Determination of the heat capacity ratios of argon and carbon dioxide at room temperature
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Transcript of Determination of the heat capacity ratios of argon and carbon dioxide at room temperature
Determination of the Heat Capacity Ratios of Argon and
Carbon Dioxide at Room TemperatureCharlotte Chaze
Abstract
The sound velocity method was used to calculate the heat capacity
ratios for argon and carbon dioxide at room temperature. A modified version
of Kundt’s tube was used to calculate the speed of sound through the gases,
and then heat capacity ratios were calculated from the speed of sound. The
experimental heat capacity ratios calculated were 1.66 ± 0.02 and 1.2869 ±
0.0009 for argon and carbon dioxide, respectively. The theoretical heat
capacity ratios due to vibrational, translational, and rotational modes using
the equipartition of energy theorem were calculated to be 1.66 and 1.1538
for argon and carbon dioxide, respectively. The theoretical heat capacity
ratio for carbon dioxide due to rotational and translational modes only is 1.4.
These results support the theory that vibrational contributions to heat
capacity ratios are negligible at room temperature, and that the equipartition
of energy theorem is therefore not applicable at room temperature.
Statistical mechanics may be used for vibrational modes to gain more
accurate predictions for heat capacity ratios. Using this method, the
theoretical heat capacity ratio for carbon dioxide is calculated to be 1.29 ±
0.02, which is much closer to the experimental value than with the prediction
from the equipartition of energy theorem.
Introduction
The purpose of this experiment is to calculate the heat capacity ratios
for argon and carbon dioxide using the sound velocity method, and to
compare these results with theoretical results from the equipartition of
energy theorem and statistical mechanics. The heat capacity of a substance
is the amount of energy required to raise its temperature by one degree
Kelvin. Absorbed heat energy causes molecules to move faster (increase
translational energy), rotate faster (increase rotational energy), and vibrate
faster (increase vibrational energy). The sound velocity method involves
measuring the speed of sound through argon and carbon dioxide in a
modified Kundt’s tube. This device is a tube that holds a speaker on one end
and a microphone on the other. The tube is filled with the gas to be
measured at a constant temperature, and is sealed to obtain a constant
pressure. A voltage-controlled oscillator (VCO), in this case, a miniature
radio, generates sound waves that travel from the speaker through the tube
that houses the gas to the microphone. The microphone picks up the sound
waves and displays the signal on an interface on a computer connected to
the microphone. The wave appears to be standing, which is a result of
interference between two waves of the same frequency traveling with the
same speed in opposite directions. The distance between nearest nodes (or
anti-nodes) is equal to λ/2. The successive nodes (or anti-nodes) are in
opposite phase (they differ in phase by 180 degrees) with maximum sound
intensity occurring at the nodes and minimum at the anti-nodes2. The signal
is interpreted as node vs. frequency. The results are then translated onto a
graph of number of nodes vs. frequency for each gas, and the slope of the
line is used to calculate the speed of sound of the gas. The number of
wavelengths in a standing wave is represented by1:
L=nλ2 (1)
where L is the length of the tube, n is the number of nodes in the standing
wave, and λ is the wavelength of the sound wave. The wavelength,
frequency, and speed of a wave can be related by the expression λ=c/v, so
equation (1) may be expressed as1:
L= nc2v
(2)
where c is the speed of sound through the gas, and v is the frequency of the
wave. Equation (2) may be further rearranged as1:
vn=( c2 L ) (3)
so that the slope of the graph (frequency vs. number of nodes) may be used
with the length of the tube to calculate the speed of sound through gas in
the tube. Once the value for c is obtained, the heat capacity ratio γ for the
gas may be calculated1:
γ=M c2
RT(4)
where M is the molar mass of the gas, R is the gas constant, and T is the
temperature at which the speed of sound is measured. Equation (4) assumes
that the gases behave ideally.
These results are then compared to the theoretical heat capacity ratios
based on the equipartition of energy theorem. This theorem shows that if the
vibrational, rotational, and translational modes could all be excited, then the
energy of a molecule of N atoms is the sum of the contribution from all three
modes1. Each molecule has 3 translational degrees of freedom. Linear
molecules such as argon gas have 2 rotational degrees of freedom, and
nonlinear molecules such as carbon dioxide have 3. The total number of
degrees of freedom is 3N. In the equipartition of energy theorem,
translational energy is equal to 3RT/2, rotational energy is equal to 2RT/2
(linear molecules) or 3RT/2 (nonlinear molecules), and vibrational energy is
equal to (3N-5)RT (linear molecules) or (3N-6)RT (nonlinear molecules). Thus,
if one considers a non-quantum mechanical approach to the contributions of
energy from each mode, monoatomic molecules such as argon gas should
have the following energy2:
Etotal=32RT (5)
which represents translational contribution, the only one present in
monoatomic molecules. Linear molecules such as carbon dioxide should
have the following energy2:
Etotal=32RT +2
2RT+ 8
2RT (6)
due to translational, rotational, and vibrational contributions. Once the total
energy from all contributions is obtained, it may be used to calculate the
constant volume molar heat capacity, given by2:
C v=( dEdT
)v
(7)
and the constant pressure molar heat capacity, which is given by2:
C p=C v+R (8)
Once Cv and Cp are obtained, they may be used to calculate γ, the heat
capacity ratio, using the equation2:
γ=C p
Cv (9)
Once the theoretical heat capacity ratio is determined using the
equipartition of energy theorem, it may be compared to experimental values.
According to statistical mechanics, each vibrational mode of the molecule
makes a contribution to the vibrational molar heat capacity by:
C v=Rθ2 e
θT
T2(eθT−1)2
(10)
where θ=hv0/kb and v0 is the fundamental absorption frequency (s-1) of the
vibrational mode and kb is the Boltzmann constant. The total vibrational
contribution is then obtained by summing the Cv term for each vibrational
mode.
The experimental heat capacity ratios may then be compared to the
theoretical heat capacity ratios using: the equipartition of energy theorem
for all three modes; the equipartition of energy theorem for translational and
rotational modes only; and the equipartition of energy theorem for
translational and rotational modes with statistical mechanics for the
vibrational modes.
ProcedureIn this experiment, the gases are Airgas ultra zero grade compressed
air, Airgas compressed carbon dioxide, and Airgas ultra high purity
compressed argon. A modified Kundt’s tube is hooked up to a 2-band radio
receiver on one end and an audio-technica microphone on the other end.
Compressed air is pumped into the tube at a constant volume and pressure,
the speaker sends radio frequency waves through the gas in the tube, and
the microphone picks up the sound waves. The data is saved and the process
is repeated for carbon dioxide and argon. Equation (3) is used to calculate
the speed of sound, and equation (4) is used to calculate the heat capacity
ratios for the gases. Equations (5-9) are used in calculating the theoretical
heat capacity ratios using the equipartition of energy principle. To calculate
the vibrational contributions using statistical mechanics, equation (10) is
utilized.
Results & DiscussionFrequency data for compressed air, argon, and carbon dioxide gases
are displayed graphically in Figures 1-3. The residuals of these data are
plotted in Figures 4-6. The residuals indicate random deviation from the
theoretical frequency at different nodes for each gas, and that a linear
regression is an appropriate mode of analysis for frequency vs. number of
nodes for each gas. The graphs of frequency vs. nodes for the gases
therefore gave linear plots with very good R2 values. The slope from each
plot (Figures 1-3) was used to calculate the speed of sound and the heat
capacity ratio (Equations 3-4) for each gas. The experimental values agree
well with the accepted values, as shown in Table 3.
Table 1 compares the experimental heat capacity ratio results with the
theoretical results using various methods of calculation. The experimental
value for the heat capacity ratio of argon is the same as the theoretical heat
capacity ratio. Carbon dioxide has an experimental heat capacity ratio that is
larger than the theoretical heat capacity ratio using the equipartition of
energy theorem for all three modes. Its experimental value is smaller;
however, than the calculated values from the equipartition of energy
theorem using only the translational and rotational modes. The experimental
value for carbon dioxide does fit the theoretical value from the equipartition
of energy theorem using the translational and rotational modes and
statistical mechanics for the vibrational modes.
Table 1. Experimental heat capacity results vs. calculated theoretical results. “Eq. E” = Equipartition of Energy Principle used in calculations. “T, R, V” = Translational, Rotational, Vibrational modes, respectively.
Slope
Speed
(m/s)
Experimental Heat
Capacity Ratio (ϒ)
Theoretical Heat
Capacity Ratio (ϒ)
from Eq. E: T, R, V
Theoretical Heat
Capacity Ratio (ϒ)
from Eq. E: T, R Only
Theoretical Heat
Capacity Ratio (ϒ)
from Eq. E: T, R only + Statistical
Mechanics: V
Argon116.
8319.8 ± 0.5
1.66 ± 0.02 1.66
Carbon
Dioxide
98.0268.3 ± 0.7
1.2869 ± 0.0009
1.1538 1.4 1.29 ± 0.02
Table 2 shows the percent errors of the experimental heat capacity
ratio values with the three calculated theoretical values for each gas. Argon
has no error associated with the experimental calculations. Carbon dioxide
has a large error of 11.54% associated with the equipartition of energy
calculations with all three modes taken into account. With only the
translational and rotational modes, the percent error drops to 8.08%. When
the vibrational mode is included but calculated using statistical mechanics
instead of the equipartition of energy principle, the percent error drops
drastically to only 0.24%. This large drop in error implies that the vibrational
modes in carbon dioxide in our experiment are better accounted for by
statistical mechanics than with the equipartition of energy theorem. These
results indicate that statistical mechanics are a better way than the
equipartition of energy theorem to predict the contribution from vibrational
modes in a molecule.
Table 2. Percent errors associated with the experimental results and all three theoretical calculated results for each gas.
Argo
n
Carbon
Dioxide
Equipartition of Energy: Translational, Rotational, Vibrational Modes
0 11.54%
Equipartition of Energy: Translational, Rotational Modes
8.08%
Equipartition of Energy: Translational, RotationalStatistical Mechanics: Vibrational Mode
0.24%
Table 3 summarizes the comparison between experimental and
accepted heat capacity ratio values. The experimental value for argon is the
same as the accepted value, but the experimental value for carbon dioxide
has an error of 11.54% (Table 2).
Table 3. Experimental Data Compared to Accepted Values.
Gas γ
1
Accepted γ
Argon1.66 ± 0.02
1.66
Carbon Dioxide
1.2869 ± 0.0009
1.1538
Figure 1 describes a linear fit of the number of nodes against the frequency
at the nodes through compressed air. The slope of the line, v/n, is used to
calculate speed of sound through compressed air, as in equation (3).
Figure 1. The frequency vs. number of nodes for compressed air at 23.1 °C measured in the Kundt’s tube.
Figure 2 is useful in obtaining the slope of the line from the frequency vs. number of nodes through compressed argon gas. This slope, which is the
0 2 4 6 8 10 120
200
400
600
800
1000
1200
1400
Fit Parameters:y=B+AxA=126.11554B=-0.80918delta A= 0.11935delta B= 0.80948
R2 = 0.99999
Y =-0.80918+126.11554 X
Figure 1. Frequency vs. Number of Nodes for Compressed Air at 23.1 C
Fre
qu
en
cy (
Hz)
Number of Nodes
Linear Fit of Data
same as v/n, may be used with equation (3) to calculate the speed of sound through argon gas.
Figure 2. The frequency vs. number of nodes for compressed argon gas at 23.1 °C measured in the Kundt’s tube.
Figure 3 represents the data of frequency vs. number of nodes in carbon dioxide. The slope of the line, or v/n, may be used with equation (3) to calculate the speed of sound through the carbon dioxide.
0 2 4 6 8 10 120
200
400
600
800
1000
1200
1400
Fit Parameters:y=B+AxA= 116.84735B= -1.33092
delta A= 8.2263 x 10-16
delta B= 5.77386 x 10-15
R2 = 1
Y =-1.33092+116.84735 X
Figure 2. Frequency vs. Number of Nodes for Compressed Argon Gas at 23.1 C
Fre
qu
en
cy (
Hz)
Number of Nodes
Linear Fit of Data
Figure 3. The frequency vs. number of nodes for compressed carbon dioxide gas at 23.1 °C measured in the Kundt’s tube.
Figure 4 represents the residuals for compressed air. This data
indicates that there is random deviation from the theoretical frequency at
different nodes for compressed air at 23.1 °C, and that linear regression is a
valid method for analysis.
0 2 4 6 8 10 120
200
400
600
800
1000
1200
Fit Parameters:y=B+AxA= 98.02177B= -2.75616delta A= 0.10606delta B= 0.71933
R2 = 0.99999
Y =-2.75616+98.02177 X
Figure 3. Frequency vs. # of Nodes for Compressed Carbon Dioxide at 23.1 C
Fre
qu
en
cy (
Hz)
Number of Nodes
Linear Fit of Data
Figure 4. Residuals for Compressed Air at 23.1 ºC
Number of Nodes
Fre
qu
en
cy
(H
z)
0.0 3.0 6.0 9.1 12.1-2.80
-1.40
0.00
1.40
2.80
Figure 4. Residuals for linear regression of frequency vs. number of nodes for compressed air at 23.1 °C.
Figure 5 represents the residuals for compressed argon gas. The random deviation here indicates that linear regression is an acceptable method for analysis of frequency vs. number of nodes. Figure 6 is similar, and thus implies the same results for compressed carbon dioxide.
Figure 5. Residuals for Compressed Argon Gas 23.1 ºC
Number of Nodes
Fre
qu
en
cy
(H
z)
0.0 3.0 6.0 9.1 12.1-3.29
-1.64
0.00
1.64
3.29
Frequency (Hz)
The experimental ratios obtained are much closer to the ideal gas
behavior theoretical values without the contributions from the vibrational
modes than with the contributions from the vibrational modes (Tables 1 & 2).
This supports the theory that the vibrational modes are not active at room
temperature. When a molecule absorbs energy or heat, it jumps to a higher
energy level in at least one of the modes of energy, but for this excitation to
occur, the energy from the heat source (RT) must be of the same order of
magnitude as the energy gap.
The vibrational modes are too quantized to apply to the equipartition
of energy theorem because the vibrational energy states are far apart, and
only the lowest energy levels are populated at room temperature. This is
expected since the vibrational modes of the molecule are capable of
absorbing more energy at higher temperatures. The rotational states are
more closely spaced and more can be populated as described by the
Boltzmann distribution principle. Translational energy gaps are extremely
small compared to the other modes, and therefore provide the prominent
contribution in monoatomic gases like argon. Larger than translational
energy gaps are rotational energy gaps, and vibrational energy gaps are the
largest. At room temperature, only the lowest vibrational energy levels may
be populated, causing their contribution to the total energy to be nearly
insignificant.
When the thermal energy kBT is smaller than the quantum energy
spacing in a particular degree of freedom, the average energy and heat
capacity of this degree of freedom are less than the values predicted by
equipartition. This explains why the experimental γ value for carbon dioxide
is much closer to the theoretical value predicted using only translational and
rotational contributions than the value using all three modes (Table 1). Even
closer is the value calculated using statistical mechanics to account for the
vibrational contributions (Table 1). This is because, while very small, there
still exists some level of contribution from the vibrational modes at room
temperature that are not necessarily negligible for nonlinear polyatomic
molecules.
Quantum mechanics tells us that the energy gap between vibrational
levels depends on the vibrational frequency (E = hv). Usually, this gap is too
large to be excited, since RT << hv and the contribution to Cv is small.
However, if the vibrational frequency is small, the gap between energy levels
is small, and there is a significant contribution to Cv2. This is why the
experimental value for heat capacity of carbon dioxide does not match the
theoretical value. CO2 is linear and has 4 vibrational modes: a symmetric
stretch, and antisymmetric stretch, and two bending modes. The symmetric
stretch and antisymmetric stretch don’t contribute much to Cv, but the two
bending modes do.
ConclusionThe sound velocity method is an accurate technique that may be used
to calculate the heat capacity ratios for argon and carbon dioxide at room
temperature. Our heat capacity ratio results support the theory that
vibrational modes are not active at room temperature, and that the
equipartition of energy theorem is therefore not applicable at room
temperature. The theory becomes inaccurate when quantum effects are
significant, such as at low temperatures. Experimental γ values for the
monoatomic gas argon agree exactly with the predicted value. The nonlinear
polyatomic molecule carbon dioxide gave results larger than the predicted γ
value from the equipartition of energy theorem. Our percent error for carbon
dioxide was significantly smaller when vibrational modes were accounted for
by statistical mechanics. These results imply that statistical mechanics are a
much more accurate approach for calculating vibrational contributions of a
molecule at room temperature.
References1. Bryant, P.; Morgan, M. Labworks and the Kundt’s Tube: A New Way to
Determine the Heat Capacities of Gases. J. Chem Edu. 2004, 81, 113-
115.
2. Garland, C.; Nibler, J.; Shoemaker, D. Spectroscopy. Experiments in
Physical Chemistry; McGraw-Hill Higher Education: New York, NY, 2009;
pp. 129-130, 320-326.
3. Physical Constants of Organic Compounds. Handbook of Chemistry and
Physics, Lide, D., Ed.; CRC Press: Boca Raton FL, 2008; 89th edition, pp.
3-4 to 3-522.