Y a
ON THE INFLATING OF BALLOONS RICHARD S. STEIN University of Massachusetts, Amherst
IT
IS a common observation that it is more difficult to start a rubber balloon inflating than to continue. This observation is predictable from the kinetic theory of rubber elasticity' as are several others. Rubber consists of long flexible randomly coiled molecules joined together by chemical bonds called crosslinks. These are introduced during vulcanization. The portion of the molecule between these crosslinks undergoes rapid Brownian motion with an energy which is proportioned to the absolute temperatnre. I n an unstretched piece of ruhber, this random motion results in a particular crosslink's experiencing tugs in all directions with equal probability. When a piece of rubber is stretched, the molecules become straightened out and the average distance between crosslinks is increased. The direction of random motion of the chains is somewhat constrained. Thermodynamically, this is described in terms of a decrease in entropy. A particular crosslink no longer experiences randomly oriented tugs, but rather more in such a direction as to tend to reduce the distance between crosslinks and return the ruhber t o the unstretched configuration. This is the origin of the restoring force. In thermodynamic terminology, because of the entropy decrease on stretching, the free energy increases. This free energy change is equal to the maximum amount of work that can be done a t constant temperature which equals the integrated product of the force times the change in distance. There is no change in the average kinetic energy of the molecules on stretching. All that changes is the resultant force due to incomplete cancellation of the Brownian motion. The situation is somewhat similar to that in an ideal gas, where the kinetic energy of the molecules does not change as the volume is decreased, but the pressure increases as a result of the greater number of collisions of the molecules with the walls. The pressure of such a gas increases as the temperatnre is raised, the kinetic energy of the molecules increases, and the collisions occur with greater force. I n a similar manner, the force on a stretched sample of rubber increases directly with absolute temperature, since the kinetic energy of the molecules and the force of the tugs becomes greater. The thermodynamic criterion of an ideal rubber is that the internal energy is independent of length; that is (bE/bl),,, = 0. This is analogous to the requirement for an ideal gas that (bE/bV), = 0. On the basis of this type of theory, it is possible to derive an equation for the force per unit area in a particnlar direction (the x-direction) on a piece of rubher stretched simultaneously in both the x and y directions':
See, far example, TRELOAR, L. R. G., "The Physics of Rubber IClasticity," Oxford University Press, New York, 1949.
VOLUME 35, NO. 4, APRIL, 1958
where N , is the number of chains per unit volume, Ic is Boltzman's constant, T is the absolute temperature, az is the ratio of the stretched length to the unstretched length in the x-direction, and a, is the corresponding ratio in the y direction. It is assumed that the volume of the sample remains constant during the deformation. For a deformation of the sort occurring on inflating a balloon,
or would then be the ratio of the length of a line on the surface of the balloon in the stretched state to that in the unstretched state. Alternatively, it could be defined as the ratio of the diameter of the inflated balloon to that of the uninflated; o would be the force acting normal to any unit area perpendicular to the surface. When the balloon is inflated, this force of elastic retraction is opposed by that resulting from the pressure of the contained gas. Let us consider the equilibrium on a circular strip dividing the balloon into two hemispherical shells. The restoring force of the ruhber will he ~ Z n r dwhere , r is the radius of the balloon and d is the thickness of the rubber in the stretched state. Since the volume of the ruhber is constant, 4/3 nr2d = 4/3 n~o~do where To and do are the unstretched radius and thickness. The gas inside the balloon may he considered to exert a force equal to the pressure, P , multiplied by the crosssectional area which the balloon would intersect on a plane passing through its center, or j = P.ar2 = P.7a2r02. By equating the two forces and solving for the pressure (using equation (2)) one finds that
The factor in brackets is a constant for a given balloon a t a particular temperature. This pressure is equal to that of the gas inside the balloon, P = nRT/V. It is interesting to note that the volume of the balloon should be independent of temperature since both the pressure of the gas and the elastic force of the rubber increase in proportion to the absolute temperature. A plot of P against a is given in the figure. One observes that there is a maximum occurring when (bP/ba) = 0 which corresponds to a = 7''" 1.4. Thus, the most difficult part of the inflation is passing this high pressure peak, occurring when the diameter of the
balloon is 40% greater than the uninflated diameter. After this, the pressure drops with increasing diameter so that subsequent inflation becomes increasingly easier. Because of the existence of this maximum, there are two possible values of a corresponding to each pressure, one on either side of the maximum. Thus, referring to the figure, balloons having diameters 1.10 and 2.26 times the uninflated diameter (corresponding to points A and B ) should be a t the same pressure and should not change diameter if connected together. A balloon blown up to 3 times the uninflated diameter (point C) will he a t a lower pressure than the one inflated to 1.10 times the uninflated diameter. If these two are connected together, the larger one will increase in size and the smaller will decrease, a surprising but correct prediction. This latter prediction accounts for the observation that when blowing up a toy balloon in the shape of an animal having ears, it is often difficult t o get the ears inflated after the body has been inflated. This occurs because the body and ears can be a t the same pressure with the body on the high a side of the maximum but the ears on the low a side. If this happens, further inflation decreases the pressure so that the diameter of the body increases while that of the ears decreases. This can he avoided at the start by preventing the body from inflating (by constricting it with one's hands) until the ears have also started to inflate so that both parts pass through the maximum approximately simultaneously. The predictions of this theory may be readily checked by connecting an open tube manometer to the balloon while inflating it (using a "T" tube). Ethylene glycol colored with a dye is a convenient manometer fluid. The a may be measured conveniently by ruling some equally spaced lines (using ink) on the surface of the uninflated balloon and following the increase in separation of these lines with inflation using a flexible plastic rule bent to follow the surface curvature. The measured pressures are plotted against a in the figure. The constant [2N,dnlcT/ro]of the theoretical curve was chosen so as to make the two curves correspond in pressure a t the maximum. It is apparent that the shape of the two curves is roughly the same a t low inflations, but above a = 3
there is an upsweep in the experimental curve which is not predicted by theory. This is a consequence of the well-known inadequacy of the kinetic theory of rubber elasticity at high elongations. It results principally from the polymer chains becoming completely stretched out at high elongations. When this happens, it becomes impossible to elongate the rubber further without breaking the chains, hence the retractive force increase. The force increase may also partly result from crystals which grow and strengthen the network a t high elongations. It is in this region of increasing pressure that breaking occurs. The or a t which the pressure starts increasing depends on the extent of vulcanization of the rubber, being lower a t higher degrees of vnlcanization. A consequence of this latter pressure increase is that, there are actually regions where there are three a's corresponding to the same pressure (points D, E, and F of the figure, for example). This may be demonstrated by connecting three balloons, all inflated to different diameters, to a common "T" tube. The unusual pressure-volume relationships encountered here provide a fertile source of unconventional problems in physical chemistry and thermodynamics involving calculation of work, entropy changes, etc., involved in changing the volume of the balloon.
JOURNAL OF CHEMICAL EDUCATION
