W 18E “Heat Capacity Ratio Γ ”

Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum

W 18e “Heat Capacity Ratio γ ”

Tasks

1 Determine the heat capacity ratio γ of air and carbon dioxide using the method of Clément and De- sormes.

2 Determine the sound velocity in air and calculate the heat capacity ratio γ. Calculate the molar heat capacities at constant pressure and constant volume for air, respectively.

3 Determine the heat capacity ratio γ for air and carbon dioxide using the resonance-tube method.

Additional task: Determine the heat capacity ratio γ of a gas using the method by Flammersfeld.

Literature

Physikalisches Praktikum, 13. Auflage, Hrsg. W. Schenk, F. Kremer, Wärmelehre, 2.0.1, 2.0.2, 2.2 Physics, P. A. Tipler, 3rd Edition, Vol. 1, 14-1, 16-4, 16-5, 16-6, 16-7, 16-8 University Physics, H. Benson, Chap. 20

Accessories

Pressure gas cylinder with CO2, air pump, digital manometer, glass bottle, function generator, electro- acoustic transducer, amplifier, oscilloscope, resonance equipment

Keywords for preparation

- Adiabatic and isothermal transformations of state, 1st Law of Thermodynamics - Heat capacities cp and cv of gases - Definition of the adiabatic exponent γ, dependence on the atomic structure of an ideal gas - Determination of γ after Clément and Desormes - Sound propagation in gases, compressibility, bulk modulus - Generation of Lissajous figures - Determination of γ by the resonance tube method

Remarks

In order to increase accuracy measurements should be repeated. Compare the measurement uncer- tainties of the different techniques. Further compare the experimental values of the adiabatic exponent γ with theoretical values calculated for the various molecules.

1 Hints to the experiment

Task 1: Determination of the adiabatic exponent γ after the method of Clemént and Desormes

General background: The adiabatic decompression might be analyzed as follows:

1. Initial pressure: pL + p1 , (pL atmospheric pressure) at T1 (room temperature).

2. Adiabatic expansion down to pL accompanied by a decrease in temperature to T0:

γ γ −1 ⎛⎞Tpp1L1⎛⎞+ ⎜⎟= ⎜ ⎟ . (1) ⎝⎠Tp0L⎝⎠

3. Equation of state immediately after adiabatic expansion:

pVL = nRT0 (2)

4. Warm up to ambient temperature T1 , pressure increase from pL to pL+p2 :

()ppVnRTL +=21 (3)

Substituting Eq.(1) in Eqs. (2) and (3) leads to

γ γγ−1 ⎛⎞Tpppp1L2L1⎛⎞⎛⎞++ ⎜⎟==⎜ ⎟⎜ ⎟ (4) ⎝⎠Tp01⎝⎠⎝⎠ p L

such that γ is given by

⎛⎞p log 1+ 1 ⎝⎠⎜⎟p γ = L . (5) ⎛ pp12⎞⎛ ⎞ log⎜ 1+−⎟⎜ log 1 + ⎟ ⎝ ppLL⎠⎝ ⎠

In order to realize the adiabatic decompression of a compressed gas a large-volume gas bottle is equipped with a vale with a large cross section (Fig. 1). This valve should turn smoothly. Before starting the measurement flush the gas bottle thoroughly with the respective measurement gas. Before using the high pressure cylinders wait for an explanation and demonstration of the gas handling system by the supervisor. Perform the measurements at five different starting pres- sures. The pressure difference between the interior and exte- rior of the gas bottle is measured with a digital manometer and should not exceed 60 hPa. After each pressure change wait for the stabilization of the new equilibrium state; this might take several minutes. It is assumed that both the outer air pressure pL as well as the room temperature T1 remain constant. Check whether the calculation of γ using the ap- proximation γ =−ppp112/( ) is sufficiently accurate. Fig. 1 Setup after Clemént and Desormes.

Task 2: Determine the adiabatic exponent γ by measuring the sound velocity

Sound oscillations are so fast compared to thermal conduction processes that there is practically no temperature equalization between the local areas of heating (due to compression) and cooling (due to expansion) which are separated by half a wavelength. The setup for sound velocity measurements consists of the following equipment (Fig. 2): - Function generator with variable frequency - Ultrasound transducer as sound emitter - Ultrasound transducer as sound receiver - Oscilloscope for the observation of Lissajous figures and for the measurement of the time delay.

Fig. 2 Setup for the sound velocity measurements

The signal from the function generator is connected to channel 1 (CH1, X) of the oscilloscope and to the sound emitter. Channel 2 (CH2, Y) of the oscilloscope is connected to the sound receiver. A varia- tion of the distance d between emitter and receiver leads to a phase shift between the voltages at the X- and Y-inputs of the oscilloscope. This can be visualized either in the time or XY modes of the oscilloscope, the latter leading to Lissajous figures (see Fig. 3). A distance change by λ leads to a phase shift by 2π.

Fig. 3 Lissajous figures for various phase angles ϕ and amplitudes (a, b).

Perform 10 measurements each at two different frequencies (in the range 38-42 kHz). Evaluate the data either graphically or by calculation. In order to enhance the measurement accuracy the distance between emitter and receiver should be varied in steps of about 5 wave lengths. Additionally, the sound velocity should be determined by a run-time technique. To this end, at a fixed distance between emitter and receiver, the time between emission and arrival of a single pulse is measured in the time mode of the oscilloscope.

3 Task 3: Measurement of the adiabatic exponent γ with a resonance tube technique

A glass tube (1, Fig. 3) that can be sealed at both ends by valves, con- tains a cylindrical oscillation bob (2) made from ferromagnetic material. Outside the vertically mounted tube a field coil (3) is attached that can be slid along the tube and that is connected to a function generator (4) with variable frequency. The coil current should not exceed 1 A (effec- tive value)! The current value is controlled by an amperemeter. If the oscillation bob is located in the range of the alternating magnetic field and if the frequency is close to the resonance frequency of the mechan- ical oscillation of the bob in the gas column, the bob will start to oscil- late around its equilibrium position with moderate amplitude. In case of resonance the oscillation amplitude is maximal. The eigenfrequency of the oscillating bob is given by 11c γγpA2 pA f == =. (6) 22ππmVmlm The unknown “gas spring constant” is obtained from Eq. (6) as γ pA2 Fig. 3 Resonance tube. c = . V

At constant temperature and pressure the determination of an unknown adiabatic exponent γx is possible with a comparison gas with known adiabatic exponent γr ; in this case one obtains from Eq. (6) 2 fx γ xr= 2 γ . (7) fr In the laboratory a calibration gas not available; the resonance tube, however, was calibrated such that the adiabatic exponent can be calculated from f 2 γ =⋅297.1(Pa s2 ) 0 . (8) pL The latter equation is valid under the assumption that the oscillation bob is located at the center of the tube and that both tube ends are sealed. The instrument constant of 297.1 Pa s2 is valid under the experimental conditions stated above, i.e. the “gas spring-constant” is given by 2c, since below and above the oscillating bob the closed gas volumes contribute to the restoring force by their alternating compression and expansion. In Eq. (8) f0 denotes the resonance frequency (unit Hz) and pL (unit Pa) the outer air pressure after pressure equilibration in the glass tube.

Hints to the experimental realization: The gas under study is let into the bottom valve of the glass tube with both valves open. By this the oscillation bob is moved to the upper end of the glass tube. The analogous procedure is done from the upper valve moving the oscillation bob down. This is repeated until the tube is thoroughly flushed by the measurement gas. In the end the pendulum bob is pressed upwards to the center of the tube, the gas inlet pipes are removed and the valves are closed. The gas pressure in the tube is now equal to the outer air pressure. After the field coil has been attached slightly below the oscillation bob, the generator frequency is slowly increased starting from 20 Hz until the bob oscillates with maximum amplitude. In order to maintain a constant volume of the gas column during the oscillation, the oscillation bob that is slowly sinking has to be brought back to the center of the tube before the start of a new measurement. The adjustment of the resonance frequency should be repeated several times.

Additional task: Gas oscillator after Flammersfeld

A cylindrical body (Fig. 4, 1) of mass m and diameter d which seals a gas volume (3) in a vertically

4 mounted precision glass tube (2), will be moved upwards in the tube, when gas streams into the gas inlet pipe (4), since overpressure builds up in the volume below the body. If the body on ascending releases the overpressure through a valve (5) (fine slit), it will sink again such that this valve is sealed. In case of a continuous gas inlet, the body moves periodically. If the gas stream is adjusted in such a way that also the unavoidable gas flow between body and tube wall is compensated for, a forced oscillation is established, since the damped oscillation is phase synchronously excited. If the body (diameter d) sinks by the small distance x below the rest position of the oscillation, the pressure p is raised by Δp, and one obtains for the pressure force F (that accelerates the body) dx2 π Fm=== Ap∆∆ d2 p. (9) dt2 4

The pressure p in the spherical glass container is equal to the sum of the outer air pressure pL and the piston pressure of the oscillation body 4mg pp=+L . (10) π d2 Since the oscillation is rather fast, it is regarded as adiabatic such that pressure and volume are re- lated by the adiabatic equation pV= const −γ . (11) Taking the derivative with respect to the volume leads to dV dp=−γ p , (12) V i.e. in case of small pressure changes Δp one has ∆γ∆ppVV=− / . (13) Inserting Eq. (13) with ΔV = π d2 x/4 in Eq. (9) yields the differential equation for the harmonic oscillator dx224γπ dp +=x 0 (14) dt2 16mV with the eigenfrequency γπ24dp ω = . (15) 0 16mV ωπ= With 0 2/T the equation for the calculation of the adiabatic exponent is obtained as 64mV γ = . (16) Tdp24

2 After being instructed by the supervisor the period (frequency) of the oscillating bob should be measured 10 times using the photoelectric barrier and a digital counter. Calculate the mean value, the standard deviation and the 5 confidence range for a confidence level of 95%. Calculate the adiabatic exponent using Eq. (16) and estimate the maximum uncertainty. 1 Abb. 4 Gas oscillator after Flammersfeld

1 Cylinder (oscillation bob, mass m=(4.575±0.005) g, diameter d=(11,90±0,04) mm) 2 Precision tube 4 3 Glass container, system volume V=(1,138±0,002)·10-3 m3 4 Capillary for the gas inlet 5 Slit 3