# Elasticity of rubber bands at varying temperatures

So what i read online was that when rubber bands can stretch (become more elastic) more when their at colder temperatures. How i understood is that entropy allows for the molecules to align up more and thus allowing it to stretch more.

What i also read was that at warmer temperatures it will stretch more then at colder ones as the molecules will be at higher states of energy and thus would stretch more and making it more elastic.

So my question is which one is it? Should they be more elastic when at colder temperatures or should they be more elastic at warmer temperatures and why?

• The behavior of elastomeric materials is highly variable. – Hot Licks Mar 7 '18 at 2:12
• In terms of rubber bands – Sir Smiles Mar 7 '18 at 2:18
• the molecules will be at higher states of energy and thus would stretch more and making it more elastic. Why do molecules at higher states of energy stretch more? Where did you read this? – sammy gerbil Mar 7 '18 at 2:32
• @SirSmiles - What makes my comment inapplicable to rubber bands? – Hot Licks Mar 7 '18 at 2:42
• (Though in general warmer rubber is more elastic, up to a point.) – Hot Licks Mar 7 '18 at 2:43

Using an ideal chain model, you can derive the force $F$ required to stretch a polymer chain by $\Delta x$; the result is

$$F=\frac{3k_B T}{N b^2} \Delta x = K_e \Delta x$$

where $N$ is the length of the polymer chain and $b$ is the size of the monomer. You will surely recognize Hooke's law; $K_e$ is called the entropic spring constant (since it comes only from entropy and not from energy) and as you can see $K_e \propto T$: therefore, it is more difficult to stretch a polymer if the temperature is higher.

This should be compared for example to metals: as the structure of metals is dominated by energy, it becomes easier to stretch them as temperature is increased. In this case, stretching the material requires displacing the atoms from their equilibrium position; since higher temperature means stronger atomic oscillations, this becomes easier when $T$ is higher.

A stretched rubber band will, when heated, rise in tension even when it does not change shape. That means that the heated rubber band takes more tension in stretching, i.e. the elastic constant of a warm rubber object is higher than that of a cold one.

Now, is 'elasticity' intended to mean the elastic spring constant? What rises with temperature, is the force that returns stretched rubber to its relaxed shape.

The reason for this odd activity is the entropy of relaxed rubber materials, which is higher than that of the same rubber when stretched (because stretching aligns molecular chains, lowering the entropy just as crystallization would). Minimization of the Gibbs free energy $$GibbsEnergy = ElasticEnergy - Temperature \times Entropy$$ implies the temperature derivative of the GibbsEnergy is zero, so the stored elastic energy goes up with temperature.