F. E. CONDON and MICHAEL FRYD hec city College, New York 31, N. Y. ..
A RECENT art,iclowith t,hist,itle by R.. S. St,einlprompts 11st,o report the resuks of n similar invest,igatiou. Our experiment,al techniynr a ~ ~treat,ment d of the data differ from Stein's. We have measured simultaneously the pressure ioside a toy balloon and its volume by means of the apparatus shown in Figure 1. The balloon was at,tached t o a one-holed rubber stopper on the end of an L-shaped glass tube and supported inside an aspirator
m .
To mono
Figure 1.
Ilpp~retu.for Mee."mm.nt of the N.t P...aun, Balloon end Its Velum. During Inflation
bottle as shown in the figure. The glass tube holding the balloon was attached via a T-tube to a U-shaped mercury manometer and, through a glass stopcock, to the reducing valve on a nitrogen cylinder. The aspirator bottle was then filled completely with water and closed with a one-holed rubber stopper carrying an eduction tube leading beneath the surface of water contained in a graduated cylinder. All rubber stoppers and rubber tubing connections were wired to prevent leakage. A constant water level in the graduated cylinder indicated pressure equilibrium in a tight system. Nitrogen was then admitted to the balloon in small portions, and readings of the mercury manometer and of the volume of water in the graduated cylinder were taken after each portion. Pressure-volume data from a typical run are plotted in Figure 2. This figure shows the early pressure peak which was described and interpreted by Stein and which is responsible for the common observation that it is harder t o start the inflation of a rubber balloon than it is t o continue it, once started. The figure also shows the pressure minimum observed by Stein and attributed by him to complete straightening out of the rubber polymer chains, making it impossible t o stretch the rubber farther without breakine the chains. ' STEIN,R. S., J. CHEM.EDUC.,35,203 (1958). See SEARS, F. W., AND M. W. ZEMANSHY, "College Physics," pp. 237-38, Addimn-Wesley Publishing Co., Inc., Cambridge, Mass., 2nd ed., 1953. We itre indebted to Profe~so~ Zemansky forpointing out this analogy. L. R. G., "The Physics of Rubber Elasticity," See TRELOAR, Oxford University Press, Nea Yark, 1949.
The balance of forces in an inflated balloon is analogous to that in a soap bubble. One can define a film tension, 7, as the force per unit length in a stretched rubber membrane, entirely analogous to the surface tension of a liquid. For an inflated spherical balloon of radius, r, the force of elastic retraction along a great circle, 2my, i< balanced by a force, rrr2P, due t o the difference in pressure, P , inside and outside the halloon.2 By equating the two forces and substituting for r from the formula for the volume, V = 4m3/3, one obtains the relationship
That is, the pressure is inversely proportional to thc cube root of the volume. The same relationship is derivable from the following thermodynamical considerations. As the balloon is inflated, an increment of pressure-volume work, d(PV), is done against the film tension, and results in a corresponding increase in film energy, analogous to surface energy, rdS, where S is the surface area of the halloon: d(PV)
=
" dS
For a regular three-dimensional figure, S is proportional to V". (For example, for a sphere, S = 4rrr2 and V = 4?rra/3, from which S = (3&r)'/'V%.) Thus during inflation of a balloon, d(PV) = rdV": from which after integration P = V-'/r (2) The foregoing derivations are based on the assumption that the film tension, y, is independent of the extension. It can be shown readily that this is equivalent to assuming a Hooke's law relationship between the forco, f, and the extension, I. That is, f = 7 X I, or f/l = 7, a constant. The assumption is consistent with theories of rubber ela~ticity.~
Fieurs 2.
Variation of Net Pr-"re
Insid. a Balloon with It. Volvme During Inflation
JOURNAL O F CHEMICAL EDUCATION
Equations (1) a11d (2) suggest that a plot of log P against log V should give a straight line with slope. Such a log-log plot of the data of Figure equal to 2 is shown in Figure 3. Most of the data do indeed fall near such a straight line4 and indicate the assumption that the film tension, 7 , is constant is valid for moderate extensions. Many interest,ing aspects of the ununual pressurevolume l~lat,ionshipst.ncount,ered during inflat,io~~ of L: balloon have been mentioned by Stein. To them may be added the possibility of evaluating. the film tension, y , and the observation of hysteresis during return of the inflated balloon to the flaccid state. 4
We an: indehted t,o Mr. A l Alaekshurg for help in preparing
the fiwrea
VOLUME 35, NO. 10, OCTOBER, 1958
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