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JAMESO.SCHRECK University of Northern Colorado Greeley, CO 80639
A Charles' Law Experiment for Beginning Students Mark G. Rockley and Natalie L. Rockley Oklahoma State University, Stillwater, OK 74078 Discussion and illustration of the gas laws is a staple of high school and first-year college chemistly courses. Many laboratory experiments are designed to give beginning chemistry students first-hand experience with measurement of these laws. Charles' Law is a t first glance a particularly simple law to demonstrate conclusively in a beginning chemistry laboratory. Indeed, a t this institution, experiments to measure absolute zero by using Charles'Law have been used many times in first-year labs--but with very modest success. Yet, it is imperative that beginning students be given the opportunity to measure accurately physical properties, especially with modest equipment. The typical Charles' Law experiment involves placing a plug of mercury into a small bore glass tube sealed a t one end so a s to trap a measurahle volume of air, inserting the apparatus into a beaker of water and heating the water. The difficulty with accurate measurements results from the small change in total volume of the gas over readily accessible temperatures (25 "C to 100 "C). Even more problematic, however, is the use of mercury, that tends to slip inside the glass tube and that also constitutes a n unnecessary laboratory and environmental hazard. This paper lays out a procedure and device for measurement of absolute zero using Charles' Law that does NOT use mercury. The procedure may be completed in under 20 min, allowing for replication in 50-min laboratories. The procedure enables very accurate determination of absolute zero (to within -1%). The apparatus is simple, does not use glass capillary, can be assembled easily by beginning chemistry students and costs about $0.50 per apparatus, assuming a normally equipped laboratory (beakers, thermometers, balances etc.). Furthermore, the quality of the data is sufficiently good and may he used to introduce students to the details a n d consequences of precision, accuracy, error propagation, and error analysis. Experimental The design of the apparatus i s not a s simple as i t might appear because many criteria have to be met. Keeping in mind that the basic experiment is to show that the volume of gas increases linearly with temperature, several factors must be considered. First, the apparatus must enable the student to relate easily laboratory observations to the written statement of the gas law. Obscure measurements leading to the gas law by inference are then ruled out. For example, the volume change should not need to be corrected for variations in hydrostatic head. This rules out using the displacement of a water column as a means for measuring the volume change (although the vapor pressure of water also precludes such devices). The measurement must lead to high accuracy, even though the skills of the experimenter are modest. This requires that the volume change must be measurable with a simple device yet with good accuracy over the temperature range examined. In par-
ticular, a polycarbonate millimeter ruler is readily available and R serves a s a useful tool to measu r e t h e volume change. Fine bore g l a s s t u b i n g should be G avoided because of the potential for breakage. The a p p a r a t u s T must be sensitive so students do not have to heat the apparatus S over a very large temperature range to obtain reliable data. V The apparatus must he efficient so a given experiment can be D completed in a very short time (say less than 20 min) because high school laboratory classes Figure 1 . The assembled are often less than 1h. If i t is ef- apparatus. ficient, t h e n more laboratory time will allow students to replicate the experiment to obtain a first glance a t experimental uncertainty. Finally, i n a n era of declining resources, the entire apparatus must be extremely inexpensive and readily available, requiring little assembly. The principle of the apparatus described here is simple. An illustration of the assembled apparatus is given in Figure 1.A 0.5-dram vial serves as the primary reservoir volume. It is connected to a small bore diameter Teflon tube containing a small plug of glycerol. The glycerol plug does not wet the tube and moves up and down, a s the gas volume in the reservoir and the tubing below the plug, increases and decreases. The volume per millimeter of the Teflon tubing is measured. In this way, a s the temperature of the gas in the reservoir is raised by placing the apparatus in a beaker of warming water, the associated volume change may be measured directly. A 0.5-dram vial (S-63246-00-A from SargentiWelch Scientific) is used as the primary reservoir volume. The volume is typically about 2.40 cc. Afew drops ofblue fountain pen ink are added to a sample of pure glycerol, and 0.01 cc of this blue-colored glycerol solution are inserted using a disposable pipet or equivalent into the end of a 150-mm section of 118-in. 0.d. x 1116-in. i.d. Teflon tubing (available from Cole Parmer Instrument Company, 7425-~orthOak Park Av,. Niles. Illinois. 60714). The internal volume of this Teflon &bing is such that a 20 'C temperature rise for the gas in the vial will cause a volume change equivalent to about 80 mm of the Teflon tubing. A 5-mm section of 0.d. 0.192 in.. 0.063 in. i.d. silicone tubing - (available from Cole-l'ennw Instrument C'onipan!: catalog no.6111-43 ii ;ittachcd to the end uV the 'li4lon tuhing into which the small plug of cl~iondgivccml has just been inserted. A few chips of' fresh, weighcd drierlte tC'aSO, are added to the vi,il. 'The silicone t u b ~ n gsection that surrr>undsthe 'fi!fllm ~~~
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tubing section is now inserted into the mouth of the vial. The silicone tubing acts as a convenient flexible airtight seal between the tubing and the vial. The entire length of the Teflon tubing is attached to a polycarbonate plastic millimeter ruler or a steel ruler. To effect this, a set of 2-mm holes may be drilled into the ruler at 3-cm spacings right next to the millimeter markings on the ruler. Copper wire (22 gauge) can be used to fasten the Teflon tubing to the ruler. The entire apparatus is then inserted into a l-L beaker of water containing 850 mL of water. Ice may be added to the water to bring the marker plug (colored glycerol) to the appropriate position on the ruler prior to beginning the experimental measurements. With constant stirring, the water is heated gradually over a range of about 20 "C. The temperature is recorded as is the position on the millimeter ruler of the bottom of the marker plug. In this way, the volume change of the vial plus the small extra volume associated with the connecting Teflon tubing up to the base of the glycerol plug is recorded as the temperature rises. Several technique considerations have considerable impact on the experiment's accuracy The drierite chips were heated moderately and cooled to room temperature prior to use. Although the volume of these chips must be subtracted from the combined volume of the vial and the connected tubing up to the starting position of the base of the marker plug, it is a small volume correction. However, the addition of drierite chips is essential. The glycerol marker plug will inevitably have some absorbed water. As the temperature rises, this water leaves the glycerol and contributes to the gas pressure in the enclosed volume. Then the observed volume increase per temperature increment appears to be greater than it should be. Drierite absorbs this excess water vapor. Despite the temptation to use the llexlble siliconetubing for the entire aooaratus. this is a bad idea. The silicone tubing outgasses t i 'the extent of a few Torr over 30 "C change in temperature. This causes anunexpected increase in gas pressure in the enclosed volume. As with the water vapor, this pressure increase is cornpensated for by an additional increase in the volume of the enclosed gas. Ultimately, these outgassing phenomena result in an erroneously high measured value for the absolute zero temperature. Consequently, the flexible silicone tubing must be used only as a seal between the vial and the Teflon tubing. Finally, the Teflon tubing must be preheated because there is a very slight shrinkage (about 3%) when the Teflon is cycled the first time through a temperature range of 30 "C to 100 "C. It is recommended that the Teflon tubing be boiled in water for about 15 min prior to first use. This one-time requirement need not be repeated prior to subsequent experiments with the apparatus. Finally, when the silicone tubing seal is inserted into the vial, the vial should be held w i t h a paper towel. This is required because the warmth of the hand holding the vial will cause the gas to expand (the purpose of the experiment) to the where the glycerol plug may he forced out the top end of the Teflon tubing. The volume of the vial is determined gravimetricaJly by filling the vial with water to the point where the silicone tubing seal ends. This determinationshould be done once the experiment is fmished because it will be difficult to completely dry the vial duringthe short time available prior to beginning the emeriment. Similarly, the Teflon tubing should be filled with water for gravimetric determination ofthe volume inside the tubing per millimeter of length. Nominally, a 150-mm length of tubing will contain about 0.3 gofwater. Therefore, to measure the volume/miIlimeter of Teflon tubing to 1%accuracy, a balance accurate to 3 mg is required. Such an a m a t e balance is not always available, in which case a longer length of 180
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Temperature (Centigrade) Figure 2. Data from a typical experiment Tdon tubing may be filled with water prior to cutting into segments. This will increase the weight change measured so that the accuracy limit of the balance does not dominate the experimental errors. Results The measured results from any experimental set always include errors. These errors need to be analyzed so the expected accuracy of the experiment may be-inferred. Furthermore, a judicious analysis of causes of error in the experiment enables the experimenter to improve upon the design. While the following analysis of results of a typical experiment involving the apparatus described here does include error analysis in some detail, it is worth noting that the experiment is independent of this analysis. The reader who does not have the time or (as, for example, a beginning student) the necessary background, is encouraged to view the results section here without paying heed to the error propagation, error sources and confidence limit discussions. The volume of the Teflon tubing after heat treating in boiling water was 1.994x lo3 f 9 x lo4 cclmm (95% c o d dence limits on all errors). This is referred to later as the linear volume. The volume of the vial plus the Teflon tubing below the glycerol plug minus the volume of the drierite (0.0333 cc for 0.0986 g at a density of 2.960 g/cc) was 2.350f 0.007 cc at 29.0°C. The least squares analysis of the data was used to back calculate the precise volume best representing the least squares fit a t 29.0°C. It is that volume which is 2.350 cc. As a result, the 95% confidence errors in the least squares values for the slope and intercept contribute 0.0043 cc to the error in the volume, in addition to the errors due to the weighing of the vial with water in it (0.002 cc) and the measurement of the length of Teflon tubing below the glycerol plug a t 29.0°C (0.001 cc). I t is the combination of all those errors that contribute to the 0.007 cc error estimate given above. The data from a typical experiment is shown in Figure 2. The temoerature raneed from 29.0 "C to 49.0 "C. The vial. tubing and millimeter ruler were immersed in water in a l-L Pyrex beaker and heated and stirred during the experiment. The glycerol plug moved 3.92 mm/"C with a 95% confidence error of 0.04 mmPC for 21 measurements. The slope was determined using a linear least squares proce-
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dure, assuming that there was random error only in the dependent variable, namely the volume. In practice, the student grade thermometer used was marked in increments of 1 "C. The linear least squares procedure presumes negligible error in the independent variable. Although each temperature was read to within 0.1 "C, the 95% confidence limit on tem~eraturemeasurement is estimated to be f0.3 "C. The pos&on of the glycerol plug was measured to the nearest millimeter z i d a 95% confidence limit estimated to be f0.5 mm. be pe; cent error in the temperature is obviously greater than the per cent error in the volume measurement. Thus, there is expected to be random error in both the dependent and independent variables. By applying the least souares ~rocedure.all the error is collected into the standard error of the slope. ~ a l ~ eand r n &eves ( 1 )show that this error in the s l o ~ can e be used to estimate the 95% confidence error in the dependent variable, ifthe errors in the dependent variable are assumed to be the same throughout the series of measurements. Using the expressions of Halpern and Reeves. the estimated error in each volume measurement correspbndsto a 95% confidence level of 1.2 mm for each point across a ranee of 80 mm. This is clearlv too large to be due solely to e m > in the ordinate. It reflects instead 95% confidence errors of 0.5 mm per point across an 80-mm range for the volume and 0.3 "C per point across a 20 'C range for the tem~erature.(1.2 mm per 80 mm range is equivalent to the square root ofthesumaofthr squares of the &timated errors divided by thc ranges for both the temperatun! a i d volume. The cadence e&r from the least squares analysis therefore confirms that while the temperature was recorded to 0.1 "C, the 95% confidence error was in fact only 0.3 "C, as one might expect. Beginning students are normally taught to read a value on a ruled device to a precision of one half the finest marked increment. For the eGeriment described here, that would correspondto a precision of 0.5 "C. Asmall amount of extra precision for each measurement in experiments reported here was obtained by observing (with attention paid to avoiding parallax error) when the expanding mercury column of the thermometer was first visible above a particular rempmiture mark. At that instant, the u!mperature and the height of the ~lvcerold u e were recorded. Tlus techniaue has theadded a2;antaie of illustrating to students how to achieve somewhat better precision with relatively modest equipment. The use of a multivariate least squares approach, while indicated here, is beyond the scope-of students doing this experiment. Figure 2 shows observed data for a typical run, least squares fitted line, and with right-side ordinate, a plot of residuals. While a correlation coefficient of 0.99974 indicates a fairly reliable fit to a straight line, a linear fit to the data is indicated only if the residuals are ao~roximatelvrandom across the ranee of data. For data given in Figure 2, the errors range fairly randomly as increments of the volume eauivalent of 1mm of tubine. to be expected because the readings were made only to thLnearest one mm. Each 1"C temperature increment took only about 20 s to occur, with the result that there was a slight temperature lag between the gas in the vial and the water surrounding the vial. This was inferred by measuring the volume versus temperature a s the water bath was cooled. The cooling curve also was linear but displaced to lower temperatures from the curve shown in Figure 2. That is, the gas apeared to occuov a meater volume when the water bath was cooling than when it was heating. The horizontal disla cement is assumed to be eaual to twice the temperature iag between the gas in the vial and the surrounding water bath. This t e m ~ e r a t u r elag contributes a systematic error that must be kbtracted From the final r&ult. The temperature lag was found to be 0.4 "C. The student grade ~
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thermometer also was found to be inaccurate, reading the temperature to be too high by 0.9 'C. This systematic error also must be subtracted from the final result. The change in temperature, AT, required for the volume to decrease to zero is the desired measurement. This change is given by:
where:
TI= a nominal starting temperature in 'C. (29.0°C for this example); S =slope of the least-squares-fittedline through the data; V = startingvolume of vial and tubing below the plug at TI; L = linear volume or volume per millimeter of the Teflon tubing.
Using the 95% confidence limits for all errors, E, and propagating the errors according to standard procedures delineated elsewhere (2):
The absolute zero temperature determined from the above data is:
After correct in^ this for the calibration error of the student grade thermometer (-0 9 ' C , and the temperature lag r ~ O . 1- C I ,thr rxprr~mrntallvdrtwnnnrd 95'i confidence limits) absolute zero temperature is: T = -272.9"C
+ 3.5T
The tvuical precision of these measurements from one experiment to t h e next gives a sample standard deviation of 1.2 OC as a n achievable goal within a 50-min laboratory period. Conclusion In eq 1,90% of the error arises from the standard error of the slope of the least squares fit of the data. Given that there are sources of error from the determination of the linear volume of the Teflon tubing and from the determination of the volume of the vial itself. this is sumrisine. Sime (3)shows that a n experiment in Ghich one source i f error dominates is a poorly designed experiment. That is the case here. The measurement error can indeed be reduced considerably by reading the position of the glycerol marker plug more precisely (to better than the nearest millimeter) or by reading the temperature more precisely (with a thermometer marked in 0.1 "C increments, for example). However, keeping in mind the audience for which this experiment is desimed and the aDDaratus available. the fact that the dominant source of error results from use of inexpensive thermometers is acceptable. This is noteworthy given that the value of absolute zero is determined to be -272.9 "C ? 3.5 "C (95% confidence limits) in a simde. ouick ex= periment!
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Literature Cited 1. Haipern, A. M.; Reeves. J. H. E x p r i m n t d Physrcol Chemistry; Scott, Foresman and Co.: Glenville. IL, 1988: p 16. 2. Bevington. P. R. Data Redudion and Error A n a l p i s for the Physicni Sciences; McGraw-Hill: NewYork,NY, 1969. p 99. 3.Sime.R. J . P h y ~ i mChm~latry:Mdhods, l T d n i q u e s , ondErpprrrnane; Holt.Rinehact and WInston,lnc.: Chicago. IL, 1990, p 149.
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