Gravimetric Factors and Calculations

Introduction

Gravimetric arithmetic begins with the solid actually weighed, not necessarily with the compound first precipitated. Its formula tells how many analyte units each mole contains. A gravimetric factor packages that relation as mass of analyte per mass of weighing form. Deriving the factor rather than memorising it prevents errors when the report asks for an element, ion or compound in different units.

Core explanation

For chloride weighed as AgCl, one mole of AgCl contains one mole of chloride. Therefore m(Cl⁻) = m(AgCl) × M(Cl)/M(AgCl). Using approximate molar masses, M(Cl) = 35.45 g mol⁻¹ and M(AgCl) = 143.32 g mol⁻¹, the factor is 0.2473 g Cl⁻ per g AgCl. The factor is dimensionless as a mass ratio, though its meaning should be written beside it. If the desired report is NaCl equivalent rather than chloride-ion mass, the numerator changes to M(NaCl), and the numerical factor changes. The original measured solid has not changed, only the reporting convention.

For sulfate weighed as BaSO₄, one mole contains one mole sulfate. The sulfate mass factor is M(SO₄²⁻)/M(BaSO₄), approximately 96.06/233.39. If reporting sulfur mass, use M(S)/M(BaSO₄), approximately 32.06/233.39. If an ignition product contains two analyte atoms per formula unit, include that coefficient. For example, one mole Fe₂O₃ contains two moles Fe, so factor for Fe mass is 2M(Fe)/M(Fe₂O₃).

The calculation sequence is: determine blank-corrected weighing-form mass; multiply by the derived factor; scale from analysed aliquot to full prepared solution if needed; divide by original sample mass or volume. If a water sample gives a precipitate from only 25.00 mL of a 100.0 mL digest, a factor of four converts analyte in that aliquot to analyte in the whole digest. Do not apply this factor if the entire digest was precipitated.

Stoichiometry assumes quantitative conversion and pure, stable product. Precipitate solubility and filtration loss bias recovered mass low; trapped salts or water often bias it high. Arithmetic cannot correct an unrecognised change in chemical composition. A calibration-like verification with a known reference sample can test the complete procedure.

Step-by-step reasoning

1. Write the exact chemical formula of the final weighing form. 2. Count analyte units per formula unit and derive n(analyte)/n(weighing form). 3. Construct the mass ratio using molar masses and coefficient count. 4. Apply the factor to blank-corrected precipitate mass. 5. Reverse aliquot factors and divide by original sample amount in the requested unit.

Visual explanation

Draw a conversion ladder: g precipitate ÷ molar mass → mol precipitate × analyte count → mol analyte × analyte molar mass → g analyte ÷ sample amount → final concentration or fraction. Circle the middle three steps as the gravimetric factor. Place the aliquot factor after analyte mass so it is easy to see which sample it scales.

Real-world analogy

If every sealed package contains two identical items, weighing packages can reveal item mass only if package weight and contents are known. A gravimetric compound is more exact: its formula fixes the mass proportion. Changing from “item mass” to “pair mass” changes the conversion factor without changing the package on the scale.

Real-world example

A mining laboratory may precipitate nickel with a selective reagent and weigh a compound of established composition. The result could be reported as nickel mass fraction in ore, while a water laboratory might report chloride in mg L⁻¹ from AgCl. The same conversion logic works only after the weighing form and original sample basis are specified.

Why?

Why should a factor be derived from the final weighing form? A precipitate may be dried as one hydrate or ignited into a different oxide. Its molar mass and analyte count can change on conditioning. Using the precursor's formula on the final mass gives a systematic error even if the balance is perfect.

Common misconception

“The precipitate mass equals analyte mass” is generally false; AgCl contains silver as well as chloride. Another mistake is using M(Cl₂) for chloride-ion mass simply because elemental chlorine occurs as diatomic molecules; the reporting unit is one chloride ion, so use its atomic molar mass per mole Cl⁻.

Worked example

An assay yields 0.5000 g pure dry BaSO₄ from a 2.000 g solid sample. Sulfur mass is 0.5000 × 32.06/233.39 ≈ 0.06868 g. Sulfur mass fraction is 0.06868/2.000 = 0.03434, or 3.434%. If the report instead asks for sulfate, use 96.06/233.39 and obtain about 0.2058 g sulfate, or 10.29% of the sample. These are different expressions of the same measured precipitate.

Quick check

1. How many moles of Fe are represented by one mole of Fe₂O₃ final weighing form? Answer: Two moles of Fe, because each Fe₂O₃ formula unit contains two iron atoms. The iron mass factor includes 2M(Fe) in its numerator.

Exam focus

Write what is weighed and what is reported. Build the factor from molar masses and coefficients, then scale aliquots only when needed. Include units at each stage and convert the original sample mass to the same mass units before calculating percent. Briefly state the purity and complete-recovery assumptions.

Advanced insight

Uncertainty in a gravimetric result is not solely balance uncertainty. Subtracting a blank, losing a small soluble fraction, imperfect stoichiometry and sample heterogeneity can dominate. If a precipitate mass is small relative to a fixed blank, relative uncertainty rises sharply; larger sample mass can help until contamination or filter handling becomes limiting.

Summary

A gravimetric factor is the mass ratio between a defined analyte and a pure final weighing form. Derive it from chemical formula and stoichiometry, multiply by corrected precipitate mass, then restore the original sample basis. Different reporting species require different factors even when the balance reading is unchanged.

Practice questions

1. What is the chloride mass factor for AgCl, using M(Cl) = 35.45 and M(AgCl) = 143.32 g mol⁻¹? Answer: 35.45/143.32 ≈ 0.2473 g Cl⁻ per g AgCl.

2. Why does a sulfate-as-BaSO₄ result use a different factor from sulfur-as-BaSO₄? Answer: The numerator is the molar mass of the requested reporting species: SO₄²⁻ for sulfate or S for sulfur. Both correspond to one mole per mole BaSO₄ but have different masses.

3. A 20.00 mL aliquot of a 100.0 mL digest is precipitated. What factor scales analyte mass in the aliquot to the whole digest? Answer: 100.0/20.00 = 5.000, assuming the digest was homogeneous and the aliquot was transferred accurately.