3.12. Temperature dependence of effective degrees of

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3.12 Temperature dependence of eﬀective degrees of freedom in gaseous CO2 329

3.12. Temperature dependence of effective degrees of
freedom in gaseous CO2
General remark: Throughout this text symbols for physical quantities are used which are
common in German scientific literature even if other symbols are more common in English
scientific literature. This is done because the German and English versions of this manual
are intended to be as compatible as possible and the dominant language in the student
lab is German.

Goal
In this experiment, the resonance frequencies of a vibrating gas column are measured to
determine the temperature dependence of the velocity of sound in CO2 . From this the
isentropic exponent and the eﬀective number of degrees of freedom are determined for a
range of temperatures.

Hints for preparation
You should know the answers to these questions before performing the experiment. They
are the basis for the discussion with your tutor before the experiment. Information on
these topics can be found in the literature listed at the end of this text.

• What is a degree of freedom?1 and which diﬀerent types of degrees of freedom do
exist?

• What are the degrees of freedom of CO2 ?

• Under which conditions is a process called isentropic?

• What is the isentropic exponent and why is it always greater than 1?

• How does a standing wave build up in a tube?

1
The term degree of freedom“ is very common and is used in different contexts with quite different
”
meanings, which unfortunately may also lead to misunderstandings.
For example, kinematic degrees of freedom“ in mechanics usually refers to the possibilities of
”
motion of a particle, while thermodynamic degrees of freedomı̈n thermodynamics means the number
of coordinates of a particle (e.g. generalized position and momentum coordinates), each of which
makes a quadratic contribution to the total energy. The latter are then sometimes called energy
”
degrees of freedom“ or squared degrees of freedom“. A molecular vibration is accordingly counted
”
as one kinematic or two thermodynamic degrees of freedom. Provided that an author thinks that the
respective meaning is sufficiently clear from the context, these adjectives will usually be omitted.
Therefore, whenever you find this term in scientific texts, make sure that the author understands
it in the same way as you do.

© Bernd-Uwe Runge, Physikalisches Anfängerpraktikum der Universität Konstanz — zum internen Gebrauch bestimmt
Diese Anleitung ersetzt NICHT den Grundlagenteil Ihres Praktikumsberichtes! Haben Sie Verbesserungsvorschläge?
last change to this section: Revision: 1577 , Date: 2022-05-03 00:51:59 +0200 (Di, 03 Mai 2022)
Gesamtversion: kompiliert am 20. Mai 2022 um 20:21 Uhr UTC

330                                                                      3. Versuche zur Thermodynamik

Parts of the setup
   • brass tube (free length inside 1 m, outer diameter 18.5 mm, wall thickness 1.5 mm)
     with thin copper tubes as gas inlet and gas outlet in watertight styrofoam tub

   • PC with sound card und microphone

   • LabVIEW program to calculate the FFT (fast Fourier transform) of the microphone
     signal

   • CO2 compressed gas cylinder with pressure regulator

   • big bucket for ice

   • insulating jar for hot water

   • wall mounted electric boiler

Basics
A detailed description of the setup is found in the publication of Cronin [Cro64].
Here only a brief description is given.

Propagation of sound as an isentropic process
Propagation of sound in gases is based on compressions and expansions on a short time
scale. Due to the high frequency of sound heat exchange can be neglected. The processes
are therefore isentropic and the isentropic exponent κ determines quite essentially the
speed of sound:
                                 "        "
                                   p         RT
                            c0 =     ·κ =        ·κ                            (3.12.1)
                                            M

with

                                   p = absolute pressure,
                                    = density,
                                  R = universal gas constant,
                                  T = thermodynamic temperature,
                                  M = molar mass.

  © Bernd-Uwe Runge, Physikalisches Anfängerpraktikum der Universität Konstanz — zum internen Gebrauch bestimmt
   Diese Anleitung ersetzt NICHT den Grundlagenteil Ihres Praktikumsberichtes! Haben Sie Verbesserungsvorschläge?
            last change to this section: Revision: 1577 , Date: 2022-05-03 00:51:59 +0200 (Di, 03 Mai 2022)
                          Gesamtversion: kompiliert am 20. Mai 2022 um 20:21 Uhr UTC

3.12 Temperature dependence of eﬀective degrees of freedom in gaseous CO2                                          331

Sound propagation in narrow tubes
If a sound wave propagates in a narrow tube, the isentropic process is disturbed by the
thermal conductivity of the wall material. This results in a reduced velocity of the wave,
an eﬀect ﬁrst described by G. Kirchhoﬀ. Quantitatively one ﬁnds:
                                                 
                                             a
                           c = c0 · 1 − √                                          (3.12.2)
                                         2r π · ν

with

                                      c = speed of sound in the tube,
                                     c0 = speed of sound in free space,
                                      r = tube radius,
                                      ν = frequency of the sound signal,
                                      a = Kirchhoﬀ constant.

Standing waves, resonance
If a sound wave propagates in a tube it will be partly reﬂected at the end of the tube due
to the impedance2 change at that position. It will propagate back in the opposite direction
superimposing the incident wave. At the other end of the tube another reﬂection occurs.
If the wavelength matches the length of the tube, a standing wave builds up. Depending
on whether the end of the tube is open or closed, there is a so called pressure node (no
variation of pressure with time) or a so called speed node (no variation of the velocity of
the particles with time) at the end of the tube. The amplitude of the standing wave can
become much larger than the amplitude of the original excitation. This is called resonance.
The wavelength of the sound signal is determined by the velocity of sound and the fre-
quency of the wave. Therefore if one measures the resonance frequencies and the length
of the tube, one can calculate the velocity of sound. This is how the experiment works.

Experimentation
      1. Carefully remove the microphone at the end of the narrow copper tube without
         damaging the connecting wires.
         Open the main valve at the CO2 compressed gas cylinder and ﬂush the tube for
         ≈ 5 min with CO2 .

      2. Start the LabVIEW program.3

      3. Choose suitable parameters to achieve a reasonable FFT display.
         Hint: You may start with
 2
     Impedance in general is a quantitative measure for the resistance against transfer of energy.
 3
     Freiheitsgrade can be started from the desktop using the shortcut Spectrum Analyzer.

     © Bernd-Uwe Runge, Physikalisches Anfängerpraktikum der Universität Konstanz — zum internen Gebrauch bestimmt
      Diese Anleitung ersetzt NICHT den Grundlagenteil Ihres Praktikumsberichtes! Haben Sie Verbesserungsvorschläge?
               last change to this section: Revision: 1577 , Date: 2022-05-03 00:51:59 +0200 (Di, 03 Mai 2022)
                             Gesamtversion: kompiliert am 20. Mai 2022 um 20:21 Uhr UTC

332 3. Versuche zur Thermodynamik

parameter value
sound device Realtek HD Audio Input
rate 48000 samples/s
resolution 16 bit
FFT resolution 1.46 Hz
lower frequency 0 Hz
upper frequency 1000 Hz
max. average no. 500

4. Try out the functions and control elements of the program, especially the possibility
to display amplitudes for speciﬁc frequencies using the cursor functions.
You can add new cursors by right clicking in the cursor table (Create Cursor / Single
Plot).

5. with microphone removed: record the frequency spectrum of the acoustic background
and save it for comparison purposes.

6. Put the microphone at the end of the tube.

7. Open the valve. The ﬂux rate should be set to a value that a sizzling sound is just
audible at the location where the gas enters the copper tube.

8. Choose several resonance maxima which are not too much distorted by ambient noise
and other interfering signals. Often the three lowest resonance frequencies may be
chosen but that may vary depending on the day of the experiment and especially
the other activities in the lab. Save the recorded FFT spectrum.

9. Determine the resonance frequencies for various gas temperatures:
a) Empty the styrofoam tub if necessary using the hand pump with the siphon
hose (it should normally be emptied after the experiment!).
b) Cautiously(!) pour ≈ 4 L of boiling water (use the electric boiler) into the
styrofoam tub.
c) Repeatedly measure and save FFT spectra while the water cools down (tem-
perature interval roughly 5 ◦ C or 10 ◦ C).
Hints:
• You need to switch the averaging function off and on again for each new
measurement.
• Note the temperature for each spectrum! Stir the water from time to time
to achieve a homogeneous temperature inside the tub. Measure the tem-
perature at different positions to make sure temperature differences stay
small.

3.12 Temperature dependence of eﬀective degrees of freedom in gaseous CO2 333

d) If the cooling rate gets too low, slowly add ice to the water in the tub.4
e) Continue with the measurement until the temperature reaches 0 ◦ C (ice/water
mixture, ice does no longer melt completely).
10. Empty the styrofoam tub using the hand pump (there shouldn’t be a relevant
amount of water remaining in the tub).

Data analysis
1. Plot the two saved FFT spectra. Please do not use Excel or similar spreadsheet
programs. Better suited programs are e. g. Origin, LabPlot, MatLab, Python, . . .
2. Calculate the actual velocity of sound in the tube using the resonance frequencies
from item 8 of the experimentation section and the length of the column in the brass
tube (see “parts of the setup” section at the beginning of this text). Make use of
the fact that at resonance the length of the tube is an integer multiple of half the
wavelength.
3. Correct the velocities using equation (3.12.2) to get the velocities of sound in free
space at the various temperatures.
For simplicity reasons a fixed value of
a ≈ 0.004 m · s− /2
1
(3.12.3)
shall be used for the Kirchhoff constant in the brass tube of the experimental setup.
4. Calculate for each temperature the isentropic exponent und the effective degrees of
freedom using the velocity value.
5. Plot the results.

Questions and exercises
1. Add the values expected from theory to the diagram from item 5. Compare experi-
mental and theoretical values.

References
The measurement principle has already been described in detail in the year 1964 in [Cro64].
At that time only one frequency at a time could be investigated, as personal computers
were not yet invented and the calculation of a Fourier transform was not readily available.
Today all frequencies can be simultaneously treated by using a PC with sound card and
stimulating resonances using a noise spectrum.
The correction of the velocity of sound in a narrow tube was ﬁrst published by G. Kirchhoﬀ
in the year 1868 [Kir68]. A slightly more detailed description was published by Kaye and
Sherratt [KS33].
4
Make sure there isn’t too much water in the tub, otherwise you need too much ice. If necessary remove
some water using the hand pump.

334                                                                      3. Versuche zur Thermodynamik

Literaturverzeichnis
[Cro64] Cronin, David J.: The temperature variation of gamma for various gases: A
        student experiment. American Journal of Physics, 32:700–704, 1964.

[Kir68] Kirchhoff, G. Annalen der Physik, 134:177, 1868.

[KS33] Kaye, G. W. C. and G. G. Sherratt: The velocity of sound in gases in tubes.
       Proceedings of the Royal Society (London) Series A, 141:123–143, 1933.

  © Bernd-Uwe Runge, Physikalisches Anfängerpraktikum der Universität Konstanz — zum internen Gebrauch bestimmt
   Diese Anleitung ersetzt NICHT den Grundlagenteil Ihres Praktikumsberichtes! Haben Sie Verbesserungsvorschläge?
            last change to this section: Revision: 1577 , Date: 2022-05-03 00:51:59 +0200 (Di, 03 Mai 2022)
                          Gesamtversion: kompiliert am 20. Mai 2022 um 20:21 Uhr UTC

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