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.

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

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

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

Derive equations: S = -(∂F/∂T)_V, N = Nk [ln (V/N_vQ) + 5/2] - ∂F_int/∂T and μ = (∂F/∂N)_T, V = -kT ln (VZ_int/N_vQ) for the entropy and chemical potential of an ideal gas.

For a diatomic gas near room temperature, the internal partitionfunction is simply the rotational partitionfunction multiplied by the degeneracy Ze of the electronic ground state.
(a) Show that the entropy in this case is
S = Nk [ln (VZeZrot/NvQ) + 7/2.
Calculate the entropy of a mole of oxygen (Ze = 3) at room tempe

Consider a two level system (such as the orientation of the magnetic moments of protons in a magnetic field, e.g. NMR). Let the spacing between the two levels be delta that is, the lower level has an energy of 0 and the other an energy of delta. Evaluate the following quantities in terms of delta and T.
a) the partition functio

I try to simplify my problem here. This is the place where I may have some misunderstanding in the adiabatic process. Consider a uniform cylinder with insulated wall cylinder that has a very thin and mass less partition at middle divide it into two sections. The left section contains gas molecules and the right side is attached

A system is composed of N one dimensional classical oscillators. Assume that the potential for the oscillators contains a small quartic "anharmonic" term
V(x) = (m*Ω^2)/2 + a*x^4
Where a*(x^4) <<< KB T and (x^4) = average value
Calculate the average energy per oscillator to the first order in a