# Statistical Mechanics - One dimensional Polymer

• YY82
In summary, we discussed the partition function, relative probabilities, and average length for a polymer formed by connecting N disc-shaped molecules into a one-dimensional chain. Thank you for considering my explanation.
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.

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.

## 1. What is statistical mechanics?

Statistical mechanics is a branch of physics that uses statistical methods to explain the behavior of a large number of particles, such as molecules, in a system. It is used to understand the macroscopic properties of a system based on the microscopic behavior of its individual particles.

## 2. What is a one-dimensional polymer?

A one-dimensional polymer is a long chain of molecules that has a linear structure and exists in one dimension, meaning it has length but no width or height. Examples of one-dimensional polymers include DNA and proteins.

## 3. What is the significance of studying one-dimensional polymers in statistical mechanics?

Studying one-dimensional polymers in statistical mechanics can provide insights into the behavior of more complex systems, such as proteins and DNA. It also allows for the development of mathematical models and theories that can be applied to a wide range of systems.

## 4. How does temperature affect the behavior of one-dimensional polymers?

At higher temperatures, one-dimensional polymers tend to have more thermal energy, causing them to move more rapidly and randomly. This can affect their shape and interactions with other molecules. At lower temperatures, they have less thermal energy and may become more ordered.

## 5. What are some applications of understanding one-dimensional polymers in statistical mechanics?

Understanding how one-dimensional polymers behave can have practical applications in fields such as biology, chemistry, and materials science. It can help in the design of new materials, the development of drug delivery systems, and the study of biological processes. It can also aid in the understanding and prediction of the properties of complex systems.

### Similar threads

• Advanced Physics Homework Help
Replies
6
Views
4K
• Advanced Physics Homework Help
Replies
3
Views
971
• Advanced Physics Homework Help
Replies
1
Views
1K
• Advanced Physics Homework Help
Replies
20
Views
3K
• Advanced Physics Homework Help
Replies
2
Views
940
• Advanced Physics Homework Help
Replies
1
Views
1K
• Advanced Physics Homework Help
Replies
2
Views
1K
• Advanced Physics Homework Help
Replies
3
Views
1K
• Advanced Physics Homework Help
Replies
1
Views
1K
• Advanced Physics Homework Help
Replies
7
Views
2K