Precipitation titrations and gravimetric analysis

Some of the most accurate measurements in chemistry are made by forming an insoluble compound. In gravimetric analysis, the analyte is converted to a pure precipitate of known composition, which is filtered, dried and weighed; the result depends only on masses and atomic weights. In a precipitation titration, the analyte is titrated with a reagent that forms an insoluble product, and the endpoint is detected when precipitation is complete. Theodore Richards's atomic weight determinations, which earned the 1914 Nobel Prize in Chemistry, were gravimetric and precipitation measurements of extraordinary care, and the chloride content of seawater was for decades defined by titration with silver nitrate. This lesson builds on the solubility equilibria from CHEM-102 and the activity and systematic treatment from lesson 6.

Precipitation titrations with silver

Silver nitrate titrations of halides and thiocyanate are known collectively as argentometric titrations. The reaction Ag⁺ + Cl⁻ → AgCl(s) has an equilibrium constant of 1/Ksp = 5.6 × 10⁹, large enough for a sharp endpoint. A titration curve plots pX = −log[X⁻] (or pAg) against the volume of titrant. Before the equivalence point, [Cl⁻] is the unreacted chloride (diluted); at the equivalence point, [Ag⁺] = [Cl⁻] = √Ksp; after it, [Cl⁻] = Ksp/[Ag⁺], where [Ag⁺] is the excess silver. An exact single expression comes from the mass balance on silver and chloride together with Ksp, as in the code below.

The size of the break depends on Ksp: AgI (Ksp = 8.3 × 10⁻¹⁷) gives a much larger break than AgCl (1.8 × 10⁻¹⁰). In a mixture of iodide and chloride, iodide precipitates first, and the titration curve shows two breaks. Because some chloride coprecipitates with AgI near the first equivalence point, the first break is slightly blurred, but potentiometric titration with a silver electrode separates the two well enough for many purposes.

Endpoints in argentometry

Three classic methods detect the endpoint of silver titrations, each named after its inventor.

The Mohr method (Karl Friedrich Mohr, 1856) adds a little chromate to the chloride solution. After all the chloride has precipitated, the first excess of silver forms red-brown silver chromate, Ag₂CrO₄, which is more soluble than AgCl. The titration must be done near neutral pH (about 7 to 10): in acid, chromate converts to dichromate and the endpoint is late; in base, silver hydroxide precipitates. Because a small excess of silver is needed to form visible Ag₂CrO₄, an indicator blank (for example, titrating calcium carbonate suspension with the same chromate) is subtracted. Chromate is a carcinogen, so the Mohr method is increasingly replaced by potentiometric or adsorption-indicator endpoints, and chromate wastes must be collected separately.

The Volhard method (Jacob Volhard, 1874) is a back titration in acidic solution. A measured excess of standard AgNO₃ is added to the chloride, and the excess silver is titrated with standard potassium thiocyanate, KSCN, using iron(III) as the indicator. The first excess of thiocyanate forms the deep red complex FeSCN²⁺. Because AgSCN is less soluble than AgCl, thiocyanate can slowly react with the AgCl precipitate itself and cause a late endpoint; the AgCl is therefore filtered off, or coated, before the back titration. Bromide and iodide, whose silver salts are less soluble than AgSCN, need no such precaution. The acidic conditions are an advantage: carbonate, phosphate and other anions that would precipitate silver in neutral solution do not interfere.

The Fajans method (Kazimierz Fajans, 1920s) uses an adsorption indicator, such as dichlorofluorescein, an anionic dye. Before the equivalence point, AgCl particles adsorb excess Cl⁻ on their surface and carry a negative charge, repelling the dye anion. After the equivalence point, the particles adsorb excess Ag⁺ and become positively charged; the dye anion is attracted to the surface, where it binds silver and changes from yellow-green to pink. Dextrin is added to keep the precipitate in a finely divided, high-surface-area colloidal form.

import math

def pX(V, Ksp, CX=0.0500, VX=50.0, CAg=0.100):
    """pX after adding V mL of Ag+; exact from mass balance and Ksp."""
    Vt = VX + V
    cx, cag = CX * VX / Vt, CAg * V / Vt
    d = cx - cag                     # [X-] - [Ag+] = cx - cag and [X-][Ag+] = Ksp
    X = (d + math.sqrt(d * d + 4 * Ksp)) / 2
    return -math.log10(X)

salts = {'AgCl': 1.8e-10, 'AgBr': 5.0e-13, 'AgI': 8.3e-17}
print("V(mL)  " + "  ".join(f"{k:>6s}" for k in salts))
for V in (10.0, 20.0, 24.0, 24.9, 25.0, 25.1, 26.0, 30.0):
    print(f"{V:5.1f}  " + "  ".join(f"{pX(V, K):6.2f}" for K in salts.values()))

# Mohr: [Ag+] needed to start Ag2CrO4 (Ksp 1.1e-12) with 2.5 mM chromate, vs [Ag+] at equivalence
Ksp_chromate, CrO4 = 1.1e-12, 2.5e-3
ag_needed = math.sqrt(Ksp_chromate / CrO4)
ag_eq = math.sqrt(1.8e-10)
Vt = 75.0                           # mL near the endpoint
extra_mmol = (ag_needed - ag_eq) * Vt
print(f"Ag+ to start Ag2CrO4: {ag_needed:.2e} M; at equivalence {ag_eq:.2e} M")
print(f"theoretical overshoot = {extra_mmol / 0.100 * 1000:.0f} uL of 0.100 M AgNO3 (before any visibility allowance)")

V(mL)    AgCl    AgBr     AgI
 10.0    1.60    1.60    1.60
 20.0    2.15    2.15    2.15
 24.0    2.87    2.87    2.87
 24.9    3.87    3.87    3.87
 25.0    4.87    6.15    8.04
 25.1    5.87    8.43   12.21
 26.0    6.86    9.42   13.20
 30.0    7.54   10.10   13.88
Ag+ to start Ag2CrO4: 2.10e-05 M; at equivalence 1.34e-05 M
theoretical overshoot = 6 uL of 0.100 M AgNO3 (before any visibility allowance)

Gravimetric analysis

A gravimetric procedure has five stages: precipitate the analyte as a sparingly soluble compound, digest and filter the precipitate, wash it free of impurities, convert it to a product of known composition by drying or ignition, and weigh it. The precipitating reagent should ideally react only with the analyte. A good precipitate is sparingly soluble (so losses are negligible), easily filtered and washed (large particles), pure, and of known, stable composition after drying.

Particle size and supersaturation

Whether a precipitate forms large, filterable crystals or a colloidal suspension depends on the competition between nucleation, the formation of new tiny particles, and particle growth, the deposition of more material on existing ones. Around 1906, Peter von Weimarn related particle size to the relative supersaturation:

where Q is the momentary concentration of solute and S its equilibrium solubility. When relative supersaturation is high, nucleation dominates and many tiny particles form; when it is low, growth dominates and fewer, larger crystals result. Analysts keep it low by precipitating from dilute, hot solutions (raising S), adding the reagent slowly with stirring (keeping Q low), and adjusting pH so that the precipitate is slightly soluble at first.

Precipitation from homogeneous solution takes this to its logical end: the precipitating reagent is generated slowly and uniformly throughout the solution by a chemical reaction, so local high concentrations never occur. A classic example uses urea, which hydrolyzes slowly in hot water to produce ammonia and raise the pH gradually and evenly: CO(NH₂)₂ + 3H₂O → CO₂ + 2NH₄⁺ + 2OH⁻. Aluminum and iron hydrous oxides precipitated this way are dense and easily filtered instead of gelatinous.

Colloids, digestion and washing

Many precipitates first form as colloids, particles 1 to 1,000 nm across that pass through filters and stay suspended because each particle carries a layer of adsorbed ions that repels the others. For silver chloride formed in excess chloride, the primary adsorbed layer is Cl⁻, with a counter-ion layer (for example Na⁺ or H⁺) farther out. Heating, stirring and adding an electrolyte compress the counter-ion layer, allowing particles to collide and coagulate into filterable masses.

Digestion, holding the precipitate in its hot mother liquor for an hour or more, lets small particles dissolve and recrystallize onto larger ones (Ostwald ripening), producing larger, purer crystals. Coagulated colloids must be washed with a solution of a volatile electrolyte, such as dilute nitric acid for AgCl, rather than pure water. Washing with water would remove the electrolyte that holds the particles together, and the precipitate would revert to a colloid and pass through the filter, a process called peptization.

Impurities: coprecipitation

Coprecipitation is the carrying down of normally soluble substances with a precipitate. It takes several forms. Surface adsorption traps ions on the surface of the particles, a serious problem for colloids with their huge surface areas; washing with a volatile electrolyte replaces adsorbed impurity ions with ones that are removed on drying. Occlusion traps impurities inside a growing crystal; slow precipitation and digestion reduce it. Mixed-crystal formation (inclusion) replaces lattice ions with impurity ions of similar size and charge, as when lead or strontium substitutes for barium in BaSO₄; it can be avoided only by removing the interfering ion beforehand. Post-precipitation, the slow formation of a second precipitate on the first, occurs when the solution is supersaturated with another compound and is avoided by filtering promptly.

Drying and ignition

The precipitate is collected in a weighed filter crucible (sintered glass or porcelain) or on ashless filter paper that is burned off. It is dried at about 110 °C if it has a definite composition at that temperature, or ignited at 500 to 1,000 °C to convert it to a stable form, and heated, cooled in a desiccator and weighed repeatedly until it reaches constant mass. Thermogravimetric analysis (TGA), which records mass continuously while heating, shows the stages: calcium oxalate monohydrate loses water near 200 °C to give CaC₂O₄, loses CO near 500 °C to give CaCO₃, and loses CO₂ near 800 °C to give CaO. The weighing form must be chosen to match a stable plateau.

Calculations

The gravimetric factor converts the mass of the weighed product into the mass of analyte:

where a and b are the numbers that balance the analyte element between the two formulas. For sulfur weighed as BaSO₄, the factor is M_S/M_BaSO₄ = 32.06/233.39 = 0.1374; for iron weighed as Fe₂O₃ and reported as Fe, it is 2 × 55.845/159.69 = 0.6994.

Dimethylglyoxime, introduced by Lev Chugaev in 1905, illustrates the selectivity of organic precipitants: in a slightly basic ammonia solution it precipitates nickel almost alone, even in steel full of iron (which is kept in solution with tartrate). Similar reagents include 8-hydroxyquinoline for aluminum and magnesium and sodium tetraphenylborate for potassium.

# Gravimetric factors and results
M = {'S': 32.06, 'SO4': 96.06, 'BaSO4': 233.39, 'Fe': 55.845, 'Fe2O3': 159.69,
     'Ni': 58.693, 'NiDMG2': 288.91, 'Cl': 35.453, 'AgCl': 143.32, 'P': 30.974, 'Mg2P2O7': 222.55}

def factor(analyte, product, a=1, b=1):
    return a * M[analyte] / (b * M[product])

cases = [('S', 'BaSO4', 1, 1, 0.6118, 0.5682),
         ('Fe', 'Fe2O3', 2, 1, 0.2543, 1.0021),
         ('Ni', 'NiDMG2', 1, 1, 0.1752, 1.1653),
         ('Cl', 'AgCl', 1, 1, 0.3317, 0.2047),
         ('P', 'Mg2P2O7', 2, 1, 0.4127, 1.2350)]
for an, pr, a, b, m_ppt, m_samp in cases:
    f = factor(an, pr, a, b)
    print(f"{an:3s} as {pr:8s}: factor {f:.4f}; {m_ppt:.4f} g -> {m_ppt * f:.4f} g {an} = {m_ppt * f / m_samp * 100:.2f} % of sample")

# effect of a 0.2 mg weighing error on each result
for an, pr, a, b, m_ppt, m_samp in cases:
    print(f"{an:3s}: relative error from 0.2 mg = {0.0002 / m_ppt * 100:.3f} %")

S   as BaSO4   : factor 0.1374; 0.6118 g -> 0.0840 g S = 14.79 % of sample
Fe  as Fe2O3   : factor 0.6994; 0.2543 g -> 0.1779 g Fe = 17.75 % of sample
Ni  as NiDMG2  : factor 0.2032; 0.1752 g -> 0.0356 g Ni = 3.05 % of sample
Cl  as AgCl    : factor 0.2474; 0.3317 g -> 0.0821 g Cl = 40.08 % of sample
P   as Mg2P2O7 : factor 0.2784; 0.4127 g -> 0.1149 g P = 9.30 % of sample
S  : relative error from 0.2 mg = 0.033 %
Fe : relative error from 0.2 mg = 0.079 %
Ni : relative error from 0.2 mg = 0.114 %
Cl : relative error from 0.2 mg = 0.060 %
P  : relative error from 0.2 mg = 0.048 %

Combustion analysis and moisture

Gravimetry includes any method that ends in a weighing. Combustion analysis, developed by Justus von Liebig in the 1830s, burned organic compounds and weighed the CO₂ and H₂O absorbed in tubes of base and drying agent; modern CHN analyzers detect the gases instrumentally but rest on the same stoichiometry. Loss on drying, the mass lost when a sample is heated to constant mass, measures moisture in grain, food and soil; ash, the residue after ignition, measures total mineral content. Both are routine in food and agricultural laboratories.