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.

1. Determination of the Heat Capacity
Ratios of Argon and Carbon Dioxide at
Room Temperature
Charlotte 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.

2. 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

3. calculate the speed of sound of the gas. The number of wavelengths in a standing wave
is represented by1:
𝐿 =
𝑛𝜆
2
(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 =
𝑛𝑐
2𝑣
(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:
𝑣
𝑛
= (
𝑐
2𝐿
) (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 =
𝑀𝑐2
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.

4. 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 =
3
2
𝑅𝑇 (5)
which represents translational contribution, the only one present in monoatomic
molecules. Linear molecules such as carbon dioxide should have the following energy2:
𝐸𝑡𝑜𝑡𝑎𝑙 =
3
2
𝑅𝑇 +
2
2
𝑅𝑇 +
8
2
𝑅𝑇 (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 = (
𝑑𝐸
𝑑𝑇
) 𝑣 (7)

5. 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
𝜃
𝑇⁄
𝑇2(𝑒
𝜃
𝑇⁄
−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.

6. Procedure
In 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 & Discussion
Frequency 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.

7. 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
Argon 116.8
319.8
± 0.5
1.66 ± 0.02 1.66
Carbon
Dioxide
98.0
268.3
± 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%

8. 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.
Argon
Carbon
Dioxide
Equipartition of Energy: Translational, Rotational, Vibrational
Modes
0 11.54%
Equipartition of Energy: Translational, Rotational Modes
8.08%
Equipartition of Energy: Translational, Rotational
Statistical 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).

9. Table 3. Experimental Data Compared to Accepted Values.
Gas γ
1Accepted
γ
Argon
1.66 ±
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).
0 2 4 6 8 10 12
0
200
400
600
800
1000
1200
1400
Fit Parameters:
y=B+Ax
A=126.11554
B=-0.80918
delta A= 0.11935
delta B= 0.80948
R
2
= 0.99999
Y =-0.80918+126.11554 X
Figure 1. Frequency vs. Number of Nodes for Compressed Air at 23.1 C
Frequency(Hz)
Number of Nodes
Linear Fit of Data
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 same as v/n,
may be used with equation (3) to calculate the speed of sound through argon gas.

10. 0 2 4 6 8 10 12
0
200
400
600
800
1000
1200
1400
Fit Parameters:
y=B+Ax
A= 116.84735
B= -1.33092
delta A= 8.2263 x 10
-16
delta B= 5.77386 x 10
-15
R
2
= 1
Y =-1.33092+116.84735 X
Figure 2. Frequency vs. Number of Nodes for Compressed Argon Gas at 23.1 C
Frequency(Hz)
Number of Nodes
Linear Fit of Data
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 12
0
200
400
600
800
1000
1200
Fit Parameters:
y=B+Ax
A= 98.02177
B= -2.75616
delta A= 0.10606
delta B= 0.71933
R
2
= 0.99999
Y =-2.75616+98.02177 X
Figure 3. Frequency vs. # of Nodes for Compressed Carbon Dioxide at 23.1 C
Frequency(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.

11. 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.
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 4. Residuals for Compressed Air at 23.1 ºC
Number of Nodes
Frequency(Hz)
0.0 3.0 6.0 9.1 12.1

12. Figure 5. Residuals for Compressed ArgonGas 23.1 ºC
Number of Nodes
Frequency(Hz)
0.0 3.0 6.0 9.1 12.1
Frequency(Hz)

13. 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

14. 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.
Conclusion
The 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

15. 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.
References
1. 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.