Two-balloon experiment

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Fig. 1. Two balloons are connected via a hollow tube. When the valve is opened, the smaller balloon shrinks and the larger balloon expands.

The two-balloon experiment is a simple experiment involving interconnected balloons. It is used in physics classes as a demonstration of elasticity.

Two identical balloons are inflated to different diameters and connected by means of a tube. The flow of air through the tube is controlled by a valve or clamp. The clamp is then released, allowing air to flow between the balloons. For many starting conditions, the smaller balloon then gets smaller and the balloon with the larger diameter inflates even more. This result is surprising, since most people assume that the two balloons will have equal sizes after exchanging air.

The behavior of the balloons in the two-balloon experiment was first explained theoretically by David Merritt and Fred Weinhaus in 1978.[1]

Theoretical pressure curve[edit]

The key to understanding the behavior of the balloons is understanding how the pressure inside a balloon varies with the balloon's diameter. The simplest way to do this is to imagine that the balloon is made up of a large number of small rubber patches, and to analyze how the size of a patch is affected by the force acting on it.[1]

The James-Guth stress-strain relation [2] for a parallelepiped of ideal rubber can be written


f_i = {1\over L_i}\left[kKT\left({L_i\over L_i^0}\right)^2-pV\right].

Here, fi is the externally applied force in the i'th direction, Li is a linear dimension, k is Boltzmann's constant, K is a constant related to the number of possible network configurations of the sample, T is the absolute temperature, Li0 is an unstretched dimension, p is the internal (hydrostatic) pressure, and V is the volume of the sample. Thus, the force consists of two parts: the first one (caused by the polymer network) gives a tendency to contract, while the second gives a tendency to expand.

Suppose that the balloon is composed of many such interconnected patches, which deform in a similar way as the balloon expands.[1] Because rubber strongly resists volume changes,[3] the volume V can be considered constant. This allows the stress-strain relation to be written


f_i = \left(C_1/L_i\right)\left(\lambda_i^2 - C_2p\right)

where λi=Li/Li0 is the relative extension. In the case of a thin-walled spherical shell, all the force which acts to stretch the rubber is directed tangentially to the surface. The radial force (i.e., the force acting to compress the shell wall) can therefore be set equal to zero, so that


\lambda_r^2 = (t/t_0)^2 = C_2p

where t0 and t refer to the initial and final thicknesses, respectively. For a balloon of radius r, a fixed volume of rubber means that r2t is constant, or equivalently


t \propto \frac{1}{r^2}

hence


\frac{t}{t_0} = \left(\frac{r_0}{r}\right)^2

and the radial force equation becomes


p = \frac{1}{C_2} \left(\frac{r_0}{r}\right)^4

The equation for the tangential force ft (where Lt \propto r) then becomes


f_t \propto (r/r_0^2)\left[1-(r_0/r)^6\right].
Fig. 2. Pressure curve for an ideal rubber balloon. When air is first added to the balloon, the pressure rises rapidly to a peak. Adding more air causes the pressure to drop. The two points show typical initial conditions for the experiment. When the valve is opened, the balloons move in the direction indicated by the arrows.

Integrating the internal air pressure over one hemisphere of the balloon then gives


P_\mathrm{in} - P_\mathrm{out} \equiv P = \frac{f_t}{\pi r^2} = \frac{C}{r_0^2r}
\left[1-\left(\frac{r_0}{r}\right)^6 \right]

where r0 is the balloon's uninflated radius.

This equation is plotted in the figure at left. The internal pressure P reaches a maximum for


r=r_p=7^{1/6}r_0\approx 1.38 r_0

and drops to zero as r increases. This behavior is well known to anyone who has blown up a balloon: a large force is required at the start, but after the balloon expands (to a radius larger than rp), less force is needed for continued inflation.

Why does the larger balloon expand?[edit]

When the valve is released, air will flow from the balloon at higher pressure to the balloon at lower pressure. The lower pressure balloon will expand. Figure 2 (above left) shows a typical initial configuration: the smaller balloon has the higher pressure. So, when the valve is opened, the smaller balloon pushes air into the larger balloon. It becomes smaller, and the larger balloon becomes larger. The air flow ceases when the two balloons have equal pressure, with one on the left branch of the pressure curve (r<rp) and one on the right branch (r>rp).

Equilibria are also possible in which both balloons have the same size. If the total quantity of air in both balloons is less than Np, defined as the number of molecules in both balloons if they both sit at the peak of the pressure curve, then both balloons settle down to the left of the pressure peak with the same radius, r<rp. On the other hand, if the total number of molecules exceeds Np, the only possible equilibrium state is the one described above, with one balloon on the left of the peak and one on the right. Equilibria in which both balloons are on the right of the pressure peak also exist but are unstable.[4] This is easy to verify by squeezing the air back and forth between two interconnected balloons

Non-ideal balloons[edit]

At large extensions, the pressure inside a natural rubber balloon once again goes up. This is due to a number of physical effects that were ignored in the James/Guth theory: crystallization, imperfect flexibility of the molecular chains, steric hindrances and the like.[5] As a result, if the two balloons are initially very extended, other outcomes of the two-balloon experiment are possible,[1] and this makes the behavior of rubber balloons more complex than, say, interconnected soap bubbles.[4] In addition, natural rubber exhibits hysteresis: the pressure depends not just on the balloon diameter, but also on the manner in which inflation took place and on the initial direction of change. For instance, the pressure during inflation is always greater than the pressure during subsequent deflation at a given radius. One consequence is that equilibrium will generally be obtained with a lesser change in diameter than would have occurred in the ideal case.[1] The system has been modeled by a number of authors,[6][7] for example to produce phase diagrams[8] specifying under what conditions the small balloon can inflate the larger, or the other way round.

See also[edit]

References[edit]

  1. ^ a b c d e Merritt, D. R.; Weinhaus, F. (April 1978), "The Pressure Curve for a Rubber Balloon", American Journal of Physics 46 (10): 976–978, Bibcode:1978AmJPh..46..976M, doi:10.1119/1.11486 
  2. ^ James, H. M.; Guth, E. (April 1949), "Simple presentation of network theory of rubber, with a discussion of other theories", Journal of Polymer Science 4 (2): 153–182, Bibcode:1949JPoSc...4..153J, doi:10.1002/pol.1949.120040206 
  3. ^ Bower, Allan F. (2009). Applied Mechanics of Solids. Taylor & Francis. ISBN 978-1-4398-0247-2. 
  4. ^ a b Weinhaus, F.; Barker, W. (April 1978), "On the Equilibrium States of Interconnected Bubbles or Balloons", American Journal of Physics 46 (10): 978–982, Bibcode:1978AmJPh..46..978W, doi:10.1119/1.11487 
  5. ^ Houwink, R.; de Decker, H. K. (1971). Elasticity, Plasticity and Structure of Matter. Cambridge University Press. ISBN 052107875X. 
  6. ^ Dreyer, W.; Müller, I.; Strehlow, P. (1982), "A Study of Equilibria of Interconnected Balloons", Quarterly Journal of Mechanics and Applied Mathematics 35 (3): 419–440, doi:10.1093/qjmam/35.3.419 
  7. ^ Verron, E.; Marckmann, G. (2003), "Numerical analysis of rubber balloons", Thin-Walled Structures 41: 731–746, doi:10.1016/S0263-8231(03)00023-5 
  8. ^ Levin, Y.; de Silveira, F. L. (2003), "Two rubber balloons: Phase diagram of air transfer", Physical Review E 69: 051108, Bibcode:2004PhRvE..69e1108L, doi:10.1103/PhysRevE.69.051108 

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