**Acoustic Gas Thermometry**

**Introduction**

In recent years, advancements in metrological technologies have led to the discovery of use of a variety of thermometry methods for different purposes. Methods such as thermodynamic gas thermometry, dielectric constant gas thermometry, black body radiation thermometry and acoustic gas thermometry are increasingly being used in the metrological sector and beyond with different objectives (Moldover et al 2016). Such approaches have been found suitable for the performance of various roles within the industry, most of which were only accomplished with lower efficiencies in the past. The modern methods are easier to use while also giving more accurate results that are reliable. For instance, reliability is determined through the comparison of results from different approaches to thermometry, and it most cases, it is realized that results obtained from two or more such modern methods are comparable. Dielectric constant gas thermometry and acoustic gas thermometry in particular, have obtained extensive use in areas where gases are involved. Especially where the objective is to determine either the molar gas constant or the Boltzmann’s constant.

Acoustic gas thermometry is applicable for a variety of roles in metrology. As such, its wide application makes it one of the most studied forms of modern thermometry as many authors find it necessary to compare the results from those obtained through other methods, formerly considered accurate. In the determination of Boltzmann’s constant, several authors apply acoustic gas thermometry as the standard of measurement. Similarly, the molar gas constant is more accurately defined through measurements obtained by acoustic gas thermometry. To understand this subject, the ensuing paper discusses various aspects in relation to acoustic gas thermometry. Its main objective is: to highlight the principles of using acoustic gas thermometry and its relevance to industry today. To accomplish this objective, the paper first highlights the principles of operation of the acoustic gas thermometers, stressing on their characteristics that enhance performance of the same.

**Acoustic Gas Thermometry Operation basics**

Moldover et al (2016) explore the operations of acoustic gas thermometry. According to their research, the use of various methods of thermometry has increased in recent years, leading to the advancement of capabilities within and above the moderate temperature ranges of 1 K to 1235 K. This is the same range within which acoustic gas thermometry finds its maximum application. This type of thermometry relies on the mechanical connections between the kinetic freedom degrees of an ideal gas under constant thermodynamic conditions. This is based on the relationship between Boltzmann’s constant Kb and the temperature of the gas, which influence the velocity of the gas molecules. Moldover and others depict this relationship as ½ m(v^{2}) = 3/2K_{b} T. From this equation, m is the atomic mass while v2 is the room mean square velocity of the gas atom. T is the temperature of the gas and Kb is Boltzmann’s constant. From this equation, it is clear that various parameters can be obtained once some are available. For instance, for an atom of known molar mass at a given temperature, velocity measurements can help to determine Boltzmann’s constant. On the other hand, if Boltzmann’s constant is known but the molar mass is unknown, it is possible to determine the molar mass through acoustic gas thermometry. This equation thus forms the basis of most of the applications of acoustic gas thermometry and the principle from which other relationships are obtained.

In most cases, using acoustic gas thermometry is challenged due to the difficulty of achieving constant gas volumes. However, the volumes of the gases have no significant impacts on the results obtained unless there is need for comparative analysis. The primary approach to AGT involves determination of thermodynamic temperatures of mono-atomic gases such as helium, argon and neon based on the velocity of sound within an enclosed cavity. The applicability and accuracy of AGT in the determination of various gas characteristics informs its relevance as a standard for the same. Most of other methods for measuring parameters such as the Boltzmann’s gas constant and the molar gas constant have been found to be less accurate than AGT hence its preference for use as a standard. For the measurement to be successful, the gases used have to possess characteristic molar masses that can be approximated to the kilogram, meter and the second. The absolute acoustic gas temperature is thus set at approximately the triple point of water which is defined as T_{tpw} (Moldover et al 2014). The gas used in the thermometry must be inert since the characteristics of the gas may to some extent affect the results. This is the basis of the application of neon, helium and argon gases. In the use of AGT, the rationale for reliance on sound is the probability of linking the speed of sound and light to that of the gases in the closed cavities used in thermometry.

AGT thermometers measure the frequencies of varied acoustic resonances while at the same time determining the frequencies of several microwave resonances pertaining to a gas filled in a cavity. The cavities used are mostly metallic while the gas can be argon, helium and neon. The frequencies measured at a particular temperature are presented as a ratio which is comparable to the ratio of the speed of sound to that of light in air at the same temperature. The ratios form the basis of temperature determination in the application of the method. In such a case, the proportionality constant is influenced by the nature of the gas in the cavity and the shape of the cavity used. The measures obtained are compared to those of the triple point of water, which is pre-determined. The determination of the unknown temperature is based on a mathematical relation between the frequency ratios and the temperature rations, highlighted as:

(Tx/Ttpw)^{1/2} = (fa/fm)_{Tx}/(fa/fm)_{Ttpw } (Moldover et al 2016)

In the equation, the first term is the ratio of the unknown temperature to the temperature of the triple point of water. The second and the last terms are the frequency ratios at the unknown temperature and that at the triple point of water respectively. Since the frequency ratios at the triple point of water are known and those at the other temperature are known, the actual measurement temperature can be obtained by inserting all the values into their respective places before finding the unknown value. The frequency ratios are neither affected by the gas pressures nor by the cavity shape changes that may result from thermal expansion, but are affected by large changes resulting from actual cavity shape differences. The thermometers work within the range of 84 to 550K. It is thus important to note that the frequency values are affected by the gas pressure due to gas differences and the cavity shape differences.

Based on the dependence of the results on the shape of the cavity, various characteristics have to be considered prior to the selection of the cavity for AGT. For instance, the cavity used must possess high Q values. This helps in ensuring that the frequency of the gas movements within the cavity is not limited by various cavity designs. Furthermore, it should have non- overlapping resonances in the microwave and the acoustic regimes. The second quality is comprehensible based on the fact that the thermometry method depends on the comparison of microwave and acoustic frequencies. Overlapping resonances can result in measurement ambiguity and lack of accuracy in the frequency. Needless to say, inaccuracies in frequency measurements result in inaccuracies in all the other applications of the thermometer. The cavity used in most cases also has to be a quasi- sphere, which is constructed through connection of two, almost hemispherical cavities (Moldover et al 2016).

While the shape of the cavity is described as almost spherical, it is not randomly selected or created. On the contrary, the cavity shape is a result of in-depth engineering design, where it is approximated to be a tri-axial ellipsoid. The axis ratios for the cavity shape axes are to be 1: (1+e): (1-e). In this case, e is defined as the error in measurement and it is supposed to be greater than 0.0005 yet less than 0.001 (Moldover et al 2016). In either case, e approximation has to be significantly large to be capable of separating the different microwave frequencies. This implies that as much as error is minimized, there is a limit beyond which the minimization cannot occur, as further error reduction would result in the overlapping of microwave frequencies. At the same time, the acoustic frequencies may also be affected due to the same issue of overlapping. The minimum error should also be small enough, to the range of the error squared. This is to help in averting the imperfections of shape which have the potential of affecting the frequency approximations negatively. The main objective in doing this is to ensure that while conducting the thermometry measurements, the results obtained are accurate and usable for standard purposes. It is possible to avert any inaccuracies and unexpected outturns through proper planning and execution.

**Specific applications of AGT for standards**

Different acoustic cavity shapes can be used with AGT to obtain different frequency results. The formation of acoustic resonance relies on the shape of the cavity and can be helpful in the determination of various gas –related constants based on the measurements carried out in dilute gases. This is usually implemented by applying radially symmetric modes of operation (Feng et al undated). According to Feng et al (undated), the key challenge in determining the frequencies through AGT is in the determination of unperturbed frequencies when using perturbed measurements. Consequently, the objective of any such practice has to be to ensure that the measurements are as stable as possible. This is particularly important in particular applications of AGT such as the determination of Boltzmann’s constant, molar gas constant and different gas characteristics. All the measurements begin from the determination of the unknown temperatures as is the objective of every approach to thermometry.

**AGT for Boltzmann’s constant determination**

Feng et al (undated) describes the most common application of the principle of AGT. According to the authors, the Boltzmann’s constant can be determined through AGT by the use of fixed length cylindrical cavities instead of any other shape of cavities. The main rationale for the use of the cylindrical cavity is that it is simple to machine as well as to install where it is to be used. This implies that it can easily be used on different sites and can also be moved from position to position without interfering with the gas characteristics. On the other hand, working with the cylindrical cavity results in the unequal distribution of admittances between different operational modes. This is however a challenge that is faced with most of the other AGT shapes as well. As such, identifying the challenges is an effective step towards addressing them and subsequently ensuring that they are well handled.

The determination of Boltzmann’s constant through AGT is described as less dependent on correction compared to the previous results including dielectric constant gas thermometry. As such, it provides accuracies above those expected from conventional methods. Gavioso et al (2010) found out through experimentation that more accurate results of the Boltzmann’s constant are obtained through the use of a single- state helium gas as it is the most inert of all the gases. Moreover, it is also light and its velocity is higher than most of the other gas molecules. In this case, the obtained frequency ratios are bound to be more accurate than any others obtained through other thermometry approaches or through acoustic cavities filled with other gases. The determination of Boltzmann’s constant relates the kinetic energy of the gas particles to the thermodynamic temperatures of the gas. This means that from the temperatures obtained through frequency ratios, the first equation can be used to determine Boltzmann’s constant. Conducting a reiterative process in this is a good way to enhance the accuracy of the results and subsequently provide a reasonable constant for the particular gas used. Higher kinetic energies automatically result in higher values for the Boltzmann’s constant. This implies that the lighter the gas, the higher the probability that it will result in a higher value for the Boltzmann’s constant. This is expected based on past records of the Boltzmann’s constant (Feng et al undated).

**Molar Gas Constant R from AGT**

Besides measuring Boltzmann’s constant, AGT has also been confirmed to be an accurate approach in the determination of the molar gas constant. Gavioso et al (2015) explored the available approaches towards this activity, with results that indicate that AGT is a more accurate method compared to others such as the black body radiation thermometry and the thermodynamic gas thermometry approaches in the determination of the molar gas constant. Acoustic thermometry as applied in AGT gives frequency ratios which can be used together with the Boltzmann’s constant to help in determining the molar gas constant. Relations that provide the associations between the molar gas constant and the characteristics of the gas such as the relative molecular mass and room mean square velocity can be applied in the chemical mole equations to give the results of the molar gas constant. The use of the inert gases ensures that the values obtained are not only accurate but also acceptable in the engineering industry. Gavioso et al (2015) report that the determination of the molar gas constant is more of a function of thermodynamic properties than it is a property of the cavity.

**Triple Point determination from AGT**

While the conventional approach in the application of AGT is to determine an unknown gas temperature, Pitre et al (2006) provide a non- common approach to its application. In their work, Pitre et al determined that by using AGT, it was possible to determine the triple point of various gases. This applies the second equation outlined previously as a starting point just like in the other equations. The authors used a quasi- spherical cavity, which was filled with neon, argon or mercury depending on the gas whose triple point was to be determined. Based on their findings, it was established that such an approach could result in more accurate results compared to other approaches. For instance, Pitre et al (2006), established that AGT resulted in uncertainties that were either less than or comparable to those realized when using a constant volume gas thermometer to determine the triple points of the targeted substances. At the same time, the results obtained were found to be comparable to the results obtained when using the dielectric constant gas thermometers in the determination of triple points of the named gases.

The principle of determination of triple points is founded on the application of a variety of frequency ratios from the initial cavity resonance determinations. Through comparison of microwave and acoustic frequencies, it will be possible to determine unknown temperatures, which align to the triple points of the gases. The main challenge can however be in the determination of the actual triple points of the gas in the cavity. This requires close monitoring of the gas and subsequent accurate measurement of the frequencies at the observed point of state transition. Temperature determination in the use of AGT for the triple point measurement depends on the availability of many temperature positions, and this can only be accomplished if the results are collected with high levels of accuracy. This comes with the need for calibration of the thermometer at different temperature values since there is no possibility of obtaining midpoint temperatures through manipulation of particular temperatures considered to enclose the desired range (Moldover et al 2016). Consequently, in determining the triple point of the gases, such calibration is mandatory as the temperatures recorded can be inaccurate. The calibrations then guide the user into the most realistic results. The differences in gas characteristics imply that different calibrations have to be used for each of the gases to be measured.

**Conclusion**

Acoustic Gas Thermometry is one of the fastest emerging trends in thermometry, especially applicable in meteorology. In comparison to other modern technologies in thermometry, AGT provides the user with the capacity to create standards pertaining to the molar gas constant, Boltzmann’s constants, triple point temperatures and unknown gas temperatures. The determination of unknown gas temperatures is the starting point of each of the applications of the AGT. In the determination of Boltzmann’s and the molar gas constants, the values are derived from the first and second equations as well as from the molar gas constant relations. On the other hand, triple point determination relies on calibrations conducted at different temperatures due to the difficulties associated with approximating temperatures based on other temperature values.

Bibliography

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Gavioso RM, Benedetto G, Albo PAG, Ripa M, Merlon A, Guianvarch D, Moro F and Cuccaro R 2010. A determination of the Boltzmann constant from speed of sound measurements in helium at a single thermodynamic state. *Metrologia*, vol. 47, no. 4, pp. 387- 409. www.iopscience.iop.org/article/10.1088/0026-1394/47/4/005

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