Lwrence Berkeley Ntionl Lbortory Lwrence Berkeley Ntionl Lbortory Title THE HOT CHOCOLATE EFFECT Permlink https://escholrship.org/uc/item/9dh21770 Author Crwford, Frnk S. Publiction Dte 2013-03-13 escholrship.org Powered by the Cliforni Digitl Librry University of Cliforni
TWO-WEEK LOAN COPY This is Librry Copy which my be borrowed for two weeks. For personl retention copy~ Tech. Division; Ext. 6782.
DISCLAIMER This document ws prepred s n ccount of work sponsored by the United Sttes Government. While this document is believed to contin correct informtion, neither the United Sttes Government nor ny gency thereof, nor the Regents of the University of Cliforni, nor ny of their employees, mkes ny wttnty, express or implied, or ssumes ny legl responsibility for the ccurcy, completeness, or usefulness of ny informtion, pprtus, product, or process disclosed, or represents tht its use would not infringe privtely owned rights. Reference herein to ny specific commercil product, process, or service by its trde nme, trdemrk, mnufcturer, or otherwise, does not necessrily constitute or imply its endorsement, recommendtion, or fvoring by the United Sttes Government or ny gency thereof, or the Regents of the University of Cliforni. The views nd opinions of uthors expressed herein do not necessrily stte or reflect those of the United Sttes Government or ny gency thereof or the Regents of the University of Cliforni.
To be submitted to Am, J, Phys, THE HOT CHOCOLATE EFFECT Frnk S. Crwford Physics Deprtment nd Lwrence Berkeley Lbortory University of Cliforni Berkeley, CA 94720 Reproduced with originls provided by the uthor.
ABSTRACT The "hot chocolte effect" ws investigted quntittively, using wter. If tll glss cylinder is filled nerly completely with wter nd tpped on the bottom with softened mllet one cn detect the lowest longitudinl mode of the wter column, for which the height of the wter column is one qurter wvelength. If the cylinder is rpidly filled with hot tp wter contining dissolved ir the pitch of tht mode my descend by nerly three octves during the first few seconds s the ir comes out of solution nd forms bubbles. Then the pitch grdully rises s the bubbles flot to the top. A simple theoreticl expression for the pitch rtio is derived nd compred with experiment. The greement is good to within the ten percent ccurcy of the experiments.
1. INTRODUCTJ.(J)N Put n ounce of dry hot-chocolte powder in mug; fill the mug with hot wter; stir. Now strt tpping on the bottom of the mug th your knuckle, Listen for note tht slowly rises in pitch, (My record: tve rise from the initil low pitch to the finl high pitch,) bout minute for the pitch to stop rising. Now stir It tke~; As th~'. spoon ccidentlly hits the inside of the mug you will her the tch descend once more nd the experiment cn be (vjith ech there is smller pitch lowering, s cn be verified by performing the experiment while sitting t.) I discovered the effect ccidentlly just before Christms, 1974, while hving hot chocolte with my friend Nncy Steiner, (She noticed H: first.) 1 2 3 4 5 Severl yers lter I found tht the effect is well known, ' It works for ny liquid into which you cn introduce gs bubbles trpped in powder, or crbon dioxide in soft drinks or beer,) After stirring, the bubbles fill the liquid volume; the velocity of sound in the bubb mixture is then reduced below tht of the bubble~free liquid nd the pitch is correspondingly reduced. As the bubbles flot to the top of the mug smller frction of the volume hs the reduced sound nd the pitch rises reching the vlue for the bubble~free liquid v1hen most of the bubbles hve floted to the top. 2. OBSERVATIONS WITH HOT WATER. THE INVERSE EFFECT. In order to hve simple experiments nd theoreticl predictions it would help if the effect worked for the simplest cse imginble-~ir bubbles in wter. I might therefore hve been discourged to red in Ref. 1 tht "lmost ny powder could produce the effect in cold or hot wter, but wter lone would not", if I hd not lredy tried it nd found
effect in wter. Perhps those uthors did not try hot tp wter. Not only does hot tp wter give fine effect, it gives something new tht I cll the "inverse" hot chocolte effect. To observe the inverse effect turn on your hot wter fucet nd wtch the wter strem s you wit for the cold wter to run off nd the wter to become hot. of smll ir bubbles). Suddenly you my see the wter become cloudy (full At tht time fill your mug rpidly, turn off the fucet nd strt tpping. For the first few seconds you my her the pitch descending. Tht is the inverse effect. After few seconds the pitch strts to rise--the "norml" effect. To see wht is hppening replce the mug by tll trnsprent glss or jr. Look t light through the glss s you tp. You will notice tht during the inverse effect (descending pitch) the wter is getting cloudier. Tht is becuse ir tht ws in solution under high pressure iri the hot wter pipe is coming out of solution nd forming bubbles. the lower the pitch. The greter the mount of ir in the form of bubbles As you continue to wtch nd tp you will see the bubbles rise under grvity, A cler region grows t the bottom of the glss, with rther shrp boundry between cler nd cloudy regions, As the boundry rises the pitch rises (the norml effect). When the boundry hs reched the top surfce of the liquid the wter is cler nd the pitch stops rising. No,effect is seen (no cloudiness) or herd (no pitch chnges) with cold tp wter. Tht is becuse the dissolved ir remins in solution. Using hot tp wter the lrgest pitch lowering tht I hve observed is fctor of bout seven in frequency-~nerly three octves. Tht ws chieved using tll grduted cylinder. With coffee mug the most I cn get out of hot tp wter is bout one octve. Tht is becuse the bubbles
hve only short distnce to flot to rech the top of the mug, nd rech the top before ll the ir hs hd time to come out of solution to form bubbles. With tll cylinder the bt::bbles cn hve floted up one coffee~mug height nd nevertheless still occupy most of the volume. Then there is time for the inverse (pitch lowering) effect to be nerly before the norml effect tkes over. The effect works on most of the hot wter fucets I hve tried., The lrgest effect (cloudiest wter nd gretest pitch lower:tng) :ts obtined w:tth the fucet vlve prtilly closed. 'The growth of the smll bubbles is pprently triggered when the supersturted hot wter flows pst constriction (th~ prtilly closed fucet) where the pressure is suddenly relesed. The bubbles re due to dissolved ir, not ir entrined t the nozzle. Tht is shown s follows: If I submerge the nozzle in wter while it is emitting hot cloudy wter there is no decrese in the cloudiness. Also, there is prcticlly no effect with cold..,.wter fucet, even though the ir entrinment should be essentilly the sme s for hot iayter. If I open the hot wter vlve completely the cloudiness goes wy nd the wter strts to run cler gin. Under these circumstnces is sometimes no effect. Sometimes there is smll delyed inverse effect where ~he wter in the rpidly filled jr remins cler with no pitch lowering for few seconds, nd then suddenly gives smll pitch lowering nd bubble formtion, but with fewer nd lrger bubbles thn with the constricted vlve. The question nturlly rises: when I get no effect with the wide open hot wter vlve, is it becuse the bubbles hve been formed fr bck in the pipe somewhere nd then for some reson become "lost" before I detect them? Or is the ir insted still in solut:ton :tn the hot wter? I tr:ted dding snd to the cler hot wter
to stimulte bubble growth, without success. It occurred to me to try sound wves. I borrowed "supersonic 11 clening device hving 6~inch by 6-inch bsin tht cn be filled three or four inches deep with wter. Sound wves of bout 11 khz fill the liquid volume nd gitte whtever is inserted there. I used 250 ml volumetric flsk to hold my wter smples. The flsk ws first filled completely nd then tipped sufficiently sidewys so tht bubbles rising in the min volume would be trpped in the domed top of the tipped min volume rther thn trveling up the neck of the flsk. When I filled the flsk rpidly with hot cloudy wter emerging from the constricted fucet the effect of the sound~wve gittion ws to rpidly colesce the smll bubbles into lrger bubbles which rpidly rose to the top nd were trpped in the dome where I could mesure the dimeter of the flttened single lrge colesced bubble. When I gin used hot cloudy wter but did not turn on the gittion I hd to wit severl minutes for the mny smll bubbles to rise nd colesce into single trpped bubbles; but I found the sme finl size for tht bubble s when I gitted the wter. When I filled the flsk with hot cler wter obtined with the fucet wide open I found tht fter witing for severl minutes I still hd prcticlly no ir bubbles, but if I gitted I immeditely strted to generte bubbles. After bout ten minutes of gittion I got no more bubbles, At tht time the totl mount of trpped ir in the single colesced bubble ws the sme size s I got from the bubbly hot wter obtined from the constricted vlve. I conclude tht the cler hot wter from the wide open fucet is still supersturted with ir nd in fct contins the sme mount of ir in solution s the cloudy hot wter contins in the form of smll bubbles. The fct tht the two methods give the sme result suggests tht in ech cse ll of the excess ir (bove tht which is in equilibrium t 1 tmosphere t the hot temperture) is coming out of solution. It would
be implusible for these two very different methods to hve the sme efficiency for getting rid of excess ir unless tht efficiency is close to 100%. (This expecttion ws lter confirmed; see Sec.7.) 3. A SIMPLE MODEL: LONGITUDINAL OSCILLATIONS Since the pitch~lowering effect works for ir bubbles in wter we should be ble to mke quntittive comprisons between experiment nd Unfortuntely, the theory given in Ref. 1 is rther sophisticted. It involves the flexing modes of the glss continer, perhps becuse those uthors tpped on the side of the mug rther thn on the bottom in to the effect. Their opredictions depend on the continer dimeter nd wll thickness nd on the elstic constnts of the glss s well s on the properties of the liquid. 1. Perhps becuse of this complexity I did not pursue the problem for severl yers. But recently (1980) my interest revived. (I drink lot of hot chocolte.) It occurred to me tht whether or not the flexing modes re present I might serch for other modes bsed on much simpler hypothesis, I mde the ssumption tht the pitch I her when I tp with soft mllet (my knuckle) on the bottom of tll cylindricl glss continer hs lmost nothing to do with the continer but is simply the pitch expected for cylindricl column of liquid undergoing longitudinl vibrtion in its lowest longitudinl mode. Since the top surfce of the liquid is free, nd the bottom, in contct with the glss, ed is fixed, I expect/\the height of the wter column to be exctly one qurter wvelength for sound wves. in the liquid, both for the cse of the bubble~ofree liquid nd for the cse where the liquid is uniformly filled with bubbles (before they hve hd time to flot to the top).. I tested this hypothesis. Since my sense of pitch cn esily be wrong
by n octve, I used microphone nd n oscilloscope. I found tht I needed suitbly softened mllet-- piece of wood with severl lyers of msking tpe over it--in order to void distrcting high frequency sounds from the glss continer. Strting with cold tp wter (20 C) in 250 ml grduted cylinder I found tht with cylindricl wter column of height 0.28 m my gentle tpping on the bottom of the cylinder excited dmped oscilltions hving frequency of bout 1300 Hz. Assuming the wter column to be one qurter wvelength gives sound velocity v = Af = 4x0.28xl300=1460 m/sec, which grees well with the hndbook vlue of 1470 m/sec. Thus my simple hypothesis ws verified. ( I lso checked tht when I poured out some of the wter the frequency incresed by the expected mount, nd tht the frequency did not depend on the dimeter of the cylinder.) Going over to hot tp wter from prtilly contricted fucet I found tht I could get frequency decrese by fctor of bout 7 during the inverse effect. When the bubbles ll rose to the top the pitch ws essentilly the sme s for cold wter. (For the sme length of wter column the frequency for the :hot wter should ctully be bout 8% higher thn for the cold wter. My oscilloscope mesurements were only good to bout ± 10% nd I did not check tht point.) Since the pitch decrese during the inverse effect hs to compete with smll simultneous pitch rise due to the rising of the bubbles, I estimte tht my experimentl pitch-lowering fctor of 7+ 0.7 should be corrected to 7.5+0.8, for the pitch-lowering I would get if the bubbles did not rise. Once I hd lerned how to detect this longitudinl mode with the tll grduted cylinder using both my er nd the oscilloscope then I ws ble to recognize it lso for wter in coffee mug, where I hd difficulty in determining the pitch by er lone. For cold wter column of height 6.8 em
I observed dmped oscilltions of 5000 Hz. Tht mkes the wter column 0,23 wvelengths high, greeing with the expected 0.25 within my mesurement errors, Tht shows tht the effect observed "nturlly" in the kitchen (i.e., in coffee mug) is the sme s the one observed with tll glss cylinders. 4. QUALITATIVE EXPLANATION OF THE LARGE PITCH RATIO The velocity of sound in ir is bout one fourth of its velocity in wter, Therefore if you completely replce the wter column by n ir column of the sme height the pitch goes down by fctor of bout 4. How cn it possibly be tht if, insted, you replce only tiny frction of the wter volume by ir bubbles, the sound velocity goes down not by tiny frction of 4 but by fctor of nerly 8? Tht is surprise, if we re expecting to find sound velocity in bubbly wter to be given by some kind of "interpoltion" between the velocities in pure ir nd'' pure wter. Further thought mkes it less surprising, Sound speed in liquid or gs depends on two physicl quntities: compressibility nd mss density. It helps to think in terms of slowness rther tht speed, Let us define slowness = 1/speed. (Slowness is mesured in units of seconds per em, or hours per mile.) The slowness of sound is greter (L e. the sound trvels more slowly), the greter the inerti (mss density) of the gs or liquid. Greter compressibility (weker "return force") lso gives greter slowness. Slowness turns out to be the squre root of the product of the density times the compressibility. Wter hs bout 800 times the density of ir, so we might expect it to hve greter sound slowness thn ir. However, ir is bout 15,000 times more compressible thn wter. Thus ir wins the "slowness rce" by fctor of the squre root of 15,000/800, which is 4.3,
nd sound trvels 4.3 times slower in ir thn in wter. Now suppose tht the wter is filled with ir bubbles distributed homogeneously throughout the liquid but occupying only smll frction of the totl liquid volume. The density will then be essentilly tht of wter. Tht gives lrge slowness contribution, But the compressibility will be prcticlly ll due to the ir in the form of bubbles. Tht lso gives lrge slowness contri~ bution. It should not surprise us tht by combining the lrge slowness contribution of the wter (its inerti) with tht of the ir (its compressi~ bility) we cn get slowness tht is greter thn tht of either wter or ir. 5. A SU1PLE QUANTITATIVE THEORY We need theoreticl expression for the velocity of sound in homogeneous mixture of wter nd ir bubbles, in order to compre with our experiments. The exct theory is rther complicted. 6 It predicts tht the sound velocity in the mixture depends on the bubble rdii. I mde only very crude mesurements of bubble rdii. However, I did mesure the frction of the liquid volume occupied by bubbles. (The method is described lter.) It turns out tht tht is ll we need to know, for our sound wve frequency nd our bubble~size regime. Wht follows is very simple theory tht cn be compred with my mesurements. The velocity of sound, v, in homogeneems liquid or gs depends on the mss density p nd the compressibility K s follows: where the compressibility K is defined s 2 1/v = Kp (1) K=(dV/dp)/V,. (2)
Here V is the volume nd dv is the volume decrese due to the pressure increse dp in the sound wve. Now consider homogeneous mixture of wter nd ir bubbles. We will only consider sound frequencies where the wvelength is lrge compred with the bubble rdii nd the spcing between bubbles. Then Eqs. (1) nd (2) should still pply. nd ir, nd no subscript for the mixture. Use subscripts w nd, for wter Since the frctionl volume occupied by bubbles is very smll we tke the density of the mixture to be tht of wter: P""P The totl volume in Eq. (2) is essentilly tht of tb.e w wter: V=V For the volume chnge dv we tke dv=dv +dv Then Eqs.(l) w w nd (2) give for the mixture 1/v 2 = (p /V )(dv /dp) + (p /V )(dv/dp). (3) w w w w w The first term in Eq. (3) is just l/vw 2, ccording to Eqs. (1) nd (2). :Multiply 2 Eq.(3) by v Then multiply the numertor nd denomintor of the scconc w term by V nd define V /V w by ir bubbles (for f, where f is the frctionl volume occupied smll compred with unity.) The second term becomes v 2 p f K, which lso equls f K /K Then Eq.(3) becomes ww w ~ 2 ~ v /v = [l+(k /K )f ] 2 = [l+v p f K ] 2 (4) w w ww We now mke simplifying (nd possibly wrong) ssumption. We neglect the possible dependence of K on bubble rdius, surfce tension, wter vpor, het of vporiztion, thcrnnl conductivity of the nir, etc., nd tke K to be the sme s one tkes for sound velocity in norml ir, which is the dibtic compressibility of dry ir. Tht gives K =1/yp, where y=l.40 is 6 2 the rtio of specific hets for dry ir nd p=l~olxlo dyne/em t 1 tmosphere 3 For wter we tke p =1.0 gm/cm nd v =1470 m/sec. w w 2 4 K /K = v p /yp = 1.49xl0 w w w Tht gives (5)
-12- Then Eq, (4) becomes 4 k v /v = [1+1.49xl0 f ] 2 w (6) For f =0, Eq,(6) sys tht the sound velocity reduces to v, s it must. w (The dissolved ir molecules hve no effect. It is only when they collect in bubbles tht they increse the compressibility.) For f =0.01, Eq. (6) predicts tht in the bubbly mixture the sound velocity is bout 1/12 of the velocity in wter, or bout 1/3 the velocity in ir, Tht grees with our qulittive discussion in Sec. 4, The formul for v /v given in the more sophisticted theory of Ref,6 w reduces to my Eqs.(4), (5), nd (6) in the limit where the sound frequency is smll compred with the nturl oscilltion frequency for rdil oscilltion of the bubbles, Tht is indeed the cse for my observtions. (See the Appendix.) 6, COMPARISON OF THEORY AND EXPERIMENT In order to mesure f nd the pitch rtio I seprted the experiment into two prts performed one fter the other with severl repetitions within few minutes, so tht the hot tp wter would not hve time to chnge its properties. For mesuring frequency rtios I used 250 ml grduted cylin~ der. This ws tll enough so tht bubbles rose by only smll frction of the height of the wter column during the "inverse" effect while they were forming. To mesure f I needed suitble "volume mgnifier", so I insted used 250 ml volumetric flsk hving tll nrrow neck (the mgnifier). I tped ruler onto the neck so tht I could red the position of the liquid meniscus. After first filling the flsk with hot bubbly tp wter I would quickly red the meniscus. This first reding ws bit difficult, since the wter ws cloudy with bubbles nd the meniscus ws "frothing" with the rrivl of new bubbles floting up from below. (An improved method described lter
eliminted the ne.ed. for this difficult first re:ding,) Nevertheless I could red it to bout ± 1 mm. After most of the bubbles hd risen to the top (it tkes two or three minutes) I found tht the meniscus hd dropped by bout 4 mm. Tht corresponded to volume decrese of 3 0,8 em ~ which I tke to be the volume of ir tht ws in the form of ir bubbles when I first red the meniscus immeditely fter fi.lling the flsk, -3 Tht gives f =0.8/250 =(3.2+ O.S)xlO Inserting this vlue of f into ~ Eq.(6) gives predicted velocity rtio v /v = 7.0+0.8. w - Since I ssume the column of liquid is one qurter wvelength, both for the pure wter nd for the mixture, the predicted velocity rtio is the sme s the predicted frequency rtio. My mesured frequency rtio ws 7.5+0.8. Thus I found good greement between my observtions nd the simple theory. (An improved mesurement off is described in Sec.7.) 7.EXPECTED VALUE OF f, Some weeks fter mesuring f to predict its vlue. it occurred to me tht I should be ble Assume the ir goes into solution t the cold wter temperture of 20 C nd reches the equilibrium concentrtion for wter t 20 C in contct with ir t 1 t pressure. After the wter gets into the pipes it sees no more ir, The dissolved ir frction remins constnt. (These ssumptions were supported by converstions with Est By Municipl Utilities District engineers.) When the wter is in my hot wter pipes it is under guge pressure of bout 50 psi. Therefore the ir remins in solution. When it emerges from the hot wter fucet it is suddenly gin t 1 t pressure nd is supersturted with ir, becuse the hot wter cnnot hold s much dissolved ir s the cold wter. If suitbly "triggered", the excess ir will come out of solution in the form of rpidly growing bubbles. When ll of the excess ir hs come out of solution the bubbles stop growing. (Tht
-14~ termintes the in'."erse effect.! We then hve clculble frction f the volume in the form of i~ Jubbles. of Here is the clcultion. According to the Hndbook of Chemistry nd Physics (35th Edition) fo-: pure N 2 t 760 mm Hg pressure in contct with wter t 20 C, the equilibric mount of N 2 in solution is 0.01545 cc of N 2 gs when reduced to STP (s~ndrd temperture nd pressure, 0 C nd 760 mm Hg) 0 for ech cc of wter. At 40 : it is 0.01184 cc (t STP) per cc of wter. Tking the difference we fine ).0145-0.01184 = 0.00361 cc (t STP) per cc of H o, tht should come mt s bubbles. The bubbles re not t 0 C 2 but t 40 C so they occupy ~rge volume in the rtio (273+40)/273=1.146. Also, when the N went into srlution the prtil pressure of N ws not 760 mm Hg 2 2 but only 78% of tht, for no~2l ir. Thus the mount in solution is reduced by fctor of 0.78. 0 Also, = 40 C the vpor pressure of wter is 55 mm Hg, The totl pressu~e of ir pluzwter vpor in the bubble should be 760 mm Hg, so the ir need o::1ly furnish -I:J0-55= 705 mm Hg pressure, Tht increses- the volume of the bubble by the :~tio 760/705=1.078. (Surfce tension my contribute n ddition~ correction :o the pressure. See the Appendix.) The mount of N 2 I expect to :ome out s bubbles is therefore -3 f (N 2 ) = 0.00361 2: Lll;6 x 0. 78 x 1.078 "" 3.5 x 10, (7) For pure oxyge-::1. t 760 mm lg pressure the equilbrium mount in solution t 20 C is 0.03102 cc of o 2 (rt STP) per cc of H 2 o. At 40 C it is 0.02306. Tking the difference, conve~ing to volume t 40 C nd 705 mm Hg, nd multiplying by the frction~?rtil pressure of oxygen in ir, 21%, we get predicted vlue f(o 2 ) = (0.03102-G02306)xl.l46 x 0.2lx 1.078= 2.1 xl0-3, (8) We cn compre the predi~ions -3 off =(3,2+0.8)xl0, - of the predicted totl (7) nd (8) with my observed vlue My obs~rved vlue ws just (57:!:-14)% - -3, f ~.6x10 given by the sum of (7) plus (8).
At first I ws delighted by this firly good greement, But then I ws struck by the fct tht my observed vlue would be in excellent greement with the predicted excess for N 2 lone, s given by (7). Ws it possible tht my bubbles were pure nitrogen, with no oxygen? I r.eceived support: for thi.s fscinting hypothesis by noticing tht t both 20 nd 40 the solubility per molecul.e of oxygen is t1:.;rice tht of nitrogen, (For exmple t 20 the rtio of the two numbers 0.03102 nd 0.01545 given bove is 2.01.) If wter "likes" oxygen twice s well s it does nitrogen, might not tht inhibit the speed of diffusion of the oxygen through the wter to rech the surfce of the growing bubble, reltive to tht of nitrogen, or inhibit its evportion into the volume of the bubble., reltive to nitrogen? Perhps I should hve just sked chemist, but I ws frid the nswer might discourge me. I needed to lern how to mesure oxygen content in wter nd ir. Fortuntely I soon contcted Prof. Dvid Jenkins nd Mr. Bruce Jcobsen of the U.C. Snitry Engineering Deprtmemt, who re experts t tht mesurement. Rther thn bring smples of wter to their lbortory I worked with Mr. Jcobsen, using smples frdm tlj.eir fucet, since their fucet gve nice cloudy hot wter (when the vlve ws constricted) nd fine pitch-lowering effect. hd the predicted mount of DO ' 0 We found tht the cold tp wter t 22 C (dissolved oxygen) for wter sturted with ir t 22 C nd 1 tmosphere. The hot wter t 56 C tht hd been "debubbled 11 by pssing it through the constricted fucet hd exctly (within mesurement errors) the reduced mount of DO expected for wter t 56 sturted with ir t 1 t. Tht is, supersturted oxygen residue ws not being left behind in the wter when the bubbles formed. This showed tht the debubbling ws prcticlly 100% efficient t removing excess oxygen, nd "shot down" my fscinting hypothesis. As further check we mesured the oxygen frction
-16~ of the gs tht cme off s bubbles from the wter emerging from the constricted hot wter fucet. The gs ws not pure nitrogen. It hd the norml frction of oxygen found in ir. This 11 drove the lst nil in the coffin" nd my fscinting conjecture ws lid to rest. Tht left unnswered the question s to why I ws only getting bout 60% of the predicted mount of ir in bubbles when I mesured f. I suspected tht I ws losing ir in lrger thn verge bubbles tht were floting up nd being lost either before I mde my first meniscus mesurement (the difficult "frothy" one) or during it. To eliminte this first meniscus mesurement I designed n improved flsk for mesuring f The flsk ws mde by joining per shped flsk t its nrrow end to 5 mm (inside dimeter)clibrted pipette. The flsk is stoppered t its brod end, which I'll cll the bottom. The pipette cn be corked t its end (the "top"). The end of the cork defines the end of the pipette volume. In use the flsk is corked t the top, inverted, filled to overflow through the bottom, stoppered to overflow t the bottom, reinverted so the top is up, immersed in hot bth t the temperture of the hot tp wter until ll the bubbles hve risen (five minutes), nd then red. Only one meniscus reding is needed becuse the cork defines the end of the pipette nd its loction replces the "frothy meniscus" mesurement. The mgnifiction is lso lrger on this flsk; the ir occupies bout 50 mm long the pipette rther thn the 4 mm of the erlier technique. With the improved flsk I find(for 20 cold nd 40 hot wter) Inserting this vlue of -3 f =(4.1 + 0.4)x10 - (9) into Eq,(6) gives predicted pitch rtio v/vw = 7.9~0.4, which is still in good greement with my mesured vlue
-17- of 7.5+0.8. My "efficieney 11 for ctching the excess ir, with the improved flsk, is found by dividing (9) by (7)+(8), to get 4.1/(3..5+2.1)""0.73. Prt of the "missing" 27% my be lost during the filling of the flsk, before it gets stoppered, since this is somewht turbulent process nd lrge bubbles relesed during the filling cn rise rpidly nd be lost before I finish filling, nd insert the stopper. Another possibility i.s tht the predictions (7) nd (B) ~re too high, becuse I hve not corrected the pressure (nd hence the volume) for surfce tension. (See the Appendix.) Whtever losses tht my occur during mesurement of f should lso occur during the filling of my grduted cylinder to mke the pitch-lowering mes ~ urements; therefore no correction fctor need be pplied to the result (9) before using it to predict the pitch rtion. 8. FURTHER OBSERVATIONS If one strts with n empty grduted cylinder nd tps the bottom with knuckle the pitch herd by er (or observed on the oscilloscope) is of course tht of the ir column, with the length of the ir column being pproximt~ly one qurter wveleng th for sound in ir. As one dds cold wter to the cylinder the ir column shortens nd its pitch rises. One cn now strt to listen for high note due to longitudinl vibrtions in the wter column. But this note is difficult to her until the cylinder is nerly full. It is msked by the much louder note from the ir column. If insted one sfrts with the cylinder completely full of wter then there is no ir column p,nd one cn esily identify the pitch of the note due to the wter column. As one now pours out wter little t time one cn keep trck of this note s its pitch scends, nd cn strt to her note from the ir column. Since the velocity of sound in ir is roughly one qurter of the velocity in wter, then, when the ir column t the top
of the cylinder is bout one qurter the height of the wter column the pitch of the wter vibrtions will equl the pitch of the ir vibrtions, For even less wter the louder nd lower ir vibrtions mke it difficult to her the wter vibrtions. In order to her the wter vibrtion it now helps to ruin the ir column by stuffing wet pper towel into the ir spce so s to dmp the ir vibrtions, Alterntively one cn use stethoscope with its detector just under the wter surfce, to enhnce the wter note. By judicious tuning of the wter level in the region where the ir column is bout one qurter of the wter column one cn, using the oscilloscope, observe bets between the simultneous ir nd wter notes when the bottom of the cylinder is tpped, Becuse of the poor impednce mtch t the wter-ir surfce it is hrd to get the wter note out of the wter, nd the wter note is wek compred with the ir note, Therefore the bets re not strongly modulted. I cn see them with microphone nd oscilloscope but cnnot her them by er. In the Appendix, I exmine the dependence of the sound velocity on bubble rdius nd conclude tht for our regime of bubble rdii nd sound frequency the ssumption of dibtic oscilltion of the ir bubbles is probbly not vlid, If tht is indeed the cse nd the oscilltions re isotherml tht would increse my predicted pitch rtio by bout 18%, Experiments more ccurte thn those I report here could settle tht question, I lso show in the Appendix tht surfce tension cn probbly be neglected for our bubble sizes. I lso show tht model where the number of bubbles remins roughly constnt while ech bubble grows with time grees both with the ppernce of the wter (incresing cloudiness during the inverse effect) nd lso the time durtion (of order 10 seconds) of the inverse effect,
-19-9.APPENDIX. DEPENDENCE OF SOUND VELOCITY ON BUBBLE DYNAMICS The rdil oscilltion of gs bubbles in wter ws investigted theoreticlly nd experimentlly by Minnert. 7 By equting the mximum potentil energy (t mximum compression) to the mximum kinetic energy (t the equilibrium rdius,), nd ssuming tht the compressibil ws the dibtic compressibility of dry ir, nmely K ;1/yp, Minnert derived the expression (10) where wr=2tifr' nd fr is the nturl oscilltion frequency. Minnert verified Eq. (10) experimentlly for bubbles hving rdii between 1.5 nd 3 mm. He lso verified the need for the fctor y=l.4 for ir bubbles, nd verified the dependence on the fctor y by using other gses besides ir. If the oscilltions hd been isotherml the fctor y=l.4 would be replced by tinity in Eq. (10). Moxe generlly. we let y denot?the 11 polytrope" index, which should equl the rtio of specific hets, 1.4, if the oscilltions re dibtic, should equl 1.0 if they re isotherml, nd my lie somewhere between if they re neither. Minnert's experiments demnd y=l.4 nd rule out l.ot for his bubble sizes. Md 6 f 1 I e w1n g1ves ormu for v vw which is the sme s our Eq.(4), except tht his expression for K depends on frequency. If we ect his dmping term (which is only importnt close to the resonnce frequency fr) his result cn be written 2 2 2 2 K =[Z /(Z -1)](3/ p w ) (11) w R where Z=wR/w, nd w=2tif, where f is the driving frequency. If we substitute Minnert's vlue of wr from Eq.(lO) we obtin. 2 2 K= (1/yp)Z /(Z -1). (12) For driving frequency f much smller thn the resonnce frequency fr we hve Z>>1. Then Eq. (12) becomes K =1/yp nd Medwin's formul reduces to my Eq. (6).
~20-. My observtions were t sound frequency f=l.3 khz. For bubble rdius 0.1 mm, Eq.(lO) predicts fr=33 khz. Most of the bubbles tht contribute to my observtions pper by eye (with pocket mgnifier) to hve rdii bout 1/50 mm. Thus my mesurements t 1.3 khz should lie in the low frequency regime of Eq. (12) nd I expect my Eq. (6) to be sufficiently ccurte t my level of experimentl ccurcy. Should one use the dibtic compressibility in clculting the resonnt frequency fr when the bubble rdius is less thn 0.1 mm? Minnert verified y=l 4 in Eq. (10) for bubbles with rdii greter thn 1. 5 mm. But for sufficiently smll bubbles nd for sufficiently low frequency f there will be time for het to flow by diffusion from the center to the surfce of the bubble during hlf period T/2 = l/2f. Air molecules cn there quickly exchnge energy with the wter. Thus for sufficiently smll bubbles we should replce y=1.4 by 1.0 in Eq.(lO), nd therefore in Eq.(l2) nd in my Eqs. (5) nd (6), If we replce y=l.4 by 1.0 in Eq.(S) tht replces the "1.49" in Eq,(6) by 2.09, nd chnges the predicted pitch lowering in my experiment from 7.9+0.4 to 9.3+0.5. I cnnot quite distinguish between those two predictions with my experimentl ccurcy; prtly tht is becuse of uncertinty in my technique for mesuring f (where is the ''missing" 27%?) nd prtly becuse it is difficult to determine the lowest frequency to better thn 10% during the simultneous frequency decrese (s bubbles form) nd increse (s they rise). At wht bubble rdii do we expect het flow to become importnt t our sound frequency? Let the collision men free pth for the ir molecules bel\ ; let the rms moleculr velocity be c, nd let the molecules diffuse for time.t. given by Then the men squre rdius R 2 of diffusion of n ir molecule is (13)
-21~ Tking A~6xl0-6 gives R=0.08 mm. em, c=300 m/s, nd t=t/2 = 3.9 x 10-4 sec for f=l.3 khz This is four times greter thn our estimted verge bubble rdius of bout 0.02 mm. Therefore for our bubbles the dibtic ssumption should brek down, wheres t Minnert's rdii of greter thn 1.5 mm the dibtic ssumption should be vlid. (I rech this sme conclusion when I substitute into the more complete formuls derived in the thorough theoreticl discussion of oscillting gs bubbles by c. Devin, Jr. 8 ) Experiments less crude thn those reported here could exmine the trnsition between dibtic nd isotherml oscilltion. Wht role does surfce tension ply? Tht depends on the bubble rdii. The guge pressure p-p 0 inside n ir bubble of rdius, when pplied to the cross sectionl re 'IT 2 of slice through bubble gret circle, must blnce the force o2 'IT 2 'IT. Tht gives due to surfce tension exerted cross the perimeter 2 (p-p )rr = o2jt, or 0 p=p +(20/)=p [1+ (2o/ p )( /)], 0 0 0 0 0 where is the finl bubble rdius. For wter we hve 0=75 dyne/em. 0 Tking =2x10-3 em (my estimte using pocket mgnifier), nd 0 6 2 = 1 t= l.oxlo dyne/em, Eq,(l4) becomes p [l + (0.075)( /)]p 0 0 Let us first pply Eq.(lS) to the predicted vlue of f. For = 0, Eq.(15) predicts tht the pressure inside the bubble is 1. 075 t. tje neglected tht (14) (15) fctor in the min text. The predicted volume is therefore reduced by fctor of 1/1.075. The sum of (7)+(8), multiplied by 1/1.075, is [(3.5+2.1)/1.075]xl0-3 = 5.2x10-3, which is to be compred with my mesured -3 vlue o (4.1± 0.4)xl0 Suppose now tht my estimte of ws bised in 0 tht I notice the lrgest bubbles most esily. If the verge vlue of is, sy, 0 0.5xio 3 em insted of 2xl0-3 em then the 0.075 should be replced by 0.3~ in Eq. (15). -3 In tht cse the predicted~~le cif f becomes 4.3xl0,in
by diffusion given by region of rdius R. 0 2 t ""R /A.c. 0 0 According to Eq.(l3) tht time is (l9) The cho.ncc tho.t during the explortion of this region the ir molecule will be cptured in the bubble should be proportionl to the time spent in the bubble volume; tht time should be proportionl to the bubble volume. Ths, without worrying bout fctors of two (the bubble volume is not constnt) we multiply Eq,(l9) by the volume rtio R 3 / 3 Tht gives our estimted 0 0 clen~out time t: t 5 3 = R / A.c. 0 0 Since the rtio R / is firly well known (it depends only on our firly 0 0 well mesured vlue of f ) but o R / =5 nd write Eq.(20) 0 0 (20) is poorly mesured, we put in our vlue s t For the men free pth A. of the ir molecule diffusing in wter we tke the edge length of the cube occupied by one wter molecule in the liquid: A.=3xlO~S em. Tking c=300 m/sec (therml velocity), nd our crude vlue = 2 x 10~ 3 em, we find n estimted 0 bubble growth time t = s 5 (2xl0-3 ) 2 /(3xl0-8 )(3xl0 4 ) = 14 sec, (21) Becuse of the lrge uncertinty in, 0 the better~thn-order-of-mgnitude greement of (21) with my observtions hs to be pure luck, But the order-of-mgnitude greement is not, nd supports the model with prcticlly constnt number of bubbles, ech growing lrger with time becuse of the cpture of diffusing ir molecules. Finlly, wht is it tht determines the initil number of bubbles per unit volume, N? (Tht is wht determines the finl bubble rdius,) v 0 I believe there is very simple concept tht would enble me to predict N. v But I hven't found the concept.
ACKNO\till,EDGEMENTS, I would like to thnk Bruce Jcobsen, Dvid Jenkins, Jrdin Kre, Phil Lubin, Terry Mst, Richrd Muller, nd Peter Tns for comments nd suggestions, This work ws supported by the Physics Di~vision the 1L S, Deprtment of Energy under contrct No. W "7~.05 ENG"L}8, REFERENCES. ~1! E. Frell, D. P, McKenzie nd R, L Prker, the Not.e Em:H tecl While Instnt Coffee, 11 Proeo PhiJ. Soc. 6j, 36 2., Jerl wlker, "The Flying Circus of :Lcs With Auswecs", John I;JJ 1977, pr, 1.22. Je:cl \.1flker, The Amteur Scientist section of ScL Am., Nov. 1977.!+, John R. Kirk, privte communiction to Jerl wlker, 1977,, "The world of Sound", (Dover, 1968), p.158, describes the ttenution of sound by bubbles in liquid. He sys "This is very e.s shown by tpping with spoon or knife tumbler contining beer or stout ~rdth lyer of fom on it. The sound is bsolutely ded: different from the tinkling sound tht the tumbler gives when empty or when LLJed with wter, The fom bsorbs the energy of the vib " he hd tpped his glss immeditely fter it ws fi1led 9 before bubb1es rose to the top, he would hve noticed the of pitch tht constitutes the hot chocolte effect, It t then hve h clled the cold beer effect. 6. IL Hedwin, Jour. Acoust. Soc. Am.2., 1100 (1974). 7. H. H:Lnnert, Phil. Mg. 1., 235 (1933). 8. C. Devin, Jr., Jour. AcousL Soc. Am. 31, 1654 (1959).