# Entropic force in polymers

According to my textbook, the elastic force in a rubber is caused to the tendency of the polymers to return to their initial disordered state of higher entropy.

But isn't this looking at entropy on a larger level (I'm hesitant to say macroscopic) than it is usually looked at? Generally, we consider the entropy of individual atoms in an ordered structure (a crystal lattice, for instance). But here, we are looking at the entropy of entire polymer chains.

Also, it just doesn't seem to make sense with the non-statistical view of entropy. The book explains it in the following way. The first law of thermodynamics,

$$dE = dQ - dW$$

Making some substitutions,

$$dE = TdS + Fdx$$

where $F$ is the elastic force, $dx$ is the displacement, $T$ is the temperature and $dS$ is the entropy change. Next, approximating $dE$ as zero, and solving for $F$,

$$F = -T\frac{dS}{dx}$$

which is supposed to tell us the that the force is proportional to the rate $dS/dx$ at which the rubber band's entropy changes. But how is there any chage in entropy when no heat is being added?

And, for that matter, if I take a string, lying all entangled, and try to straighten it, why doesn't it exert an entropic force and try to go back to its higher entropic state?

• Entropy is a molar quantity i.e. the total entropy is proportional to the number of molecules present. In a piece of rubber there are an enormous number of molecules, but with a piece of string you have just one molecule of string. The entropy change when you straighten the string is too small to produce a detectable force - if you had $6.023 \times 10^{23}$ pieces of string it might be a different matter. – John Rennie Mar 8 '14 at 8:03
• @JohnRennie: But the string itself is composed of a large number of molecules, isn't it? – Gerard Mar 8 '14 at 8:23
• Yes, but you aren't changing the entropy of the molecules in the string. You're only changing the entropy of the string itself. – John Rennie Mar 8 '14 at 8:39
• So, in the rubber, we are changing the entropy of the polymer chains. The entropic force is only significant because of their large number, am I right? – Gerard Mar 8 '14 at 14:03
• Yes. The point is that you are stretching the rubber and therefore changing the conformation of all the polymers in it. You aren't stretching the string, you are just rearranging it. – John Rennie Mar 8 '14 at 14:57