$1.$ Consider the following simple molecular model for a rubber band. We imagine that a rubber

band is a polymer made up of N segments $($monomers$).$ Each monomer can be in one of two

conformations, a long conformation with length l$1$ or a short conformation with length l$2,$ but just

how many you have of each of these conformations can vary. Assume there are N $1$ long segments

and N$2$ short segments, and that both conformational possibilities for a monomer have the same

energy.

a$.$ Give expressions for the total number of segments N as a function of N$1$ and N$2,$ and for

the total length L as a function of N$1,$ N$2,$ l$1,$ and l$2.$

b$.$ It turns out that the entropy, S$,$ of the rubber band can be approximated as

S$/$k b $=$ N ln N $$ N $1$ ln N$1$ N$2$ ln N$2$

where kb is Boltzmann$$s constant. If there were no forces applied to the rubber band, it

would have a preferred length Leq $.$ Using the $2$nd law of thermodynamics and the results

from part a$,$ calculate this length as a function of N$,$ l$1,$ and l$2.$ Remember that l$1>$ l$2.$

c$.$ It also turns out that the tension can be calculated using the laws of thermodynamics

as

$=-$T $($S$/$L$)$ N

Using your work in part b$,$ give an expression for as a function of N$,$ L$,$ T$,$ l$1,$ and l$2.$

d$.$ What value do you get for the tension if you substitute L$=$Leq from part b$?$

e$.$ Based on physical reasoning alone, what can you say about the sign of the derivative

$($$/$L$)$ N$?$ Check your prediction by computing $($$/$L$)$ N from your answer to part c$.$