Mechanical waves are
waves which propagate through a material medium (solid, liquid or gas)
at a speed which depends on the elastic and inertial properties of the
medium. There are two types of waves,
and
In this experiment we
used a simple device to calculate the speed of sound through air and through
carbon dioxide. We then used our measured speed of sound for carbon dioxide
to determine the number of degrees of freedom for carbon dioxide.
A monatomic gas such
as helium has three degrees of freedom, one for each dimension. diatomic
gases such as oxygen and nitrogen (Main components of air) have five degree
degrees of freedom: Three for each dimension and two due to vibration.
Carbon dioxide has a potential for seven degrees of freedom: three
due to each dimension and four additional due to vibration.
The apparatus we used to measure the speed of sound is shown below:
Using a tone generator,
connected to a speaker we sent sound waves into the tube, which contained
water. A microphone picked up the sound wave as well as the reflection
of the sound wave in the tube and sent this data to an oscilloscope, where
it could be observed. We altered the length of the tube by adjusting
the water level. We adjusted the water level by raising and lowering
a jug which was connected to the glass tube via flexible tubing.
When the length of the tube was just right, a large amplitude standing
wave resulted. This is also called resonance. A Standing wave
is vibrating, but in a stationary pattern.
The increase in amplitude
was visible as an increase in the size of the sine wave on the oscilloscope.
Directly before the actual increase in amplitude, the wave flattened out;
we took our measurements at this point because it was easier to see.
Also, the change could be heard in the sound of the tone; it got
slightly louder.
As we raised and lowered
the jugs we placed pieces of tape on the tube, at the distances where a
standing wave occurred. We determined that the distance between these
points where the standing wave occurred was equivalent to half the wavelength
of the sound wave being put out by the tone generator. This can be
shown by looking at a diagram of the wave in the tube:
We took these measurements
for both air and carbon dioxide. For the carbon dioxide we could
not go back and check our measurements because as the water level rose,
it pushed the carbon dioxide out of the tube. So, our carbon dioxide
measurements were not as precise as our air measurements.
We then used the half
wavelength measurements to find the full wavelength of the sound wave.
We checked the frequency of the sound wave using a volt meter, where one
volt was equal to 100 Hz; we also used a frequency counter to doublecheck
the frequency. (The difference between these measurements being only
about 0.5%.) Using the frequency and the wavelength, we determined
the speed of sound. This calculation was done as shown below, where
c represents the speed of sound.
The speed of sound for
air was calculated to be 336 m/s +/- 1.73 m/s (334, 338). The speed
of sound for carbon dioxide was calculated to be 269m/s +/- 19.3 m/s (250,288).
Notice there is a lot more uncertainty in the measurement for the speed
of sound of carbon dioxide. It would have been a good idea to repeat
the experiment for the speed of sound of carbon dioxide, but due to time
constraints we were unable to do so. Most of the error probably lies
in human error in determining the exact resonance points. At any
rate, the speed of sound for carbon dioxide is lower because carbon dioxide
molecules are more massive, so it takes longer for sound to move through
them.
In class we derived
an expression for the speed of sound, based on the expression we had learned
earlier for angular frequency. We used this expression to calculate
a constant temperature model for the speed of sound. The derivation
is shown below:
The constant temperature model speed of sound was calculated to be 293 m/s for air and 238 m/s for carbon dioxide. Both of these differ from the measured speeds of sound. We learned that the reason for this is that the constant temperature model is inaccurate. Sound waves are adiabatic because they are so fast. As a result the gas molecules inside the tube get hotter as they are jostled by the sound wave. Room temperature is not an accurate representation of the temperature inside the tube. An adiabatic factor has to be used to calculate the speed of sound. In class, we derived and expression for calculating , the adiabatic constant. The derivation is shown below:
We calculated the
g
for air to be 1.32 and the g
for carbon dioxide to be 1.28. Calculating the uncertainties,
we found that the g
for air was 1.32 +/- 1.028% , which gives a range of (1.31, 1.33).
The g for carbon dioxide
was 1.28 +/- 14.36%, which gives a range of (1.10, 1.46).
We used the value of
g
to determine the degrees of freedom of carbon dioxide. We learned
in class that g
is equivalent to Cp/CV(Heat capacity at constant
pressure/Heat capacity at constant volume). Utilizing this information
and the first law of thermodynamics we determined that g
is also equal to (degrees of freedom +2)/degrees of freedom. So,
for a monatomic gas g
= 5/3 = 1.67; for a diatomic gas g=
(7/5) =1.4. For a gas which has 4 additional degrees of freedom due
to vibration, as carbon dioxide may potentially have, g
= 11/9 = 1.22
We know that air consists
primarily of diatomic molecules N2 and O2.
So, we would expect the g
to be 1.4. Our measured g
had a range of 1.31 to 1.33 with a mean of 1.32. So, we were close
to the expected, but although we had a high degree of precision in our
measurements, our accuracy was not very good.
Our precision for carbon
dioxide was even worse. Based on our data, we can’t rightly say that
carbon dioxide has 4 extra degrees of freedom. The expected g
if this were the case (1.22) falls into our range of 1.10 to 1.46, but
so does the expected g
for a diatomic molecule (1.4). However, even though our data was
not very good, I know that some other lab groups did have data which supported
carbon dioxide having four extra degrees of freedom. The only way
to know for sure would be to repeat the experiment several times being
more careful in taking measurements for the half-wavelength.
For other experiments done by the Physics II class
of 2001 at Warren Wilson College: