1) Polymers, like rubber, are made of very long molecules, usually tangled up. Another very crude model of a rubber band is that each link is either crumbled up or stretched. If it is crumbled up, its length is negligible and it is in the lowest energy state, call it E = 0. If it is stretched, then its length is L and it is in its high energy state of E = Eo. Consider a rubber band made of N links. The total length, Ltot, of the rubber band is the net displacement from the beginning of the first link to the end of the last link.

a) How many microstates are there when the energy of the system is nEo. Here, n is an integer that is less than the total number of links, N.

b) What is the length of the rubber band if it has N links and the total energy is nEo?

c) If N = 3 and the temperature is T, what is the partition function for this system? Your answer can depend (but it doesn't have to depend on all of these) on any of the quantities already given in this problem such as L, T, Eo, N, and n (the number of stretched polymers).

d) What is the average length of rubber band when it has N links and is at a temperature of T? This answer requires you to do a summation. Just set up the summation symbolically; do not try to do it. Just give a symbolic formula and explain what it means and give your reasoning. Your answer can depend (but it doesn't have to depend on all of these)on any of the quantities already given in this problem such as L, T, Eo, N, and n (the number of stretched polymers).