International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04 9

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International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04 9 Investigation on a Practical Model to Explain the Temperature Change of a Deflating Balloon Yongkyun Lee*, Yongnam Kwon

1 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04 9 Investigation on a Practical Model to Explain the Temperature Change of a Deflating Balloon Yongkyun Lee*, Yongnam Kwon Korean Minjok Leadership Academy, Gangwondo, Korea *Corresponding author ykhl1itj@naver.com Abstract When air quickly escapes from a balloon, the temperature of the surface drops significantly. Despite the simplicity of the phenomenon, there exist little previous research about the temperature change of a balloon. Scientists have mainly studied the temperature change of a rubber band, which is one or two dimensional, following its expansion and contraction. Because of the complicated explanations based on the entropy, it is extremely difficult for high school students to understand most existing researches. Therefore, in this paper we explain a practical model to predict the temperature change of the deflated balloon according to the volume change and the position on the balloon. Then, we examine the validity and the reliability of the model by showing the experimental data. Physics educational value of this research is also discussed. Index Term deflating balloon, temperature, pressure, entropy, rubber, polymer I. INTRODUCTION Rubber warms up, when stretched rapidly. On the other hand, it cools down when contracted. This unusual characteristic of rubber was first officially by John Gough ( ), an English natural and experimental philosopher. One can observe this phenomenon easily with an ordinary balloon. As air escapes from an inflated balloon, it becomes cooler to touch. More theoretical and experimental researches on natural and synthetic rubber were conducted in the middle of 20 th century, as the tire industry was boomed up. S. L. Dart and his colleagues observed the temperature changes with synthetics and natural rubbers. They plotted hysteresis curves of the changes on fast stretching of rubbers[1]. R. S. Stein modelled and verified the unusual pressure-volume relationships of inflating balloon. It has provided a fertile source of unconventional problems in physical chemistry and thermodynamics involving calculation of work, entropy changes, etc., applied to the volume change of a balloon[2]. F. Rodriguez suggested demonstrations whereby theoretical predictions can be illustrated either in a qualitative or a quantitative manner[3]. The demonstration includes considerations of uniaxial and biaxial stretching, as well as simple shear. D. R. Merritt, and F. Weinhaus derived an equation between the internal pressure and the radius of a rubber balloon. The theoretical pressure curve was shown to be experimentally verifiable in the case of low to moderate extensions, as long as the effects of hysteresis are ignored[4]. R. J. Farris provided an idealized mechanical and thermodynamic analysis of the rubber cycle and compared it to an equivalent cycle wherein a gas is the working fluid. His analysis demonstrated that elastomers are ideally suited for energy conversion when only small temperature differences are available[5]. D. Roundy, and M. Rogers described a way to measure the tension of a rubber band as a function of temperature and length and how to use the Maxwell relation to find the change in internal energy and entropy of an isothermal stretch. They gave us insights to experimentally check the entropic spring model of elastomers and observe that entropy does indeed decrease when a rubber band is stretched[6]. Most previous researches are insightful, yet their approaches are very difficult to understand for many people. In this paper, we build easily understandable theoretical model to predict the temperature change of the surface of a deflating balloon. And we verify our model by experiments. II. THEORETICAL APPROACH Most existing researches have some limitations. Many early stage researches focused on the observation of the phenomenon. Newer researches, motivated by rubber (tire) industry, tried to build temperature change models of rubber that is stretched only in one or two directions. In this paper, we theoretically model the temperature drop of the surface of a three dimensionally deflating balloon. A. Model for Restoring Force of Rubber by Molecular Approach The restoring force of rubber could be analyzed by rubber s molecular model. Figure 1 describes the molecular structure of rubber. Fig. 1. The molecular structure of rubber From [7], according to the first law of thermodynamics, the restoring force can be expressed as

2 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04 10 In an ideal rubber, which can be assumed, total internal energy is constant even if the length of rubber changes, thus (1) To figure out the, the restoring force, we need to know the entropy change according to the length of rubber. To do so, first, entropy of rubber molecules of Figure 1 is calculated. In Figure 1, the number of possible molecular distribution is And apply Tayler series, By Stirling s approximation, for large N we have (3) Apply this to, then we can obtain an equation for entropy: Since polymer, as a macro molecular material, is consisted of enormous number of chains, we can safely assume. So equation (3) can be reduced to -2x ( = -. Consequently, the restoring force of rubber is (4) where, is the Boltzman constant. Then we put the entropy S in equation (1), (2) We can get the restoring force if we know and. From Figure 1,,, That means the restoring force of rubber is proportional to the temperature. This explains why rubber expands when the temperature goes down and Hooke s law is correct. When the rubber expands in two directions (x-axis, y-axis), we have to consider the polymer chains aligned in x and y direction. B. Theoretical Temperature Model for Deflating Balloon Though a real inflated balloon looks much like an egg, we assume that the balloon is a perfect sphere, consisted of rings with infinitesimal width as in Figure 2. We can calculate the change of temperature through linear integral of the restoring force, as shown in the previous section., Thus, and are: Fig. 2. Theoretical model of a deflating balloon. Then we replace them in equation (2), after let Due to the symmetry of sphere, the temperature of all the points on a ring is the same. The net temperature change is the sum of its change due to x-direction deformation and y- direction deformation. In other words, the temperature change depends on the initial radius of the balloon and latitude on the surface.

International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04 11 Let s begin with the decrease of temperature due to the x- direction contraction.

3 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04 11 Let s begin with the decrease of temperature due to the x- direction contraction. ) (5) Similarly, the temperature change of a ring due to the y- direction contraction can be obtained. is different from the general spherical coordination system because we set the direction of as parallel as the ring surface. Fig. 3. The restoring force according to expansion ratio. According to Figure 3, the restoring force of rubber is linearly related to the ratio of expansion. Based on the slope of the graph,, or is measured as 45.3 when the rubber is in the 18 water. In room temperature, 25, will be 49.8 since the coefficient is proportional to temperature. The second experiment is to measure the temperature change of a point on the surface of a deflating balloon. Figure 4 is the experimental setting. We used two infrared thermometers to point the same position on the balloon before and after deflation. Let, in as Then the temperature change by retracting rubber is: (6) III. EXPERIMENTS To verify the temperature change model, Equation (7), we conducted multiple experiments. First, we found, then we measured the temperature drop of the surface of a deflating balloon. The first experiment is to find the restoring force coefficient,, of Equation (7). We find by expanding a piece of rubber cut from a balloon. Then, based on the value of, we can mathematically calculate (7) Fig. 4. Measurement of temperature of deflating balloon Figure 5 demonstrates the temperature drops. The solid line shows the measured values. The dotted line shows the theoretically calculated values. The theoretical values are obtained by the Equation (7). We used the coefficient measured in the first experiment. Since the restoring force of rubber is proportional to the width of rubber, we can obtain using the ratio of the length and width of each ring. The location, which the thermometer targets as in Figure 4, is the point where (of Figure 2) is. Thus, the width in x-axis is and the width in y-axis is., the area of measurement (a circle of 1cm radius) of the infrared thermometers can be calculated by the second law of cosine. After all, of a balloon with radius

4 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No: cm is. is specific heat capacity of rubber(=1.35). other words, when the square is closer to the equator of the balloon. The difference between the measured values and the calculated values is mainly due to the disparity between the assumption that the balloon is a perfect sphere and the reality that the balloon is an oval shape. Also, the coefficients in the main balloon temperature change equation were calculated by the experiment results of rubber band experiments. Yet, the context of a two dimensional rubber band and the balloon is largely different, inevitably causing the difference in theoretical and experimental values. Nevertheless, the tendency of the both values correspond, demonstrating the validity of the model. Fig. 5. Temperature drop of deflating balloon As we see in Figure 5, the measured temperature drops are about the same as theoretical values of our model. In other words, our theoretical model for temperature drop on the surface of deflating balloon is valid. When the initial volume of the balloon is larger, temperature drops more because the change of entropy increases. The difference between the measured value of temperature decrease and theoretically calculated value is mainly due to the effect of adiabatic process of a deflating balloon and the possible coefficient error. The third experiment is to measure the temperature drop on various points on the surface of a deflating balloon of which the initial radius is 13cm. Figure 6 shows the decrease of temperature on various points on the balloon s surface. Again the solid line shows the measured values. The dotted line shows the theoretical values. Fig. 6. Temperature drops on various points on balloon's surface In Figure 6, we can find the temperature drops vary according to the position on the surface following the trends we predict with our theoretical model. The biggest drop is found in the middle of balloon, because those areas experience the greatest change of entropy. This phenomenon can be observed by drawing an equal size square on two different parts of a balloon and blowing the balloon. The size of the square after inflation is large when its longitude is small, in IV. CONCLUSION When the balloon is deflated, its temperature decreases. There exist numerous factors that affect its temperature. We proposed another theoretical model to predict the surface temperature of a deflating balloon. It depends on the restoring force and specific heat capacity of balloon material and the position on the balloon surface. Our model assumes the balloon to be a symmetrical sphere. We verified our model with a series of experiments by comparing the measured temperature drops with those calculated based on our theoretical model. By revisiting this unusual phenomenon, our contribution could be found in educational purposes. Temperature drop in deflating balloon is easy to experience but hard to understand. In this study, we predicted the temperature changes from the rubber sample to three dimensionally deflating balloon. To build our theoretical model, we used basic calculus only. Thus, undergraduate students could easily and scientifically understand the entropy concept and its relation to the temperature drop in deflating balloon. There exist numerous rubber products, like tires and buffers. If these objects temperatures change when they are deformed, the products may become more prone to crack, and its functionality may decrease. Based on the molecular approach of rubber contraction or expansion, a new type of rubber whose temperature is kept constant even during deformation can be created. For further research, we need to integrate the effect of adiabatic process of deflating balloon. And there needs to be more elaborate experiment setting to obtain more accurate coefficients, as well as temperature data. ACKNOWLEDGMENT This research has been advised by Yonsoo Kim (Korean Minjok Leadership Academy) and Yongseob Park (Kyung Hee University). Also, it has been financially supported in part by the Education Office of Gangwondo. REFERENCES [1] S. L. Dart, R. L. Anthony, and E. Guth, "Rise of temperature on fast stretching of synthetics and natural rubbers," Industrial & Engineering Chemistry, vol. 34, no. 11, pp , 1942.

5 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04 13 [2] R. S. Stein, "On the inflating of balloons," Journal of Chemical Education, vol. 35, pp , [3] F Rodriguez, "Demonstrating rubber elasticity," Journal of Chemical Education, vol. 50, pp , [4] D. R. Merritt, and F. Weinhaus, "The pressure curve for a rubber balloon," American Journal of Physics, vol. 46, no. 10, pp , [5] R. J. Farris, "Rubber heat engines, analyses and theory," Rubber Chemistry and Technology, vol. 52, no..1, pp , [6] D. Roundy, and M. Rogers, "Exploring the thermodynamics of a rubber band," American Journal of Physics, vol. 81, no. 1, pp , [7] L. R. G. Treloar, The physics of rubber elasticity. Oxford University Press, pp.28-41, 1975.

2014 Problem 12: Cold Balloon As inflated, a rubber balloon s surface becomes cooler to the touch

2014 Problem 12: Cold Balloon As inflated, a rubber balloon s surface becomes cooler to the touch 33 As it is clear from figure 17, height of water and cola Radius mm drops increased and height of syrup and salty water decreased during solidification. Because of vertical expansion of water and cola