writing a lab report for physical chemistry lab. " speed of sound in different gases"
rehab
HeatCap_inter.pdf
This experiment was recently re-‐developed for use with an iPad
by Brad Pearlman (’16), Ian Wyse (’16) and Prof. Varberg.
Okay, you’ve been learning in class about the heat capacity ratio γ (= Cp/Cv) of a gas. This property is important in the design of practical devices such as refrigerators and air conditioners, where a
refrigerant gas is cooled by adiabatic expansion. In this experiment, you will measure this quantity with high accuracy, if you’re careful in your work. You will measure γ indirectly, by employing the fact that a different property, the speed of sound (u) in a gas, is proportional to its square root. By measuring u, you obtain γ. You will determine the speed of sound in a rather nifty way, by sending white noise (sound of all frequencies, like that of a waterfall) into one end of a tube containing the gas, and
recording the resonant frequencies of the tube at the other end. This is exactly how a saxophone produces sound, which your ears detect as euphonious combinations of harmonic frequencies. Instead of using your ears, though, you will record the sound digitally as an interferogram, which is a plot of the sound amplitude as a function of time (in s). All of the frequencies present in the sound interfere to produce this signal. By taking the Fourier transform of the interferogram, you will generate a spectrum of the amplitude as a function of frequency (in s–1). This is the same method that our FTIR and FT-‐NMR spectrometers use.
Introduction
he molar heat capacity at constant pressure, Cp, and at constant volume, CV, are two fundamental properties of a pure substance, as is the heat capacity ratio, 𝛾 = 𝐶p 𝐶V. The
heat capacities of gases per unit volume are rather small, so that accurate measurements of Cp or CV for gases are not easy to make. However, the heat capacity ratio of a gas is easily measured to a very high accuracy, as this experiment demonstrates.
The acoustic theory of gases shows that γ can be obtained from a measurement of the speed of sound in a gas. For the case of an ideal gas, thermodynamics shows that the difference in the heat capacities is given by 𝐶p − 𝐶V = 𝑅. (Indeed, the most accurate value obtained for the gas constant R was made by measuring the speed of sound in pure argon.) The experiment you will perform uses acoustic interferometry to measure the speed of sound in the following four gases: helium, nitrogen, carbon dioxide, and 1,1,1,2-‐tetrafluoroethane (CF3CH2F). The last compound, which is also known as R134a, is the refrigerant that is used in automobile air conditioners in the U.S., replacing a chlorofluorocarbon that damages the ozone layer. It is also used as the propellant gas in Airsoft guns. As you know from studying the Carnot engine or refrigerator, the heat capacity ratio is an important property to be measured for any refrigerant gas.
Theory
The relationship between γ and u, the speed of sound in a particular gas, is given by
𝛾 = − ! !! !m!
!!m !" !
[1]
T
B. Heat capacity ratio by acoustic interferometry
26 ⎢ B. Heat capacity by acoustic interferometry
where M is the molar mass of the gas. You should be able to show that for an ideal gas, Eq. [1] reduces to
𝛾 = !! !
!" [2]
For real gases, γ depends slightly on the pressure. You will carry out your measurements at ambient pressure. (Questions: How does γ depend on temperature? On molar mass? Don't respond too quickly! Will γ be significantly different for hydrogen, 1H2, and deuterium, 2H2?)
We have not derived Eq. [2], but note that this result is consistent with our expectations. We can invert Eq. [2] to obtain the following equation for the speed of sound:
𝑢 = !"# ! [3]
Compare this equation to one derived from the kinetic theory of gases that gives the mean speed 𝑐 of an atom or molecule in a gas at thermal equilibrium:
𝑐 = !!" !" [4]
Thus the speed at which a sound wave propagates through a gas (u) is proportional to the average speed of the gas molecules (𝑐), which is what one would intuitively expect.
If we can measure the speed of sound u in a gas accurately, then we can use Eq. [2] to determine the heat capacity ratio, assuming ideality. Usually, u is determined indirectly, by simultaneously measuring the frequency ν and wavelength λ of audio waves in a gas:
𝑢 = 𝜈𝜆 [5]
In a cylindrical tube, we can measure the properties of standing waves that resonate in an acoustic cavity. Within such an interferometer, audio waves of most frequencies destructively interfere, reducing the amplitude of the sound wave at those frequencies. However, for a few frequencies, constructive interference occurs, and the amplitude is enhanced. Such constructive interference occurs only for acoustic waves that just “fit” into the cavity with nodes at each end cap, so that a half-‐integral number of wavelengths is equal to the length of the cavity, L:
𝑛 ! ! = 𝐿, 𝑛 = 1,2,3,… [6]
or
𝜆 = !! ! , 𝑛 = 1,2,3,… [7]
where n is called the order. This concept is demonstrated in the figure below for the n = 1 and 2 standing waves.
B. Heat capacity by acoustic interferometry ⎢ 27
Combining Eqs. [5] and [7] we obtain an expression for these resonant frequencies
𝜈res = !" !! , 𝑛 = 1,2,3,… [8]
Therefore, by measuring a set of resonant frequencies, the speed of sound can be determined. A plot of 𝜈res versus n should produce a straight line with a slope of 𝑢 2𝐿. You should be able to measure a large number of resonant frequencies corresponding to consecutive values of n. Since L is known, the speed of sound is determined.
Procedure
The experimental apparatus is fairly simple, consisting of a meter long Plexiglas tube with Plexiglas end caps. The left end cap contains a radio earphone that will act as a speaker, while the right end cap contains a small microphone. The plan is to generate white noise, which is sound containing most or all of the audible frequencies (a waterfall is a good example). We do this by detuning the radio so as to produce “static.” You will pass this white noise down the tube, and record the resulting sound with the microphone at the other end. The tube acts like an organ pipe, in that only certain resonant frequencies are supported by the tube. The interferogram you record is a graph of the amplitude of the sound as a function of time (in s). To determine the resonant frequencies, you need to then take a Fourier transform of the interferogram to create a spectrum that plots the sound amplitude as a function of frequency (in s–1 or Hz). You will use an iPad app called SignalScope to record your data. This program digitizes the variation in sound amplitude with time and then performs a Fast Fourier Transform to report the variation in sound amplitude with frequency. This same approach to data collection lies at the heart of two of the most important pieces of chemical instrumentation, NMR and infrared spectrometers.
The length of the tube has already been carefully measured for you using an accurate, metal meter stick. It was determined to be L = 999.4 ± 0.2 mm. You should record the temperature of the lab today in your notebook, as you need to know T for your analysis.
To make your interferometric measurements, first flush the tube with the gas to be studied. Use rubber tubing to connect the left (entrance) port to your gas source. Flush the tube at a moderate rate—put your finger briefly over the exit port to detect the momentary build up of pressure—until you are confident you have removed all of the previous gas; this should take no more than a minute or two. Note that CF3CH2F comes in a metal container packaged for filling your car’s air conditioner. The compound CF3CH2F boils at –26°C, and its vapor pressure is about 6 atm at room temperature. We don’t regulate this pressure, but rather just restrict the flow of the gas with a small metering valve.
The ports on the 1-‐m tube are check valves that should prevent room air from getting into the system. However, experience has taught us that it is best to maintain a slow flow of gas throughout the course of the data collection. (This is particularly true for the speedy helium.) If you listen carefully when you introduce a new gas, you will hear the pitch change as the new gas flows out through the exit port. The tube has been constructed with the entrance and exit ports 180° apart. Rotate the entire tube in its mount to situate the ports appropriately when refilling with a new gas. (What is the best way to orient the tube if you are replacing a heavier gas with a lighter gas, or vice versa? Ask your instructor or TA if you don’t know.) A good way to check that the tube is completely flushed is to record an interferogram and measure the frequency (see below) of a high-‐n peak. Then flow the gas for another minute and re-‐record its spectrum. If the frequency of the chosen peak did not shift significantly (say, by less than a few Hz), then the tube is fully flushed and you can measure all of the resonant frequencies with confidence.
28 ⎢ B. Heat capacity by acoustic interferometry
Detune the radio on the AM band so that all you hear is static. Turn the volume about halfway up with the earphone plugged in. There is a small amplifier for the microphone located along the cord running from the microphone to input jack on the computer. Leave this amplifier OFF.
Launch SignalScope on the iPad. You will see five icons along the top. Click on the Tool Options icon. Select FFT Analyzer and check that the following settings are correct.
Frequency Resolution: Set to 0.5 Hz Averaging: Set to Exp Averages: Set to 10 Vertical Axis: Set to dB Autoscale: Set to Auto Horizontal Axis: Set to Lin Frequency Units: Set to Hz
When you are ready to start collecting data, click the Start Recording button. Momentarily click on the Oscope icon along the bottom. The app will then display the real-‐time signal as an interferogram (amplitude as a function of time). Then click on the FFT icon. Now you are seeing the Fourier transform of that data (amplitude as a function of inverse time, or frequency). You should observe that many peaks are present, each one corresponding to a frequency that matches the resonance condition of Equation [8].
With Averaging set to Exp, the app will continuously take data, updating the screen about every two seconds. After you have collected data for 30–60 seconds, the signal-‐to-‐noise ratio may not be changing much, so you can click on the Pause button. Stop the flow of gas by closing the valve on the regulator. Now you are ready to measure the frequencies of the peaks. First, you may notice for some gases that at high frequency, a different looking set of peaks (with a narrower frequency spacing) begins to show up. We are not sure what these peaks arise from, but you should not measure them.
Use a two-‐finger Pinch or Zoom motion to zoom in on the first 3 or 4 peaks at low frequency. Using one finger, move the cursor to the center of each peak and record these peak frequencies, together with their values of n, in a spreadsheet or in your notebook. The frequency of the cursor position is shown in the upper left corner of the screen. Note that the center of a peak is not necessarily the point of highest amplitude—it is best to place the cursor at what seems to be the center of the overall line shape. Also note that there will be occasional noise spikes and perhaps a few imposter lines to the left of the n = 1 line. (The n = 2 line should be at exactly twice the frequency of n = 1.) The precision and accuracy of this experiment is high if you measure your lines carefully. Use a two-‐finger swipe to translate the spectrum to the next set of 3 or 4 peaks to be measured. Continue measuring peaks to higher frequency until the signal is too weak, or the bogus set of high frequency peaks appear.
You only need to record one trial for each of the four gases.
Clean up
When you are completely finished, please turn off the radio and the microphone’s amplifier. Make sure that all the gas cylinder and regulator valves are closed.
