Physics of the fire piston and of the fog bottle

Transcription

1 Physcs o the re pston and o the og bottle J Güémez Departamento de Físca Aplcada, Unversty o Cantabra, E-3900 Santander, Span C Folhas, and M Folhas Departamento de Físca and Centro de Físca Computaconal Unversty o Combra, P Combra, Portugal Abstract We analyze two rreversble adabatc processes occurrng n two thermodynamcal devces: the re pston and the og bottle In the rst, an abrupt adabatc compresson o ar nsde a pston causes the burnng o a small paper pece, snce the nal temperature reaches 700 K In the second, an abrupt adabatc expanson o a gas at hgh pressure leads to a temperature decrease wth condensaton o water vapour and og ormaton We show that, the two processes were carred out reversbly, the nal temperature would always be lower than n the rreversble processes Introducton Count Rumord 1 was the rst to recognze that mechancal methods could be equvalent to heat We may conrm ths wth an easy experment: a small alumnum bar rctoned wth ron wool may heat up so that we get burned we touch t wth a bare hand The re pston s a thermal devce used n popular demonstratons 3 The experment done wth t shows the converson o mechancal energy nto heat or, more precsely, nternal energy, exemplyng the Frst Law o Thermodynamcs 4 The process s more controllable than that wth the metallc bar On the other hand, producng og n a bottle s another nterestng experment, where the eect s opposte An abrupt expanson o ar n a vessel leads to a temperature decrease, whch causes condensaton o the water vapor present n the ar We descrbe here n detal these experments, as done by us, and present the calculatons assocated wth the two adabatc rreversble processes We show that the processes were reversble the nal temperatures n both cases would be lower than n the actual experments 1 E C Watson, Reproductons o prnts, drawngs and pantngs o nterest n the hstory o physcs 10 Gllray' s carcature o Count Rumord, Am J Phys 8, 10 (1940); Marcy S Powell, Count Rumord: solder, statesman, scentst, Am J Phys 3, 161 (193) S C Brown, The dscovery o convecton currents by Benjamn Thompson, Count o Rumord, Am J Phys 1, 73 (1947) 3 A Serrano, Fre pston, The Phys Teacher 44, 613 (006) 4 M E Loverude, C H Kautz, P R L Heron, Student understandng o the rst law o thermodynamcs: relatng work to the adabatc compresson o an deal gas, Am J Phys 70, (00) M Talmage Graham, Cloud ormaton, The Phys Teacher 4, (004) 1

2 The re pston For the re pston 6, we used a transparent lucte tube, wth heght h = 179 cm and nternal dameter d = 10 cm, tghtly mounted vertcally on a wood platorm A small pece o cotton or a very thn bt o paper was placed nsde the tube whch was closed by means o a steel pston To better close the ar nsde the tube, an approprate O-rng was mounted around the steel bar 7 To orgnate a lame at the bottom o the cylnder, we smply pulled downwards the pston very quckly We placed the apparatus on a bathroom scale to estmate the orce exerted on the pston Usng a small camera we made a move whch we vewed rame by rame Fgure 1 shows two pctures obtaned n ths way A thn and small pece o paper burns at T F = 41 ºF (usng the amous novel Fahrenhet 41 by the scence-cton wrter Ray Bradbury 8 ), equvalent to T = 38 ºC 10 K Fgure 1: Fre pston The ar nsde the cylnder s compressed n adabatc condtons On the rght, the cotton placed n the nteror burns due to the hgh temperature The scale measures the orce 6 H Hayn, S C Bard, Adabatc compresson n a re syrnge pston, The Physcs Teacher 3, (198) 7 Fre Syrnge Re FIR-10 n Educatonal Innovatons Inc, see wwwteachersourcecom 8 Ths novel descrbes a socety n whch books are orbdden, beng burned by the polce The temperature at whch the paper s burned s 41 ºF

3 The ormaton o the lame s explaned usng the Frst Law We rst show that the ar compressed nsde the cylnder reaches the gnton temperature o the paper The varaton o the nternal energy s gven by ( ) U = n c T T, where T 300 K s the ntal temperature o the ar, T ts nal temperature, n s the 1 number o moles and c R 0 J mol 1 K s the heat capacty o the ar at constant volume 9 1 ( 8314 J mol 1 R = K s the deal gas constant) The volume occuped by the ar s = = π ( d / h, so that the number o moles s gven by ) P = = RT 3 n mol The Frst Law states U = Q + W, where Q stands or heat and W or work The system s contaned n plastc wall and the process s so ast that there s no heat exchange, e t s adabatc: Q = 0 The conguraton work can be expressed as a uncton o the pressure and the volume Assumng that the work s done at constant external pressure P 10 E we have W = F L = PE E, where F E t s the external orce, L s the pston dsplacement and = A L s the (negatve) varaton o the volume o the gas ( A = π ( d / ) s the area o the nternal secton o the tube) Accordng to our measurement, we take F E = 0 N as a body o kg were placed on the system The correspondng pressure s Ths s also the nal pressure o the gas The work done on the gas s FE 0 P E = = = Pa A π ( 10 / ) W = P ) E( 9 Accordng to K Raznjevc, Tables et Dagrammes Thermodynamques, Edtons Eyrolles, Pars (1970), at 0 ºC one has or the ar c P = 17 kj kg 1 K 1 and γ = 140 Assumng the molecular weght 896 g mol 1, we have c = c P / γ = 38 J mol 1 K 1 10 C E Mungan, Irreversble adabatc compresson o an deal gas, The Phys Teacher 41, (003) It s the external pressure that matters The process s non-statc and rreversble so that the pressure o the system s not even dened 3

4 The ntal volume o the ar s d = hπ = 141 m 3 10 Usng U = W = nc T T ) and snce (we accept that ar s an deal gas) ( P T = and nr T = P nr we obtan T = T c + R ( PE / P ) c + R The nal volume s gven by P T c ( P / PE ) + R = = P T c + R E As P E, the volume does not vansh but approaches ts mnmum value, gven by (we use the relaton c = c R ) P + mn = = cp /R 7 For ar ntally at 300 K, and takng P E /P = 63, one obtans T = 1 T = 743 K, correspondng to the temperature varaton T = 44 K Ths value explans the paper gnton 11 The nal volume s then = = 61 m 3 We may calculate the temperature varatons an adabatc reversble processes would brng the gas rom the same ntal state to a nal state n whch one o the state varables s the same as n the nal state o the rreversble adabatc process The temperature correspondng to the nal state (P = Pa,, T ) ater the adabatc process s (1 γ ) / γ P 08 T = T = = 073 K P 11 H Hayn, S C Bard, Adabatc compresson n a re syrnge pston, The Physcs Teacher 3, (198) 4

5 Ths temperature s lower than the nal temperature o the rreversble process The correspondng nal volume, whch can be obtaned rom the equaton o state or rom the equaton o the adabatc process, s = m 3 I, nstead, we consder the nal state ( P, = m 3, T ) reached rom the ntal state through another adabatc reversble process, the nal temperature s γ T = T = 300 = 434 K, 6 also lower than the temperature reached n the rreversble process The nal pressure s now P = Pa Let us consder the work n an adabatc process: P P W = Pad d = γ 1 For the two reversble adabatc processes, one obtans W = 43 J and W = 1 7 J Both are smaller than the work done or the rreversble process, W = 3 10 (61 141) 10 3 J, 04 I = explanng the smaller temperature ncrease n reversble processes The process descrbed here s the bass o the Desel engne, n whch the mxture o Desel ol and ar become nlamed spontaneously when puttng t under a hgh pressure, wthout a spark plug Fog bottle An experment closely related to the re pston 1 s the producton o og n a bottle 13 We are stll n the presence o an adabatc rreversble process, but now we have an expanson 14 Usng an ar pump the pressure s ncreased nsde a bottle (the og bottle) wth a lateral entrance (the walls o the bottle should be resstant enough to avod blowng up o the glass wth the pressure ncrease) and wth a plastc cork rmly attached to ts mouth A T-juncton allows the connecton o a manometer In our experment, when the pressure was around 4 bar, the cork o bottle suddenly led (the bottle was rmly mounted n a rgd structure to prevent ts backwards moton) We observed a cloud nsde the bottle, whch lasted or a ew seconds (see Fgure ) 1 R D Russell, Demonstratng adabatc temperature changes, The Phys Teacher, (1987) 13 J J Coop, A demonstraton o og producton, Am J Phys 9, 4 (1941) 14 Z Blaszczak, P Gauden, Applcaton o a laser beam n demonstraton o adabatc gas decompresson, Am J Phys 8, (1990)

6 Fgure : Fog n a bottle The pressure s ncreased nsde the bottle When the cork les apart rom the bottle, the abrupt decrease o the pressure orgnates a cloud The experment works better ew drops o water are placed nsde the lask (as suggested by Coop, see ootnote 13) The volume nsde the bottle s = 0 cm 3, and the gas (supposed to be deal) s ntally at pressure P = 4 10 Pa and temperature T = 300 K The abrupt expanson aganst the atmosphere lowers the pressure to P 0 = 1, Pa, and the work made by the gas s W = P 0, where = s the ncrease o the gas volume The expanson s so quck so that t may be consdered adabatc The nternal energy varaton s U = W = nc ( T T ), wth T = P 0 / nr and T = P / nr The nal temperature s whle the nal volume s T c + R ( P0 / P ) T = T, c + R γ c ( P / P0 ) + R = c + R We have T T /γ as P And the volume tends to nnty as P 0 / P The external pressure can be low but never zero For P 0 = 0, the work would be zero, the expanson would be ree (Brad Baker, An easy to perorm but oten counterntutve demonstraton o gas expanson, 6

7 For ar at 300 K, and takng P 0 /P = 1/4, one obtans the nal temperature T = 078 T = 37 K, so that the temperature decrease s T = 4 K Ths temperature reducton explans the ormaton o og nsde the bottle For an ntal volume 0 cm 3 the number o moles s n = = mol The vapour pressure at the nal temperature may be calculated rom the Clausus- Clapeyron equaton 17 The vapour pressure o water at temperature T s gven by h T0 T P = ( T ) P0 exp, R T0T where h = J mol 1 s the molar enthalpy varaton or the vaporzaton o water (assumed to be constant), and T 0 = 3731 s the bolng pont o water at normal pressure, P 0 = Pa Thus, at 0 ºC, P (93) = 003 bar, P (T) = 003 bar or a relatve humdty o 70%, and the number o moles o water molecules s n = = mol Snce at such low temperatures the vapour pressure s bascally zero, n agreement wth the Clausus-Clapeyron equaton, all water vapour condensates, gvng rse to og Ths phenomenon s better observed wth the bottle lled wth smokng partcles and a ew drops o water The nal ar volume s gven by P T P = P T γ P In our experment, 3 As beore, let us now evaluate the temperature varatons the gas were submtted to a reversble adabatc expanson between the same ntal state (P = 4 10 Am J Phys 67, 71 (1999)) and there would be no temperature varaton or an deal gas (J-O Goussard and B Roulet, Free expanson or real gases, Am J Phys 61, (1993)) 16 For an rreversble adabatc compresson, the temperature goes to nnty as the pressure ncreases, but the volume tends to a mnmum, whereas n an rreversble adabatc expanson, the volume goes to nnty as the pressure decreases and the temperature reaches a mnmum For reversble adabatc processes, leadng to lm T P = γ 1 P γ T = T ; P P = P lm = ; lm = 0 ; and T = 0 ;, n contrast wth the rreversble P lm P 0 processes 17 S elasco, J Faro, and F L Román, An experment or measurng the low temperature vapor lne o water, Am J Phys 68, (000) P 0 1 γ 7

8 Pa, = m 3, T = 300 K) and a nal state or whch one o the state varables s the same o the rreversble process The nal temperature T o the state (P = Pa,, T ) reached by an adabatc reversble process s gven by 1 γ γ P 08 T = T = = 00 K P Ths temperature s lower than that obtaned ater the rreversble process For the volume one nds = 4 = m 3 On the other hand, the nal temperature T o the state ( P, 3 = m 3, T ) reached by an adabatc reversble process s gven by γ 1 04 T = T = = 1908 K Ths temperature s also lower than that obtaned at the end o rreversble process For the pressure one nds P = P = Pa 3 For the reversble process between the ntal state (P = Pa, = m 3, T = 300 K) and the rst nal state the work s W = 8 J, whch s hgher, n absolute value, than the work n the rreversble process: 3 ( 07 0) 10 = 0 W = J I Ths explans the larger temperature decrease For the reversble process between the ntal state and the second nal state, the work s W = 90 6 J, agan hgher n absolute value than n the rreversble case In concluson, n spte o the partcular character o the descrbed experments both the re pston and the og n a bottle they are examples o a general, probably counter-ntutve concluson: the nal temperature n any adabatc reversble process s always lower than n the rreversble one wth equal ntal and nal dsplacement coordnates (pressure or volume) Ths result s qute general snce the Second Law or a PT system mposes: 8

9 () n any rreversble process the system only can reach states to the rght o the lne o the adabatc reversble process, n a P dagram Ths s the Carathéodory Prncple 18 () c P > c and γ > 1, mplyng that the slope o the adabatc lne n a P dagram s always smaller (larger n absolute value, but negatve) than the slope o the sothermal at the same pont Ths s the Le Châteler Prncple Mark W Zemansky, Kelvn and Caratheodory A Reconclaton, Am J Phys 34, (1966) 19 Robert Glmore, Le Châteler recprocal relatons and the mechancal analog, Am J Phys 1, (1983) 9