled 500 mL borosilicate filtration flasks connected to a water aspirator vacuum manifold via thick-walled gum-rubber vacuum tubing, fitted with conical neoprene adapter collars to form hermetic vacuum seals with the Gooch crucibles.
• Convection Drying Oven & Vacuum Desiccator: Thermostatically regulated laboratory forced-air drying oven operating at 110.0 ± 1.0 °C for calcining isolated precipitates to constant mass, paired with a heavy ground-glass vacuum desiccator charged with activated cobalt-chloride-indicating silica gel desiccant.
• Four-Decimal Analytical Balance: High-precision electromagnetic force restoration balance with 0.1 mg (0.0001 g) resolution, internal calibration motor, and anti-draft glass shielding to record empty crucible tares and gross dry precipitate masses.
• Constant initial aliquot of 30.00 mL of 0.1000 M BaCl₂ (3.000 mmol Ba²⁺) acidified with 1.0 mL of 6 M HCl and diluted with 50 mL deionized water.
• Titration and digestion temperature maintained at 85.0 ± 1.0 °C on a thermostated ceramic magnetic stirrer to facilitate Ostwald ripening.
• Vacuum filtration conducted using fine-porosity Grade 4 sintered glass Gooch crucibles (pore size 1.0–1.6 µm) pre-calcined to constant mass.
• Precipitate washed with boiling deionized water until filtrate tests negative for residual chloride ions using 0.1 M AgNO₃ reagent.

4. Theory and graph choice

Quantitative chemical gravimetry is founded upon the principle of stoichiometric conservation of mass and the selective precipitation of sparingly soluble ionic compounds from homogeneous solution. When aqueous solutions containing barium cations (Ba²⁺) and sulfate anions (SO₄²⁻) are combined, they undergo a rapid metathesis reaction yielding an extraordinarily insoluble crystalline salt, barium sulfate (BaSO₄), whose thermodynamic solubility product constant at 25 °C is K_sp = [Ba²⁺][SO₄²⁻] = 1.08 × 10⁻¹⁰ (molar solubility s ≈ 1.04 × 10⁻⁵ mol/L, corresponding to mere 0.0024 g/L). In this classic analytical determination, an excess quantity of barium chloride (30.00 mL of 0.1000 M BaCl₂, representing 3.000 mmol of Ba²⁺) is held constant, while variable volumetric increments of standardized sodium sulfate (0.1000 M Na₂SO₄, from 4.00 to 24.00 mL, delivering 0.400 to 2.400 mmol of SO₄²⁻) are systematically introduced. Throughout this entire operational range, sulfate acts strictly as the limiting reactant: every mole of sulfate added reacts quantitatively with one mole of barium ions to precipitate exactly one mole of barium sulfate. Plotting the isolated mass of dry precipitate m against the volume of sulfate solution added V generates a highly linear gravimetric isotherm whose slope m_slope = C · M / 1000 directly yields the stoichiometric molar mass of barium sulfate (M_BaSO₄ = 233.390 g/mol). Understanding the physicochemical factors that govern crystalline nucleation versus particle growth, supersaturation dynamics as formalized by von Weimarn's law, Ostwald ripening through thermal digestion, and the mechanisms of coprecipitation (occlusion, surface adsorption, and isomorphous replacement) is indispensable for achieving sub-milligram accuracy in modern quantitative analytical chemistry.

The mathematical regression fits the stoichiometric linear model m_BaSO₄ = [C_SO₄ · M_BaSO₄ / 1000] · V_SO₄ + m₀, where V_SO₄ represents the volume of 0.1000 M Na₂SO₄ added in mL, m_BaSO₄ represents the dry precipitate mass in g, C_SO₄ represents titrant molarity (0.1000 mol/L), and m₀ represents the baseline intercept in g (theoretically m₀ = 0.0000 g). The slope m_slope directly relates to the molar mass of barium sulfate via M_BaSO₄ = (m_slope / C_SO₄) · 1000, while constant crucible tare artifacts and minor transfer residues are absorbed into the intercept.

5. Method

1. Using a calibrated 30.00 mL Class A volumetric transfer pipette, quantitatively dispense 30.00 mL of standardized 0.1000 M BaCl₂ solution (3.000 mmol Ba²⁺) into a clean, labeled 250 mL borosilicate digestion beaker. Dilute the aliquot with approximately 50 mL of deionized water (18.2 MΩ·cm) and deliver 1.0 mL of 6.0 M HCl using a graduated cylinder. The hydrochloric acid suppresses hydroxide, carbonate, and phosphate ion activities, preventing the coprecipitation of insoluble BaCO₃ and Ba₃(PO₄)₂ while optimizing the ionic strength.
2. Place the reaction beaker onto the ceramic hotplate stirrer, insert a PTFE-coated magnetic stir bar, and heat the acidified barium solution to 80–85 °C under gentle continuous stirring (300 rpm). Fill a clean 50.00 mL Class A burette with standardized 0.1000 M Na₂SO₄ solution. Slowly dispense the designated aliquot (4.00, 8.00, 12.00, 16.00, 20.00, or 24.00 mL) dropwise into the vortex of the heated barium solution over a period of 4 to 5 minutes. The dropwise addition at elevated temperature minimizes local supersaturation (von Weimarn ratio (Q - S)/S), suppressing explosive spontaneous nucleation and promoting ordered crystal growth.
3. Following complete sulfate addition, cover the beaker with a ribbed watch glass and maintain the suspension at 85–90 °C just below boiling for a minimum of 45 minutes on the hotplate with gentle stirring. During this digestion phase, Ostwald ripening occurs: small, high-energy, sub-micron BaSO₄ crystallites with high surface curvature and increased solubility dissolve back into solution and recrystallize onto the faces of larger crystals, coarsening the precipitate from a colloidal haze into dense, rapidly settling, easily filterable microcrystalline aggregates.
4. Clean a Grade 4 sintered glass Gooch crucible (1.0–1.6 µm pore size), dry it in the laboratory convection oven at 110 °C for 60 minutes, cool it inside a sealed desiccator for 25 minutes, and record its tare mass (m_tare) on the four-decimal analytical balance to ±0.0001 g. Repeat heating and cooling cycles until constant mass (agreement within ±0.0002 g) is verified. Seat the tared crucible into the neoprene adapter ring of the vacuum filtration flask and activate the water aspirator vacuum to establish a steady differential pressure of ~20 kPa.
5. Decant the warm supernatant liquid through the Gooch crucible under vacuum. Quantitatively transfer the digested barium sulfate precipitate into the crucible using a fine stream of boiling deionized water from a wash bottle and a PTFE rubber policeman to dislodge adhering crystals from beaker walls. Wash the precipitate with six successive 10 mL portions of boiling deionized water. Collect drops of filtrate passing through the crucible stem into a test tube and test with 3 drops of 0.1 M AgNO₃ reagent acidified with HNO₃. Continue washing until no white opalescence or AgCl cloudiness is detected, proving complete removal of adsorbed chloride and sodium ions.
6. Place the crucible containing washed BaSO₄ precipitate into the drying oven at 110 °C for 60 minutes. Transfer immediately to the vacuum desiccator, cool to ambient room temperature for 25 minutes, and record the gross mass on the analytical balance. Repeat the 30-minute drying and desiccation cycle until consecutive readings match within ±0.0002 g. Subtract m_tare to isolate dry precipitate mass m. Perform spot tests on the original filtrate by adding 3 drops of 0.1 M Na₂SO₄ to verify residual excess Ba²⁺ ions. Tabulate mass data across all sulfate aliquots, plot m versus V, and apply ordinary least-squares regression to extract the empirical molar mass of BaSO₄ and verify theoretical percentage yield.

6. Observations and calculation table

All gravimetric observations were recorded using a calibrated 4-decimal electronic analytical balance enclosed within a draft shield chamber. Volumetric aliquots were dispensed from a Class A burette. Residuals are computed relative to the ordinary least-squares regression line.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Circular markers represent empirical gravimetric observations. The solid line represents the ordinary least-squares linear regression fit.

9. Uncertainty and comparison

Percentage difference = |estimate − theoretical value| / |theoretical value| × 100 = 0.0110%.

For n = 7 gravimetric observations, Σresidual² = 3.4286e-09; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.7496e-06. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.7255e-05.

Propagating the fit parameters yields an experimental density standard error of 0.01750 g/mol, with an ideal coefficient of determination R² = 1.000000.

In quantitative gravimetric precipitation, measurement uncertainty is governed by the stoichiometric relation m_BaSO₄ = (C_SO₄ · M_BaSO₄ / 1000) · V_SO₄ + m₀. A rigorous analytical uncertainty budget partitions experimental variance into distinct chemical, physical, and metrological components: (1) Volumetric Dispensing Tolerances: The Class A 50.00 mL burette used for delivering sodium sulfate titrant possesses a certified tolerance of ±0.05 mL, and the 30.00 mL volumetric pipette possesses a tolerance of ±0.02 mL. Random meniscus alignment uncertainty is approximately s_V = ±0.015 mL. For a mid-range aliquot of 16.0 mL, the relative volumetric uncertainty is δV / V = 0.015 / 16.0 ≈ 9.4 × 10⁻⁴ (0.094%). (2) Analytical Balance Gravimetric Resolution: The electronic analytical balance features a readability of ±0.0001 g and a certified repeatability of ±0.00015 g. For the tared Gooch crucible gross weighings (masses ~0.3734 g of BaSO₄), the relative random weighing uncertainty is δm / m = 0.00015 / 0.3734 ≈ 4.0 × 10⁻⁴ (0.040%). (3) Thermodynamic Solubility Loss: Barium sulfate has an equilibrium solubility product K_sp = 1.08 × 10⁻¹⁰ at 25 °C. In the presence of excess barium ions ([Ba²⁺] ≈ 0.01 M), the common ion effect suppresses equilibrium sulfate solubility to ~1.1 × 10⁻⁸ M, making precipitate loss during reaction negligible (<0.001 mg). However, subsequent washing with deionized water (where [Ba²⁺] is zero) dissolves up to 0.24 mg of BaSO₄ per 100 mL of wash liquor. Restricting wash volume to exactly 60 mL limits dissolution loss to ~0.14 mg, which is bounded within the balance readability tolerance. (4) Coprecipitation and Foreign Ion Occlusion: Surface adsorption of excess BaCl₂ onto positively charged crystal facets would lead to a negative mass bias if unwashed, while occluded sodium chloride in rapid crystal growth pockets would cause a positive bias. Maintaining slow dropwise addition at 85 °C and digesting for 45 minutes promotes Ostwald ripening and desorbs impurities, confirmed by negative silver nitrate spot tests on the filtrate. Ordinary least-squares linear regression across six independent volumetric settings (4.0 to 24.0 mL) extracts the empirical slope m_slope = 0.0233364 g/mL with a standard error SE(m) < 4.0 × 10⁻⁶ and an ideal coefficient of determination R² = 1.00000, isolating the experimental molar mass M(BaSO₄) = 233.36 g/mol within 0.011% of the theoretical reference.

10. Result and discussion

The experimental observations provide rigorous quantitative verification of the Law of Limiting Reactants, Gravimetric Precipitation Thermodynamics, and von Weimarn's Nucleation Dynamics. In aqueous solution, the metathesis reaction between barium chloride and sodium sulfate proceeds via the net ionic equilibrium Ba²⁺ (aq) + SO₄²⁻ (aq) <===> BaSO₄ (s). Because barium sulfate possesses an exceptionally low solubility product constant (K_sp = 1.08 × 10⁻¹⁰ at 25 °C), the precipitation reaction proceeds to virtual completion. Throughout all six trials, the initial quantity of barium ions was maintained constant at 3.000 mmol (30.00 mL of 0.1000 M BaCl₂), while sodium sulfate was added in quantities ranging from 0.400 mmol (4.00 mL) to 2.400 mmol (24.00 mL). In every trial, the stoichiometric ratio between Ba²⁺ and SO₄²⁻ is 1:1, meaning that sulfate was completely consumed, acting as the limiting reactant that dictated the theoretical yield of precipitate. Unreacted barium ions remained in excess in the supernatant, which was qualitatively confirmed by adding sodium sulfate spot reagent to the filtrate and observing renewed white BaSO₄ cloudiness. Across the six trials, the isolated mass of barium sulfate increased in strict linear proportion to the volume of sulfate added, generating a regression line with an ideal R² = 1.00000. The slope of the regression line, m_slope = 0.0233364 g/mL, corresponds to an experimental molar mass of M = 233.36 g/mol, matching the true formula weight of 233.390 g/mol to within 0.011%. Precipitation at elevated temperature (85 °C) under dilute conditions with dropwise addition minimized relative supersaturation (Q - S)/S according to von Weimarn's principle. Thermal digestion for 45 minutes enabled Ostwald ripening, allowing high-solubility sub-micron crystallites to redissolve and recrystallize onto larger crystal faces. This coarsening prevented colloidal losses through the Grade 4 sintered glass fritted disc (pore size 1.0–1.6 µm). Acidification with 1.0 mL of 6 M HCl successfully prevented the coprecipitation of barium carbonate (BaCO₃) and barium phosphate (Ba₃(PO₄)₂), while thorough washing with boiling deionized water eliminated adsorbed chloride ions, as proven by negative silver nitrate tests on the effluent washings.

11. Precautions and limitations

• Acidify the barium chloride solution with 1.0 mL of 6 M HCl before precipitation to prevent coprecipitation of insoluble barium carbonate and barium phosphate.
• Dispense the sodium sulfate titrant dropwise from a burette into the vortex of the heated, vigorously stirred barium solution to keep localized supersaturation low and prevent colloidal formation.
• Perform thermal digestion at 85 to 90 °C for at least 45 minutes to facilitate Ostwald ripening and ensure complete crystallite aggregation prior to filtration.
• Never skip testing the filtrate washings with 0.1 M AgNO₃ reagent; continue washing with boiling deionized water until no silver chloride turbidity is observed.
• Always cool heated Gooch crucibles inside a sealed desiccator for 20 to 25 minutes prior to balance placement to prevent thermal convection currents on the analytical balance pan.
• Exercise appropriate laboratory safety precautions when handling soluble barium salts, as barium cations are toxic; wear nitrile gloves, splash goggles, and dispose of heavy-metal wastes in designated toxic containers.

The gravimetric determination of barium sulfate stoichiometry is subject to fundamental thermodynamic and mechanical boundary conditions: (1) Intrinsic Solubility Loss: Although barium sulfate is sparingly soluble (K_sp = 1.08 × 10⁻¹⁰ at 25 °C), approximately 0.24 mg of BaSO₄ dissolves per 100 mL of aqueous wash water, setting an asymptotic thermodynamic limit on quantitative recovery. (2) Coprecipitation and Occlusion: Barium sulfate exhibits a pronounced propensity to occlude alkali nitrates, alkali chlorates, and foreign multivalent cations within its crystal lattice; because our system utilizes pure BaCl₂ and Na₂SO₄ in dilute HCl, occlusion is minimized but can still account for ±0.1% mass variance. (3) Creeping and Colloidal Penetration: Sub-micron crystallites that escape Ostwald ripening can occasionally creep up crucible walls or penetrate pores larger than 1.6 µm, causing slight negative yield errors if filtration vacuum is applied too aggressively. (4) Balance Sensitivity to Ambient Static: Sintered glass crucibles can accumulate electrostatic surface charges in low-humidity environments, necessitating antistatic ionizers or balance grounding. The gravimetric stoichiometric model assumes complete precipitation of the limiting sulfate anion according to the 1:1 net ionic equilibrium Ba²⁺ (aq) + SO₄²⁻ (aq) <===> BaSO₄ (s). Because the thermodynamic solubility product of barium sulfate is K_sp = 1.08 × 10⁻¹⁰ at 25 °C, residual dissolved sulfate in a 100 mL reaction mixture with an excess barium concentration of [Ba²⁺] ≈ 0.01 M is suppressed by the common ion effect to [SO₄²⁻] = K_sp / [Ba²⁺] = 1.08 × 10⁻⁸ M, representing an unprecipitated fraction <0.0003%. Thermal digestion at 85 °C promotes Ostwald ripening, shifting the crystallite size distribution from sub-micron colloids to coarse aggregates readily captured by 1.0–1.6 µm Gooch crucibles. Dilute hydrochloric acid (1 mL of 6 M HCl) is added to maintain pH ≈ 1, effectively protonating carbonate (CO₃²⁻) and phosphate (PO₄³⁻) impurities into soluble species and preventing the coprecipitation of BaCO₃ and Ba₃(PO₄)₂.

12. Conclusion

This laboratory investigation successfully demonstrated the quantitative principles of limiting reactants and gravimetric stoichiometry through the precipitation of barium sulfate. Through systematic volumetric additions from V = 4.0 mL to 24.0 mL of 0.1000 M Na₂SO₄ into excess BaCl₂, we extracted an experimental molar mass of M(BaSO₄) = 233.36 g/mol with a fit standard error of 0.01750 g/mol and an ideal R² = 1.00000. The experimental value agrees with the theoretical chemical formula weight (233.390 g/mol) to within 0.0110% error. The investigation verified von Weimarn's law of supersaturation, demonstrated the thermodynamic necessity of Ostwald ripening during thermal digestion, and proved that multi-point ordinary least-squares regression provides an exceptionally robust analytical methodology for stoichiometric verification.