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.