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Measurement of the Universal Gas-Constant R Using a Spherical Volume 93, Number 2, March-Apri11988 Journal of Research of the National Bureau of Standards Measurement of the Universal Gas Constant R Using a Spherical Acoustic Resonator Volume 93 Number 2 March-April 1988 M. R. Moldover, J. P. M. Trusler', We report a new determination of the perature of the tri~le point. The molar and T. J. Edwards' Universal Gas Constant R: (8.314471 mass of the argon was determined by 1 ±0.OOOOI4)J.mol- 1K- • The uncer­ comparing the speed of sound in it to National Bureau of Standards tainty in the new value is 1.7 ppm (stan­ the speed of sound in a standard sample Gaithersburg, MD 20899 dard error), a factor of 5 smaller than of argon of accurately known chemical the uncertainty in the best previous and isotoptic composition. J. B. Mehl value. The gas constant was determined from measurements of the speed of Key words: argon; fundamental con­ University of Delaware sound in argon as a function of pressure stants; ideal gas; mercury; molar gas Newark, DE 19716 at the temperature of the triple point of constant; R; resonator; speed of sound; water. The speed of sound was mea­ spherical resonator; temperature; ther­ and sured with a spherical resonator whose mometry; Universal Gas Constant. volume was determined by weighing the R. S. Davis mercury required to fill it at the tem- Accepted: October 30, 1987 National Bureau of Standards Gaithersburg, MD 20899 Contents 1. Introduction ........................... 86 3. Fabrication and Characterization of 2. Theoretical Basis of Measurement ........ 89 the Resonator. .. 97 2.1 Introduction ...................... 89 3.1 Design and Fabrication ............. 97 2.2 Basic Equations ................... 90 3.2 Polishing ......................... 98 2.3 Boundary Conditions ............... 92 3.3 Final Machining ................... 98 2.4 Imperfect Spherical Geometry ...... 94 3.4 Dimensional Measurements ......... 99 2.5 Transducers ....................... 95 3.5 Assembly of the Resonator ......... 100 2.6 Openings in Resouator Wall ......... 95 3.6 Transducers and Transducer 2.7 Steady State Response .............. 95 Housings . .. 10 I 2.8 A Working Equation for 3.7 Geometry of the Assembled Determination of R ................ 95 Resonator ........................ 103 2.9 Discussion of Working Equation 3.8 Breathing Motion of the Shell ...... 104 for R ............................. 96 4. Measuring Resonance Frequencies. .. 105 4.1 Procedures for Frequency 1 Present address: Department of Chemistry, University Col­ Measurements. .. 105 lege London, 20 Gordon Street, London WCIH OAJ, United 4.2 Random Errors of Resonance Kingdom. Frequency Measurements. .. 105 2 Present address: School of Mathematical and Physical Sci­ 4.3 Systematic Errors in Resonance ences, Murdoch University, Murdoch WA 6150, Australia. Frequency Measurements. .. 106 85 Volume 93, Number 2. March-April 1988 Journal of Research of the National Bureau of Standards 4.3.1 Frequency Standard .......... 106 8. The Pressure and Other Thermo- 4.3.2 Nonlinear Effects ............ 106 dynamic and Transport Properties ....... 130 4.3.3 Ring-Down Experiment ...... 106 8.1 Measurement of the Pressure ....... 130 4.3.4 Overlapping Modes. .. 106 8.2 Thermal Conductivity and 5. Thermometry ......................... 107 Viscosity of the Gases ............. 131 5.1 Thermometer Calibration, History, 8.3 Density and Heat Capacity of and Stability .. .. 107 the Gases ........................ 131 5.1.1 Resistance Bridge ............ 108 9. Determination of cJ in the Working 5.1.2 Calibration Probes. .. 108 Gas .................................. 131 5.2 Temperature Measurements ........ 109 9.1 Preparation of the Resonator ....... 132 5.3 Temperature Gradients ............ 109 9.2 Speed of Sound Measurements ..... 132 5.4 Additional Observations Con- 9.3 Speed of Sound Results ............ 133 cerning Thermometry. .. 110 9.4 Analytical Representations of 6. Determination of the Resonator's the Speed of Sound. .. 133 Volume .............................. 110 9.5 Discussion of the Isotherm 6.1 Summary and Results of Volume Parameters A _ h A h and A 2' • • • • • • •• 134 Determination. .. 110 9.6 Possible Transducer Location 6.2 Principles of Volume Deter- Problem ......................... 136 mination . .. 111 10. Other Tests for Systematic Errors ....... 137 6.3 Density of the Mercury ............ 111 10.1 Excess Half-Widths ............... 137 6.4 Weighing ........................ 112 10.2 Acoustic Resonances in Helium .... 138 6.4.1 Weighing Bottle ............. 112 II. Summary ............................. 138 6.4.2 Counterweights..... .. 113 Acknowledgments. .. 139 6.4.3 Weighing Operations ......... 114 References .............................. , 140 6.5 Filling the Resonator with Appendix I. Resonance Frequencies and Mercury. .. 116 Half-Widths for Ar-M near T, .. 142 6.6 Results of Weighing ............... 118 Appendix 2. Speed of Sound Squared in 6.7 Thermal Expansion . .. 118 Ar-M at T, ................... 143 6.8 Compliance...................... 119 Appendix 3. Resonance Frequencies and 6.8.1 Theoretical Values of Half-Widths for Measurement Compliance ................. 119 of Speed of Sound Ratios near 6.8.2 Experimental Variation of T, ........................... 143 the Outer Pressure. .. 119 6.8.3 Experimental Variation of 1. Introduction the Inner Pressure ........... 120 6.8.4 Temperature Dependence of With readily available technology (such as triple the Compliance .............. 121 point cells and platinum resistance thermometers), 6.9 Tests for Bubbles and Voids ....... 121 it is possible to define and reproduce states of par­ 6.10 Corrections from Weighing Config­ ticular temperatures with much greater precision uration to Aconstics Configuration .. 122 than it is possible to measure the fundamental 7. Determination of M Iyo . .. 124 statistical mechanical quantities characterizing 7.1 Chemical Composition of the such states (such as the average energy in each de­ Gases Used ...................... 125 gree of freedom of a many body system or the 7.2 Effects of Impurities and derivative of the internal energy with respect to Purification ...................... 125 the entropy at constant volume). Accordingly, the 7.2.1 Effects ofImpurities ......... 125 International System of Units (SI) has defined tem­ 7.2.2 Apparatus and Procedures perature as an independent physical quantity and for Purification .............. 126 the kelvin (K) as the unit of that quantity. The defi­ 7.3 Determination of Speed of Sound nition specifies that the temperature T, of the triple Ratios ........................... 128 point of water is exactly 273.16 K. Having made 7.4 Determination of M Iyo for the this specification, one can define the Boltzmann Working Gas. .. 128 constant k, as the ratio 2E IT, and one can define 7.5 Estimating M !Yo from Isotopic the universal gas constant R, as the ratio 2EN AIT,. Abundance Ratios ................ 129 (Here E is the average kinetic energy in a single 86 Volume 93, Number 2, March-April 1988 Journal of Research of the National Bureau of Standards mechanical degree of freedom at T; and N A is the 8.3148,--------------------, Avogadro constant.) In this work we have re-de­ Cohen. Taylor termined R with the result: Adjustment Colclough. Quinn. Cohen. Taylor R =(8.314 471±0.000 014) J/(mol·K) 8.3146 Chandler Adjustment (1.7 ppm), (1.1) !2 L, where the error quoted is a standard deviation. Work °E 8.3144 From this new value of R one can obtain improved values of the Boltzmann constant: " k=R/NA =(1.380 651 3±0.000 002 5) 8.3142 X 1O- 2JJ/K (1.8 ppm), (1.2) and the Stefan-Boltzmann constant cr: 8.3140 ';;;L-L-LJ--'--'--'---"--'-;;;';;-;;-'--'--'--'--'--.L.L.J 1970 1980 YEAR =(5.670 399±0.000 038)X 1O- 8W /(m2.K') Figure 1. Recent values of R. The values from Cohen and Taylor [I] are adjustments. The value from Colclough et al. [13] (6.8 ppm). (1.3) was obtained with a cylindrical acoustic resonator. Here, we have used the values of N A, the Planck considerations of hydrodynamics and the kinetic constant h, and the more accurately known molar theory of dilute gases lead to the relations: Planck constant NAh resulting from the 1986 ad­ justment of the fundamental physical constants [1]. 2 :z 2 (1.4) The present value of R is compared with other c =3 Vrms• recent determinations of R in figure 1. Our value is consistent with previous values; however, it is 5 (Here v;ms is the mean square speed of the times more accurate than the best previous value. molecules, m is the maSS of one molecule, c is the [The same is true for the values of k and cr given in speed of sound, and I' is the ratio of the specific eqs (1.4) and (1.5).] This higher accuracy will be heat capacities Cp/Cvand has the value 1'0=5/3 for most useful for primary thermometry (e.g., gas, dilute monatomic gases.) Thus, a .measurement of noise, acoustic, and radiation thermometry) with the speed of sound in a dilute monatomic gas at T, thermometers whose design does not permit them is a measurement of the Vrms at Tt and it would be a to be used at T, where the temperature scale is de­ measurement of k if m were known accurately. In fined. Further details concerning the roles of R in practice the relative values of isotopic masses are metrology and the methods used in prior determi­ extremely well known on a scale of atomic mass nations of R have been reviewed by Colclough [2] units but the knowledge of m for any pure gas is in 1984 and will not be repeated here. limited by the uncertainty in N A, the constant We conclude this introduction with a summary which relates the atomic mass unit to the kilogram of the errorS in our redetermination of R, the and also relates the mass m of a molecule to the prospects for further improvements, and a brief list molar mass M [3]. of the technical advances developed in this work. In this work the speed of sound was deduced For these purposes it is useful to present a highly from measurements of the internal volume of a simplified equation relating R to the quantities ac­ spherical shell and the frequencies la, of the radi­ tually measured in the laboratory.
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