MEASURING THE RATIO OF SPECIFIC HEATS OF A GAS Thermodynamic Experiments Involving Harmonic Motion JOSEPH A. SCHUFLE New Mexico Institute of Mining and Technology, Socorro, New Mexico
Om of the methods most frequently used in physical chemistry laboratory courses for the measurement of 7, the ratio of specific heats of a gas, is that of Clement and Desormes.' Because of several inherent difficulties in this method its over-all accuracy is often of the order of *5%, a factor somewhat discouraging to students. Equilibrium is reached slowly, so that leakage of the apparatus and room temperature fluctuations are two important causes of error. In the two methods described here the time for measurement is short, and the two causes for error mentioned above are minimized. Considering the number of times the harmonic oscillator and wave motion are encountered in modern theories of chemistry and physics, it would be helpful to the student of chemistry to have laboratory experiments in which these principles are used. The first experiment described here uses an equation of motion for a harmonic oscillator. The secoud experiment uses principles of wave motion. MODIFIED ASSMANN METHOD
This method is an old one, originated over 100 years ago by C. A ~ s m a n n . ~His apparatus consisted of a U-tube containing mercury which oscillated in the tube between two large volumes of gas connected to the ends of the U-tube. This has been modified and improved many times in the intervening years. One of the simplest improvements was suggested by E. Riichardta in 1928. For the mercury he substituted a steel ball, and for the U-tube, a vertical tube of uniform inside dia~neter.~ The apparatus used in our laboratory (Figure 1) was adapted from Riichardt, substituting a steel cylinder for the steel ball. Such a cylinder may be turned out easily on a lathe to fit the tubing closely, whereas fitting a steel ball to tubing size is a difficult operation. A 4-foot length of standard Pyrex tubing was used for the vertical tube. Special glass tubing ground to
' For example: S ~ I N B A C0. H ,F., AND C. V. KING, "Experiments in Physical Ch~mistry,"American Book Co., New York, 1950, p. 41. ASSMANN, C., (fJber E r w i m u n g und Erkiiltung von Gasen und dureh plijteliche Vohuniinderung," Annala d m Ph& Chemie, 85, 1-36 (1852). R~~CELAEDT, E., P h y i k . Z.,30,58-9 (1929). 'Such an apparatus is available commercially from E. Leybold's Nachfolger, in K d n , or their U. S. representative, J. Klinger, 82 160th Street, Jamaica, Nea York.
uniform bore can be purchased, but we have found that, with a little selection, pieces of sufficiently uniform bore can be taken from stock tubing. The cylinder is dropped from the top of the tube and oscillates up and down on the air spring provided by a large volume of enclosed gas attached to the bottom end of the glass tube. The fundsmental equation can b~ derived as follons, n-here: f
= --..forre
m = mass of metal cylinder a = sccelerstion =
q z
= croas sectional area of tube and cylinder = distance through ahich t,he cylinder moves = time "...." = volume of gas ~nclosedin system = frequency of oscillation = llu = period of oscillation
1
I' u
r
f
pressure of the enclosed gas = P.k,.
mg +P
P
r = C,/Cr, ratlo of specific heats of the gas Py = force upward on cylinder
=
Differentiating: df = p dP dt' =
* dz
For an adiabatic compression: PV7? = constant, (k) Rearranging and differentiating: d P / d V = k(-r)i-(-?-l)
Substituting (1)in (5)and rearranging: dP =
- yP di, i-
Substituting (3) in ( 6 , : d P = - rP (q dz)
Substituting (7) in (2!: df = Integration of
(8)
- %9
1-
dr
gives:
But since: d% f=ma=m--, dl
(9) can be rewritten as:
Defining: u = qPrP -, and z' = Vm
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(10) simplifies to:
Equation (11) can be seen to have the form of a second-order differential equation which defines z as a harmonic function of 1. A d n t i o n of such s. differential eqostion of motion is: 2 =
.4 sin ~ $ 1 ,
(12)
4;
m-here is the period of oarillation. But for such a harmonic oscillation, the period is also equal to 2 s times the frequency, v . Therefore:
The time for one oscillation,
r, is
given by:
Therefore:
The quantities V , m, and y are easily determined constants of the apparatus. P is evaluated for each experiment. The metal cylinder is dropped from the top of the tuhe and allowed to oscillate as many times as the apparatus \\ill permit. At least three or four oscillations should he obtained, and the time .r for a single oscillation is determined. Better results are usually obtained if the first oscillation is disregarded. The measurement of r is the biggest source of error. A variation of 0.01 sec. in r causes a variation of 0.03 in the value of y. Parodis describes methods for improving the measurement of r . Homver, results of sufficieut accuracy to show the significance of y and its variation from one gas t,o another can be obtained by students using a stopwat,ch. Typical determinations of y for air at 25'C., with the use of two different tubes aud cylinders, are given in Table 1. Gases other than air can he investigat,ed readily. The apparatus is evacuated and filled ~ ~ ithe th TABLE 1 Determination of y by Modified Assmann Apparatus P .
.4ir Argon CU-
...
."
4200 18.91 1,733 4200 18.50 1.784 4200 18.50 1.784 4200 18.42 1 784
64.5 64.8 61.5 64 5
.
.
0.926 0902 0.817 0.920
10 20 6
R
1.42 1.38 1.68 1 32
i
1. Modified Asamann Apparatus
velocity with which sound travels through a gas can be expressed by the equation:
n here:
,I
P d
velocity of soirnd pressure of the gas = density oi the gas = C,,/L', for the gas = =
Methods using the usual I(uudt tuhe for measuriug the velocity of sound in gases halve been described in laboratory manuals such as that of Mack and F r a n ~ e . ~ We h a w tried a modification of the Kundt tube, using a 25-watt loudspeaker unit to generate t,he sound at known frequencies instead of the usual brass rod. An auuaratus using: a similar loudsueaker source MACK,E., AND \V. G. F ~ A N C "Laboratory F, Manual of Elementary Phydral Chemistry," 2nd ed., D Van Nostt-and Co., Inc., Nea. York, 1934.
gas to be tested a t a pressure slightly above atmospheric pressure. The top end of the glass tubing is closed aith a rubber stopper during the filling operation. The stopper is removed just before the run, an instant allowed for the pressure to drop to atmospheric pressure, and the metal cylinder inserted and dropped as usual. Typical data a t 25°C. for carbon dioxide and argon are given in Table 1. MODIFIED KUNDT METHOD
The second method for measuring y also will help the chemistry student to visualize wave motion. The PARODI, M., Compl. R e d . , 218, 311-13 (1944).
VOLUME 34, NO. 2, FEBRUARY, 1957
Fagure 2.
Us* of Loudspeaker w i t h K u n l t Tube
79
of excitation was described by Reynolds.' The apparatus, shown in Figure 2, consists of a Lucite tube about 1 m. long and about 5 cm. in diameter, closed a t both ends with a metal seal. The metal caps are provided with inlet and outlet tubes for introducing gases other than air. A movable piston is provided as usual a t one end. This can be adjusted so that the distance from the sound generator to the piston is an even number of half wave lengths. The loudspeaker unit is attached to the other end of the tube and is connected through an amplifier to an audio oscillator which can be set a t known frequencies. The frequency scale on the oscillator can be calibrated with tuning forks. A little lycopodium powder is sprinkled through the tube. The speaker is set a t a known frequency and the piston moved until the powder piles into little heaps a t the vibration nodes. The average distance between nodes gives the half wave length. Velocity is given by frequency times wave length. Data obtained a t 2S0C. are given in Table 2.
' REYNOLDS, M. B., J. CHEWEDUC., 20,121 (1943).
TABLE 2 Determination of r by Kundt's Method Y =
Frequency
uad
average u
-
33,820 34,3i0 36,500
34,563
1.39
58.0 18.80 14.24 9.62
2i,500 26,800 26,970 27,300
27,143
1.31
33.3 22.20 1680 5.53
32,230 32,180 32,420 32,000
32,208
1.69
A
11
(em.)
(cm./sec.)
23.70 18.12 12.48
P
Air ( y = 1.40): 1424 1896 2844 CO? ( y = 1.30): 4i4 1424 1896 2844 Argon (? = 1.6i): 966 1450 1930 2895
The apparatns is very useful for demonstrations of standing waves. I t is more interesting in this respect than the usual Kundt tube, because the effect of varying frequency on the u-ave length can be demonstrated more easily.
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