Exploring the thermodynamics of a rubber band David Roundy Department of Physics, Oregon State University, Corvallis, OR 97331 Michael Rogers Department of Physics, Ithaca College, Ithaca, NY 14850 We describe an upper-division experiment in thermal physics where students measure the tension of a rubber band as a function of temperature and length, and use a Maxwell relation to find the change in internal energy and entropy for an isothermal stretch. This allows students to experimen- tally check the predictions of the entropic spring model for elastomers and observe that the entropy does indeed decrease as a rubber band is stretched. I. INTRODUCTION First Law of Thermodynamics, combined with the defini- tion of work as the dot product of force and displacement. Rubber bands provide an appealing subject for a ther- We could use the differential relation given in Eq. (1), but mal physics laboratory experiment. In contrast to most since we are working at constant T , the Helmholtz free solids and liquids, which respond weakly to small changes energy F provides a more useful starting point in temperature, rubber's tension noticably increases with dF = τdL SdT: (2) increasing temperature. This Gough-Joule effect|which − is contrary to student expectations|is a standard exam- The corresponding Maxwell relation ple in courses in statistical mechanics. Combined with low cost and familiarity to students, these factors make @S @τ the lowly rubber band an almost ideal material for a ther- = ; (3) − @L @T mal physics experiment. T L Numerous laboratories and classroom demonstrations tells us that we can determine how entropy changes with address how the tension of rubber bands increases with length at fixed temperature by measuring how the ten- 1{6 temperature, or to measure mechanical properties of sion changes with temperature at fixed length. At the 7,8 rubber bands at room temperature. However, few of same time, measurement of the tension reveals how the these experiments involve carefully measuring a rubber free energy varies with isothermal changes in length band's equation of state. The majority of the exper- iments that demonstrate the surprising dependence of @F τ = : (4) tension on temperature make a purely qualitative demon- @L stration in which the temperature is changed, but not T measured.2{5 Savarino and Fisch describe a laboratory Thus, measurements of τ and (@τ=@T )L, as a function of experiment that involves measuring the tension in a rub- length, allow us to find ∆F , ∆S, and ultimately ∆U by ber band as a function of temperature and extension, integration. using a space heater to change the temperature of the rubber band.6 Their experiment is used to teach data analysis methods by using multiple approaches to extract III. EXPERIMENTAL ARRANGEMENT the parameters in an equation of state for the rubber band; however, there is no thermodynamic analysis of Figure 1 shows a photo and a schematic of the exper- the resulting values, nor do they address the applicabil- imental setup. Students stretch a rubber band from a ity of the equation of state. In this paper, we introduce hook in a stopper at the bottom of a glass tube to a a rubber band experiment that uses Maxwell relations to chain connected to a force meter at the top of the tube. measure changes in entropy and internal energy. This setup allows the rubber band to be completely sub- merged when the tube is filled with water. It also allows students to quickly alter the length of the rubber band II. THEORETICAL BACKGROUND by changing which link of the chain is hooked onto the force meter. The thermodynamic identity for a rubber band is During the experiment, students adjust the tempera- ture by pouring water into the tube. They then measure dU = T dS + τdL; (1) tension as a function of rubber band length for different water temperatures. By emptying the water out of the where T is the temperature, τ is the tension, U is the top of the tube, students can efficiently replace the wa- internal energy of the rubber band, S is its entropy, and ter and change the temperature using water prepared by L is its length. This relation follows naturally from the mixing boiling water, room-temperature water, and ice. 2 force meter 3.5 3.0 chain 2.5 2.0 1.5 Tension (N) thermometer 1.0 hook 0.5 67.0◦C 1.4◦C 0.0 rubber 24 26 28 30 32 34 36 38 40 stopper Length (cm) FIG. 1: A photograph and schematic of the experimental ap- FIG. 2: (color online) Tension versus length at the highest paratus. and lowest temperatures measured. The solid lines are the tension as the rubber band is stretched and the dashed lines are the tension as it relaxes. The difference between the two Once the water is added to the system, the temperature curves is due to hysteresis. is monitored with an electronic thermometer while stu- dents make their measurements; several measurements can be made for different lengths before the temperature The hysteretic behavior of the rubber band is eas- has dropped (or risen) by as much as 1 ◦C. ily visible in Fig. 2. The tension during stretching is Unfortunately, deviations from ideal elastic behavior 0:1 N greater than the tension relaxing (at the maxi- present a major challenge. Ideally, a rubber band should mum∼ length there is no hysteresis because this point was return to its original state when released. However, rub- only measured once). Although this change in tension ber deviates from elastic behavior in two ways. First, is less than 10% of the measured value, it is comparable rubber bands only slowly return to their original state to the change in tension due to temperature. By mea- after being released, leading to hysteresis in tension suring the tension for the same sequence of lengths|and measurements|a different tension is measured when a in the same order|students can achieve reproducible re- rubber band is stretched to a given length than when sults that distinguish between the effects of temperature it is stretched further and relaxed to the same length. and hysteresis. The second deviation is due to plasticity|if a rubber Using the same data, students next plot tension versus band remains stretched for too long, it never returns to temperature for each of the measured lengths, as shown its original length. Avoiding very high tensions (greater in Fig. 3. The trend of increasing tension with increasing than about 6 N) and maintaining a relaxed state between temperature is again evident. Because we plot all tem- tension measurements (e.g. while changing the tempera- peratures and lengths that were measured in this plot, it ture) reduces plastic behavior. is apparent that experimental uncertainties in tension are significant. Some of these uncertainties come from hys- teretic effects, which can be seen by comparing the solid IV. ANALYSIS and dashed lines, which correspond respectively to ten- sions measured while stretching and relaxing the rubber As shown in Fig. 2, students plot tension versus length band. for a few different temperatures. The data in this figure Because most errors are due to rate-dependent hys- was taken starting at the shortest length (24.5 cm), mov- teretic and plastic effects—which tend to be systematic, ing to the longest length (38.8 cm), and returning again rather than statistical|error analysis is challenging in to the shortest length. Solid lines show the tension as this experiment. These effects depend on the time the the rubber band is stretched and dashed lines show the rubber band spends at each length before the tension tension as it is subsequently relaxed. The close proximity measurement is taken, which is challenging to control. of the measured curves shows that the variation of the The thermal equilibrium tension should be measured af- tension due to temperature ( 0:2 N) is small compared ter waiting for the tension to stabilize after changing the with its variation with length (∼ 2 N). Figure 2 also shows length. However, the tension takes a long time to fully that the tension increases with∼ increasing temperature, stabilize|too long for the experiment to be completed contrary to student expectations. in a few hours|and waiting for this to happen would 3 3.5 1.0 − 3.0 1.5 − 2.5 2.0 (mN/K) − L 2.0 ∂τ ∂T 2.5 − 1.5 − Tension (N) = T 3.0 1.0 − ∂S ∂L 38.8 cm 28.1 cm 3.5 0.5 35.2 cm 24.5 cm − 31.7 cm 0.0 4.0 0 10 20 30 40 50 60 70 − 24 26 28 30 32 34 36 38 40 Temperature (◦C) Length (cm) FIG. 3: (color online) Tension versus temperature for each FIG. 4: (@S=@L)T versus length, as extracted from the slopes measured length. As in Fig. 2, the solid lines represent mea- found from Fig. 3. The solid and dashed lines respectively surements taken with increasing length, while dashed lines correspond to slopes taken from the data as the rubber band correspond to decreasing length. is stretched and relaxed. over the range of temperatures studied. Once students Potential Change (J) T ∆S −0:09 have created Fig. 4, they can numerically integrate with ∆F 0.22 respect to length to find the entropy change for their ∆U 0.13 isothermal stretch @S ∆S = dL: (6) TABLE I: Results for changes in thermodynamic potentials jT @L found by integrating the curves in Figs. 2 and 4. This data Z T corresponds to an isothermal stretch of 14 cm at 30 ◦C.