~uchardt'sMethod for Measuring the Ratio of Heat Capacities of eases A Laboratory Experiment in Physical Chemistry S. Walter Orchard and Leslie Glasser University of the Witwatersrand, WITS 2050, South Africa
The heat capacities of a gas at constant pressure (Cp)and at constant volume (Cv) are quantities of fundamental importance, as is their ratio Cp/Cv= 7. In the case of an ideal gas, Cp = Cv R, where R is the gas constant, so that the value of y uniquely defines Cp and CV.lMoore has recently referred to various methods for measuring heat capacity ratios and has reported an improved version of a commonly used laboratory experiment based on an adiabatic expansion process (1). Moore demonstrates that his method for measuring y is accurate to within 3% for nitrogen, carbon dioxide, and argon, but for helium the experimental value is always about 10%too low. We have developed a student experiment in which y is measured using Riichardt's method (2).This experiment is attractive for a number of reasons: the results are good, and for helium are more accurate and reproducible than those by Moore's method; the experiment is visually appealing and popular with students; there is a meaningful, yet simple, interfacing of experimental equipment to a microcomputer; students gain hands-on experience in nonlinear curve-fitting; and the experiment uses less gas than does Moore's method. The procedure is based on a demonstration experiment described by Jolls and Prausnitz (3) but differs significantly in that we have made use of a microcomputer for data acquisition and treatment. Consider the closed apparatus containing the gas under study at pressure po,shown in Figure 1. The upper bulb of volume Vu is connected to the lower bulb of volume Vl by a vertical length of precision-bore tubing. A steel ball of mass m and cross-sectional area A is just small enough to move in the tube without touching the sides. Since the fit of the ball in the tube is not gas-tight, it will tend to settle due to gravity as gas leaks up past it. However, if the position of the ball is briefly disturbed (e.g., by a magnet), it will commence to oscillate about the mean settling path. Subject to certain simplifying assumptions (notably that the expansions and contractions of the gas are small, reversible, and adiabatic), it is possible to derive an equation to describe the damped oscillations of the ball (see Appendix for the derivation). As the ball oscillates, so do the pressures in the upper and lower bulbs, p u and pi, respectively. The pressure difference between the bulbs, (pu - pi), is proportional to the displacement of the ball from its mean position,
+
The IUPAC recommended value of R has recently been reworked (6) as R = 8.314510 J molU1KV1.Interestingly, this value is derived from the heat capacity ratio itself, using the defining relation
where Mis the molar mass of argon, cothe velocity of sound in argon, and T is the absolute temperature. The new definition of R further demonstrates the importance of 7.
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Journal of Chemical Education
Steel ball, mass m, cross- sectional area A
1-
1
bore
Figure 1. Ruchardt'smethod: underlying principles.
and this quantity is thus also periodic. I t can readily be shown that the time dependence of p u - pl = B is given by B = Byexp (-^t/2) sin ( w t
+ 6) + C
(1)
Here Bo is an amplitude; Q is a friction factor necessary to account for the observed damping of the oscillations; t is time; w, the angular frequency, is related to the period of the oscillation, T, by w = I&; 6 is a phase angle; and C is a constant representing the time-averaged value of B. The angular frequency u is a property of the gas in the apparatus and is related to y by eq 2:
In typical experiments w2 is so much greater than p2/4that an error of only about 0.1% in 7 is introduced by neglecting the latter. Thus it is necessary to characterize the oscillation behavior only in terms of w (or T)to determine y; our procedure in fact determines all five constants in eq 1,viz., Bo,Q, u, 6, and C, and so enables w to be determined to a high precision. Apparatus and Procedure The apparatus used in our implementation of Ruchardt7smethod is shown in Figure 2. In this version, the "upper bulb" is the atmosphere, so that Vu = and po = atmospheric pressure. The ball settles onto a mild steel spring, so as to avoid cracking the glass when the ball is accidentally (but frequently!) dropped down the precision-bore tubing. To obtain a good fit between ball and tube, a
II A
precision bore i. d.)
tube
Spring
-1 1 1
c
11 w -
Electrical connection to ADALAB interface Pressure transducer
Figure 3. A part of the experimentaldata for a C02 "run" (Â¥fitted to eq 1. The experimental points are taken at 0.1-s intervals. (The constants used in fitting the full data set of this run to eq 1 are Bo = 68.6 (arbitrary units), 8 = 0.24 s l , w = 5.805 radiansls, 6 = 2,448 radians, C = 1.45 (same units as Bo).)
5 liter bulb
Gas inlet
eat Capacity Ratios at 295 K Class wool filter
Gas
fils-'
He Ar N2 air C02
0.64 0.43 0.36 0.34 0.26
This Worka pnlkPa
Figure 2. Ruchardt's method: approaches used in the present experiment.
slightly oversized steel ball bearing can be etched to size in 50% nitric acid. By repeating a cycle consisting of a brief immersion in the acid, rinsing, drying, and testing the fit, a satisfactory fit is readily obtained. In our apparatus, a ball of mass 5.3494 g was obtained, which settled in the 11mm-i.d. tube at a rate of about 2 cmls. To minimize damping effects, there should be no major constrictions in the path from the bulb through the precision-bore tubing and the J tube to the atmosphere. The various sections connect with greased ground-glass joints, held in place by springs. Both the precision-bore tubing and the ball should be kept clean and grease-free to ensure unrestricted movement of the ball. The J tube serves as a buffer zone to reduce mixing between the gas in the apparatus and the air outside. The pressure transducer (Honeywell Micro Switch 143PCOlD) senses the pressure differential p,, - pl, when stopcock C is closed. The transducer is powered from the 12-V supply on the motherboard of an Apple IIe microcomputer, and its output is sampled at 0.1-s intervals through the AID converter on an Adalab (Interactive Microware, Inc.) interface card in the Apple. To eliminate the dc component of the transducer signal, a 10-nF coupling capacitor was connected in series with the input line to the A/ D converter. The signal sampling is controlled by software described below. With stopcocks A and C open, the apparatus is flushed with the gas to be studied until a volume of at least 25 L (five times the volume of the apparatus) has been displaced. During this process the outlet at the J tube may be connected to a flowmeter. The gas supply is then shut off, the flowmeter is disconnected, and A is closed. The ball is lifted to the top of the precision tubing using a strong magnet, and the stopcock C is carefully closed. The ball is then released, and data collection begins as the ball oscillates in the tube. Useful data can be collected for about 10-15 s, after which time the oscillations have been fully damped out or the ball has come to rest. The entire experiment can be done in less than 10 min for a single gas, so it is possible to study several gases in a fairly short time. Additional time should be allocated to the curve-fitting procedures and other calculations below.
Software
Data collection is under control of an Applesoft Basic program, R DATACOLL, which utilizes the QUICK110 routine supplied with the Adalab interface. Values of (pu - pi) may be collected for up to 20 s at 0.1-s intervals. The data are scaled and filed on a floppy disk for subsequent processing and curve-fitting. R HANDFIT is a program that utilizes data from a data file created by R DATACOLL. I t utilizes the high-resolution
83.31 83.31 82.99 82.99 83.31
Published 7 Values Moore ( 1) Literature (5)
-Y"
1.569 k 0.002 1.626 k 0.005 1.391 & 0.002 1.390 k 0.003 1.285 A 0.004
1.50 A 0.05 1.63 k 0.03 1.39 A 0.01
1.67 1.67 1.40
1.30 k 0.02
1.29
-
-
m2; = kg; A = 9.354 X a Using apparatus of Figure 2, where rn = 5.349 X 5.182 X m3. Values quoted for y are the mean and standard deviation of two values obtained from separate "runs".
*
graphics of the Apple to display the data points, plotted as values of B (i.e., pu - pi) versus time. The student enters trial values of Bo,6, u, 0, and C, based upon interactive measurements on the graphic display of the data, in an attempt to fit eq 1to the data points. The curve of B vs. t corresponding to the trial values is then calculated by the program and superimposed on the data points. New trial values may be selected in order to improve the fit. It is usually possible, after a little familiarization, to obtain an excellent fit within about five trials. A third program, R AUTOFIT, permits further refinement of the fit of eq 1 to the experimental data. This program takes the experimental data file and the best trial values of the five constants as input and, utilizing the Newton-Raphson method ( 4 ) , calculates the values of the five constants that best fit the data according to the leastsquares criterion. Results and Discussion
Figure 3 is based on a print-out of the graphics display of R HANDFIT. For clarity in showing the excellent fit of the experimental data to eq 1, only a limited range of the data obtained in the "run" is shown. The values of u and 8 obtained in fitting eq 1 to the experimental data can then be used to calculate 7 via eq 2. Some typical results obtained in this way are presented in the table. For comparison we have included the experimental results reported by Moore ( I ) , as well as other literature values. For most gases this version of Ruchardt's method gives results of high precision and of similar accuracy to those of Moore (1).For helium, Ruchardt's method appears to give rather better results. The good precision is achieved despite the rather small differences in oscillation frequency for the different gases; the success of the method depends on the Volume 65
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September 1988
825
very accurate determination of w, which is possible through our curve-fitting methods. The friction parameter ,6' is greatest for helium, for which gas the error in 7 is also greatest. As Moore (1)has pointed out, the asumption of adiabaticity during expansions is least valid for a monatomic gas of high thermal conductivity, such as helium. Argon is the only other gas that we studied for which the error in 7 is substantial, and here, too, the value of ,f3 is high. It appears then that the errors in this method are largely attributable to nonadiabatic behavior that is reflected in an unusually large value of 6. Examination of the table indicates that, for our apparatus, good values of 7 are obtained when 8 is less than 0.4 sV1. Our students are required to calculate theoretical values for y (and, hence, for Cv) using the molecular vibrational frequencies. They generally find the good agreement between experiment and theory to be very satisfying. The sight of the ball bouncing on a cushion of gas is fascinating too, and the mathematical treatment of the phenomenon as well as the curve-fitting are interesting. While stopwatch timing of a number of cycles of the oscillation is a possible alternative method for obtaining w, the results obtained are inferior. We have also made use of a two-bulb system that can be set up by replacing the J tube in Figure 2 with a second bulb of similar volume to the lower bulb. I t is then possible to use different pressures, close to and below atmospheric. However, a more complicated evacuation and filling procedure was involved, leading to occasional student mistakes and breakage of the glassware. There are no particular hazards associated with this experiment, aside from those due to the possible toxicities of the gases that might be used. For students who are unfamiliar with the use of compressed gases and this type of glassware, it is prudent to provide careful instruction a t the beginning of the laboratory session. Thereafter our third-year students manage well, with an occasional need for help with running the curve-fitting programs. Students are required to do the experiment on helium and on carbon dioxide. Following the curve-fitting procedures, they then calculate y and hence obtain Cvfor each gas. The experimental value of Cv is then compared with a value calculated as the sum of the translational, rotational, and vibrational contributions. The students' experimental results for helium and carbon dioxide are generally indistinguishable from the corresponding results reported in the table. Our procedure involves fairly expensive equipment, and the relationship between 7 and experimental variables is not particularly simple. However, the method has enough attractive features to merit its serious consideration as an alternative to the better known one described by Moore (1). For a nominal fee we can supply a copy of the student instructions and a disk containing the three programs and a typical data file. Please send a check for $15.00 payable to "Chemistry Department, University of the Witwatersrand" and address correspondence to S. W. Orchard. Acknowledgment
We thank K. R. Jolls and J. M. Prausnitz for permission to use and cite their unpublished work, on which this experiment is based.
,
Appendix: Derivation of Equations 1 and 2 (See Ref 3) Consider the apparatus shown in Figure 1.The steel ball of mass m and cross-sectional area A moves as a piston in the tube (apart from an inevitable slight leakage of gas) and its movement thus alters the pressures pu and p l . Suppose that the magnet exerts a force on the ball that counteracts its weight and holds it in position while the pressures pu and p l are equal at the value of p o . If the magnet is now removed, there will be a net force on the ball, and it will accelerate downward. At any instant, if friction is not considered, the net upward force F is given by
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Journal of Chemical Education
where p l and pu are the instantaneous pressures in the lower and upper bulbs, respectively, and g is the gravitational acceleration. If we assume that the movement of the ball results in the gas following a reversible adiabatic path (frictionless motion, negligible heat transfer) we can make use of the relation pV7 = constant
(4)
If the ball is displaced a distance y from the point at which pu= p i = PO, eq 4 leads to the results
and
Here Vuoand Vloare the upper and lower volumes (bulb plus part of tube) when y = 0. From eqs 5a and 5b we get
Assuming that the volume Ay displaced by the oscillation of the ball is small in relation to the volumes Vuoand Vlo, the bracketed terms in eq 6 can be simplified using the series approximation
Equation 3 can then be written as
We now introduce a term to account for the force of friction, which is proportional to the velocity and acts in the opposite direction to the motion of the ball. Equation 8 is then replaced by
where 13 is a positive constant with units of reciprocal time. Since F = m d2y/dt2, eq 9 can be rewritten as
The complete solution to this equation is
where
+
+
and a1 and a2 are constants. Now a1 sin a t 0 9 cos a t = as sin (at O ) , where as and 6' are new constants. Also, the displacement y is proportional to p i - pu (compare eqs 3 and 8), the differential pressure sensed by the transducer. Thus the transducer signal is a sinusoidal function of angular frequency a and can be represented by eq 1. Equation 12 can be rewritten as eq 2, to give an expression for y.
Literature Cited 1. Moore, W. M. J. Chem. Educ. 1984,61,119. 2. Ruchardt, E. Physik. Zeitschr. 1929,30,58. 3. Jolls, K. R.; Prausnitz, J. M. "Laboratory Demonstrations for Teaching Chemical Thermodynamics",preprint of paper for presentation at the 1983 Annual Meeting of the American Institute of Chemical Engineers, Washington, DC. The derivation of eqs 1 and 2, as presented in the Appendix, and the apparatus shown in Figure 2 are adapted with only minor modifications from this reference. 4. Brink, G.; Glasser, L.; Hasty, R. A.; Wade, P. J. Chem. Educ. 1983,60,564. 5. Zemansky, M. W. Heat and Thermodynamics, 4th ed.; McGraw-Hill: New York, 1957; p 130. This book quotes accurate experimental values that are within 1%of values calculated from molecular properties. 6. Fundamental Physical Constants, Report of the CODATA Task Group on Fundamental Constants, CODATA Bulletin No. 63,1986.
