Problem — Rubber and heat

Problem

Suddenly pull an elastic band held between your lips: it warms up. Let it shorten: it cools down. Explain this behaviour, opposite to that of a gas, in terms of the configurational entropy of the polymer chains.

Solution

Rubber is made of long, tangled polymer chains. At rest each chain can take on very many folded configurations: the configurational entropy is high.

When you pull (stretch rapidly). The chains unfold and align in the direction of the stretch. The available configurations collapse, so the configurational entropy $S$ decreases. Since the stretching is rapid (adiabatic) and thermodynamics imposes the link $T\,\dd S$ with heat, the reduction in configurational disorder at fixed internal energy must be compensated: the “missing” disorder is released as thermal agitation, and the elastic band warms up. If you hold it stretched at constant temperature, the elastic band releases heat to the surroundings.

When you let it shorten. The chains go back to tangling up: the configurational entropy increases again. To regain disorder at the expense of thermal agitation, the material absorbs energy from its own molecular motion, and the elastic band cools down (absorbs heat from the surroundings).

Why it is opposite to a gas. In an ideal gas the energy is entirely kinetic: compressing it (adiabatically) warms it up because work is done on the molecules. In rubber, instead, the elastic force is entropic in origin, not energetic: stretching the chains does not increase a bond potential energy but reduces the entropy, and it is the system’s tendency to return to disorder that provides the restoring force. Hence the reversed sign of the heat exchange. $\ev{\text{entropic elasticity: stretching}\to S\downarrow,\ \text{warms};\quad \text{releasing}\to S\uparrow,\ \text{cools}}$

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Topics: Kinetic theory of gases Concepts: Entropy · Temperature Skills: Micro-macro interpretation