Sherril D. Christian and Eric Enwall The University of Oklahoma
Norman. 73019
I I
Bubble Pressure and Volume A demonstration experiment
Beginning.chemistry students rarely learn much surface chemistry or physics; even in t h e physical chemistry course, t h e treatment of surface phenomena is usually cursory.'l'his is unfortunate, since there a r e many striking a n d beautiful experiments which can h e performed t o show quantitatively t h e effects of surface forces.' An interesting phenomenon, which illustrates several imnortant nrincioles. . . is t h e deoendence of t h e size of a soan huhhle on pressure. 1.pt us introduce this subject hy describing t h e followine "hlack-box" e x ~ e r i m e n t .Students are shown t h e apparatus diagrammed in kigure 1;they are able to see the ~ r o t r u d i "n e" elass tubes (4m m a d . ) a n d s t o ~ c o c k s .t h e attached micrometer huret, a n d t h e sensitive pressure gauge a t t h e ~ i e h tT. h~e students a r e unable t o see behind t h e oDaaue screen, which hides all of the apparatus t o the left of s&&k A. T h e atmaratus is assembled a n d readv t o use before the students&rive t o watch t h e demonstratibn. To begin the experiment, stopcocks A and R are opened, and the observers can see that the pressure reaches a value of zero on the differential oressure "eanee " .Le... AP = 0). Now..sto~eock . A is elwed. and with stopcock R still open, the volume of the air in the micrometer huret is slowly decreased. Sets of values of the pressure difference and the corresponding huret volume are recorded. At first, as the volume of the air in the pipet (V) is decreased, AP increases. This seems reasonable enough, since the air in the system is being compressed. Obviously samething in the hidden part of the apparatus offers some resistance to the flow of air into the system. However, as V is further decreased, AP begins to level off, then reaches a maximum value, and finally becomes smaller as V is reduced by sizeable fractions of a milliliter. Figure 2 shows data ohtained with the apparatus in a given experiment. The curve indicates that after a certain value of V is reached, further "compressionn of air leads to a decrease in AP. The observers are asked to repeat the experiment themselves, varying V in bath directions from the point at which A P has its maximum value. They can at any time open stopcock A, note that @returns to zero, close stopcock A again, and once more start the compression cycle. (Students are asked not to increase the air volume after re-zeroing AP, for fear that they might blow the experiment.) Eventually, the experiment ends spontaneously, when it is noted that APgaes to zero and stays there regardless of the position of the pipet plunger. At this point, everyone is probably willing to concede that the hidden region of the apparatus has sprung a Leak, or has otherwise became opened to the atmosphere. Lively discussions can now be had reearding the nature of the unseen Dart of the~, svstem.. which must have ronrmned a mystrrlouz marrrial that firat allwed t h pressure ~ dit'fwenr~alto increase and then to drcrease as the air volume on the right hand side of the apparatus was reduced. Unless the students have already studied the properties of surface films, this discussion is not apt to lead to any profound conclusions, and it is orobablv time to remove the screen and exoose the whole aooara.. tus. The hidden part of the apparatus (shown in Fig. 3) turns out to be quite simple. Only an open stopcock ( C ) and an open tube (D) are visible, so it is certainly true that the apparatus is now in communication with the atmosphere. We can reload the apparatus by dipping a wire ring into a soap bubble solution and transferring part ofthe film spanning the ring to the end of tube D. (This is conveniently done by passing the ring over the end of the tube and breaking the film remainine on the rine" aeainst stoocock C.1 To increase the lifetime of " the tilm, it is worthwhile placi& an ,,pen henker, partly fdled wirh water, around the tube, with the liquid lwel one or two em hdow the end of tube D. I t is now instructive to repeat the measurement of values of V and AP, while watching the growth ofthe soap bubble. As the film changes from the plane surface (at zero pressure) to progressively smaller radii 538 / Journal of Chemical Education
Syringe Figure 1. Damstration
a
an
apparatus.
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3
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+.I Figure 2. Bubble pressure versus volume.
u
Figure 3. Exposed apparatus.
of curvature, A P increases; finally, as the bubble becomes still larger (passing beyond the hemispherical shape) A P decreases.
AP decremring
A P increasing .I
7
AP = 0
APm.x
'For discussions of the properties of surface films and other curved interfaces see: Adamson, A. W., "Physical Chemistry of Surfaces," 2nd. ed., Wiley Interscience, New York, 1967: Boys, C. V., "Soap Bubbles and the Farces Which Mould Them," reprint edition, Dover, New York, 1958: van Olphen, H., and Mysels, K. J., "Physical Chemistry: Enriching Topics from Colloid and Surface Science," IUPAC Commission 1.6, Theorex, La Jolla, Calif., 1975. 2The pressure gauge is a temperature stabilized transducer assembly manufactured hy Validyne Engineering Corporation, Northridge, California. In the differential mode, it is capable of detecting changes in pressure on the order of 2-5 X torr. The buret is an RGI 2-ml microburet capable of delivering a given volume to within f0.002 ml.
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Ll' Figure 4. Two bubble experiment.
Most observers will concede that there is some connection between the curvature of the film and AP-the greater the curvature (i.e., the smaller the radius of curvature) the greater the pressure. The Laplace equation' can he derived a t this point, or simply presented in the form A P = 4ylR (1) where y is the surface tension and R is the radius of curvature of the bubble. (Note that A P = 4rlR. not 2rlR as in hlawine a bubble in a liquid, in which ease there iionly oneinterface.) 1fro;;thc radiusof thc inside of the g l ~tube, ~ s we can we that H 3 romd that L1I'reaches its maximum value as R r a A number of interesting calculations can be made based on data such as those displayed in Figure 2. First of all, y can be calculated if ro is known; thus from AP,,
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r = AP,,.,
.ro/4
The value ofro can of course be obtained by direct measurement, hut i t can also he inferred from ooints of the APversus I AVI, curve. Corresponding to a given value of 31' (e.g.. 0.20 Iarr) one can read two different vnluesof A V . (In Figure 2, the pressure U.20 tcm is reached at (A\'( = 0.052 ml and 0.285 m l ~ The ium of these twovolumes ia the volume of s complete sphereof radius R; this can be seen by examining the accompanying diagram ~
whpre R ir the radiuscl rurvaturecalculated fm thr hubbleat pressure 1P. If the volumed thegos in the total apparatus is fairly small (-10 ml or less) it can he shown that i t is a good approximation t o equate I AVI measured with the micropipet to the volume of the map bubble. Thus. the entire curve in Figure 2 can be calculated from the equation for the volume of a sohe&al seement and the e x ~ r e s s i o nA P = AP-.. r ~J R, . . where --,,~ . R $inferred f;om the observed value of AV and where roand U ' , , are the only fitting parameters.The solid curve in Figure 2 has been calculated using ro = 0.35: cm and lPm. = 0.242 tom. I t can he seen that this model provides a good fit of all the data, have not been optimized by although these values of ro and AP,., least squares analysis. Using a nonlinear least squares p r o g ~ a mwe ,~ have found that the best values of the oarameters are ro = 0.353 0.002 em and AP-.. = 0.241 z t 0.003 tar;. from which the value of the surface tension, 7 = 28.4 0.4 dyne cm-'. can be calrulaId. The rmt mean square deviation in L1P is 0 OOfi torr for this set uf parameters; the cslculaled curve shown in F i p r e 1 is nlmosr indiitin~uiahahlr visually from thp curve hmed on the lea31 squares con it ant^.^ Although many addnional eaperimentncnn be performed with the amaratus. onlv one more demomitratton will be described. F~rst,the &i'crometer pi& is removed, and soap bubbles of unequal size are blown on the two tubes. as is shown in Fieure 4. Staococks A and B are closed and C isopen, st, that the value of U'correqwnding to the preaaulr differential acrmr the curved surface d the amall huhhle can he measured. Now, stopcock B is cracked open ever so slightly, so that air can begin to flow through i t from t h e high presswe to t h e low pressure srde. The obvious questions t o have asked before opening B are, "What will happen t o the two hubhles, and how will @vary as the pressure equalizes?" Students who have been paying close attention should respond that the small bubble will become smaller, pumping up the big bubble. They should also he able to predict that the observed AP will go through a maximum, and then decrease rapidly to a fairly small value as the bubble a t the right practically disappears. The final film on the right hand side will have the same radius of curvature as the large bubble
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Hoth huhbles together have a volume equal to 4rRV3. The authors wish 141thank Mr. Dnvld C. Thomas ior several helpful discussions. At the given pressme, the two buhhlen must have tht. same radiosof curvature, R , so that the volume of the smaller bubble is just equal to the volume of thespherical segment the larger bubhle lacks tomake a complete sphere. Therefore the equation
IAVIl+ IAVlz=0.337ml= 4rR3/3 can be solved for R, giving R = 0.432 em. Now y can he calculated directly from 0.2 X 0.432 X 1.013 X 106= 28,8 y=AP.R14= 4 X 760 Knowing the maximum pressure, we can calculate ro from
3NLLSQ-A nonlinear least squares program written locally using an algorithm given by Marquardt, D. W., J. Soc. Indust. Appl. Math., 11,431 (1963). 41n performing the least squares calculations, corrections are made for the effect of finite system volume on AV. Thus, ( = I AV,,dI - V,,, where V., = VSmernX AP1760 (assuming that 760 ton is the ambient pressure). For a 10 cc system volume a t a = 0.003 ml. The approximate system volpressure of 0.24 tom, V, ume can be determined by a) closing stopcocks C and A and opening B (see Fig. 3) and h) observing the change in pressure caused by chaneine .. ,. the buret volume bv a known mount. Thus. if lAVl = 0.100 ml causes a pressure changebf 5.110tom, I he vdumr bf ;he system to ,,.,,'\ = 760 X O.l/$ = 15.2 ml. the w h t ofstopcock A is ~
ro = RAPIAP-,
= 0.432 X 0.2 = 0,357
0.242
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Volume 55, Number 8, August 1978 1 537
