Air density (CIPM-2007)
Density of moist air at the balance from barometric pressure, temperature, relative humidity or dew point and the CO₂ fraction, with the full CIPM-2007 equation: saturation vapour pressure, enhancement factor and compressibility factor. The OIML R 111-1 approximation is shown beside it as a labelled secondary check when the inputs are in its range.
Air density · CIPM-2007
v0.1.0 · Metrologia 45 (2008) 149Air density ρa1.19931 kg/m³= 0.00119931 g/cm³ · xv = 0.0115893, Z = 0.99961477, psv = 2339.16 Pa, f = 1.0040256
Secondary check · OIML R 111-1 eq. (E.3-1)1.19929 kg/m³CIPM-2007 differs by 16.3 × 10⁻⁶ relative; the approximation’s stated uncertainty is 2 × 10⁻⁴.
xCO2 was assumed to be 0.0004. The paper (p. 152) advises basing it on a survey of your laboratory and adding an uncertainty component, ≈ 0.4 × u(xCO2) relative.
Wording for a record
Air density ρa = 1.19931 kg/m³ calculated with the CIPM-2007 equation (Picard et al., Metrologia 45 (2008) 149) from p = 1013.25 hPa (mbar), t = 20 °C, RH = 50 %, xCO2 = 0.0004. The equation itself contributes u(ρa)/ρa = 2.2 × 10⁻⁵.
ρa = p·Ma/(Z·R·T)·[1 − xv(1 − Mv/Ma)] · R = 8.314 472 · β = 3.14 × 10⁻⁸ Pa⁻¹
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 |
|---|---|---|---|---|
| AD-01 | pressure = 101325, temperature = 20, relativeHumidity = 50 | 1.1993138954744933 kg/m³ | ± 1e-12 | Picard, Davis, Gläser and Fujii, "Revised formula for the density of moist air (CIPM-2007)", Metrologia 45 (2008) 149–155 (NIST copy: nist.gov/sites/default/files/documents/calibrations/CIPM-2007.pdf), eqs. (1), (4), (A1.1)–(A1.4) with the paper’s constants, evaluated in 50-digit decimal arithmetic (Python decimal), independently of this library. The paper gives no worked example; psv and Z are also checked against the values Davis 1992 prints (AD-21 to AD-27). |
| AD-02 | pressure = 101325, temperature = 20, relativeHumidity = 0 | 1.204557341628052 kg/m³ | ± 1e-12 | Picard, Davis, Gläser and Fujii, "Revised formula for the density of moist air (CIPM-2007)", Metrologia 45 (2008) 149–155 (NIST copy: nist.gov/sites/default/files/documents/calibrations/CIPM-2007.pdf), eqs. (1), (4), (A1.1)–(A1.4) with the paper’s constants, evaluated in 50-digit decimal arithmetic (Python decimal), independently of this library. The paper gives no worked example; psv and Z are also checked against the values Davis 1992 prints (AD-21 to AD-27). |
| AD-03 | pressure = 100000, temperature = 23, dewPoint = 15 | 1.169159339448989 kg/m³ | ± 1e-12 | Picard, Davis, Gläser and Fujii, "Revised formula for the density of moist air (CIPM-2007)", Metrologia 45 (2008) 149–155 (NIST copy: nist.gov/sites/default/files/documents/calibrations/CIPM-2007.pdf), eqs. (1), (4), (A1.1)–(A1.4) with the paper’s constants, evaluated in 50-digit decimal arithmetic (Python decimal), independently of this library. The paper gives no worked example; psv and Z are also checked against the values Davis 1992 prints (AD-21 to AD-27). |
| AD-04 | pressure = 60000, temperature = 27, relativeHumidity = 100, xCO2 = 0.001 | 0.6810875206597737 kg/m³ | ± 1e-12 | Picard, Davis, Gläser and Fujii, "Revised formula for the density of moist air (CIPM-2007)", Metrologia 45 (2008) 149–155 (NIST copy: nist.gov/sites/default/files/documents/calibrations/CIPM-2007.pdf), eqs. (1), (4), (A1.1)–(A1.4) with the paper’s constants, evaluated in 50-digit decimal arithmetic (Python decimal), independently of this library. The paper gives no worked example; psv and Z are also checked against the values Davis 1992 prints (AD-21 to AD-27). |
| AD-05 | pressure = 110000, temperature = 15, relativeHumidity = 0 | 1.330491272386205 kg/m³ | ± 1e-12 | Picard, Davis, Gläser and Fujii, "Revised formula for the density of moist air (CIPM-2007)", Metrologia 45 (2008) 149–155 (NIST copy: nist.gov/sites/default/files/documents/calibrations/CIPM-2007.pdf), eqs. (1), (4), (A1.1)–(A1.4) with the paper’s constants, evaluated in 50-digit decimal arithmetic (Python decimal), independently of this library. The paper gives no worked example; psv and Z are also checked against the values Davis 1992 prints (AD-21 to AD-27). |
| AD-06 | pressure = 101325, temperature = 20, relativeHumidity = 50, xCO2 = 0.0008 | 1.1995113813111424 kg/m³ | ± 1e-12 | Picard, Davis, Gläser and Fujii, "Revised formula for the density of moist air (CIPM-2007)", Metrologia 45 (2008) 149–155 (NIST copy: nist.gov/sites/default/files/documents/calibrations/CIPM-2007.pdf), eqs. (1), (4), (A1.1)–(A1.4) with the paper’s constants, evaluated in 50-digit decimal arithmetic (Python decimal), independently of this library. The paper gives no worked example; psv and Z are also checked against the values Davis 1992 prints (AD-21 to AD-27). |
| AD-07 | pressure = 101325, temperature = 20, relativeHumidity = 50, output = xv | 0.011589340130234807 mol/mol | ± 1e-17 | Picard, Davis, Gläser and Fujii, "Revised formula for the density of moist air (CIPM-2007)", Metrologia 45 (2008) 149–155 (NIST copy: nist.gov/sites/default/files/documents/calibrations/CIPM-2007.pdf), eqs. (1), (4), (A1.1)–(A1.4) with the paper’s constants, evaluated in 50-digit decimal arithmetic (Python decimal), independently of this library. The paper gives no worked example; psv and Z are also checked against the values Davis 1992 prints (AD-21 to AD-27). |
| AD-08 | pressure = 101325, temperature = 20, relativeHumidity = 50, output = Z | 0.9996147675251258 1 | ± 1e-15 | Picard, Davis, Gläser and Fujii, "Revised formula for the density of moist air (CIPM-2007)", Metrologia 45 (2008) 149–155 (NIST copy: nist.gov/sites/default/files/documents/calibrations/CIPM-2007.pdf), eqs. (1), (4), (A1.1)–(A1.4) with the paper’s constants, evaluated in 50-digit decimal arithmetic (Python decimal), independently of this library. The paper gives no worked example; psv and Z are also checked against the values Davis 1992 prints (AD-21 to AD-27). |
| AD-09 | pressure = 101325, temperature = 20, relativeHumidity = 50, output = psv | 2339.1632301967875 Pa | ± 1e-9 | Picard, Davis, Gläser and Fujii, "Revised formula for the density of moist air (CIPM-2007)", Metrologia 45 (2008) 149–155 (NIST copy: nist.gov/sites/default/files/documents/calibrations/CIPM-2007.pdf), eqs. (1), (4), (A1.1)–(A1.4) with the paper’s constants, evaluated in 50-digit decimal arithmetic (Python decimal), independently of this library. The paper gives no worked example; psv and Z are also checked against the values Davis 1992 prints (AD-21 to AD-27). |
| AD-10 | pressure = 748.1, pressureUnit = mmHg, temperature = 22.3, relativeHumidity = 37 | 1.17194 kg/m³ | ± 0.00012 | NIST SOP 2 (2019), Recommended Standard Operating Procedure for Applying Air Buoyancy Corrections (NISTIR 6969), Appendix B, Example 1: P = 748.1 mmHg, t = 22.3 °C, U = 37 % → ρa = 1.171 94 × 10⁻³ g/cm³ from SOP 2’s approximate eq. (11). |
| AD-11 | pressure = 612.3, pressureUnit = mmHg, temperature = 23.4, relativeHumidity = 23 | 0.95632 kg/m³ | ± 0.000096 | NIST SOP 2 (2019), Recommended Standard Operating Procedure for Applying Air Buoyancy Corrections (NISTIR 6969), Appendix B, Example 3 (laboratory well above sea level): P = 612.3 mmHg, t = 23.4 °C, U = 23 % → ρa = 0.956 32 × 10⁻³ g/cm³. |
| AD-12 | pressure = 101325, temperature = 20, relativeHumidity = 50 | 1.1992943050311118 kg/m³ | ± 0.00024 | OIML R 111-1:2004 Annex E, eq. (E.3-1): ρa = [0.348 48 × 1013.25 − 0.009 × 50 × exp(0.061 × 20)]/293.15 = 1.199 294 305 kg/m³; the Recommendation gives its relative uncertainty as 2 × 10⁻⁴. |
| AD-13 | pressure = 100000, temperature = 23, relativeHumidity = 40 | 1.1717566835048907 kg/m³ | ± 0.00024 | OIML R 111-1:2004 Annex E, eq. (E.3-1) at 1000 hPa, 23 °C, 40 % = 1.171 756 684 kg/m³; relative uncertainty 2 × 10⁻⁴. |
| AD-14 | pressure = 59000, temperature = 20, relativeHumidity = 50 | OutOfRangeError | exact | Picard et al. 2008, Appendix A.3 (600 hPa ≤ p ≤ 1100 hPa, 15 °C ≤ t ≤ 27 °C) and A.1 (0 ≤ h ≤ 1). |
| AD-15 | pressure = 101325, temperature = 28, relativeHumidity = 50 | OutOfRangeError | exact | Picard et al. 2008, Appendix A.3 (600 hPa ≤ p ≤ 1100 hPa, 15 °C ≤ t ≤ 27 °C) and A.1 (0 ≤ h ≤ 1). |
| AD-16 | pressure = 101325, temperature = 20, relativeHumidity = 101 | OutOfRangeError | exact | Picard et al. 2008, Appendix A.3 (600 hPa ≤ p ≤ 1100 hPa, 15 °C ≤ t ≤ 27 °C) and A.1 (0 ≤ h ≤ 1). |
| AD-17 | pressure = 101325, temperature = 20, dewPoint = 21 | OutOfRangeError | exact | Physical limit: the dew point cannot exceed the air temperature (h ≤ 1, Picard et al. 2008 A.1). |
| AD-18 | pressure = 101325, temperature = 20, dewPoint = 10 | OutOfRangeError | exact | Picard et al. 2008, Appendix A.3 (600 hPa ≤ p ≤ 1100 hPa, 15 °C ≤ t ≤ 27 °C) and A.1 (0 ≤ h ≤ 1). The paper gives no separate dew-point range, so psv(td) is held to the same 15 to 27 °C. |
| AD-19 | pressure = 101325, temperature = 20, relativeHumidity = 50, dewPoint = 15 | RangeError | exact | Picard et al. 2008, eq. (A1.3): xv comes from h or from td, not both. |
| AD-20 | pressure = 101325, temperature = 20, relativeHumidity = 50, xCO2 = 0.002 | OutOfRangeError | exact | Picard et al. 2008, p. 152: xCO2 = 0.001 is the extreme case discussed; the neglected term 0.26ρa·xv(0.0004 − xCO2) grows beyond it. |
| AD-21 | pressure = 100000, temperature = 20, relativeHumidity = 50, output = psv | 2339.2 Pa | ± 0.05 | R S Davis, "Equation for the determination of the density of moist air (1981/91)", Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column, ITS-90, xCO2 = 0.0004. CIPM-2007 keeps eqs. (A1.1) psv and (A1.4) Z and all their constants unchanged from CIPM-81/91 (Picard et al., Metrologia 45 (2008) 149–155, Appendix A.1 and A.2), so the printed value applies to CIPM-2007 as it stands. |
| AD-22 | pressure = 100000, temperature = 15, relativeHumidity = 90, output = psv | 1705.7 Pa | ± 0.05 | R S Davis, "Equation for the determination of the density of moist air (1981/91)", Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column, ITS-90, xCO2 = 0.0004. CIPM-2007 keeps eqs. (A1.1) psv and (A1.4) Z and all their constants unchanged from CIPM-81/91 (Picard et al., Metrologia 45 (2008) 149–155, Appendix A.1 and A.2), so the printed value applies to CIPM-2007 as it stands. |
| AD-23 | pressure = 60000, temperature = 25, relativeHumidity = 50, output = psv | 3169.8 Pa | ± 0.05 | R S Davis, "Equation for the determination of the density of moist air (1981/91)", Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column, ITS-90, xCO2 = 0.0004. CIPM-2007 keeps eqs. (A1.1) psv and (A1.4) Z and all their constants unchanged from CIPM-81/91 (Picard et al., Metrologia 45 (2008) 149–155, Appendix A.1 and A.2), so the printed value applies to CIPM-2007 as it stands. |
| AD-24 | pressure = 100000, temperature = 20, relativeHumidity = 50, output = Z | 0.999619 1 | ± 5e-7 | R S Davis, "Equation for the determination of the density of moist air (1981/91)", Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column, ITS-90, xCO2 = 0.0004. CIPM-2007 keeps eqs. (A1.1) psv and (A1.4) Z and all their constants unchanged from CIPM-81/91 (Picard et al., Metrologia 45 (2008) 149–155, Appendix A.1 and A.2), so the printed value applies to CIPM-2007 as it stands. |
| AD-25 | pressure = 110000, temperature = 20, relativeHumidity = 10, output = Z | 0.999608 1 | ± 5e-7 | R S Davis, "Equation for the determination of the density of moist air (1981/91)", Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column, ITS-90, xCO2 = 0.0004. CIPM-2007 keeps eqs. (A1.1) psv and (A1.4) Z and all their constants unchanged from CIPM-81/91 (Picard et al., Metrologia 45 (2008) 149–155, Appendix A.1 and A.2), so the printed value applies to CIPM-2007 as it stands. |
| AD-26 | pressure = 100000, temperature = 15, relativeHumidity = 90, output = Z | 0.999555 1 | ± 5e-7 | R S Davis, "Equation for the determination of the density of moist air (1981/91)", Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column, ITS-90, xCO2 = 0.0004. CIPM-2007 keeps eqs. (A1.1) psv and (A1.4) Z and all their constants unchanged from CIPM-81/91 (Picard et al., Metrologia 45 (2008) 149–155, Appendix A.1 and A.2), so the printed value applies to CIPM-2007 as it stands. |
| AD-27 | pressure = 60000, temperature = 25, relativeHumidity = 50, output = Z | 0.999769 1 | ± 5e-7 | R S Davis, "Equation for the determination of the density of moist air (1981/91)", Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column, ITS-90, xCO2 = 0.0004. CIPM-2007 keeps eqs. (A1.1) psv and (A1.4) Z and all their constants unchanged from CIPM-81/91 (Picard et al., Metrologia 45 (2008) 149–155, Appendix A.1 and A.2), so the printed value applies to CIPM-2007 as it stands. |
| AD-28 | pressure = 100000, temperature = 20, relativeHumidity = 50 | 1.1835569088741702 kg/m³ | ± 5e-7 | Composed: R S Davis, Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column (ρa = 1.183472 kg/m³), converted to CIPM-2007 with the two constants Picard, Davis, Gläser and Fujii, Metrologia 45 (2008) 149–155 changed (p. 152 and §4): Ma 28.9635 → 28.965 46 g/mol (eq. (4)) and R 8.314 510 → 8.314 472 J/(mol·K), with xv from eqs. (A1.1)–(A1.3) in 50-digit decimal arithmetic; Z cancels because it is unchanged. |
| AD-29 | pressure = 110000, temperature = 20, relativeHumidity = 10 | 1.3066762725922156 kg/m³ | ± 5e-7 | Composed: R S Davis, Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column (ρa = 1.306582 kg/m³), converted to CIPM-2007 with the two constants Picard, Davis, Gläser and Fujii, Metrologia 45 (2008) 149–155 changed (p. 152 and §4): Ma 28.9635 → 28.965 46 g/mol (eq. (4)) and R 8.314 510 → 8.314 472 J/(mol·K), with xv from eqs. (A1.1)–(A1.3) in 50-digit decimal arithmetic; Z cancels because it is unchanged. |
| AD-30 | pressure = 100000, temperature = 15, relativeHumidity = 90 | 1.2024940798717143 kg/m³ | ± 5e-7 | Composed: R S Davis, Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column (ρa = 1.202408 kg/m³), converted to CIPM-2007 with the two constants Picard, Davis, Gläser and Fujii, Metrologia 45 (2008) 149–155 changed (p. 152 and §4): Ma 28.9635 → 28.965 46 g/mol (eq. (4)) and R 8.314 510 → 8.314 472 J/(mol·K), with xv from eqs. (A1.1)–(A1.3) in 50-digit decimal arithmetic; Z cancels because it is unchanged. |
| AD-31 | pressure = 60000, temperature = 25, relativeHumidity = 50 | 0.6942113658468131 kg/m³ | ± 5e-7 | Composed: R S Davis, Metrologia 29 (1992) 67–70 (NIST copy: nist.gov/system/files/documents/calibrations/metv29i1p67-2.pdf), Table 2, 1981/91 column (ρa = 0.694162 kg/m³), converted to CIPM-2007 with the two constants Picard, Davis, Gläser and Fujii, Metrologia 45 (2008) 149–155 changed (p. 152 and §4): Ma 28.9635 → 28.965 46 g/mol (eq. (4)) and R 8.314 510 → 8.314 472 J/(mol·K), with xv from eqs. (A1.1)–(A1.3) in 50-digit decimal arithmetic; Z cancels because it is unchanged. |
| ADO-01 | pressure = 101325, temperature = 20, relativeHumidity = 50 | 1.1992943050311118 kg/m³ | ± 1e-12 | OIML R 111-1:2004 Annex E, eq. (E.3-1) (unchanged by Amendment 2025), evaluated in 50-digit decimal arithmetic. |
| ADO-02 | pressure = 100000, temperature = 23, relativeHumidity = 40 | 1.1717566835048907 kg/m³ | ± 1e-12 | OIML R 111-1:2004 Annex E, eq. (E.3-1) (unchanged by Amendment 2025), evaluated in 50-digit decimal arithmetic. |
| ADO-03 | pressure = 101325, temperature = 20, relativeHumidity = 85 | OutOfRangeError | exact | OIML R 111-1:2004 Annex E.3: valid for hr < 80 %. |
| ADO-04 | pressure = 89000, temperature = 20, relativeHumidity = 50 | OutOfRangeError | exact | OIML R 111-1:2004 Annex E.3: valid for 900 hPa < p < 1100 hPa. |
| ADO-05 | pressure = 101325, temperature = 31, relativeHumidity = 50 | OutOfRangeError | exact | OIML R 111-1:2004 Annex E.3: valid for 10 °C < t < 30 °C. |
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