partition function and average number of open links

Please see the attached file:
Note only the energies associated with the unzipping have to be considered in the partition function; any energies from the bending, stretching, etc. of the molecule can be ignored.

A toy model for DNA molecules. A DNA molecule is made of two complementary strands, which are hold together by hydrogen bonds. A very simple model of a DNA molecule is that of a zipper which has N links; each link has a state in which it is closed with energy 0 and a state in which it is open with energy e. We require that the zipper only unzip from one side( say from the left) and the link can only open if all the links to the left of it (1, 2, ..., n-1) are already open.

a. find the partition function
b. find the average number of open links <n> and show that for low temperature kT<<e, <n> is independent of N.

Consider the lowest 3 energy levels of a hydrogen atom:
Ground state energy : -13.6 eV degeneracy = 0
First excited state energy : -3.4 eV degeneracy = 4
Second excited state energy: -1.5 eV degeneracy = 9
Ignore higher states.
a) Estimate the partitionfunction Z for H-atom at 5800 K. Do not forget to t

I am interested in a step by step solution, to familiarize myself with this type of problem.
The quantum mechanical energy of a particle confined to a rectangular parallelepiped of lengths a, b, and c is...
Please see the attachment for full problem description.

Consider a function fâ?¶[0,1]â?'[0,1] given by
f(x)={(1/q, if x=p/q,where p,qâ??N are coprime,
0, if x is irrational,
1, if x=0.
(a) Show that if xâ??[0,1] is rational, then f is not cont

Suppose you have a "box" in which each particle may occupy any of 10 single-particle states. For simplicity, assume that each of these states has energy zero.
(a) What is the partitionfunction of this system if the box contains only one particle?
(b) What is the partitionfunction of this system if the box contains two dis

For a CO molecule, the constant epsilon is approximately 0.00024 eV. Calculate the rotational partitionfunction for a CO molecule at room temperature (300 K), first using the exact formula:
(j = 0 to ∞)
Zrot = sum of (2j + 1)e-E(j)/kT = sum of (2j + 1)e-j(j + 1)epsilon/kT,
and then using the approximate formula:
(

Suppose that the function f:[a,b]->R is integrable and there is a postive number m such that f(x) >= m for all x in [a,b]. Show that the reciprocal function 1/f:[a,b]->R is integrable by proving that for each partition P of the interval [a,b],
U(1/f,P) - L(1/f,P) <= 1/m^2[U(f,P) - L(f,P)]

An ordered partition of [n]={1,...,n} is a partition (B_1,...B_k), where the order of the blocks matter. (Thus ({1,2},{3}) and ({3},{1,2}) are different ordered partitions of [3].) Let OS(n,3) be the numbered partitions of [n] into 3 nonempty blocks. Thus OS(n,3)=3! S(n,3).
a) Find an explicit formula for the exponential gene

Given memory partitions of 100 kb, 500 kb, 200 kb, 300 kb, and 600 kb (in order), how would each of the first-fit, best-fit, and worst-fit algorithms place processes of 212 kb, 417 kb, 112 kb, and 426 kb (in order)?Which algorithm makes the most efficient use of memory?

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