Heat. LD Physics Leaflets. Determining the adiabatic exponent c P /c V of various gases using the gas elastic resonance apparatus P2.5.3.
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1 WZ Heat Kinetic theory of gases Specific heat of gases LD Physics Leaflets P..3. Determining the adiabatic exponent c P /c V of various gases using the gas elastic resonance apparatus Experiment Objectives Determine the resonance frequency of the system. Calculate the adiabatic exponent of the gas used. Use different gases. Principles In a precision tube with scales for measuring volume, a gas column with the cross-section A, whose volume V, pressure p and type of gas can be varied, is brought to resonant vibration. A magnetic piston inside a glass tube is moved through an electromagnetic alternating current field as a suspended mass m. This periodically compresses and expands the gas (adiabatic constitutional change). If the frequency of the alternating current field is equal to the resonance frequency of the system the oscillation amplitude of the piston will be maximized (the minimum off-resonance due to friction can be neglected). The time of vibration T for the glass tube closed on both sides and the piston in the middle is: With V = l * A and = f it follows that k: l = Length of the gas column (0.8 m), see Fig. 1 A = Cross-section of the gas column ( m²) m = Mass of the piston ( kg) p = gas pressure = Air pressure (in the example Pa) f 0 = resonance frequency Fig 1: Glass flask of the gas elastic resonance apparatus If the inherent values due to the apparatus design are used, we obtain 1
2 P..3. Apparatus 1 Gas elastic resonance apparatus Digital Multimeter Function generator P Stand base V-shape, small Fine regulating valve Minican Neon Minican Carbon dioxide way-valve Silicone tubing Rubber tubing Connector Connecting lead 0 cm, blue Connecting lead 100 cm, red/blue, pair Derivation of equation () The force equation applies to the tube opened on one side: Because this vibration is an adiabatic process, the equation = constant (Poison's law) applies. It follows that: and with (4) Setup Before inserting the gas resonance apparatus into the experiment arrangement, turn the tube with opened valve by 180 in the opposite direction of the operating position, so that the piston can slide to the upper end of the tube; close the valve immediately and set up the experiment as shown in Fig. For instructions on how to fill the tube with different gasses see the end of this leaflet. Let the piston (representing the end of the gas columns) slide in the center position needed for the experiment by briefly opening the valves. Attach the coil so that the upper edge is at the same level as the lower edge of the piston (Fig. ). Important The inside of the glass tubing must definitely be free from any type of dirt, so that the piston can move freely. It can be cleaned with a bottle cleaner moistened with alcohol. Definitely avoid touching the surface of the piston with bare skin. Notes If the glass tubing is very clean, then it can also be used in a horizontal arrangement. This has the advantage that the piston remains in one position and does not sink slowly to the bottom, thereby making it more difficult to read the measured values. With and and the linear force we obtain the following: For the glass tube enclosed on both sides and the piston in the middle, the direction force D is twice as large: From the relation for the time of vibration of a harmonic vibration: and the relationship (6) for the direction force D* we obtain the frequency: () follows from this. Fig : Setup of the experiment
3 P..3. Carrying out the Experiment Safety note Caution: Never let any substantial pressure build up in the glass during filling with other gases, the glass tube might explode First choose a frequency of approx. 0 Hz and set the AC current to 1 A using the amplitude adjustment of the function generator (ordinary multimeters show correct values only for 0/60 Hz). Take a note of the position the amplitude setting and do not exceed it during further experimentation for longer times. Watch the overload LED of the function generator. To find the resonance frequency, slowly increase the generator frequency from 0 Hz onwards, until the maximum amplitude of the oscillating piston just passed; then lower the frequency until the maximum amplitude as seen before is achieved again. After each setting of a new frequency, the piston will need some time to reach the final amplitude. If the friction inside the tube is very low, the amplitude of the piston in resonance might be to large. Reduce the AC current amplitude appropriately. Adjusting the last digit of the function generators frequency will require a calm hand. It is very hard to adjust the last digit stepwise, sometimes it is easier to do a jump forward and backward to set the desired frequency. For infinite vertical operation without the piston sagging, it is necessary to add a small DC current to the coil using the DC offset switch of the function generator, but the additional force will also influence the oscillation frequency of the piston and therefore the measured values. Therefore this is more for demonstration than for precise student s measurement. Evaluation From the measuring example, we can calculate: air CO Ne s 1.9 Hz Hz s s Theoretical values 3.0 Hz 1. 3 According to kinetic gas theory, the adiabatic exponent is (f+)/f, With f being the degrees of freedom inside the gas molecule. air ; diatomic, f=, 3 translation + rotation CO 1.9 ; linear triatomic, f=7 Ne , monoatomic f=3, 3 translation Measurement Example Using the constant dimensions of the apparatus we derived the formula f0 97,1 Pa s 97,1 f p 1,07 10 Pa 3,89310 s f0 And for example we measured: 0 Resonance frequency of air: f 0 = 1.9 Hz Resonance frequency of CO : f 0 = 1.0 Hz Resonance frequency of Ne: f 0 = 3.0 Hz 3
4
5 P..3. How to perform the experiment with other gas fillings After finishing the setup (Fig. 1) place the gas apparatus horizontally on the table (Take the whole apparatus out of the stand base). Step 1: Open all valves and let the piston slide to the end shown in the Fig. 1. To do see whether the piston slides freely to the end tilt the apparatus. 1 Step : Close the open end of the T-valve between the Minican and the gas resonance apparatus. Step 3a: Open very carefully the valve of the Minican! Assuming that the check of sliding was successful (step 1) the piston will start to move towards the other end of the gas resonance apparatus. The speed of the piston can be controlled easily by valve of the Minican. Open the valve only to that extend (reduced gas stream) to establish a slow speed of the piston! Caution!!! If the valve is opened fast the piston will be accelerated towards the other end and can result in damaging the gas resonance apparatus. Thus the air inside the glass tube is pushed out. Note: If the piston gets stuck you may knock gently at the glass tube to achieve a further movement. LD DIDACTIC GmbH Leyboldstrasse 1 D-034 Huerth Phone: (033) Fax: (033) info@ld-didactic.de LD DIDACTIC GmbH Technical alterations reserved
6 P Step 3b: Decrease the gas pressure, i.e. close the valve of the Minican slightly when the piston has reached the other end of the glass tube. Caution!!! Not decreasing the gas pressure when the piston is at the other end may result in damaging the gas resonance apparatus. Step 4: After you have reduced the gas pressure in step 4 close first the end of the glass tube with the piston (4a) and then immediately the valve of the Minican (4b). 4 4a 4b Step : Open shortly the T-valve to expand the gas to normal pressure. Thus the gas pressure becomes the same as the air pressure. Step 6: Close the valve of the gas resonance apparatus opposite of the piston. LD DIDACTIC GmbH Leyboldstrasse 1 D-034 Huerth Phone: (033) Fax: (033) info@ld-didactic.de LD DIDACTIC GmbH Technical alterations reserved
7 P Step 7: Position the gas resonance apparatus vertically to perform the experiment. The piston will gradually move towards the middle of the glass tube. While this movement downwards the resonance frequency can be adjusted and measured as a test. To do so you have to move the coil towards the piston. The most accurate measurement is done when the piston is approximately in the middle of the tube. 7 LD DIDACTIC GmbH Leyboldstrasse 1 D-034 Huerth Phone: (033) Fax: (033) info@ld-didactic.de LD DIDACTIC GmbH Technical alterations reserved
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