Optimizing acoustic measurements of the Boltzmann constant

Abstract

We review the progress in acoustic metrology of gases that has occurred since the 1988 measurement of the universal gas constant R using a spherical acoustic resonator. The advances in understanding resonators and in calculating the thermophysical properties of helium ab initio suggest that today one could determine Boltzmann's constant k B from acoustic measurements using either helium or argon with a relative uncertainty less than 10 −6 . RésuméProgrès dans la détermination de la constante de Boltzmann par voie acoustique L'article récapitule les avancées effectuées en métrologie acoustique dans les gaz depuis la mesure de la constante universelle des gaz R effectuée en 1988 en utilisant un résonateur acoustique sphérique. Des progrès substantiels ont été réalisés, tant pour mieux comprendre le fonctionnement des résonateurs acoustiques que pour calculer ab initio, avec une fiabilité accrue, les caractéristiques thermo-physiques de l'hélium. Ils permettent d'envisager désormais une détermination de la constante de Boltzmann k B par voie acoustique avec une incertitude relative meilleure que 10 -6 , en utilisant soit l'argon, soit l'hélium.________________________________ 1. Introduction.

Measurement of the Boltzmann Constant k B Using a Quasi-Spherical Acoustic Resonator

“…The difference is the factor 2 in the denominator, which was omitted in [15] and in [35]. Additional corrections are generated by the two capillary tubes that carry gas into and out of the QSR.…”

Cylindrical Acoustic Resonator for the Re-determination of the Boltzmann Constant

“…We have not completed an analysis appropriate to the two-cylinder method; therefore, we now outline the complications and conclude by guessing the uncertainty contribution from fitting the data. First, we recall, that in the context of re-determining k B using argon in a quasispherical cavity resonator, one of us (MRM) observed that the lowest uncertainties can be obtained by measuring acoustic resonance frequencies in the range from 100 kPa to 500 kPa [20]. At still lower pressures, the perturbations from the thermo-acoustic boundary layer and the accommodation coefficients diverge; at higher pressures, the perturbations from the elastic response of the shell become excessive.…”

“…In reference [20], one of us (MRM) estimated the relative uncertainties of the viscosity u r (η Ar ) ≈ 0.00025 and the thermal conductivity u r (λ Ar ) ≈ 0.00025 of argon in the limit of zero density at 273.16 K. Using these estimates in Eqs. 4 and 5 leads to an estimated relative uncertainty of the thermo-acoustic boundary layer correction: u r ( f b ) ≈ 0.00013.…”

On the dimensionality of the Avogadro constant and the definition of the mole

“…The best practical determination of F is by silver coulometry, and gives a relative uncertainty of 1.3 × 10 -6 [21,58,59], almost three orders of magnitude higher than u r (α 2 ). 10 The current value of R comes from direct measurements of the speed of sound in argon, and has u r = 1.7 × 10 -6 [21,58,60,61]: although there are ongoing efforts to measure k directly [62,63], rather than to determine it by k = R/N A (23) these have yet to reach the necessary precision even to supersede the value obtained from equation (22) [64,65]. 10 Indeed, ur(α 2 ) is more than an order of magnitude lower than the current relative uncertainties [44] in Ar( 14 In short, any additional reduction in the measurement uncertainty of fundamental constants by the redefinition of the mole in terms of a fixed numerical value of N A , over and above that achieved by fixing the numerical values of h, e and k, would be completely insignificant at the practical level.…”