# Entropy of a Rubber Band

## Materials

Box of large rubber bands

## Procedure

Hand out rubber bands to the class. Ask them to stretch the rubber band, then place it on their upper lip. They will feel heat. Release the tension on the rubber band and touch it to their upper lip, the rubber band will feel cooler.

## Discussion

Entropy change (stretching and contracting at constant T):

Heat is produced in the rubber band as it is stretched. If we stretch slowly in air at room temperaure, we have a reversible, isothermal process with q< 0 (heat is lost to the surroundings). Then:

$$\ce{\Delta S_{stretch} = \frac{q_{rev}}{T} < 0}$$

in the stretching process. Since entropy is a state function:

$$\ce{\Delta S_{contract} = \frac{q_{rev}}{T} > 0}$$

for the reverse process of slowly allowing the rubber band to contract. The rubber band has lower entropy, (is more ordered) when stretched. Under tension, the molecules in a rubber band line up and the arrangement becomes much more ordered, lowering the entropy. The criterion for spontaneity for an isolated system is $$\ce{\Delta S_{sys} > 0}$$ . Suppose the system is the rubber band and the air in a large room. Therefore, if a constraint keeping the rubber band stretched is removed, it will spontaneously contract, with ∆Ssys, contract > 0

Enthalpy change (with heat source):

There is another "PV"-like term in the enthalpy, namely "-FD", where F is the force and D is the distance . The "-" sign is there because the force is taken to be positive (as is the pressure) but the force of the rubber band is in the opposite direction of the force on a piston, pulling in instead of pushing out. Thus:

H=E + PV - FD è q + w + PV - FD If we heat the rubber band with a weight attached, the force F is constant, but the band contracts. The work w is F∆D-P∆V, (the P∆V term is very small), and ∆H= q + F∆D-P∆V-(F∆D-P∆V) = q. Since we heated the band up, ∆H= q>0. This can be done reversibly, so that:

$$\ce{\Delta S_{heat} = \frac{\Delta dq_{rev}}{T} > 0}$$

This entropy change toward more disorder is perhaps not so surprising as in the isothermal case for contraction, since here it has been heated to a higher T.

LeChatelier: Heating the rubber band contracts it. Therefore, according to LeChatelier's principle, stretching the rubber band MUST increase it temperature, because by getting hotter it tends to contract, resisting the stress of stretching.

## References

So here is my attempt to interpret the rubber band experiment.

The simplest way (to my mind) is to focus on the band as it is allowed to relax. So stretch it and hold it there for a bit, then let it relax (quickly) and feel whether it cooled or heated by touching the band to the upper lip.

The discussion of an enthalpic spring is that when it is stretched one did work to get it stretched, so that energy was put in it. Now when you relax it the energy must come out as heat. (No work is done relaxing the rubber band).

So this predicts that the energy must be in the rubber band as potential energy (it is exothermic in relaxation) and so the rubber band should be hotter as it is allowed to relax. If this is how a spring should work, and the comparison with the actual rubber band produced the opposite result, maybe our logic is faulty. We need a way to turn the temperature change totally around.

Now consider that a rubber band is composed of a bunch of polymer strands (looking like a bowl of cooked spaghetti). When stretched the polymer strands should be more aligned and straighten out along the direction of the stretching. When you relax it the strands go in all directions, or become more disordered.

So the entropy increases (the entropy change is positive) when the rubber band can relax. Entropy is related to heat as the amount of heat transferred (in a reversible fashion) at constant Temperature divided by the temperature:

$$\ce{\Delta S = \frac{q_{rev}}{T} > 0}$$

For this change the entropy increases, which means the heat is positive. So if the system (aka rubber band) heat is positive, heat must go into the strands of polymer. If the heat goes into the rubber band, where does it come from? The heat comes from the environment; so heat going into the rubber band comes from your lip, so it feels cold. The newly relaxed rubber band is poised to take heat from the room or your lip or whatever will provide the heat.

In summary: The rubber band is an entropic machine.