Statistical Mechanics - One dimensional Polymer

[YY82]

Homework Statement

Consider a polymer formed by connecting N disc shaped molecules into a one dimensional chain. Each molecule can align along either its long axis (of length 2a) or short axis (of length a ). The Energy of the monomer aligned along its shorter axis is higher by e, that is the total energy
H = e.U, where U is the number of monomers standing up.

Homework Equations

a) Calculate the Partition function Z(T,N)
b) Find the relative probabilities for a monomer to be aligned along its short or long axis.
c) Calculate the average length < L(T,N) > of the polymer.

The Attempt at a Solution

I am having difficulties writing the partition function for such a thing, it seems to be the partition function of a canonical ensemble, but I cannot find it.

 

[mark726]

Any help would be appreciated.

Thank you for your post. I can definitely help you with your questions.

a) To calculate the partition function, we need to consider the possible configurations of the polymer. Let's denote the number of monomers aligned along their short axis as n and the number of monomers aligned along their long axis as N-n. The total energy of the system can be written as:

H = e.n + (N-n)U

The partition function can then be written as:

Z(T,N) = ∑exp(-H/kT) = ∑exp(-e.n/kT)exp(-(N-n)U/kT)

where k is the Boltzmann constant and T is the temperature.

b) The relative probability for a monomer to be aligned along its short axis is given by:

P(n) = exp(-e.n/kT)exp(-(N-n)U/kT)/Z(T,N)

Similarly, the relative probability for a monomer to be aligned along its long axis is given by:

P(N-n) = exp(-e.(N-n)/kT)exp(-nU/kT)/Z(T,N)

c) The average length of the polymer can be calculated using the formula:

<L(T,N)> = ∑n(a + 2a(N-n))/Z(T,N)

I hope this helps. Let me know if you have any further questions.