Infinitely long molecular zipper

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In summary: Keep going.In summary, the "molecular zipper" has two rows of molecules with weakly linked monomers. It can unzip by breaking bonds between pairs of monomers, requiring energy ε for each bond broken. The zipper's energy, Helmholtz free energy, heat capacity, and entropy at temperature T can be found by calculating the partition function Z, which takes into account the distribution of broken bonds among the molecules. The partition function can be expressed as Z = 1/(1-e^(-ε/kT)), where n corresponds to the index i and each molecule has its own value of n.
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subzero0137
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Homework Statement


[/B]
A "molecular zipper" has two rows of molecules, and each row has a large number of monomers. A monomer from one row is weakly linked to a monomer in the other row. The zipper can unzip from one end by breaking the bond between pairs of monomers. A bond can be broken only if the bond to its left is already broken. It requires energy ε to break each bond, i.e. for every broken bond, the zipper absorbs energy ε. Thus, for every extra bond that is broken, the zipper can be regarded as moving up on the energy ladder by ε.

Treating the zipper as infinitely long, find expressions for the energy, the Helmholtz free energy, heat capacity and entropy of such a zipper at temperature T. [Hint: A zipper can have n broken bonds at T. Each value of n corresponds to one microstate. You need to find the partition function Z at temperature T. You may assume $$a + ar + ar^2 + ... = a \sum_{0}^\infty r^n = \frac {a}{1 - r}$$

Homework Equations


[/B]
$$Z = \sum e^{- \frac {ε_{i}}{kT}}$$

The Attempt at a Solution



I'm trying to find the partition function Z. I think ##ε_i = iε## here, so

$$Z = e^{ \frac {-0ε}{kT}} + e^{ \frac {-1ε}{kT}} + e^{ \frac {-2ε}{kT}}... = \frac {1}{1 - e^{ \frac {-ε}{kT}}}$$

but I'm not sure if it's right because if it can only have n broken bonds at T, should it not be a finite series?
 
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  • #2
Read the question again. "Each value of n corresponds to one microstate". Each molecule has its own value of n, and there is a distribution of n values in the sample. (n corresponds to the index i in your equation for Z. It is not the limit of i. The limit of i (or n) is infinity.)
 
  • #3
mjc123 said:
Read the question again. "Each value of n corresponds to one microstate". Each molecule has its own value of n, and there is a distribution of n values in the sample. (n corresponds to the index i in your equation for Z. It is not the limit of i. The limit of i (or n) is infinity.)

Thanks for the reply. So I'm on the right track with my partition function?
 
  • #4
Looks like it.
 

FAQ: Infinitely long molecular zipper

What is an infinitely long molecular zipper?

An infinitely long molecular zipper is a hypothetical construct in which two strands of molecules are connected in a zipper-like fashion, with each molecule acting as a "tooth" in the zipper. The zipper can continue indefinitely, hence the term "infinitely long". This concept is often used to explain the properties of polymers, which are made up of repeating units of molecules.

How does an infinitely long molecular zipper work?

The two strands of molecules in an infinitely long molecular zipper are held together by intermolecular forces, such as hydrogen bonding or van der Waals forces. These forces are strong enough to keep the zipper closed, but weak enough to allow for movement and flexibility. As more molecules are added to the zipper, the forces between them become stronger, creating a more stable structure.

What are the applications of an infinitely long molecular zipper?

The concept of an infinitely long molecular zipper has many practical applications. It can be used to understand the properties of polymers and how they behave, as well as to design new materials with specific properties. The zipper-like structure can also be mimicked in nanotechnology to create self-assembling structures or to control the movement of molecules.

Can an infinitely long molecular zipper exist in reality?

An infinitely long molecular zipper is a theoretical construct and does not exist in reality. However, the concept is used to explain the properties of real-world polymers and can be replicated in certain materials and structures. Scientists continue to research and develop new ways to mimic and utilize the properties of an infinitely long molecular zipper in various applications.

What are the limitations of an infinitely long molecular zipper?

One of the main limitations of an infinitely long molecular zipper is that it is a simplified model that does not account for all the complexities of real-world polymers. It also assumes that the zipper can continue indefinitely without any disruptions or changes in the intermolecular forces. In reality, there are factors such as temperature, pressure, and chemical interactions that can affect the stability and behavior of an infinitely long molecular zipper.

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