The Speed of Sound in Air

Transcription

1 Experiment #21 The Speed of Sound in Air References 1. Your first year physics textbook e.g. Resnick, R., Halliday, D. and Krane, K.S., Physics, Fifth Edition, Wiley, Lord Rayleigh, The Theory of Sound, Dover, 1945 (QC 223 R262). 3. Feynman, R.P., Leighton, R.C. and Sands, M, The Feynman Lectures on Physics, Vol. 1, Addison-Wesley, 1963 (QC 23 F47) 4. Tabor, D., Gases, Liquids and Solids, Cambridge, 1991 (QC 173 T25) 5. Zemansky, M.W. and Dittman R.H., Heat and Thermodynamics, McGraw Hill Chapter 5 in the 7th Edition, Introduction Sound waves propagate through a gas by the motion of individual molecules in the gas and the collisions between these molecules. Thus we expect the sound velocity to depend directly on the molecular velocity, and be roughly equal to it. The relationship between these two velocities is given by γ v s = 3 v rms, (1) where v s is the speed of sound, v rms is the root-mean-squared (rms) velocity of the gas particles, and γ is the ratio of specific heats at constant pressure and constant volume for the gas. Since the temperature of a gas is really only a measure of the rms gas particle velocity one can derive a relation between the sound velocity and temperature. This is given by γ R T v s =, (2) M where R is the universal gas constant, M is the molar mass of the gas and T is the absolute temperature. The speed at which a wave travels can be determined if the wavelength, λ, and frequency, ν, of the wave are known v s = λ ν. (3) Thus one method of measuring the speed of sound in a gas is by measuring resonant frequencies in a column of gas. At resonance, the wavelength of the wave is related to the length of the column.

2 In this experiment, we will study resonant frequencies in a column of air in a copper tube in which one end is closed and the other is open. In this situation, resonance occurs when the length of the tube L is equal to 1λ, 3λ, 5λ, 7 λ,... The quarter-wavelength resonance is the strongest and is called the fundamental. At the fundamental resonance for a tube closed at one end, Eq. 3 can be written v s = λ ν R = 4 L ν R. (4) The other resonances occur at higher frequencies. In general, the speed of sound in a tube of air closed at one end is given by v s = 4 L ν R n n, (5) where n = 1 for the fundamental resonance and n = 3,5,7,... for the harmonics and ν Rn is the frequency of the n th resonance. Prelab Questions 1. Does the theoretical expression for the velocity of sound as a function of temperature apply to real gases or only to ideal gases? 2. Explain the meaning of specific heats at constant pressure and constant volume. What is the theoretical ratio γ for a diatomic gas? You can ignore vibrations of the gas molecules (why?). 3. Using a diagram showing nodes and antinodes for the standing wave set up in the tube, give a brief explanation of Eqs. 4 and 5. Caution Please read Appendix Liquid Nitrogen before you attempt the optional section of the experiment. Apparatus copper tube loudspeaker audio oscillator frequency counter dewar temperature probe digital voltmeter

3 immersion heater ice dry ice and alcohol bath liquid nitrogen (optional) Experiment 1. Connect a loudspeaker to an audio oscillator and place it over the end of, but not quite touching, the copper tube. Vary the frequency of the oscillator until you hear a resonance. 2. At room temperature, measure all the resonant frequencies discernible. Plot your results to verify the applicability of Eq. 5. You should be able to plot your data so as to get a straight line whose slope is related to v s. Determine v s and calculate γ. Compare your results with accepted values. 3. Determine the temperature dependence of the velocity of sound in air by immersing the tube in baths of hot water, ice and water, and solid CO 2 and alcohol. Compare the results with Eq. 2. The temperature may be monitored by means of the temperature probe provided. Determine γ and compare the result with the accepted value. Optional: To extend the temperature range further, use a nitrogen bath as well. Analyze your data keeping end effects in mind. Further Questions Here are some more questions to keep in mind while you are doing the lab or during your analysis. 1. Using the temperature probe, should you measure the temperature of the copper tube itself, or of the air inside the copper tube? 2. Does the temperature of the air vary along the length of the tube? If so, how could you estimate the average temperature and its uncertainty? 3. At liquid nitrogen temperatures, you will get oxygen and nitrogen condensing in the tube. Assuming the same temperature, would each resonant frequency be increased, decreased, or stay the same as if there were no liquid layer at the bottom of the tube? 4. Is there a range of frequencies over which each resonance is detected? How could you estimate the uncertainty in each resonant frequency recorded? 5. Does the output of the audio generator stay constant as you vary the frequency over the audible range, or are there natural resonances in the generator itself?

4 6. The theory presented in the Introduction assumes that the standing waves have a node at the closed end of the tube and an antinode at the open end. The location of the antinode is sensitive to the air pressure at the open end and the air flow in and out of the tube. Do you see evidence for end effects in your data? UG2/2007, PCH/2010, NA/2010

5 Appendix: Liquid Nitrogen Liquid nitrogen should be treated with great respect as it is capable of giving you a severe burn. If you handle it properly, however, you will find it to be a very useful constant temperature bath. Its normal boiling point is 77 K and heat of vaporization is 161 J/cm. Its cost is a consideration, so you should not waste it. Liquid nitrogen is available in 5 litre transfer dewars, which will be provided by your lab instructor. Pour some liquid from the transfer dewar into your experimental dewar. Pour only a small amount initially, to cool the dewar, then fill to the desired level, about 2/3 full. This reduces waste due to evaporation. The liquid will evaporate slowly and it may be necessary to top it up occasionally. The dewars you are using have a large open surface area and consequently a large evaporation rate. To reduce this, place some poorly conducting material over the exposed liquid; this will reduce losses due to both heat conduction and air convection. Cautions: Do not permit extended contact of liquid nitrogen with your skin, or you will get a severe burn. A few drops sprinkled on you will evaporate quickly and will cause you no harm. Do not hold a piece of metal which is in contact with liquid nitrogen. Your hand will become frozen to it. Do not contain liquid nitrogen in a pressure-tight vessel without provision for a way for the gas to escape. Otherwise the pressure will build up with evaporation and an explosion may result.