picard_2008#
A. Picard, R. S. Davis, M. Gläser and K. Fujii, “Revised formula for the density of moist air (CIPM-2007)”, Metrologia 45, 149 (2008)
Source record · Catalogue · Review register
DOI: 10.1088/0026-1394/45/2/004
Source kind: compilation
Recorded technique: The CIPM-2007 equation of state for moist air, approved by the 96th CIPM meeting (November 2007): the CIPM-81/91 equation with two constants amended, the mole fraction of argon (from the KRISS mass-spectrometric and LNE gas-chromatographic determinations against gravimetric standards) and the molar gas constant (CODATA), and an uncertainty evaluation following the GUM. The composition of dry air is a reference (conventional) composition, the mole fractions of the US Standard Atmosphere 1976 with new argon, the default carbon dioxide and the oxygen that follows from it (pp. 149-151).
Materials in structured entries: Ar, CH4, CO, CO2, H2, He, Kr, N2, N2O, Ne, O2, Xe
Dataset quantities: mole_fraction
Contents: 12 datasets; 3 formulas.
Curation notes#
Identity checked against the first page (Metrologia 45 (2008) 149-155, published 18 February 2008). Page numbers are the journal’s: p. = pdf page + 148.
THE COMPOSITION (Table 1, p. 151; image checked digit by digit). Mole fractions of the twelve constituents of dry air, recorded as mole_fraction datasets with parameters.mixture = ‘dry air (CIPM-2007)’, one per constituent (all twelve are registered gases; no material is registered for air, which is a mixture, TC-543). N2 0.780848, O2 0.209390, Ar 0.009332, CO2 0.00040, Ne 18.2e-6, He 5.2e-6, CH4 1.5e-6, Kr 1.1e-6, H2 0.5e-6, N2O 0.3e-6, CO 0.2e-6, Xe 0.1e-6. Every printed contribution x_i M_i is reproduced by the printed x_i and M_i to the last digit (e.g. 0.780848 x 28.0134 = 21.874207), which confirms each digit of the major fractions; sum x_i = 1 - 2.9e-6, as the paper says (p. 150: ‘presumably due to minor rounding errors in [12]’, hence the normalisation in Eq. 3); M_a = sum x_i M_i/sum x_i = 28.96546 g/mol, as printed (computed 2026-09-27). Which values are whose: N2, O2, CO2 and Ar are set by this paper (Ar measured by KRISS and LNE, N2 the remainder, CO2 the conventional default, O2 = S - x_CO2); the eight minor constituents are the US Standard Atmosphere 1976’s, unchanged (reproduced). Water vapour is not a constituent of dry air: moist air enters through x_v (Eq. 1, Appendix A).
THE MOLAR MASSES of Table 1 (10^-3 kg/mol: N2 28.0134, O2 31.9988, Ar 39.948, CO2 44.01, Ne 20.18, He 4.0, CH4 16.0, Kr 83.8, H2 2, N2O 44, CO 28, Xe 131) are IUPAC 2005’s [17], ‘rounded for convenience’ from CO2 on (p. 150); no quantity, kept here and in the formula eq3_molar_mass.
UNCERTAINTY (Table 2, p. 152, standard uncertainties per the GUM, assuming x_CO2 known exactly). Parameter, standard uncertainty (1e-6 of its unit), relative contribution to u(rho_a)/rho_a (1e-6): R, 15 (in 8.314472), 1.7; M_N2, 200 (in g/mol), 5.4; M_O2, 300, 2.2; M_Ar, 500, 0.2; M_CO2, 500, 0.0; x_Ar, 3 umol/mol, 1.2; S = x_O2 + x_CO2, 60 umol/mol, 8.3; Delta x_O2, 60, 8.3 (type A); Delta x_CO2, 60, 8.3 (type A); Z, 15 (1e-6), 15. Quadratic sums 18.2 (type B), 11.7 (type A); combined standard uncertainty of the formula itself 22e-6 relative. Only x_Ar has a standard uncertainty of its own (the dataset x_Ar); the oxygen components are in x_O2’s notes. The sensitivities to the measured inputs (Eq. 6, p. 152): +1e-5 Pa^-1 u(p), -4e-3 K^-1 u(T), -9e-3 u(h), -3e-4 K^-1 u(t_d), +0.4 u(x_CO2) (relative).
THE DENSITY FORMULA (Eqs. 1-5, p. 150-151; Appendix A, pp. 153-154) is recorded as formulas without a form: the composition’s molar mass (Eqs. 3-4), the density (Eqs. 5a/5b), and Appendix A’s saturation vapour pressure (A1.1), enhancement factor (A1.2), water-vapour mole fraction (A1.3) and compressibility factor (A1.4), valid for 600-1100 hPa and 15-27 degrees C (Appendix A.3). R = 8.314472(15) J/(mol K), ‘recommended by CODATA 2006 [15]’ (p. 150; the reference list’s [15] is Mohr and Taylor’s CODATA 2002, whose R is the same number). The formula omits a term of about 0.26 rho_a x_v (0.0004 - x_CO2) (p. 152). CIPM-2007 against CIPM-81/91 (3.48349 in Eq. 5b): +72e-6 relative, from x_Ar (+67.7e-6) and R (+4.6e-6) (p. 152); 3.483740/3.48349 - 1 = 71.8e-6 (computed).
OTHER VALUES (notes only): the gravimetric-minus-formula air density difference that prompted the revision, 6.4e-5 with standard uncertainty 1.2e-5 (p. 149; Picard et al. 2004, Euromet project 144, Madec et al. 2007).
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
picard_2008/x_N2#
{
"id": "x_N2",
"material": "N2",
"file": "N2/picard_2008__x_N2.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"abscissa": "none",
"use": "primary",
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_N2 = 0.780848 mol/mol in the reference composition of dry air of the CIPM-2007 equation. N2: ‘not directly measured. It is simply determined as the remainder when the mole fractions of all other components of dry air have been subtracted from 1’ (p. 150); it changed with the new argon value. No uncertainty of its own is given (Table 2 budgets S, x_Ar and the molar masses).
picard_2008/x_O2#
{
"id": "x_O2",
"material": "O2",
"file": "O2/picard_2008__x_O2.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"abscissa": "none",
"use": "primary",
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_O2 = 0.209390 mol/mol in the reference composition of dry air of the CIPM-2007 equation. O2: x_O2 = S - x_CO2 with S = x_O2 + x_CO2 = 0.20979 mol/mol (Eq. 2, p. 150; Appendix B), the value of S accepted from CIPM-81 and -81/91 [1, 2], so 0.209390 = 0.20979 - 0.00040 (checked). Uncertainty (Table 2, Appendix B): u(S) = 60 umol/mol (‘each determination is within 50 umol/mol of the accepted value for S’, KRISS’s two air samples of different continents and elevations, each with u = 60 umol/mol) and a further u(Delta x_O2) = 60 umol/mol for oxygen changes not anticorrelated with CO2 (standard uncertainties; the earlier estimate of 170 umol/mol rested on unconfirmed nineteenth-century data). No single u(x_O2) is printed: not recorded as the dataset’s uncertainty.
picard_2008/x_Ar#
{
"id": "x_Ar",
"material": "Ar",
"file": "Ar/picard_2008__x_Ar.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"abscissa": "none",
"use": "primary",
"uncertainty": {
"absolute": 3e-06,
"level": "standard"
},
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_Ar = 0.009332 mol/mol in the reference composition of dry air of the CIPM-2007 equation. Ar: ‘x_Ar = 9.332(3) mmol/mol’ (pp. 150, 153), the new value of CIPM-2007: ‘A weighted average gives essentially the KRISS result’ (KRISS 2002, Park et al. [11]: 9.332(3) mmol/mol by mass spectrometry; LNE 2006, Sutour et al. [14]: 9.330(16) by gas chromatography; both calibrated with gravimetric synthetic standards). CIPM-81 and -81/91 used 9.17 mmol/mol; atmospheric science had kept Moissan’s 1903 9.34. The change raises the CIPM air density by 67.7e-6 (p. 152). Standard uncertainty 3 umol/mol (Table 2, the GUM evaluation).
picard_2008/x_CO2#
{
"id": "x_CO2",
"material": "CO2",
"file": "CO2/picard_2008__x_CO2.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"abscissa": "none",
"use": "primary",
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_CO2 = 0.00040 mol/mol in the reference composition of dry air of the CIPM-2007 equation. CO2: ‘the average mole fraction of carbon dioxide in laboratory air, x_CO2, has a default value of 0.00040’ (p. 150), ‘as in the previous CIPM formulas’: a conventional reference value, not a measurement. The formula is to be used with a measured x_CO2 where available (Eq. 4); otherwise an uncertainty for the assumption must be added (Eq. 6, 0.4 u(x_CO2) relative in density): Euromet project 503 found 440 umol/mol on average among twelve laboratories, with maxima averaging 660 and reaching 1000 umol/mol (p. 152). No uncertainty given for the default.
picard_2008/x_Ne#
{
"id": "x_Ne",
"material": "Ne",
"file": "Ne/picard_2008__x_Ne.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"scale": 1e-06,
"abscissa": "none",
"use": "reproduced",
"reproduced_from": [
"US Standard Atmosphere 1976 (Washington, DC: US Government Printing Office) pp. 3, 33"
],
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_Ne = 18.2 x 10^-6 mol/mol (printed with its factor; scale 1e-6), a minor constituent of the CIPM-2007 reference composition of dry air, taken unchanged from the mole fractions of [12], the US Standard Atmosphere 1976 (p. 150: ‘based on the mole fractions found in [12] with the following exceptions’: argon, carbon dioxide and oxygen). No uncertainty given; the minor constituents’ molar masses are rounded (p. 150). Adopted by CIPM-2007 as part of its definition of dry air; the US Standard Atmosphere is not in the library.
picard_2008/x_He#
{
"id": "x_He",
"material": "He",
"file": "He/picard_2008__x_He.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"scale": 1e-06,
"abscissa": "none",
"use": "reproduced",
"reproduced_from": [
"US Standard Atmosphere 1976 (Washington, DC: US Government Printing Office) pp. 3, 33"
],
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_He = 5.2 x 10^-6 mol/mol (printed with its factor; scale 1e-6), a minor constituent of the CIPM-2007 reference composition of dry air, taken unchanged from the mole fractions of [12], the US Standard Atmosphere 1976 (p. 150: ‘based on the mole fractions found in [12] with the following exceptions’: argon, carbon dioxide and oxygen). No uncertainty given; the minor constituents’ molar masses are rounded (p. 150). Adopted by CIPM-2007 as part of its definition of dry air; the US Standard Atmosphere is not in the library.
picard_2008/x_CH4#
{
"id": "x_CH4",
"material": "CH4",
"file": "CH4/picard_2008__x_CH4.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"scale": 1e-06,
"abscissa": "none",
"use": "reproduced",
"reproduced_from": [
"US Standard Atmosphere 1976 (Washington, DC: US Government Printing Office) pp. 3, 33"
],
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_CH4 = 1.5 x 10^-6 mol/mol (printed with its factor; scale 1e-6), a minor constituent of the CIPM-2007 reference composition of dry air, taken unchanged from the mole fractions of [12], the US Standard Atmosphere 1976 (p. 150: ‘based on the mole fractions found in [12] with the following exceptions’: argon, carbon dioxide and oxygen). No uncertainty given; the minor constituents’ molar masses are rounded (p. 150). Adopted by CIPM-2007 as part of its definition of dry air; the US Standard Atmosphere is not in the library.
picard_2008/x_Kr#
{
"id": "x_Kr",
"material": "Kr",
"file": "Kr/picard_2008__x_Kr.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"scale": 1e-06,
"abscissa": "none",
"use": "reproduced",
"reproduced_from": [
"US Standard Atmosphere 1976 (Washington, DC: US Government Printing Office) pp. 3, 33"
],
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_Kr = 1.1 x 10^-6 mol/mol (printed with its factor; scale 1e-6), a minor constituent of the CIPM-2007 reference composition of dry air, taken unchanged from the mole fractions of [12], the US Standard Atmosphere 1976 (p. 150: ‘based on the mole fractions found in [12] with the following exceptions’: argon, carbon dioxide and oxygen). No uncertainty given; the minor constituents’ molar masses are rounded (p. 150). Adopted by CIPM-2007 as part of its definition of dry air; the US Standard Atmosphere is not in the library.
picard_2008/x_H2#
{
"id": "x_H2",
"material": "H2",
"file": "H2/picard_2008__x_H2.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"scale": 1e-06,
"abscissa": "none",
"use": "reproduced",
"reproduced_from": [
"US Standard Atmosphere 1976 (Washington, DC: US Government Printing Office) pp. 3, 33"
],
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_H2 = 0.5 x 10^-6 mol/mol (printed with its factor; scale 1e-6), a minor constituent of the CIPM-2007 reference composition of dry air, taken unchanged from the mole fractions of [12], the US Standard Atmosphere 1976 (p. 150: ‘based on the mole fractions found in [12] with the following exceptions’: argon, carbon dioxide and oxygen). No uncertainty given; the minor constituents’ molar masses are rounded (p. 150). Adopted by CIPM-2007 as part of its definition of dry air; the US Standard Atmosphere is not in the library.
picard_2008/x_N2O#
{
"id": "x_N2O",
"material": "N2O",
"file": "N2O/picard_2008__x_N2O.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"scale": 1e-06,
"abscissa": "none",
"use": "reproduced",
"reproduced_from": [
"US Standard Atmosphere 1976 (Washington, DC: US Government Printing Office) pp. 3, 33"
],
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_N2O = 0.3 x 10^-6 mol/mol (printed with its factor; scale 1e-6), a minor constituent of the CIPM-2007 reference composition of dry air, taken unchanged from the mole fractions of [12], the US Standard Atmosphere 1976 (p. 150: ‘based on the mole fractions found in [12] with the following exceptions’: argon, carbon dioxide and oxygen). No uncertainty given; the minor constituents’ molar masses are rounded (p. 150). Adopted by CIPM-2007 as part of its definition of dry air; the US Standard Atmosphere is not in the library.
picard_2008/x_CO#
{
"id": "x_CO",
"material": "CO",
"file": "CO/picard_2008__x_CO.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"scale": 1e-06,
"abscissa": "none",
"use": "reproduced",
"reproduced_from": [
"US Standard Atmosphere 1976 (Washington, DC: US Government Printing Office) pp. 3, 33"
],
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_CO = 0.2 x 10^-6 mol/mol (printed with its factor; scale 1e-6), a minor constituent of the CIPM-2007 reference composition of dry air, taken unchanged from the mole fractions of [12], the US Standard Atmosphere 1976 (p. 150: ‘based on the mole fractions found in [12] with the following exceptions’: argon, carbon dioxide and oxygen). No uncertainty given; the minor constituents’ molar masses are rounded (p. 150). Adopted by CIPM-2007 as part of its definition of dry air; the US Standard Atmosphere is not in the library.
picard_2008/x_Xe#
{
"id": "x_Xe",
"material": "Xe",
"file": "Xe/picard_2008__x_Xe.csv",
"table": "tbl_b2ed47a88f5d",
"quantity": "mole_fraction",
"unit": "1",
"scale": 1e-06,
"abscissa": "none",
"use": "reproduced",
"reproduced_from": [
"US Standard Atmosphere 1976 (Washington, DC: US Government Printing Office) pp. 3, 33"
],
"parameters": {
"mixture": "dry air (CIPM-2007)"
}
}
Table 1 (p. 151): x_Xe = 0.1 x 10^-6 mol/mol (printed with its factor; scale 1e-6), a minor constituent of the CIPM-2007 reference composition of dry air, taken unchanged from the mole fractions of [12], the US Standard Atmosphere 1976 (p. 150: ‘based on the mole fractions found in [12] with the following exceptions’: argon, carbon dioxide and oxygen). No uncertainty given; the minor constituents’ molar masses are rounded (p. 150). Adopted by CIPM-2007 as part of its definition of dry air; the US Standard Atmosphere is not in the library.
Formulas#
picard_2008/eq3_molar_mass#
{
"id": "eq3_molar_mass",
"equation": "Eqs. (2)-(4), p. 150, with Table 1",
"printed": "M_a = sum x_i M_i / sum x_i (Eq. 3) = 28.96546 x 10^-3 kg mol^-1; with a measured x_CO2: M_a = [28.96546 + 12.011 (x_CO2 - 0.0004)] x 10^-3 kg mol^-1 (Eq. 4); x_O2 + x_CO2 = 0.20979 mol mol^-1 = S (Eq. 2)",
"variable": "x_CO2: mole fraction of carbon dioxide in dry air (mol/mol); M_a: molar mass of dry air",
"range": "dry air of the Table 1 composition, x_CO2 near 0.0004",
"coefficients": {
"M_a": "28.96546e-3",
"dM_dxCO2": "12.011e-3",
"x_CO2_ref": "0.0004",
"S": "0.20979"
},
"check": "Table 1's x_i and M_i give sum x_i M_i/sum x_i = 28.965456 x 10^-3 kg/mol (sum x_i = 0.9999971), the printed 28.96546; 12.011 = M_CO2 - M_O2 = 44.01 - 31.9988 = 12.0112 (checked 2026-09-27)."
}
The molar mass of the CIPM-2007 dry air; Eq. (4) moves CO2 against O2 at fixed S (Eq. 2). Air is not a material: this formula is recorded for review (solprop’s air).
picard_2008/eq5_density#
{
"id": "eq5_density",
"equation": "Eqs. (1), (5a), (5b), pp. 150-151",
"printed": "rho_a = (p M_a/(Z R T)) [1 - x_v (1 - M_v/M_a)] (Eq. 1); rho_a/(10^-3 kg m^-3) = [28.96546 + 12.011 (x_CO2 - 0.0004)]/8.314472 x (p/(Z T)) (1 - 0.3780 x_v) (Eq. 5a) = [3.483740 + 1.4446 (x_CO2 - 0.0004)] (p/(Z T)) (1 - 0.3780 x_v) (Eq. 5b); R = 8.314472(15) J mol^-1 K^-1; M_v = 18.01528(17) x 10^-3 kg mol^-1; 1 - M_v/M_a = 0.3780",
"variable": "p: pressure in Pa; T = 273.15 + t/degrees C, the thermodynamic temperature in K; x_v: mole fraction of water vapour (Appendix A, formula appendix_a); Z: compressibility factor (Appendix A); rho_a in 10^-3 kg m^-3 as written (i.e. g m^-3 for p in Pa)",
"range": "600-1100 hPa, 15-27 degrees C (Appendix A.3)",
"coefficients": {
"M_a": "28.96546",
"dM_dxCO2": "12.011",
"R": "8.314472",
"R_sigma": "0.000015",
"c0": "3.483740",
"c1": "1.4446",
"x_CO2_ref": "0.0004",
"water": "0.3780",
"M_v": "18.01528"
},
"check": "28.96546/8.314472 = 3.483740 (5b's constant, checked); 12.011/8.314472 = 1.4446; 1 - 18.01528/28.96546 = 0.37803 (0.3780 printed). At 20 degrees C, 101 325 Pa, x_v = 0, x_CO2 = 0.0004, with appendix_a's Z = 0.99964: rho_a = 1.20456 kg/m^3 (computed; the paper prints no worked example). Checked 2026-09-27."
}
Combined relative standard uncertainty of the formula itself 22e-6 (Table 2), to which the measured inputs’ uncertainties are added with Eq. (6)’s sensitivities. The coefficient 0.3780 is treated as constant; the missing term is about 0.26 rho_a x_v (0.0004 - x_CO2) (p. 152). The density of moist air, not an optical quantity: recorded for the air model.
picard_2008/appendix_a#
{
"id": "appendix_a",
"equation": "Eqs. (A1.1)-(A1.4), Appendix A, pp. 153-154",
"printed": "p_sv = 1 Pa x exp(A T^2 + B T + C + D/T), A = 1.2378847 x 10^-5 K^-2, B = -1.9121316 x 10^-2 K^-1, C = 33.93711047, D = -6.3431645 x 10^3 K (A1.1); f = alpha + beta p + gamma t^2, alpha = 1.00062, beta = 3.14 x 10^-8 Pa^-1, gamma = 5.6 x 10^-7 K^-2 (A1.2); x_v = h f(p, t) p_sv(t)/p = f(p, t_d) p_sv(t_d)/p (A1.3); Z = 1 - (p/T)[a0 + a1 t + a2 t^2 + (b0 + b1 t) x_v + (c0 + c1 t) x_v^2] + (p^2/T^2)(d + e x_v^2), a0 = 1.58123 x 10^-6 K Pa^-1, a1 = -2.9331 x 10^-8 Pa^-1, a2 = 1.1043 x 10^-10 K^-1 Pa^-1, b0 = 5.707 x 10^-6 K Pa^-1, b1 = -2.051 x 10^-8 Pa^-1, c0 = 1.9898 x 10^-4 K Pa^-1, c1 = -2.376 x 10^-6 Pa^-1, d = 1.83 x 10^-11 K^2 Pa^-2, e = -0.765 x 10^-8 K^2 Pa^-2 (A1.4)",
"variable": "T in K, t in degrees C, p in Pa; h the relative humidity (0-1), t_d the dew-point temperature",
"range": "600-1100 hPa, 15-27 degrees C (Appendix A.3)",
"coefficients": {
"A": "1.2378847e-5",
"B": "-1.9121316e-2",
"C": "33.93711047",
"D": "-6.3431645e3",
"alpha": "1.00062",
"beta": "3.14e-8",
"gamma": "5.6e-7",
"a0": "1.58123e-6",
"a1": "-2.9331e-8",
"a2": "1.1043e-10",
"b0": "5.707e-6",
"b1": "-2.051e-8",
"c0": "1.9898e-4",
"c1": "-2.376e-6",
"d": "1.83e-11",
"e": "-0.765e-8"
},
"check": "No worked values are printed. Computed (2026-09-27): p_sv(20 degrees C) = 2339 Pa; Z(20 degrees C, 101 325 Pa, x_v = 0) = 0.999643. The constants are unchanged from CIPM-81/91 (p. 153); not checked against a printed value."
}
The interpolating formulas for x_v and Z of the CIPM-81/91 equation, ‘listed here for completeness’ (p. 153). Z’s uncertainty is dominated by the second virial coefficient of dry air (15e-6, Table 2).
Corrections#
No dataset errata are recorded in this source record.
Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.