Air buoyancy correction

Correct a weighing for air buoyancy. Comparison: a test weight against a reference of known conventional mass, with the exact and first-order correction factor C and the R 111 test for whether it may be neglected. Balance reading: the mass of a sample of any density weighed on a balance adjusted with 8000 kg/m³ weights, as in NIST SOP 2.

Air buoyancy correction

v0.1.0 · OIML D 28 · NIST SOP 2

WeighingAir density during the weighing ρa (kg/m³)Mass unitReference conventional mass m_crIndication difference Δm_w (test − reference)Reference weight density ρr (kg/m³)Test weight density ρt (kg/m³)U of the result (optional, for the neglect test)

Conventional mass of the test weight m_ct1000000.4422 mgBuoyancy correction m_cr·C = 0.4422 mg; C = 0.0000004422 (first order 0.0000004420); without it 1000000.0000 mg

m_ct = m_cr(1 + C) + Δm_w · C = (ρr − ρt)(ρa − ρ0)/[(ρr − ρ0)(ρt − ρa)] · ρ0 = 1.2 kg/m³

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How this is calculated

Equation: Comparison: m_ct = m_cr(1 + C) + Δm_w, C = (ρr − ρt)(ρa − ρ0)/[(ρr − ρ0)(ρt − ρa)] (first order: C ≈ (ρa − ρ0)(1/ρt − 1/ρr)); negligible when |C| ≤ U/(3m0). Balance reading: M_x = M_R(1 − ρa′/ρc)/(1 − ρa/ρx)
Source: OIML D 28:2004 eqs. (8)–(11), reference conditions clause 4. Balance reading: NIST SOP 2 (2019) eq. (22) and Appendix B Examples 3 and 4.
Valid range: Air density 0.5 to 1.5 kg/m³; densities of weights and samples 100 to 25 000 kg/m³; masses > 0, all in one mass unit
Rounding: Full precision inside; masses shown to 4 decimal places by default, rounded half up, corrections and C to 4 significant figures.
Version: v0.1.0 · changelog
NCSLI MII taxa: Measure.Mass.Conventional Measure.Mass.True Source.Mass.Conventional Source.Mass.True

Calculated value, not a certified result. Accredited laboratories must validate any software they use (ISO/IEC 17025 clause 7.11).

Test vectors for this tool

The reference cases this calculator is tested against on every change. Use them to check your own software. All test vectors · Download JSON

ID	Inputs	Expected	Tolerance	Source
BC-01	referenceConventionalMass = 1, airDensity = 1.1, referenceDensity = 7820, testDensity = 8100, output = CFirstOrder	4.4e-7 1	± 5e-9	OIML D 28:2004, Conventional value of the result of weighing in air, clause 7, Example 2: ρr = 7820, ρt = 8100, ρa = 1.1 kg/m³ → \|C\| = 4.4 × 10⁻⁷ from eq. (10).
BC-02	referenceConventionalMass = 1, airDensity = 1.1, referenceDensity = 7820, testDensity = 8100, output = C	4.421726643283747e-7 1	± 1e-21	OIML D 28:2004, Conventional value of the result of weighing in air, eq. (9): C = (ρr − ρt)(ρa − ρ0)/[(ρr − ρ0)(ρt − ρa)], hand arithmetic in 50 digits.
BC-03	referenceConventionalMass = 1, airDensity = 1.16, referenceDensity = 7800, testDensity = 8400, output = CFirstOrder	3.7e-7 1	± 5e-9	OIML D 28:2004, Conventional value of the result of weighing in air, clause 8, Example 3: ρr = 7800, ρt = 8400, ρa = 1.16 kg/m³ → \|C\| = 3.7 × 10⁻⁷.
BC-04	referenceConventionalMass = 1, airDensity = 1.16, referenceDensity = 7800, testDensity = 8400, U = 0.0000015, output = negligible	true	exact	OIML D 28:2004, Conventional value of the result of weighing in air, Example 3 with eq. (11) \|C\| ≤ U/(3m0): class F1 1 kg, U ≤ 1.5 mg — “A buoyancy correction is not necessary according to equation (11)”.
BC-05	referenceConventionalMass = 1, airDensity = 1.1, referenceDensity = 7820, testDensity = 8100, U = 5e-7, output = negligible	false	exact	OIML D 28:2004, Conventional value of the result of weighing in air, Example 2: for class E2 with U ≤ 0.5 mg “a buoyancy correction has to be applied”.
BC-06	referenceConventionalMass = 15, airDensity = 0.95632, referenceDensity = 8000, testDensity = 7780, output = correctionFirstOrder	-0.00001292 g	± 5e-9	NIST SOP 2 (2019), Recommended Standard Operating Procedure for Applying Air Buoyancy Corrections (NISTIR 6969), Appendix B, Example 4: 15 g (0.956 32 × 10⁻³ − 1.2 × 10⁻³ g/cm³)(1/7.78 − 1/8.0 cm³/g) = −12.92 µg.
BC-07	reading = 15.000358, airDensity = 0.95632, sampleDensity = 2329.1	15.004726 g	± 5e-7	NIST SOP 2 (2019), Recommended Standard Operating Procedure for Applying Air Buoyancy Corrections (NISTIR 6969), Appendix B, Example 3: (15.00 g + 0.000 358 g)(1 − 0.956 32 × 10⁻³/8.0)/(1 − 0.956 32 × 10⁻³/2.3291) = 15.004 726 g.
BC-08	reading = 15.000358, airDensity = 0.95632, sampleDensity = 2329.1	15.004725743093063 g	± 1e-12	NIST SOP 2 (2019), Recommended Standard Operating Procedure for Applying Air Buoyancy Corrections (NISTIR 6969), Appendix B, eq. (22) with Example 3’s inputs, hand arithmetic in 50 digits.
BC-09	reading = 15, airDensity = 2, sampleDensity = 2329.1	OutOfRangeError	exact	Metrology Bench limit: air at a balance is 0.5 to 1.5 kg/m³ (CIPM-2007’s own range gives about 0.68 to 1.33 kg/m³).
BC-10	referenceConventionalMass = 1, airDensity = 1.2, referenceDensity = 8, testDensity = 8000, output = C	OutOfRangeError	exact	Metrology Bench limit: densities 100 to 25 000 kg/m³.

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