In the Laboratory
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LabWorks and the Kundt's Tube: A New Way To Determine the Heat Capacities of Gases Philip A. Bryant and Matthew E. Morgan* Department of Chemistry, U.S. Air Force Academy, Colorado Springs, CO; *
[email protected] This laboratory adds computer-assisted data acquisition to a classic physical chemistry experiment to determine the heat capacities of gas (1). In the past, students manipulated sound frequency and found standing waves using an oscilloscope. This experiment employs a computer and the LabWorks interface to automatically record the frequencies at which standing waves occur. The biggest advantage of using the LabWorks interface is that it allows students to take data quickly, giving them more time to perform calculations, refine their experimental technique, and rerun the experiment if necessary. Experiment The heat capacity ratio for a gas, γ, can be obtained by measuring the pressure change during an adiabatic expansion. The rapid expansion and compression of gas molecules produce sound waves. This expansion and compression occur so quickly that heat energy does not transfer between the gas molecules during the sound vibration, and the generation of sound waves can be considered adiabatic. The formula for determining heat capacity from sound velocity is, γ =
Mc 2 RT
(1)
where M is the molar mass of the gas, and c is the speed of sound (not the speed of light) in the gas medium. Students measure the speed of sound of various gases and use these values to find the heat capacities for the gases. The device used to measure the speed of sound in the gas is called a Kundt’s tube. It is a Plexiglas tube that houses a speaker and a microphone. The tube is filled with the gas to be measured and then sealed. A voltage-controlled oscillator (VCO) generates specific sound waves that play through the speaker in the tube. A microphone picks up the sound waves and displays the signal on the LabWorks interface. By increasing the frequency of the sound, the number of wavelengths in the standing wave increases. The number of wavelengths in a standing wave is represented by,
Equation 3 can be rearranged to solve for the frequency: ν =
c n 2L
In this experiment students generate a graph of sound frequency versus the number of wavelengths in the standing wave. The slope of this line is c兾2L. Multiplying the slope of the line by 2L gives the speed of sound in the gas. Once the students find c, they can substitute it in eq 1 to obtain γ, and from there find the molar heat capacities at constant volume, Cv,m. Experimental Setup The speed of sound equipment is a Kundt’s tube, 1.18 meters in length (Figure 1). The tube is sealed, except for openings that allow sample gases to flow in and out. The experiment setup generates a signal and measures a response using a LabWorks data acquisition interface (2). The interface is connected to a personal computer, the microphone in the Kundt’s tube, and to a device called a voltage-controlled oscillator (VCO) (3). The LabWorks interface uses a 12-bit digital-to-analog converter (DAC) to generate specific voltage values between 150 and 1200 mV. These voltages are sent to the VCO, which converts the millivolt input to a corresponding frequency, which in turn activates the speaker in the Kundt’s tube. The LabWorks interface is also connected to the microphone in
gas flow out
gas flow in
microphone
nλ L = 2
speaker
(2) LabWorks interface
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 are related by the expression, λ = c兾ν, and eq 2 can be expressed as: L =
nc 2ν
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(4)
(3)
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DAC
voltagecontrolled oscillator (VCO)
Figure 1. Experimental setup to measure the speed of sound in gas.
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In the Laboratory
Table 1. Frequency Data for Ar, N2, and CO2
1200
y = 135.11x + 143.57 Ar Frequency/ Hz
N2 Frequency/ Hz
CO2 Frequency/ Hz
1
279
160
236
2
416
307
353
3
551
454
459
4
683
600
572
5
808
742
688
6
958
903
800
7
1093
1052
919
1000
Frequency / Hz
Number of Nodes
R 2 = .9997
800 600 400 200 0
0
1
2
3
4
5
6
7
8
Number of Nodes Figure 2. Graph of frequency versus number of nodes for argon gas.
Table 2. Experimental Data Compared to Accepted Values Slope
Speed/ (m/s)
γ
Cv,m/ (J/K mol)
Accepted Cv
Error (%)
Ar
135.1
318.8
1.644
12.909
12.471
3.52
N2
148.4
350.3
1.391
21.279
20.810
2.25
CO2 113.5
267.9
1.278
29.890
28.795
3.80
the Kundt’s tube and measures and records electrical current signals from the microphone. The computer records the frequency and the corresponding current. In many LabWorks experiments students need to calibrate the sensors used to obtain data. Since students are setting DAC values and reading raw electrical current, no calibration is necessary in this experiment. Students write a data acquisition program using the LabWorks Experiment Builder.
1200
y = 148.43x + 8.571
1000
Frequency / Hz
Ga s
R 2 = .9998
800 600 400 200 0 0
1
2
3
4
5
6
7
8
Number of Nodes Figure 3. Graph of frequency versus number of nodes for nitrogen gas.
Hazards
Results and Discussion Frequency data were obtained for argon, nitrogen, and carbon dioxide gases. Sample data and graphs are shown in Table 1 and Figures 2–4, respectively. The graphs of frequency versus number of nodes for the gases gave linear plots with good R values. The slope from each plot was used to find the speed of sound and the heat capacity (eqs 1–4). Table 2 shows the calculated values and comparisons to accepted heat capacities (4). The experimental values agreed well with the accepted values.
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1000
y = 135.5x + 121.86
Frequency / Hz
The gases used in this experiment, argon, nitrogen, and carbon dioxide, are unreactive. The extreme pressures inside the cylinders, however, could produce catastrophic results if a valve or the cylinder ruptures. Students must make sure the cylinders are always chained to a lab bench while they are being used. Students need to wear safety goggles at all times.
R 2 = .9998
800
600
400
200
0 0
1
2
3
4
5
6
7
8
Number of Nodes Figure 4. Graph of frequency versus number of nodes for CO2 gas.
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In the Laboratory
Conclusion
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This method for measuring the speed of sound and heat capacity of different gases is accurate, reproducible, and results in a high success rate for students. The time required for data acquisition is drastically reduced compared to older methods. The only potential pitfall to this experiment is that the students see the apparatus as a black box, obtaining their data without fully understanding what processes occur in the Kundt's tube. However, appropriate instructor involvement can keep this from happening. Acknowledgments We would like to thank the Air Force, the United States Air Force Academy, and the Dean of Faculty for giving us the time, materials, and opportunity to complete this research. We also thank Peter deGraff and SCI Technologies for their advice and support in troubleshooting hardware problems.
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Supplemental Material
Instructions for the students and notes for the instructor are available in this issue of JCE Online. Literature Cited 1. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 6th ed.; McGraw-Hill Book Co.: New York, 1996; pp 112–116. 2. LabWorks was made by SCI Technologies, Bozeman, MT. However the company has gone out of business. 3. The VCO was constructed by Peter de Graaf, The MITRE Corporation. Email:
[email protected] 4. Accepted Cv values were obtained by subtracting 8.314 from Cp values in Atkins, P.; de Paula, J. Physical Chemistry, 7th ed.; W. H. Freeman & Co.: New York, 2002; Table 2.6, pp 1078– 1084.
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