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I am doing speed of sound measurements in $\ce{CO2}$ using ultrasonic pulses at 125 kHz. $\ce{CO2}$ is particularly difficult because it strongly absorbs ultrasound due to its multiple vibrational modes. The problem we have discovered is that absorption increases substantially as temperature rises. Is this expected, and if so is there a simple relationship between temperature and absorption coefficient?

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    $\begingroup$ It is not clear to me how the ultrasound translates into the excitation of vibrational modes (though apparently there is a way). That being said, yes, it is expected for more modes to "unfreeze" as the temperature rises, and no, there is no simple relationship. $\endgroup$ – Ivan Neretin May 24 at 9:21
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    $\begingroup$ Molecular vibration energy is quantized with quite big quantization steps, so it occurs significantly at elevated temperatures only. The energy needed to increase vibration is kind of composed in your case from thermal energy of molecules and the energy of sound wave. As result, higher the temperature = higher probability the sound wave will cause absorption of sound energy, causing molecule transition to higher vibration mode. It may be difficult to quantize it, more easier can be to get experiemntal data or empirical expressions. Try engineringtoolbox. $\endgroup$ – Poutnik May 24 at 9:22
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    $\begingroup$ The acoustic pulses must be made up of several frequencies but all these seem very low compared to vibrational frequencies of molecules, possibly you are causing rotational levels to be excited, these are far more numerous than vibrational levels and also of much lower frequency and so their population is very sensitive temperature changes. However, absorption even with rotational levels is in the terahertz region. $\endgroup$ – porphyrin May 24 at 10:41
  • $\begingroup$ @porphyrin Even so, CO2 is a strong absorber of sound all across the acoustic spectrum, becoming extreme at frequencies above about 300kHz At 125kHz we are getting a 5x attenuation over 120mm $\endgroup$ – Dirk Bruere May 24 at 10:49
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    $\begingroup$ I don't doubt your experimental data. I'm no expert here, but as the sound wave passes does it not cause compression and rarefaction and these cause energy loss (heating?) if there are intermolecular interactions in the gas caused by by van-der-waals /dispersion forces. $\endgroup$ – porphyrin May 24 at 11:20
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The acoustic wave is accompanied by a temperature oscillation. To a first approximation, the temperature oscillation is adiabatic and reversible in the bulk, but irreversible near solid boundaries. In many cases, boundary losses dominate (for example, in musical wind instruments), but as the frequency increases, and the surface-to-volume ratio decreases, bulk irreversibility may become important. Bulk losses in single-component polyatomic gasses are almost always due to loss of local thermodynamic equilibrium when the temperature oscillation describing internal degrees of freedom lags the temperature oscillation describing translational degrees of freedom.

A full discussion of these effects is given in the book "Ultrasonic Absorption" by A.B. Bhatia. His figure 5.1 shows that the attenuation for $\ce{CO2}$ peaks at about 30 kHz (STP conditions), and his figure 6.3 shows the velocity dispersion that occurs at about the same frequency. The dispersion is a large effect, with the velocity increasing by about 4% as the frequency goes from below to above the attenuation peak. The attenuation and dispersion occur when the acoustic frequency matches the rate at which the vibrational mode temperature relaxes to the translational temperature. The rotational modes relax much faster than the vibrational modes, so they do not play an important role under these conditions.

The relaxation time for the vibrational temperature will generally decrease with increasing temperature, so the attenuation peak will move to higher frequency. I am not sure what temperatures you are exploring, but if your frequency is above the attenuation peak, then you will see increasing attenuation as the temperature increases.

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