Significant Figures in Gravimetric Analysis: From Precipitate Mass to Percent Composition

Gravimetric analysis is one of the most accurate classical methods in analytical chemistry. A sample is dissolved, an analyte is precipitated, the precipitate is filtered, dried or ignited, and weighed. The mass of the precipitate is then related to the mass of analyte in the original sample. Because the entire calculation rests on a few mass measurements, significant figures and rounding rules are not cosmetic—they determine how much confidence the final percent composition deserves.

This reference page explains how significant figures propagate from precipitate mass to percent composition. It is part of our precision-and-rounding reference; the calculator is a useful check, but the rules, standards, and examples below are the authority.

Rule Statement: Significant Figures in Gravimetric Analysis

Core rule: Every final result is limited by the least precise measurement used in the calculation. In gravimetric analysis, the critical measurements are masses: sample mass, tare mass, crucible plus precipitate mass, and sometimes ignition residue mass. Formula masses and stoichiometric coefficients are exact or carry more digits than the measurements, so they should not be allowed to limit the result prematurely.

Two rules govern:

• Addition and subtraction: Round to the least number of decimal places. This applies to mass by difference: m_precipitate = (m_crucible+precipitate) – m_empty crucible.
• Multiplication and division: Round to the least number of significant figures. This applies to moles, formula ratios, and percent composition.

Exact factors such as 100 in percent composition, stoichiometric coefficients, and defined molar masses do not limit significant figures. Atomic weights from IUPAC are measured quantities, but in routine gravimetric work they are used with enough digits—typically one more than the measured mass—so they are not the limiting factor. For high-accuracy work, propagate uncertainty per the GUM rather than relying on significant figures alone.

ASTM E29-22 and ISO 80000-1:2009 both treat significant figures as a rounding and reporting convention, not as a substitute for a full uncertainty budget.

Worked Examples: From Precipitate Mass to Percent Composition

Example 1: Percent CaO from ignited precipitate

A 0.5000 g limestone sample yields 0.2843 g CaO after precipitation and ignition. Calculate % CaO.

1. Molar mass CaO = 56.0774 g/mol (more digits than needed).
2. Moles CaO = 0.2843 g / 56.0774 g/mol = 0.005070 mol.
3. % CaO = (mass CaO / sample mass) × 100 = (0.2843 / 0.5000) × 100 = 56.86%.

Significant figures: 0.2843 has 4 sig figs; 0.5000 has 4 sig figs; 100 is exact. Report 56.86% CaO (4 sig figs). If sample mass were 0.500 g (3 sig figs), report 56.9%.

Example 2: Mass by difference and decimal places

Crucible + precipitate = 24.3672 g; empty crucible = 23.9876 g. Precipitate mass = 24.3672 – 23.9876 = 0.3796 g. Both masses have 4 decimal places, so the difference is kept to 4 decimal places. The result 0.3796 g has 4 sig figs. If the empty crucible were recorded as 23.987 g, the difference would be 0.3802 g before rounding, then rounded to 3 decimal places: 0.380 g (3 sig figs).

Example 3: Percent chloride from AgCl

A 0.2000 g sample gives 0.3521 g AgCl. Calculate % Cl.

1. M_Cl = 35.45 g/mol; M_AgCl = 143.32 g/mol.
2. Mass Cl = 0.3521 g × (35.45 / 143.32) = 0.08709 g.
3. % Cl = (0.08709 / 0.2000) × 100 = 43.55%.

The limiting factor is 0.3521 g or 0.2000 g, both 4 sig figs. The atomic-mass ratio is used with at least 4 sig figs. Report 43.55% Cl.

Counter-Examples: Common Sig-Fig Errors in Gravimetric Calculations

• Calculator-copy error: Reporting 56.8642% because the calculator displays eight digits. The measured masses support only 4 sig figs: 56.86%.
• Tare decimal-place error: Subtracting 24.3672 g – 23.987 g and keeping 0.3802 g. Addition/subtraction demands 3 decimal places, so 0.380 g is correct before further calculation.
• Treating 100 as limiting: In % = (mass analyte / sample mass) × 100, the 100 is exact and does not reduce sig figs.
• Premature atomic-weight rounding: Using Cl = 35 g/mol and AgCl = 143 g/mol can limit the result to 2 sig figs, even when masses have 4 sig figs. Use sufficient digits and round at the end.
• Double rounding: Rounding moles to 0.00507 mol and then using that rounded value in a second multiplication can shift the last digit. Keep extra digits in intermediate steps.
• Sig figs instead of uncertainty: For regulatory or metrological work, ASTM E29-22 and the GUM require an uncertainty statement, not just a sig-fig count.

Convention Comparison Table

Standards Citation

Gravimetric results are often used for conformance and reference materials, so the following standards are directly relevant:

• ASTM E29-22, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications, Sections 5–7. It specifies rounding and significant-digit procedures for test data and conformance decisions.
• ISO 80000-1:2009, Quantities and units — Part 1: General, Clause 6.5. It gives general rules for rounding and presenting numerical values.
• JCGM 100:2008 (GUM), Evaluation of measurement data — Guide to the expression of uncertainty in measurement, Clause 7. It defines how to report uncertainty and how many digits to retain in the uncertainty statement.
• NIST TN 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Section 7. It provides practical NIST guidance for reporting uncertainty.
• IUPAC atomic weights are periodically revised; use the current table and retain more digits than the limiting mass measurement.

Common Mistakes

• Rounding the precipitate mass before subtracting the tare mass.
• Using the addition rule for percent composition, which is a multiplication/division calculation.
• Letting a low-precision atomic weight limit a high-precision mass.
• Forgetting that mass by difference inherits the decimal-place limit, not the sig-fig limit of the larger mass.
• Reporting too many digits in the final percent because the calculator output is long.
• Assuming significant figures alone satisfy ISO 17025 or GUM uncertainty requirements.

Practice Problems

1. A 0.4000 g sample gives 0.2156 g precipitate. The precipitate is 12.34% analyte by formula mass. What is the percent analyte in the sample? Answer: Mass analyte = 0.2156 × 0.1234 = 0.02660 g; % = (0.02660 / 0.4000) × 100 = 6.650%.
2. Crucible + precipitate = 18.7654 g; empty crucible = 18.421 g. What is the precipitate mass to correct significant figures? Answer: 18.7654 – 18.421 = 0.3444 g before rounding; least decimal places is 3, so 0.344 g.
3. A 0.2500 g sample yields 0.1523 g AgCl. Calculate % Cl. Answer: Mass Cl = 0.1523 × (35.45 / 143.32) = 0.03767 g; % Cl = (0.03767 / 0.2500) × 100 = 15.07%.

Quick Reference Table

Related Rules

• Significant Figures in Addition and Subtraction
• Significant Figures in Multiplication and Division
• Rounding Rules and Rounding Methods
• Exact Numbers and Significant Figures
• Error Propagation in Measurements
• Measurement Uncertainty and Reporting
• ASTM E29: Significant Digits and Conformance
• ISO 80000: Quantities, Units, and Rounding