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Gravimetric analysis

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Statement

Quantifying an analyte by the mass of a precipitate.

Why it matters

Gravimetric analysis is one of the oldest and most fundamentally reliable quantitative techniques in analytical chemistry: it determines the amount of an analyte using nothing more than a precipitation reaction and an accurate mass balance, with no instrument calibration curve required at all. Where calibration-standard-addition and later instrumental methods rely on a calibrated response (a signal that must be related to concentration via a calibration curve, and can drift or be subject to interference), gravimetric analysis is an absolute method: the measured mass is directly, stoichiometrically related to the analyte, making it a standard reference technique against which other analytical methods are historically validated.

Hypotheses
The analyte can be quantitatively converted into a precipitate of known, fixed, and reproducible stoichiometric composition.If the precipitate's composition is variable (non-stoichiometric) or if precipitation is incomplete, the measured mass no longer corresponds to a definite, known fraction of the original analyte, and the entire calculation in the Proof below breaks down. The precipitate can be isolated free of contamination (coprecipitated impurities, residual mother-liquor solutes, or occluded solvent) and brought to a constant, reproducible mass by digestion, washing, and ignition/drying.Coprecipitation of a second species, or failure to reach truly constant mass on ignition (organic or volatile residues still present), introduces a systematic mass error that is not visible from the measurement itself — only a comparison against an independent method, or a wash-blank/replicate check, typically reveals it.
Proof
1
\[ \text{Analyte } A \xrightarrow{\text{precipitating reagent}} \text{precipitate } P \ (\text{fixed, known stoichiometry}) \]
A reagent is chosen that reacts with the analyte to form a precipitate of very low solubility (a large, favourable solubility-product equilibrium lying essentially entirely toward the solid), ensuring the precipitation reaction goes effectively to completion under the conditions used. A
2
\[ \text{Digestion: warming the precipitate with its mother liquor promotes crystal growth and reduces surface area and occluded impurity.} \]
Freshly formed precipitates are often colloidal or finely divided, with large surface area prone to adsorbing impurities and difficult to filter cleanly; digestion (ageing at elevated temperature) allows smaller crystallites to redissolve and redeposit onto larger ones (Ostwald ripening), yielding a purer, more easily filtered, more reproducibly washed solid. A
3
\[ \text{Ignition/drying to constant mass: } m_{n} - m_{n-1} \to 0 \text{ across repeated heating/weighing cycles.} \]
The isolated precipitate is heated (or, for less thermally robust precipitates, dried) and reweighed repeatedly until successive weighings agree within an acceptable tolerance, confirming that all residual solvent, occluded mother liquor, or (if intended) volatile decomposition products have been removed and a genuinely fixed, reproducible chemical form has been reached. A
4
\[ \text{Gravimetric factor } GF = \frac{a}{b}\cdot\frac{M_A}{M_P}, \qquad m_A = m_P\times GF \]
Given the balanced stoichiometric relationship \(a\,A \rightleftharpoons b\,P\) between moles of analyte and moles of the weighed precipitate form, the mass of analyte originally present is recovered from the measured precipitate mass by multiplying by this fixed gravimetric (conversion) factor, built entirely from molar masses and the balanced stoichiometry. A
Result
\[ m_{\text{analyte}} = m_{\text{precipitate}} \times \frac{a}{b}\cdot\frac{M_{\text{analyte}}}{M_{\text{precipitate}}} \]

Reading. A single, carefully measured mass of a purified, stoichiometrically well-defined precipitate is converted directly into the mass of the original analyte using a fixed gravimetric factor, requiring no calibration curve or reference standard beyond the balance itself.

Scope. Requires quantitative, selective precipitation (Hypotheses); this makes gravimetric analysis best suited to determinations where a sufficiently selective, low-solubility, well-characterised precipitate is available — it is generally slower and less suited to trace-level analysis than modern instrumental methods, but remains a benchmark for accuracy at macroscopic analyte quantities.

Corollaries & converses
  • Because the technique is absolute (traceable only to the mass balance and known molar masses, not to a calibration curve), gravimetric analysis is a standard method for independently validating or certifying reference materials later used to calibrate faster, indirect instrumental techniques (calibration-standard-addition).
  • error-propagation applies directly to the gravimetric factor calculation: since \(m_A\) is a simple product of the measured mass and a fixed constant, its relative uncertainty is, to first order, just the relative uncertainty of the mass measurement itself, making gravimetric analysis' accuracy ultimately limited chiefly by balance precision and by how completely and purely the precipitate was isolated.
  • chromatography-partition offers an alternative separation-based approach when no sufficiently selective precipitating reagent exists for a given analyte in a complex mixture; the two techniques are often complementary rather than competing.
Fails without
  • Drop the quantitative, fixed-stoichiometry precipitation hypothesis: if precipitation is incomplete, or the solid formed has variable, non-stoichiometric composition, the measured mass no longer stands in the fixed ratio to the analyte that the gravimetric factor of Step 4 assumes, and the calculated analyte mass is systematically wrong in an unpredictable direction.
  • Skip digestion or wash the precipitate insufficiently (Step 2): coprecipitated impurities or occluded mother liquor add mass not accounted for by the stoichiometric factor, inflating the apparent analyte mass above its true value.
Common errors
  • Confusing the molar mass of the weighed (ignited/dried) precipitate form with that of the original analyte in the gravimetric factor; these are frequently chemically different species (Step 4), and using the wrong molar mass anywhere in the ratio corrupts the entire result.
  • Insufficient digestion time, leaving a precipitate contaminated with coprecipitated impurities or occluded mother liquor that adds spurious mass not accounted for by the stoichiometric factor.
  • Stopping the ignition/drying cycle before truly constant mass is reached (Step 3), introducing a systematic negative or positive bias depending on what residual material remains.
  • Applying the technique to an analyte for which the precipitating reagent is insufficiently selective, so that other species present in the sample coprecipitate or precipitate alongside the analyte, inflating the apparent mass.
Discussion

Gravimetric methods predate essentially all modern instrumental analytical chemistry, and classic determinations — sulfate as barium sulfate, chloride as silver chloride, nickel as the nickel–dimethylglyoxime complex — were, for well over a century, the standard reference methods against which faster techniques were validated. Even as spectroscopic and chromatographic instrumental methods have largely displaced gravimetry for routine, high-throughput analysis, it remains valued precisely for its status as an absolute method requiring no external calibration.

The choice of precipitating reagent is itself a substantial subfield of classical analytical chemistry: organic chelating reagents (such as dimethylglyoxime for nickel) were developed specifically because they can offer far greater selectivity for a single metal ion than simple inorganic anions, minimising the coprecipitation interferences that limit accuracy in complex real samples.

Common misconception: that gravimetric analysis directly weighs the analyte itself. In nearly every real procedure the analyte is chemically converted into a different compound (the precipitate, often after further ignition to yet another compound) before weighing; the gravimetric factor (Step 4) is precisely what corrects the measured mass of this different, weighed species back to the mass of the original analyte.

Worked examples
1
\[ \text{Determine \%S in a sample as } \text{BaSO}_4:\quad \text{SO}_4^{2-}+\text{Ba}^{2+}\to\text{BaSO}_4(s)\downarrow \]
Sulfate is precipitated quantitatively as barium sulfate (very low solubility product), digested, filtered, ignited to constant mass, and weighed; here \(a=b=1\) mole of sulfur per mole of \(\text{BaSO}_4\), with \(M_S=32.07\,\text{g/mol}\) and \(M_{\text{BaSO}_4}=233.4\,\text{g/mol}\). A
2
\[ GF = \frac{M_S}{M_{\text{BaSO}_4}} = \frac{32.07}{233.4} = 0.1374; \qquad m_{\text{precipitate}}=0.4813\,\text{g} \Rightarrow m_S = 0.4813\times0.1374 = 0.0661\,\text{g} \]
Applying the gravimetric factor to a measured precipitate mass of \(0.4813\,\text{g}\) recovers the mass of elemental sulfur originally present in the sample. A
\[ m_S = 0.0661\,\text{g}\ \text{(from } 0.4813\,\text{g BaSO}_4\text{)} \]

Reading. A single weighed mass of the ignited, purified precipitate directly yields the mass of the original analyte, once the correct gravimetric factor for the chosen precipitation chemistry is applied.

Scope. The identical logic applies to any analyte with a sufficiently selective, low-solubility precipitating reagent and a well-defined, constant-mass ignition product.

Further example. A 0.6250 g sample of a metal alloy is dissolved in acid and the manganese is precipitated quantitatively as manganese(II) oxalate, \(\text{MnC}_2\text{O}_4\). The precipitate is filtered, washed, and ignited in air to convert it to manganese(III) oxide, \(\text{Mn}_2\text{O}_3\), which is then weighed. The mass of the ignited \(\text{Mn}_2\text{O}_3\) is found to be \(0.1845\,\text{g}\). Calculate the mass percent of manganese in the original alloy. Use \(M_{\text{Mn}}=54.94\,\text{g/mol}\) and \(M_{\text{Mn}_2\text{O}_3}=157.9\,\text{g/mol}\).

1
\[ 2\,\text{Mn (as MnC}_2\text{O}_4) \xrightarrow{\text{ignition in air}} 1\,\text{Mn}_2\text{O}_3 \]
The stoichiometry of the ignition process dictates that two moles of manganese atoms are incorporated into one mole of the final weighed oxide \(\text{Mn}_2\text{O}_3\). This establishes the mole ratio \(a/b = 2/1\) required for the gravimetric factor. A
2
\[ GF = \frac{a \cdot M_{\text{Mn}}}{b \cdot M_{\text{Mn}_2\text{O}_3}} = \frac{2 \times 54.94}{1 \times 157.9} = \frac{109.88}{157.9} \]
The gravimetric factor (GF) converts the mass of the weighed form to the mass of the analyte. It is derived from the ratio of the molar masses multiplied by the stoichiometric coefficients of the analyte in the weighed compound. A
3
\[ GF = 0.6959 \]
Calculating the numerical value of the factor: \(109.88 / 157.9 \approx 0.69588\). Retaining four significant figures for intermediate calculation yields \(0.6959\). A
4
\[ m_{\text{Mn}} = m_{\text{precipitate}} \times GF = 0.1845\,\text{g} \times 0.6959 = 0.1284\,\text{g} \]
Multiplying the measured mass of the ignited oxide by the gravimetric factor yields the mass of pure manganese originally present in the sample. A
5
\[ \% \text{Mn} = \frac{m_{\text{Mn}}}{m_{\text{sample}}} \times 100 = \frac{0.1284}{0.6250} \times 100 = 20.54\% \]
The mass percent is calculated by dividing the mass of the analyte by the total mass of the initial sample and multiplying by 100. A
\[ \% \text{Mn} = 20.54\% \]

Reading. The alloy contains 20.54% manganese by mass. The method relies on the complete conversion of the intermediate precipitate to a stable, well-defined oxide form with a known stoichiometry.

Scope. This approach is valid for any metal that can be precipitated and subsequently ignited to a stable oxide with a fixed oxygen-to-metal ratio, provided the ignition temperature is sufficient to remove all volatile components (like water or carbon dioxide from oxalates) without volatilizing the metal oxide itself.

Problems
  1. A 0.5000 g sample containing chloride is treated with excess \(\text{AgNO}_3\), precipitating \(\text{AgCl}\) (\(M=143.3\,\text{g/mol}\)); after digestion, filtration, and drying, \(0.2870\,\text{g}\) of \(\text{AgCl}\) is obtained. Find the mass and mass percent of chloride (\(M_{\text{Cl}}=35.45\,\text{g/mol}\)) in the sample.
    Solution\(GF = M_{\text{Cl}}/M_{\text{AgCl}} = 35.45/143.3 = 0.2474\). \(m_{\text{Cl}} = 0.2870\times0.2474 = 0.0710\,\text{g}\). Mass \% \(= (0.0710/0.5000)\times100 = 14.20\%\).
  2. Explain why a precipitate that is only partially digested before filtration would typically lead to a mass measurement that is too high, referencing Steps 1–3 of the Proof.
    SolutionInsufficiently digested precipitate remains finely divided with high surface area, more prone to adsorbing (coprecipitating) other solutes from the mother liquor and to trapping (occluding) mother liquor within its structure (Step 2); this additional, unwanted material adds mass beyond the true precipitate stoichiometry, and if not fully removed by washing or driven off during ignition/drying to constant mass (Step 3), it inflates the measured mass above the true value predicted by the gravimetric factor in Step 4.
  3. An analyst determines iron in a sample by precipitating it as \(\text{Fe(OH)}_3\), then igniting the precipitate to \(\text{Fe}_2\text{O}_3\) (\(M=159.7\,\text{g/mol}\)) before weighing, obtaining \(0.1425\,\text{g}\) of \(\text{Fe}_2\text{O}_3\). Given \(M_{\text{Fe}}=55.85\,\text{g/mol}\) and that each mole of \(\text{Fe}_2\text{O}_3\) contains two moles of iron, find the mass of iron in the sample.
    Solution\(GF = \dfrac{2\times M_{\text{Fe}}}{M_{\text{Fe}_2\text{O}_3}} = \dfrac{2\times55.85}{159.7} = \dfrac{111.7}{159.7} = 0.6994\). \(m_{\text{Fe}} = 0.1425\times0.6994 = 0.0997\,\text{g}\). This illustrates the \(a/b\) factor of Step 4 explicitly, since here \(a=2\) moles of analyte correspond to \(b=1\) mole of the weighed precipitate form.
  4. A 0.4500 g sample of a fertilizer is analyzed for phosphorus. The phosphorus is precipitated as magnesium ammonium phosphate hexahydrate, \(\text{MgNH}_4\text{PO}_4\cdot 6\text{H}_2\text{O}\), and then ignited to form magnesium pyrophosphate, \(\text{Mg}_2\text{P}_2\text{O}_7\). The mass of the ignited \(\text{Mg}_2\text{P}_2\text{O}_7\) is \(0.2150\,\text{g}\). Calculate the mass percent of phosphorus in the fertilizer. Use \(M_{\text{P}}=30.97\,\text{g/mol}\) and \(M_{\text{Mg}_2\text{P}_2\text{O}_7}=222.0\,\text{g/mol}\).
    Solution First, determine the stoichiometry. The formula \(\text{Mg}_2\text{P}_2\text{O}_7\) contains 2 moles of phosphorus per 1 mole of the compound. Calculate the gravimetric factor (GF): \[ GF = \frac{2 \times M_{\text{P}}}{M_{\text{Mg}_2\text{P}_2\text{O}_7}} = \frac{2 \times 30.97}{222.0} = \frac{61.94}{222.0} \approx 0.2790 \] Calculate the mass of phosphorus: \[ m_{\text{P}} = 0.2150\,\text{g} \times 0.2790 = 0.0600\,\text{g} \] Calculate the mass percent: \[ \% \text{P} = \frac{0.0600}{0.4500} \times 100 = 13.33\% \]
  5. Explain why the precipitate in gravimetric analysis must be ignited or dried to "constant mass" before weighing, and describe how failing to do so would affect the calculated concentration of the analyte.
    Solution Precipitates often retain adsorbed water, solvent, or volatile impurities (such as unreacted reagents or decomposition products) on their surface or within their crystal lattice. "Constant mass" ensures that all such volatile components have been completely removed. If the precipitate is weighed before reaching constant mass, it will contain excess mass from these volatiles. Since the gravimetric factor assumes the weighed mass corresponds strictly to the stoichiometric formula of the pure compound, this extra mass leads to an overestimation of the moles of precipitate, and consequently, an overestimation of the mass and concentration of the analyte.