An Experiment to Measure Gas Imperfection W. P. Baskett University of Oxford Physical Chemistry Laboratory. South Parks Road, Oxford OX1 3QZ,England G. P. Matthewsl Chemistry Division, Plymouth Polytechnic, Drake Circus, Plymouth, Devon PL4 8AA, England Gas imperfection is a subject of considerable importance in physical chemistry. However, i t is a quantity that is difficult to measure, so experiments on the subject rarely feature in experimental courses for students. ~ e r e ~ w present e a method of measuring gas imperfection that has heen recently inrroduced in our laboratow and that consistentlv oroduces results of acceptable accuracy. I t involves the mekkrement of the deviations of carbon dioxide from perfect gas behavior by means of p V measurements a t pressures of up to 20 atm a t room temperature ( I ) . Experimental Methods The experimental methods available for the measurement of gas imperfection all rely on accurate measurements of pressure, volume, and temperature and may be divided into three main categories: (1) absolute methods (21, (2) relative methods (3-5), and (3) the Burnett expansion and related methods (6). Of these, the Burnett expansion method has proved to be one of the most successful and reliable. As a student experiment, the method has a number of further advantages, in that it avoids the need for (1) accurate volume determinations, (2) direct mass or normal volume measurements, (3) precise temperature measurement, (4) the construction of glass gas burets, and (5) the use of mercury as a confining fluid. Theory The Virial Equation of State At low densities, the behavior of most gases approximates closely to the perfect gas law At higher densities, however, the nonzero size of the molecules in a real gas, and the fact there are long-range attractive and short-ranze rewulsive forces between the molecules, causes appreciable deviations from perfect behavior. One common method of expressing the nonperfect behavior of a real gas is to use the uirial equation of state pVInRT = I + B nIV+ C n 2 / V 2+. . .
(2)
where the coefficients B and C are called the second and third uirial coefficients. In the present experiment it is more convenient to use an expansion in terms of pressure rather than reciprocal molar volume pVlnRT = 1
where B' = BIRT and C' = (C - B2)l(RT)2.The coefficient C' and higher coefficients are too small to be accurately measured a t the pressures used in the experiment, and we therefore ignore them. The compression factor Z of a gas is pVlnRT, and thus from eqn. (3), Z = pVInRT = I
+ B'p
(4)
The second virial coefficient B may be determined by plotting the compression factor Z = pVlnRT against p, from which a straight line is obtained of slope B' = BIRT (eqn. (4)).
The Modified Burnen Expansion Method (2) The apparatus for the Burnett expansion method comprises a sample vessel connected to an expansion vessel (or several expansion vessels), as shown in Figure 1. The sample vessel is filled with the test gas at high pressure, and the expansion vessel is evacuated. The tap between the sample vessel and expansion vessel is then opened so that the pressures equalize. out and the nrocedure Then the exwansion vessel is numned . repeated seGeral times. The second virial coefficiknt is calculated from measurements of the Dressure of the "eas at each stage. In this experiment we use a modification of Bumett's method in that the expansion vessel is larger than the sample vessel, Figure 2, and the expansion is carried out in many stages such that the pressure in the expansion vessel isuever allowed to be above 1 atm. These alterations allow more measurements to be made tor a given starting prthssure. The basis of the present method is to measure the amount of gas contained inthe sample vessel of volume VBat various values of the pressurep. The number of moles of gas, n , cannot be measured directly, and so we assume that a t pressure up to atmospheric deviations from perfect behavior are negligible and that we may therefore apply the perfect gas law to each of the small quantities of gas which are allowed to escape from the sample vessel into the expansion vessel. Suppose that there are initially no moles of gas in the sample vessel a t pressure poS.If the volume of the expansion vessel A
' Author to whom correspondence should be addressed.
+ B'p + C'p2 t
0
Figure 2. Modified Burnen expansion apparatus used for the present expertFigure 1. Apparatus tor Burnen expansion memad.
ment.
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is Ve and t h e gas from the sample vessel raises its pressure from zero t o ple, then by the perfect gas law the number of moles transferred is n l = p,eVelRT. T h e number of moles remaining in t h e sample vessel is now (no - n l ) and the nressure in i t is reduced to D,*. If the exnansion vessel is kvacuated again, and more sample gas let into i t t o raise its DP. then the numher of moles in t h e nressure from zero t o . sample vessel will fall to (no - n l - nz), where nz = pzeVelRT, and t h e pressure in the sample vessel will fall t o pZ8.Continuing in this way we obtain values of t h e numher of moles i n the sample vessel, (no - n l - nz . . .), corresponding t o measured values of pa. T o find no, t h e number of moles initially in the sample vessel, the experiment is continued until it is found that at the m t h expansion t h e pressure in t h e expansion vessel does not rise above atmospheric, hut t h a t with the connection between t h e two vessels fully open, t h e pressure in hoth only reaches t h e value pme.T h e numher of moles in the sample vessel is now pmeVSIRT, and no is found by adding together this quantity and the numher of moles from each of the previous expansions into the expansion vessel,
-
where pierepresents the pressure rise in the expansion vessel due t o the i t h expansion. After r expansions, the numher of moles n, left in the sample vessel is
Therefore after t h e r t h expansion,
T h e volume ratio of the vessels, V W a , can be calculated from an expansion a t low pressure. Then for each value of P,~, p,sV8fn,RT is calculated from eqn. (7). Finally a graph of p,gVgln,RT against prgyields B' (eqn. (4)), and hence B.
Apparatus The apparatus is as shown in Figure 2 The line must be entirely ofmrtal or rrinforrrd plastic, rated to:IOatm. Cas~ssupplirdtothe line from a ras rvlmder via reaularing valw lurked at ?O arm maximum. The &pie and expansion vessels are commercial gas cylinders, one about double the volume of the other. The cylinders should be painted with several layers of anticorrosive and rubberized paint, and anticorrosion agent should be added to the water hath. The pressures are measured by means of calibrated pressure transducers (e.g., CEC Instrumentation type BHL 4100-00-01MO and 4807-0000-03M0 absolute) linked to digital read-outs in atm. Pressure transducers are used to avoid the inaccuracies due to hysteresis effects which occur with Bourdon pressure gauges. When gases expand, their temperatures change due to the work of expansion and kinetic effects, and the gas vessels must therefore be contained in a thermostatically controlled water hath. Procedure When describing the procedure to the students, it should first be emphasized that they should never open tap A until they have checked
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that tap B is closed, because the low-pressuretransducer is only rated up to 1 atm pressure. Secondly, they should never switch off the vacuum pump until they have opened the air leak to the atmosphere, otherwise oil may he sucked hack into the appartus. Set Up First the temperature of the water bathshould be checked, and if necessary adjusted to 25°C. Next the students should turn on the oressure transducers. Thev should check whether the hieh-oressure kmsducer reads less than.1 atm and, if it does not,follow the procedure in the paragraph below. Then the zero of the pressure transducers should be checked as follows. Turn on the rotary vacuum pump (and leave it switched on throughout the experiment). Check that the air leak is closed, and then evacuate hoth vessels (taps B and C open). After 10 min the readings on both pressure transducers should br stable and close to zero. If they are not, then the offset controls should be adjusted. If, at the start of the experiment,the pressure in V8is much g r e a h than 1atm. tao B should not he onened. Instead.,the oumo . . should he turned on and only the expansion vessel evacuated by opening tap C. The experiment is then continued as described below. Leave the pump switched on during the rest of the experiment. Filling of Sample Vessel Close off the expansion vessel and pump (taps B and C). Open tap Agently and fill the sample vessel with COz to a pressure of not more than 20 atm. (The high-pressure transducer reading tends to go off scale while tap A is open, so close the tap from time to time to read the true pressure, until the right pressure is reached). Shut off the supply by closing tap A firmly, and then allow a few minutes for the reading to stabilize. Note this reading (pea). Note the temperature of the water hath. Expansion of Sample Gas Admit gas into the evacuated expansion vessel by slowly opening tap B. Continue to admit gas until the pressure reading on the lowpressure transducer is about 1atm. Close tap B. Allow a few minute? for the gases to reach thermal equilibrium and for their pressures to stabilize. Then measure the pressure ple in the expansion vessel and the pressure of gas in the sample vessel, PI'. Then re-evacuate the expansion vessel by opening tap C and pumping it out for about 5 min. Once again, the pressureplSonthe high-pressure transducer should he noted. If there is a discrepancy of more than a few hundredths of an atmosphere between this and the previous reading, then not enough time was allowed for thermal equilibrium, or taps A and B may not be firmly closed. When the expansion vessel has been evacuated, close tap C, and repeat the expansion procedure to obtain pze and pzs. Continue repeating the procedure until, after the mth expansion, the pressure in both vessels with tap B fully open is equal bmB = pme)and less than 1atm. Measurement of the Volume Ratio Ve:V' To measure the volume ratio, evacuate both vessels (taps B and C open). Switch off the vacuum pump and immediately open the air leak, so that bothvesselsfill withair toatmosoheric oressure (o.+-). Close the air leak and tao . B.. switeh on the o;mo., , the e v a c u s ~ e ~ h e rxpansion vessel. Cloie tap C . Opm tap n and allow the prrssurr i n horh veasels torqualizr. U'P the hwpresrurr trnnsdurer todrter~nine this pressure, (p'). Then
~.~
~~~~~~~~~~
so that
The students should repeat this determination several times and use the average value of VelVSin their calculation. Shut Down On finishing the exprriment, theatudrntsshould open tap(: and evacuate hoth wssela. They should make sure that all thrrr mrtll taps are rlosrd, then switch uff the t,acuunl pump and immediately o p m the air leak, and finally switch off the pressure transducers.
Percentage Deviations of the Second Virlai Coelficient of Carbon Dioxide from Literature Values
TI K
&,,/cms mol-'
B,,/cm3 mol-'
Student resuns 298
-124.5
-109.9 -125.9 -137.7
Our results 276.5
-147.5
-134.11 -137.83 -156.53
absolute deviationlcmS mol-'
peroentage deviationm
Figwe 3. Percamage deviations of secand virial coefficient of &n dioxide lrom iitwature values as a funnion of temperature (0=our results, A = sbdent results).
*h prcemase deviationsd mean B,,taeach 0.0. -3.2, 20.6. 5.8, and 12.8. respscllwly.
d me iive oetsdexperimants werd
Results
Some results are shown in the tahle, expressed as percentage dev~ationagainst temperature. The percentage deviation is of the form [(Be,, - B i , d f B ~ ]X lM), where the literaturevalue B i , t is taken from the compilation by Dymond and Smith (7). The deviations are shown as a function of temperature in Figure 3. The accuracy of the results is limited by the accuracy of the pressure measurement. It can he seen that using CEC Instrumentation pressure transducers, the experiment yields results within 35%of the true virial coefficient, and that the accuracy of the experiment may he improved considerably by taking the average value of a series of determinations.
Virial coefficients are notoriously difficult to measure, especially at low temperatures where~ithecomes increasingly difficult to estimate the effect of adsorptiun of the gas on to t h ~wallsof . the vrsaels. With this in mind. the accurnev of the results from this experiment is very acceptable. Acknowledgement
The authors would like to thank J. Hatton, of the Clarendon Laboratory, Oxford, and E. B. Smith, for their advice. Literature Cited 11) Maaherva, G: P., " w n t d Wcsl,-," Clsuendm h. Wnd, in (21 Bette1heiribF.A..'"Ex~perimental PhyalcalChernistry,"W. 8. Saunden, Philadelphia, 1971, p: 37. (3) Botfon!ley, G. A,. Reeve. C. G., and Whitelawdray, R.. Pmc. R o y Soc., A246, 5M
,."-",.
,,PrjJ
(41 (5) (6) 17)
Kapullo, W.,Lund,N., andScbafu, K., Z Phys. Chem, 3%196 (1963). Hamann, S. D., and Pesrce, J. F., Tmm. For. Sac., 48,101 (19521. Burnett, E. S.. J Appl. M ~ c h .S8,A136, , (19361. h a n d . J. H.. and Smith,E B.B'BTh~Vii~iIC~mmiitetaf ~u~ueo~e~d~iitututu~ Clsrendon PMS. Oxford. 1980, p. 51.
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