Here is an expansion on what John Mallinckrodt said, and also
explanation of some other problems.
The usual formula for the speed of sound in a gas that behaves as an
ideal gas (in terms of PV = nRT) is...
V = sqrt(gamma*R*T/M)
R is the gas constant 8.314 J/mol.K
T is Kelvin temperature
M is the molar mass in kg. For CO2 this is 0.04401 kg/mol
Ignore gamma for a minute.
If Vo is the reference velocity at the reference temperature (To),
then...
V/Vo = [sqrt(gamma*R*T/M)]/[sqrt(gamma*R*To/M)]
V/Vo = sqrt(T/To) assuming gamma, R, and M are constant.
V = Vo*sqrt(T/To)
Which the formula John M. provided.
Gamma is the ratio of specific heat at constant pressure to specific
heat at constant volume. Theoretically Cv is f/2*R where f is the
degrees of freedom for the molecule. Cp is Cv+R. Since CO2 is a linear
molecule, it has three degrees of translational freedom and two degrees
of rotational freedom, for a total of five degrees of freedom. That
means Cv should be 5/2*R and Cp should be 7/2*R and gamma should be 7/5
= 1.40
The problem is that we haven't included any vibrational degrees of
freedom because the initial assumption is that room temperature is not
high enough for there to be appreciable population of anything but the
ground vibrational states. In actual experiments conducted in many
physical chemistry labs (including those done by my students) gamma
comes out to be about 1.28 at room temperature, which is clearly showing
more than 5 degrees of freedom. That means room temperature is hot
enough to yield some population of vibrational degrees of freedom. At
0C gamma is about 1.30 which shows that the vibrational contribution is
slightly less at 0C than at 25C. At 100C gamma is about 1.26. Thus, if
you are varying the temperature from 0C to 100C you have a gamma change
of about 3%. Maybe you won't detect that and it is okay to use the
formula John provided. We can detect this change in our lab because we
measure the speed of sound quite accurately because our goal in
physical-chemistry lab is to determine gamma for several gases.
Most likely the water-tube method for determining the speed of sound
(which we use in other classes) will only give v to about 1 or 2%
accuracy and that is not likely going to show the gamma differences, so
the discussion in the previous two paragraphs may be of no significance
for the experiment that prompted the discussion.
On the other hand a problem that is significant is getting pure CO2 in
the tube and keeping it there. In our experiments we continually feed a
stream of CO2 from a compressed cyclinder into our resonance tube. We
use an oscilloscope in x-y mode to display a microphone signal and an
oscillator signal so we can see the phase difference of the sound
generator and the reflected signal as we vary the length of the
resonance tube. It takes a lot longer than you might think to purge the
air from the tube, and it takes a steady stream of CO2 to keep the air
out. It is difficult for me to imagine that generating CO2 with
Alka-Seltzer tablets will be sufficient CO2 generation. You are going
to have a mix of air and CO2 in the tube, and that makes it rather
useless to be worried about temperature affects or published speeds of
sound for CO2, etc. You'll clearly see the speed of sound difference
with some CO2 (M=0.044) mixed with air (M=0.029), but the speed you
measure will not be the published speed for CO2 because you don't have
pure CO2.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu