Technology Report Cite This: J. Chem. Educ. XXXX, XXX, XXX-XXX
pubs.acs.org/jchemeduc
Determining the Speed of Sound and Heat Capacity Ratios of Gases by Acoustic Interferometry Thomas D. Varberg,* Bradley W. Pearlman, Ian A. Wyse, Samuel P. Gleason, Dalir H. P. Kellett, and Kenneth L. Moffett Department of Chemistry, Macalester College, 1600 Grand Avenue, St. Paul, Minnesota 55105, United States S Supporting Information *
ABSTRACT: In this paper, we describe an experiment for the undergraduate physical chemistry laboratory in which students determine the speed of sound in the gases He, N2, CO2, and CF3CH2F. The experimental apparatus consists of a closed acrylic tube containing the gas under study. White audio noise is injected into one end of the tube, and the sound amplitude is recorded as a function of time at the other end. The data are recorded and Fourier transformed in real time with a spectrum analyzer application on an Apple iPad. The resulting frequency spectrum of the cavity standing waves is used to determine the speed of sound in the gas by least-squares fitting, with experimental values that fall within about 0.2% or less of the accepted values. The speed of sound is related to the heat capacity ratio in the ideal gas limit, providing students with quantitative evidence of the nonideality of the gases at ambient pressure and temperature. The experiment demonstrates the power and accuracy of interferometry and Fourier analysis using a modern tablet computer. KEYWORDS: Upper-Division Undergraduate, Laboratory Instruction, Physical Chemistry, Hands-On Learning/Manipulatives, Gases, Heat Capacity, Fourier Transform Techniques, Laboratory Equipment/Apparatus
W
In our experiment, we use white noise to probe an acoustic cavity of fixed length and record the audio signal as an interferogram, which is a graph of the amplitude against time. A Fourier transform of the interferogram produces a frequency spectrum of the standing waves of the acoustic cavity. Such a time-domain measurement imparts an important multiplex (or Fellgett) advantage in that all frequencies are simultaneously recorded, resulting in a significant increase in the observed signal-to-noise ratio.10 We developed this laboratory experiment to use an inexpensive spectrum analyzer app that runs on an Apple iPad. The tablet application provides a modern, userfriendly interface for students to record their data and display the Fourier transform in real time. Three articles describe acoustic interferometry experiments that are similar to ours. Steel et al.11 provide useful background on theoretical aspects of such an experiment, but the apparatus they describe uses custom-assembled electronics to generate the white noise and digitize the microphone signal, with the required Fourier transform accomplished using a Basic program compiled on a computer. Martin12 describes an apparatus similar to ours utilizing a Vernier LabPro interface. DeLomba et al.13 describe an interferometric experiment in which white noise is generated in situ by the flow of sample gas, with
e have developed a physical chemistry laboratory experiment in which students measure the speed of sound in various gases rapidly and accurately using a spectrum analyzer application on an Apple iPad. The speed of sound is directly related to the ratio of the heat capacity measured at constant pressure and at constant volume, γ = Cp/CV, which is an important thermodynamic property of a gas. The experiment demonstrates to students the use of interferometry as a powerful experimental method, which can be implemented on a modern tablet computer. The use of an acoustic cavity to measure the speed of sound (c) in a gas dates to the work of Kundt1 in the mid-19th century, and the determination of c (and thus γ) has become a classic physical chemistry laboratory experiment. In the traditional method, a sample of gas is held in a sealed tube, and sound waves produced from an audio sine wave generator are introduced at one end. A microphone at the other end is used to determine the resonant wavelengths of the cavity, identified by observing an acoustic antinode at the microphone. Variations of this method involving either a fixed audio frequency with an adjustable cavity length or a fixed cavity with a variable frequency have been described in this journal2−6 and in physical chemistry laboratory textbooks.7−9 However, these traditional scanning methods are slower and less accurate than the interferometric method we describe here, for they measure the response of the acoustic cavity one frequency at a time. © XXXX American Chemical Society and Division of Chemical Education, Inc.
Received: July 16, 2017 Revised: September 16, 2017
A
DOI: 10.1021/acs.jchemed.7b00526 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Technology Report
Figure 1. Screen shot of the SignalScope application, showing the frequency spectrum between 0 and 5 kHz recorded with nitrogen in the acoustic cavity. The first 14 standing wave resonances are observed. The movable cursor with its blue cross hairs is located at the peak of the n = 5 resonance, and the cursor frequency of 1749 Hz is displayed at the upper left of the display. A few extraneous peaks are seen near the n = 1 and 2 resonances, which arise from an unidentified source of noise.The Q factor of the cavity, which is defined as the ratio of the frequency of a resonance line to its width (full width at half-maximum), is estimated to be ∼30.
■
EXPERIMENTAL THEORY The relationship between γ and c, the speed of sound in a particular gas, is given by16
students collecting data using a Vernier LabQuest2 hardware interface with subsequent Fourier transformation on a personal computer. However, the frequency response of their apparatus is not flat, producing undesirable oscillations in the frequency spectrum. Our values for the speed of sound are about a factor of 10 more accurate than those determined in refs 12 and 13 when compared to accepted values taken from the NIST Fluid Properties webpage.14 Another advantage of our experiment is that the Fourier spectrum is generated instantaneously by the iPad app, so that students can see the frequency spectrum in real time as they are collecting data. The high sensitivity and flat frequency response of the microphone and cavity generates a large signal-to-noise ratio and highly accurate data. Molek et al.15 describe in this journal an alternative method for determining the speed of sound in gases in which a shock wave is generated by the detonation of nitrocellulose in a tube containing a sample of gas. The time delay of the resulting sound wave between two fixed microphones in the tube is measured with an oscilloscope and used to determine the speed of the wave directly as the ratio of the distance traveled to the time delay. The measured speeds of sound are about a factor of 6 less precise than those we report here. One pedagogical goal of this experiment is to introduce students to the use of the Fourier transform in chemistry. Such methods lie at the heart of two of the most important pieces of chemical instrumentationinfrared and nuclear magnetic resonance spectrometersand so this experiment serves as a useful introduction, via the acoustic domain, to such multiplex signal acquisition techniques. A second pedagogical goal is to demonstrate to students the use of tablet computers such as iPads as robust data acquisition and analysis tools.
γ=−
c 2M ⎛ ∂Vm ⎞ ⎜ ⎟ Vm2 ⎝ ∂p ⎠T
(1)
where Vm and M are the molar volume and molar mass of the gas, respectively. For gases obeying the ideal equation of state (pVm = RT), eq 1 reduces to
γ=
Mc 2 RT
(2)
which enables us to determine γ from a measurement of c, assuming ideality. We can further determine the heat capacity at constant pressure, Cp, if we combine the ideal relationship Cp − CV = R with the definition of γ = Cp/CV to obtain Cp =
Rγ γ−1
(3)
The speed of sound c can be determined by simultaneously measuring the frequency v and wavelength λ of audio waves in a gas
c = vλ
(4)
In this experiment, a closed cylindrical tube containing a sample of gas is used to measure the wavelengths of standing waves that resonate within the acoustic cavity. Within such a tube, audio waves of most wavelengths destructively interfere with themselves when they reflect off an end because the incident B
DOI: 10.1021/acs.jchemed.7b00526 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Technology Report
and reflected waves are not in phase with each other. However, for those wavelengths for which an antinode falls at each end of the cavity, constructive interference occurs.11 The resonance condition is that a half-integral number of wavelengths must be equal to the length of the cavity, L: ⎛λ⎞ n⎜ ⎟ = L , n = 1, 2, 3, ... ⎝2⎠
(5)
where n is called the order. By combining eqs 4 and 5, we find that for the resonant frequencies v=
■
⎛ c ⎞ ⎜ ⎟n , n = 1, 2, 3, ... ⎝ 2L ⎠
(6)
EXPERIMENTAL DETAILS The experimental apparatus is simple, consisting of a 0.5 m long acrylic tube with acrylic end-caps. An earbud is placed in a hole drilled in the center of one end-cap. The earbud is connected to a detuned AM radio, so as to produce “static” or white noise of all frequencies. A small microphone is placed in the center of the other end-cap. Each cap also has a plastic hose barb installed to permit gas flow in and out via latex tubing. Gas is slowly flowed through the apparatus for a few minutes to flush out the previous sample, and the flow is maintained during data collection. The length L of the tube can be determined in advance by the instructor and provided to the students. We calibrated our device with argon, for which the speed of sound is accurately known, in order to determine the value of L to four significant figures. It can also be carefully measured using a machinist’s metal rule. The temperature should be measured with a platinum high-accuracy (±0.1 °C) digital thermometer. The probe can be placed into the exhaust port of the interferometer or simply be used to measure the ambient temperature near the apparatuswe find these temperatures usually agree within 0.2 °C. The audio interferogram is collected by the microphone and digitized on an Apple iPad using the SignalScope spectrum analyzer app available on the App Store for $25. (Similar apps are available for Android tablets.) The SignalScope software can be set to signal average over multiple acquisitions and can display either the interferogram or its Fourier transform in real time. The sampling rate is 48 kHz, which is high enough to prevent the observation of any aliased signals above the Nyquist frequency.11 We have students record the average of 25 acquisitions, which requires 25 s. An example of a frequency spectrum recorded with nitrogen is displayed in Figure 1. The iPad software allows students to measure the peak frequencies with a cursor, which we have them determine for n = 1 to 10. In a spreadsheet, students plot v versus n, producing a straight line with a slope of c/2L, together with its estimated uncertainty. A typical plot is shown in Figure 2. Because L is provided to the students, the speed of sound c is directly determined from eq 6. We have students record data for He, N2, CO2, and CF3CH2F. The last compound is the refrigerant gas R134a used in American automobile air conditioners and can be purchased from automotive supply shops. We no longer use SF6 due to its high global warming potential. The data collection takes about 1.5 h to complete. Our students work in teams of two and rotate through a variety of experiments over the course of a semester, so that only one team is using the apparatus during the same lab period. In a typical lab section, we will have four
Figure 2. Plot of resonant frequency versus n for nitrogen at 22.3 °C and 740 Torr. The correlation coefficient of the least-squares fit has the value R2 = 0.999998. The derived values of the slope, c, γ, and Cp are shown in Table 1.
pairs of students make measurements with the apparatus. Additional experimental notes and drawings of the apparatus are provided in the Supporting Information.
■
RESULTS In Table 1, we list the results of a student team that carefully completed the experiment, together with literature values taken from the NIST Fluid Properties webpage.14 The experimental values for the speed of sound of the four gases are highly accurate, with deviations from literature values of less than 0.2% in each case. The experimental heat capacity ratios are calculated from eq 2.5 These values show a greater deviation from the literature values, due to the assumption of ideality in deriving eq 2. The heat capacity at constant pressure is determined from eq 3. Because this equation was derived by assuming ideal behavior twice, it is not surprising that the values differ more from the literature values than those for the heat capacity ratio. One measure of the nonideality of gases is the deviation of the quantity Cp − CV from its ideal value of R. We have students calculate Cp − CV for each gas from the values given in ref 14 as a way of quantifying the extent of nonideality. As expected, the deviation from R grows in the order He, N2, CO2, and CF3CH2F, in the same manner that the magnitude of the error in the experimental γ values grows. (For our data, helium is the exception to this behavior, caused by the fact that the % error in its speed of sound measurement is the largest.) This trend quantitatively demonstrates to students the increasing nonideality of gases as the strength of their intermolecular forces increases. In our physical chemistry laboratory, we have the students present their results each week in either written or oral form. We expect written reports to follow the format of a research article such as one might submit to the Journal of Physical Chemistry A. Students present their work orally in a 15 min format, as if they were giving a talk at a scientific conference. These reports include an introduction, an experimental section, and a presentation and discussion of the results, including comparisons to the literature. C
DOI: 10.1021/acs.jchemed.7b00526 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Technology Report
Table 1. Student Results for the Slope, Speed of Sound, and Heat Capacity Ratio of Various Gases quantity
He
N2
CO2
CF3CH2F
T, °C slopea,b speed of sound, cexperimental valuea speed of sound, cliterature valuec error, % heat capacity ratio, γexperimental valuea heat capacity ratio, γliterature valuec error, % constant pressure, Cpexperimental valuea constant pressure, Cpliterature valuec error, %
22.9 1010.2(29) 1010.9(29) 1012.8 −0.18 1.662(7) 1.667 −0.28 20.88(23) 20.79 0.44
22.3 350.1(6) 350.4(6) 350.5 −0.03 1.400(3) 1.401 −0.09 29.10(24) 29.17 −0.24
22.5 267.5(6) 267.7(6) 267.6 0.02 1.283(4) 1.295 −0.98 37.74(55) 37.33 1.08
23.1 161.2(3) 161.3(3) 161.1 0.14 1.077(3) 1.120 −3.81 115.6(47) 86.5 33.7
Numbers in parentheses are standard deviations of the quantities in units of the last reported digit. bValues of the slope (from eq 6) are in s−1, speed of sound values are in m s−1, heat capacity ratios are unitless, and heat capacities are in J K−1 mol−1. cLiterature values are taken from the NIST Fluid Properties webpage and are adjusted to the recorded temperature of the experiment.14 a
■
HAZARDS Helium, nitrogen, and carbon dioxide are nonflammable gases and are not toxic at the concentrations generated in this experiment. The compound CF3CH2F is not flammable in mixtures with air at temperatures below 100 °C. It has very low acute toxicity by inhalation,17 but we recommend exhausting this gas to a fume hood by attaching tubing to the exit port of the acoustic cavity.
material is based upon work supported by the National Science Foundation under Grants CHE-1265741 and CHE-1565969.
■
■
SUMMARY In this experiment for the physical chemistry laboratory, students measure the speed of sound of various gases by the use of an acoustic interferometer. The results are highly accurate, with speeds determined within 0.2% of the accepted values. The heat capacity ratios can then be derived under the assumption the gases behave ideally and compared to accepted values, enabling students to observe deviations from ideal behavior. Through this experiment, students gain an appreciation for the power of interferometry and Fourier analysis to rapidly and accurately determine a frequency spectrum.
■
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.7b00526. Laboratory instructions for the students (PDF, DOCX) Experimental notes for instructors (PDF, DOCX) Sample spreadsheet file (XLSX)
■
REFERENCES
(1) Kundt, A. Acoustic Experiments. London, Edinburgh Dublin Philos. Mag. J. Sci. 1868, 35, 41−48. (2) Colgate, S. O.; Williams, K. R.; Reed, K.; Hart, C. A. Cp/Cv Ratios by the Sound Velocity Method Using a Spherical Resonator: A Modification of a Common Physical Chemistry Laboratory Experiment. J. Chem. Educ. 1987, 64, 553−556. (3) Tennis, R.; Bailey, R.; Henderson, G. Vibrational Spectra and Heat Capacity of Methane, and the Speed of Sound. J. Chem. Educ. 2000, 77, 1634−1636. (4) Bryant, P. A.; Morgan, M. E. LabWorks and the Kundt’s Tube: A New Way To Determine the Heat Capacities of Gases. J. Chem. Educ. 2004, 81, 113−115. (5) Halpern, A. M.; Liu, A. Gas Nonideality at One Atmosphere Revealed through Speed of Sound Measurements and Heat Capacity Determinations. J. Chem. Educ. 2008, 85, 1568−1570. (6) Aristov, N.; Habekost, G.; Habekost, A. Kundt’s Tube: An Acoustic Gas Analyzer. J. Chem. Educ. 2011, 88, 811−815. (7) Garland, C. W.; Nibler, J. W.; Shoemaker, D. P. Experiments in Physical Chemistry, 8th ed.; McGraw-Hill: New York, 2009; pp 114− 118. (8) Halpern, A. M.; McBane, G. C. Experimental Physical Chemistry: A Laboratory Textbook, 3rd ed.; W.H. Freeman: New York, 2006; pp 2-1−13. (9) White, J. M. Physical Chemistry Laboratory Experiments; PrenticeHall: Englewood Cliffs, NJ, 1975; pp 168−174. (10) Skoog, D. A.; Holler, F. J.; Crouch, S. R. Principles of Instrumental Analysis, 6th ed.; Thomson Brooks/Cole: Belmont, CA, 2007; pp 204−211. (11) Steel, C.; Joy, T.; Clune, T. Teaching FFT Principles in the Physical Chemistry Laboratory. J. Chem. Educ. 1990, 67, 883−887. (12) Martin, B. E. Measuring the Speed of SoundVariation on a Familiar Theme. Phys. Teach. 2001, 39, 424−426. (13) DeLomba, M. J.; Hernandez, M. D.; Stankus, J. J. Speed of Sound in Gases Measured by in Situ Generated White Noise. J. Chem. Educ. 2016, 93, 1961−1964. (14) Thermophysical Properties of Fluid Systems; http://webbook. nist.gov/chemistry/fluid (accessed August 2017). (15) Sinclair Molek, K.; Reyes, K. A.; Burnette, B. A.; Stepherson, J. R. Measuring the Speed of Sound through Gases Using Nitrocellulose. J. Chem. Educ. 2015, 92, 762−766. (16) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954; pp 369−370. (17) Alexander, D. J.; Libretto, S. E. An Overview of the Toxicology of HFA-134a (1,1,1,2-tetrafluoroethane). Hum. Exp. Toxicol. 1995, 14, 715−720.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Thomas D. Varberg: 0000-0002-7731-5110 Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS The corresponding author thanks Joseph Brom (University of St. Thomas) for the original idea for this experiment, and Keith Kuwata (Macalester College) for helpful discussions. This D
DOI: 10.1021/acs.jchemed.7b00526 J. Chem. Educ. XXXX, XXX, XXX−XXX
