In the Laboratory
Cp /CV Ratios Measured by the Sound Velocity Method Using Calculator-Based Laboratory Technology
W
Mario Branca* Dipartimento di Chimica, Università di Sassari, Via Vienna 2, 07100, Sassari, Italy; *
[email protected] Isabella Soletta Istituto di Istruzione Superiore “G. Manno”, Via Carlo Alberto 92, 07041Alghero (Sassari), Italy
In a reversible adiabatic process the perfect gas obeys the equation PV γ = constant
The heat capacity ratio is linked to the velocity of the sound as follows
(1)
where P and V are the pressure and volume of the gas, respectively, and γ is the heat capacity ratio, defined as the ratio between the specific heat at constant pressure (Cp ) and the specific heat at constant volume (CV ). As we are considering an ideal gas, the following relationship exists Cp = CV + R
(2)
where R is the universal gas constant. The γ value expected for a gas is γ = 1 + (2兾j )
(3)
where j is the number of degrees of freedom of its particles (1). For a monatomic gas, there are only three translational degrees of freedom and γ = 1 + (2兾3) = 1.66. For a diatomic gas, such as nitrogen, oxygen, or air (a mixture of diatomic gases), there are five degrees of freedom (three translational and two rotational) and γ = 1 + (2兾5) = 1.40. For a linear, tri-atomic molecule, such as carbon dioxide, there are five degrees of freedom (three translational and two rotational) and, therefore, γ = 1 + (2兾5) = 1.40. University students taking courses in physical chemistry often determine the value of the heat capacity ratio, γ, experimentally. There are many different ways of obtaining the value of γ, such as the Clement–Désormes method (2– 4), the Rüchardt method (1, 5, 6), and also measuring the velocity of sound (5, 7, 8) in the gas.
γ = v2
M RT
(4)
where v = velocity of sound (m s᎑1), R = gas constant (8.314 J mol᎑1 K᎑1), T = temperature (K), and M = molar mass of the gas (kg mol᎑1). This equation is valid for gases in ideal conditions, but will also give approximate results for real gases at atmospheric pressure and room temperature. As the values of M and R are well-known, all that needs to be measured is the temperature, T, and the velocity of sound in the gas. The velocity of sound is usually measured indirectly, using the resonance phenomena inside a tube (Kundt’s tube) (5, 7, 8). However, it can also be measured directly with a microphone using Calculator-Based Laboratory (CBL) technology. Materials • Texas Instruments CBL interface • TI-92 graphic–symbol calculator • Vernier MCA-CBL digital microphone • A thin sheet of metal, such as a jar lid • PVC tube with one end capped: length = 1.5 m, diameter = 5 cm • Thermometer • Cylinders of argon, oxygen, nitrogen, and carbon dioxide
Apparatus
Figure 1. Apparatus to measure the velocity of sound in a gas: (a) gas inlet, (b) jar lid, (c) microphone, (d) CBL, and (e) TI-92 graphing calculator.
462
Journal of Chemical Education
•
The apparatus to measure the velocity of sound in air, oxygen, nitrogen, argon, and carbon dioxide is shown in Figure 1. We did the experiments at atmospheric pressure and room temperature (293 K). One end of the tube is open and the other closed with a cap. The cap has a small hole (3-mm diameter) through which the gas, at slightly more than atmospheric pressure, can be slowly injected. Once in the tube, the gas returns to atmospheric pressure. We placed a microphone connected to a data collection system at the open end of the tube (Figure 1). The data acquisition system consisted of a Vernier MCA-CBL digital microphone, a Texas Instruments CBL interface, and a TI92 graphic–symbol calculator (9). The data acquisition software was PHYSICS (Vernier Software, version 9/9/02).
Vol. 84 No. 3 March 2007
•
www.JCE.DivCHED.org
In the Laboratory
Method A short sound signal is produced at the open end of the PVC tube near the microphone. The sound travels along the tube until it reaches the capped end, is reflected, and returns to the open end. The microphone records the signal and the echoes. Since we know the length of the tube, we can calculate the velocity of the sound by measuring the time interval between the sound production and the echo return. In air, the time that the sound takes to reach the end of the tube and be reflected is about 9 × 10᎑3 s. To distinguish the original signal from the echo, the duration of the impulse must be at least ten times smaller, that is, around 10᎑3 seconds. We generated a short, clear signal by pressing the center of a jam jar lid (10). The data acquisition system triggers automatically when the sound is generated. The experiment takes 2.3 × 10᎑2 seconds and, in this time, 99 data points are acquired. The tube must be mounted in such a way that the mouth can be either pointing downwards or upwards depending on whether the gases being measured are, respectively, less or more dense than air. The experiment can be carried out in 2 hours with the students in groups of two or three. At the end of the experiment, the students compare their results and draw conclusions from them in a guided discussion under the supervision of the instructor. Hazards
by a flat surface changes by 180⬚ (12). So the form of the wave reverses on each reflection. This is why the first and third signals are similar while the second is the mirror image of the other two on the horizontal axis. Sometimes the first group of signals is not clear, owing to the trigger effect. To measure the time interval, on the graph we identify the clearest peak in the second signal and then find the corresponding peak in the third signal. In the example of Figure 2, peak B from the second and third signal groups was chosen. Once we have established the time interval, we can calculate the velocity of sound in gas from the following equation: v = 2l 兾∆t (5) In the case of air, shown in Figure 2, l = 1.49 m ± 0.01 m, ∆t = 0.00864 s ± 0.00023 s and, thus, v = 345 ± 9 m s᎑1. There was a margin of error of about 3%, but the results were close to 343 m s᎑1, the accepted value at 20 ⬚C. We can now calculate the heat capacity ratio γ from eq 4: v = 345 m s᎑1; T = 293 K; M = 28.96 × 10᎑3 kg mol᎑1; R = 8.314 J mol᎑1 K᎑1 2
γ = (345) ×
28.96 × 10 −3 = 1.41 8.314 × 293
(6)
The expected value for air is γ = 1.4. Students’ Response
The only danger involved is the use of compressed gas cylinder. Students must not be allowed to use compressed gas cylinders unless they have received appropriate instruction and training in their use (11). Analysis and Elaboration of Data The signals we obtain are similar to those shown in Figure 2. This shows the shape of sound perturbation versus time in the case of air. Different groups of signals are shown. The first is the sound emitted, the others are the echoes reflected from the ends of the tube. The vertical axis shows the intensity of the sound: the units are arbitrary. The horizontal axis is time. The scale varies from zero to 0.0231 s, which is the total time for data acquisition. This value is predefined by the data acquisition mode used (WAVEFORM/TRIG). The phase of a sound wave reflected
As the students had never used this type of apparatus, a brief training preliminary session was necessary. The students were given the measuring apparatus (microphone, CBL, TI92 graphic calculator), an explanation on how to use it for the experiment, and also a detailed instructions of the method (see the Supplemental MaterialW). The students tested the equipment, recording other sounds (such as a click of their fingers, a whistle, etc.) with the graphic calculator. Within about 30 minutes, they were all capable of using the equipment with confidence and could carry out the experiment to verify the velocity of sound. The instructor occasionally had to help the students to identify which peaks to use to determine the time interval. TI Device Explorer software can be used to transfer the data to a PC for further numerical and graphical elaboration. This software can be downloaded free (13).
Table 1. Determination of Heat Capacity Ratio, γ Number of Gas Data Sets Median
Figure 2. Shape of sound perturbation vs time in air.
www.JCE.DivCHED.org
•
γ Range
Expected
Standard Deviation
Ar
20
1.73
1.84–1.66
1.66
0.051
O2
16
1.45
1.64–1.33
1.40
0.065
N2
19
1.44
1.55–1.37
1.40
0.035
Air
17
1.47
1.77–1.33
1.40
0.096
CO2
19
1.33
1.44–1.27
1.40
0.035
NOTE: Data collected at 293 K. The expected values were calculated from eq 3.
Vol. 84 No. 3 March 2007
•
Journal of Chemical Education
463
In the Laboratory
Results and Conclusions The students’ results for γ (Cp 兾CV ratios) are shown in Table 1. The table shows the results for the different gases and for air. The data compare well with that reported in the literature (3, 5, 8). W
Supplemental Material
Detailed procedure instructions for the setup and use of the CBL and calculator are available in this issue of JCE Online. Literature Cited 1. Torzo, G.; Delfitto, G.; Pecori, B.; Scatturin, P. Am. J. Phys. 2001, 69, 1205–1211. 2. Spitzer, R. J. Chem. Educ. 1986, 63, 251–252. 3. Moore, W. M. J. Chem. Educ. 1987, 61, 1119–1120.
464
Journal of Chemical Education
•
4. Kadir, O.; Richards, R.; Warrington T. J. Chem. Educ 1988, 65, 374. 5. Schufle, J. A. J. Chem. Educ. 1957, 34, 78–80. 6. Gill, S. J. J. Chem. Educ. 1960, 37, 586. 7. Riley, S. A.; Noble, A.; Crabb, J.; Walkup, T; Jones, D.; Nishimura, A. M. Chem. Educator 1998, 3 (4), 1–9. 8. Colgate S. O.; Williams K. R. ; Reed K.; Hart C. A. J. Chem. Educ. 1987, 64, 553–556. 9. Gastineau, J.; Appel, K.; Bakken, C.; Sorensen, R.; Vernier, D. Physics with CBL; Vernier Software: Portland, OR. 10. Rafanelli, M. La Fisica Nella Scuola XXII 1999, 90–94. 11. Edith Cowan University. Procedures–Use of Gas Cylinders. http://www.chs.ecu.edu.au/org/osh/documents/procedures/ GasCylinders.pdf (accessed Dec 2006). 12. Halliday, D.; Resnik, R. In Fisica Generale; Casa Editrice Ambrosiana: Milano, 1966; Chapter 19. 13. TI Connect Software Home Page. http://education.ti.com/us/ product/accessory/connectivity/features/software.html (accessed Dec 2006).
Vol. 84 No. 3 March 2007
•
www.JCE.DivCHED.org
