may_2009#

E. F. May, M. R. Moldover and J. W. Schmidt, “Reference measurements of hydrogen’s dielectric permittivity”, Mol. Phys. 107, 1577 (2009)

Source record · Catalogue · Review register

DOI: 10.1080/00268970902980052

Source kind: measurement

Recorded technique: The apparatus of may_2008: a quasi-spherical microwave cavity resonator (four modes, 2.6-7.2 GHz, n^2 from frequency ratios, Eq. 5) and a cross capacitor (epsilon_r from capacitance ratios at 2 kHz) in two pressure vessels in one thermostatted stirred-oil bath, filled from one gas supply and pressure balance. Normal H2 (supplier’s purity 99.9999 %) along two isotherms, 273.203 and 293.213 K (5 mK, ITS-90), 0.1-6.5 MPa in rising and falling steps of 0.5 or 1 MPa, 2.5 h equilibration; pressure to 0.1 kPa. Both instruments calibrated with helium against its ab initio properties (Schmidt et al. 2007; densities from Hurly and Mehl’s B and Garberoglio and Harvey’s C; D refitted), so the H2 results are relative to helium’s ab initio epsilon_r. The resonator’s n^2 (ten times more precise than the capacitor, and agreeing with it to < 2e-6) is converted to epsilon_r with the calculated magnetic term A_mu rho (at most 1.34e-7) (pp. 1577-1582).

Materials in structured entries: H2

Dataset quantities: molar_refractivity, polarizability

Contents: 6 datasets; 2 formulas.

Curation notes#

Identity checked against the first page (Mol. Phys. 107, 1577-1585 (2009), received 24 February 2009). Page numbers are the journal’s: pdf page 1 is the publisher’s cover, so p. = pdf page + 1575. ‘All uncertainties reported here are one standard uncertainty’ (abstract, p. 1577). The paper hohm_2013 cites as its ref. [43] (the H2 static polarizability, re-pointed 2026-09-27).

DEFINITIONS (p. 1578). Molar polarizability P = [(eps_r - 1)/(eps_r + 2)]/rho = A_eps (1 + b_eps rho + c_eps rho^2) (Eq. 1) and A_eps = (N_A/3 eps0)[alpha_e(T) + mu_e^2/(3 k_B T)] (Eq. 2), mu_e = 0 for H2. NOTE THE FORM: b_eps here (cm^3/mol) is the second dielectric virial coefficient RELATIVE to A_eps, not the B_eps = A_eps b_eps (cm^6/mol^2) of bose_1970 and schmidt_2003 (for H2, B_eps = 2.031 b_eps); it has no vocabulary unit (refractivity_virial_coefficient takes cm^6/mol^2) and is kept in the formulas and notes. The static (2 kHz and 2.6-7.2 GHz, no frequency dependence detected, p. 1581) polarizability of H2 is its electronic polarizability averaged over the populated rotational states of normal H2 (it rises with T by centrifugal stretching: A_tau).

RESULTS (pp. 1582-1585). Method 1 (Eq. 9, fitted to all of Table 1 with densities from Leachman et al.’s equation of state in REFPROP 8.0; c_eps not significant, set to 0): A_eps,273K = 2.03075 +- 0.00001 cm^3/mol, b_eps = 0.0636 +- 0.0015 cm^3/mol, A_tau = 0.00599 +- 0.00017 cm^3/mol (statistical uncertainties); with Kunz et al.’s (GERG-2004) equation of state instead, 2.03042, 0.0650 +- 0.0019 and 0.00627 +- 0.00017: the choice of equation of state dominates (p. 1583). Method 2 (no equation of state: the difference of the H2 and He polynomial coefficients d_1, Eq. 8, with Lach et al.’s A_eps(He) = 0.51725419 cm^3/mol): A_eps = 2.0307 +- 0.0001 (273 K), 2.0311 +- 0.0001 (293 K), hence A_tau = 0.0061 +- 0.0014; from d_2 (Eqs. 10-11) B - b_eps = 13.87 +- 0.03 (273 K) and 14.31 +- 0.03 cm^3/mol (293 K), and with Patkowski et al.’s ab initio B(T) the weighted mean b_eps = 0.003 (standard deviation 0.02) cm^3/mol. RECOMMENDED (the weighted average of the two methods, ‘the uncertainty bound we assign ... encompasses the standard uncertainties obtained by each method’, p. 1584): A_eps,273K = 2.03065 +- 0.00017 cm^3/mol, A_tau = 0.0061 +- 0.0014 cm^3/mol, alpha_e = (8.9568 +- 0.0008) x 10^-41 F m^2 (3 eps0 A_eps/N_A = 8.95682e-41, checked), b_eps = 0.03 +- 0.05 cm^3/mol (273-293 K); dalpha_e/dT = alpha_e x (1.1 +- 0.3) x 10^-5 K^-1 (abstract; = A_tau/(273.16 K A_eps)), against the calculated 1.0e-5 K^-1. For representing Table 1’s epsilon_r(p, T) the authors recommend Method 1 with Leachman’s equation of state (p. 1584).

COMPARISONS quoted (reproduced datasets): Harvey and Lemmon’s alpha_e = (8.9566 +- 0.0026) x 10^-41 F m^2 at 273.16 K from Rychlewski’s ab initio polarizabilities averaged over J = 0-11 (the bound is the omitted non-adiabatic part, about 0.03 %), and their A_tau = 0.0056 cm^3/mol (p. 1585; notes only); Rychlewski’s 293 K normal-H2 average 8.958 x 10^-41 F m^2 (A_eps = 2.0309 cm^3/mol), 0.25 % above Maryott and Buckley’s recommended 8.981 x 10^-41 (293 K, 1 atm, from radio, microwave and optical data) (p. 1578); Orcutt and Cole’s b_eps = 0.03 +- 0.10 cm^3/mol at 322 K (p. 1585: orcutt_1967’s H2 row prints 0.03 +- 0.10 as its second dielectric virial coefficient in cm^6/mol^2, i.e. B_eps, which would be b_eps = 0.015 in this paper’s form; May et al. quote the number unchanged as b_eps, notes only); Harvey and Lemmon’s b_eps 0.09 from Stewart’s para-H2 data. Fig. 4 (A_eps against T with Orcutt and Cole’s and Newell and Baird’s values) and Figs. 2, 3, 5 (deviations, b_eps estimates) are not digitised.

CHECK (2026-09-27): Method 1’s Eq. (9) (formula eq9_method1) reproduces Table 1’s epsilon_r - 1 to 1-90 ppm at 1.0 MPa (273.203 K), 225-240 ppm at 1.5 MPa (293.213 K) and about 400 ppm at 2.5 MPa, and to 0.04-0.14 % (the printed four digits and u(epsilon_r) = 3.4e-7) at 0.1 MPa, with densities from p = rho RT(1 + B rho) and B = (B - b_eps) + b_eps from the paper’s Method 2 values: the growth with pressure is the neglected third density virial coefficient of H2, which the check omits (the paper uses Leachman’s full equation of state).

TABLE 1 (p. 1582, 40 points: epsilon_r - 1 of normal H2, x 10^4, against pressure at 273.203-273.204 K (27 points, 0.14-6.24 MPa, rising then falling) and 293.213 K (13 points, 0.10-6.49 MPa)) is NOT extracted: the vocabulary has no quantity for a gas’s permittivity (or refractivity) at a stated pressure (refractivity is n - 1 against wavelength at one state), as for may_2008’s Table III. Uncertainty of epsilon_r: [1.7 + 2.7 (293 K/T)(p/6 MPa)] x 10^-7 from u(T) = 5 mK and 2.6e-7 (293 K/T) from u(p) = 0.1 kPa, combined 3.4e-7 at 293 K and 3 MPa (p. 1582); deviations from Harvey and Lemmon’s correlation below 0.9e-6, mean +0.3e-6 (p. 1582, Fig. 3). A permittivity-against-pressure quantity would let these data be fitted directly (reported).

AGAINST may_2008 (O2), the same apparatus and methods: the same two methods (Method 1 = may_2008’s Eq. (19) regression, Method 2 = its Eq. (18) coefficient differences) and the same recommendation (their weighted mean, with a bound encompassing both). Differences: H2 is diamagnetic, so n^2 and epsilon_r agree within 2e-6 and the (more precise) resonator values are used, where for paramagnetic O2 the capacitor’s epsilon_r was used and the resonator measured the magnetic susceptibility; the H2 run adds a second-order coefficient d_2 analysis for b_eps (Eqs. 10-11); O2 was measured at three isotherms (273, 293, 323 K), H2 at two (the 323 K run was lost to a pressure-transducer failure, p. 1580).

Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets#

may_2009/method2_H2#

{
  "id": "method2_H2",
  "material": "H2",
  "table": "text",
  "quantity": "molar_refractivity",
  "unit": "cm3/mol",
  "abscissa_unit": "nm",
  "use": "primary",
  "excluded_rows": {
    "273": "averaged into recommended_H2 (the paper's recommended 273 K value)"
  },
  "uncertainty": {
    "column": true,
    "level": "1 sigma"
  },
  "parameters": {
    "temperature_k": 293.213,
    "ortho_para": "3:1 (normal H2)"
  }
}

2 inline points in the source record.

p. 1583: A_eps from Method 2 (Eq. 8, the difference of the H2 and He resonator coefficients d_1, with A_eps(He) = 0.51725419 cm^3/mol, independent of the equation of state) at each isotherm (273.203 and 293.213 K; rows are the temperatures as printed, ‘273 K’, ‘293 K’), one standard uncertainty. The 293 K value carries the temperature dependence (A_tau = 0.0061 +- 0.0014 cm^3/mol). As may_2008’s eq18_O2; the dataset’s temperature_k is the second isotherm, the rows differ.

may_2009/alpha_e_H2#

{
  "id": "alpha_e_H2",
  "material": "H2",
  "table": "text",
  "quantity": "polarizability",
  "unit": "SI",
  "scale": 1e-41,
  "abscissa_unit": "nm",
  "use": "check",
  "uncertainty": {
    "column": true,
    "level": "1 sigma"
  },
  "parameters": {
    "temperature_k": 273.16
  }
}

1 inline points in the source record.

pp. 1577, 1585: alpha_e = (8.9568 +- 0.0008) x 10^-41 F m^2 at 273.16 K, recommended_H2 through Eq. (2): the same value, not independent (3 eps0 A_eps/N_A = 8.95682e-41, checked). F m^2 = C m^2/V. = 5.4324 +- 0.0005 au (hohm_2013 reprints it so).

may_2009/harvey_lemmon_2005_H2#

{
  "id": "harvey_lemmon_2005_H2",
  "material": "H2",
  "table": "text",
  "quantity": "polarizability",
  "unit": "SI",
  "scale": 1e-41,
  "abscissa_unit": "nm",
  "use": "reproduced",
  "reproduced_from": [
    "A. H. Harvey and E. W. Lemmon, Int. J. Thermophys. 26, 31 (2005) (with J. Rychlewski, Mol. Phys. 41, 833 (1980))"
  ],
  "uncertainty": {
    "column": true,
    "level": "bound"
  },
  "parameters": {
    "temperature_k": 273.16
  }
}

1 inline points in the source record.

pp. 1578-1579, 1585: Harvey and Lemmon’s effective alpha_e = (8.9566 +- 0.0026) x 10^-41 F m^2 of normal H2 at 273.16 K, a population-weighted average of Rychlewski’s ab initio polarizabilities over J = 0-11 (v = 0); ‘the uncertainty bound represents the small (~0.03 %) non-adiabatic contributions omitted from Rychlewski’s calculation’. Theory; a bound. Not in the library.

may_2009/rychlewski_1980_H2#

{
  "id": "rychlewski_1980_H2",
  "material": "H2",
  "table": "text",
  "quantity": "polarizability",
  "unit": "SI",
  "scale": 1e-41,
  "abscissa_unit": "nm",
  "use": "reproduced",
  "reproduced_from": [
    "J. Rychlewski, Mol. Phys. 41, 833 (1980)"
  ],
  "parameters": {
    "temperature_k": 293
  }
}

1 inline points in the source record.

p. 1578: Rychlewski’s ab initio effective alpha_e of normal H2 at 293 K, averaged over J = 0-5 (v = 0), 8.958 x 10^-41 F m^2, ‘corresponding to A_eps = 2.0309 x 10^-6 m^3 mol^-1’; relative uncertainty about 0.03 % (omitted non-adiabatic contributions). Not in the library (rychlewski_1983 is a different paper).

may_2009/maryott_buckley_1953_H2#

{
  "id": "maryott_buckley_1953_H2",
  "material": "H2",
  "table": "text",
  "quantity": "polarizability",
  "unit": "SI",
  "scale": 1e-41,
  "abscissa_unit": "nm",
  "use": "reproduced",
  "reproduced_from": [
    "A. A. Maryott and F. Buckley, Tables of Dielectric Constants and Electric Dipole Moments of Substances in the Gaseous State (National Bureau of Standards, Washington, 1953)"
  ],
  "parameters": {
    "temperature_k": 293
  }
}

1 inline points in the source record.

p. 1578: ‘the value alpha_e = 8.981 x 10^-41 F m^2 recommended by Maryott and Buckley for H2 at 293 K and 1 atm ... based on measurements at radio, microwave, and optical frequencies’; Rychlewski’s value is 0.25 % smaller. A compilation value; static or optical not separated. Not in the library.

Formulas#

may_2009/eq9_method1#

{
  "id": "eq9_method1",
  "material": "H2",
  "equation": "Eq. (9), p. 1582, with the Method 1 parameters, p. 1583",
  "printed": "(eps_r^calc - 1)/(eps_r^calc + 2) = A_eps,273K rho (1 + b_eps rho + c_eps rho^2) + A_tau rho (T/273.16 K - 1); A_eps,273K = 2.03075 +- 0.00001 cm^3 mol^-1, b_eps = 0.0636 +- 0.0015 cm^3 mol^-1, A_tau = 0.00599 +- 0.00017 cm^3 mol^-1 (c_eps = 0, not significant)",
  "variable": "rho: molar density in mol/cm^3 from Leachman et al.'s equation of state for normal H2 (REFPROP 8.0); T in K; static (2 kHz-7.2 GHz) relative permittivity",
  "range": "273.203 and 293.213 K, 0.1-6.5 MPa (Table 1)",
  "coefficients": {
    "A_eps_273K": "2.03075",
    "A_eps_273K_sigma": "0.00001",
    "b_eps": "0.0636",
    "b_eps_sigma": "0.0015",
    "A_tau": "0.00599",
    "A_tau_sigma": "0.00017",
    "c_eps": "0"
  },
  "check": "Reproduces Table 1's epsilon_r - 1 to 1-90 ppm at 1.0 MPa (273.203 K), 225-240 ppm at 1.5 MPa (293.213 K), about 400 ppm at 2.5 MPa, and 0.04-0.14 % at 0.1 MPa (the four printed digits), with densities from p = rho RT(1 + B rho), B = 13.87 + b_eps and 14.31 + b_eps cm^3/mol (the paper's Method 2 B - b_eps): the rise with pressure is the third density virial coefficient the check leaves out. Checked 2026-09-27."
}

The authors’ recommendation for representing the epsilon_r(p, T) of Table 1 (p. 1584). The uncertainties are statistical; the parameters depend on the equation of state (with Kunz et al.’s GERG-2004: A_eps,273K = 2.03042, b_eps = 0.0650 +- 0.0019, A_tau = 0.00627 +- 0.00017 cm^3/mol, p. 1583), on the number of free parameters and on the objective function. b_eps is relative to A_eps (Eq. 1): B_eps = A_eps b_eps.

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.