Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Advanced Physics Homework Help
One-dimensional polymer (Statistical Physics)
Reply to thread
Message
[QUOTE="CharlieCW, post: 6149739, member: 649017"] [h2]Homework Statement [/h2] Consider a polymer formed by connecting N disc-shaped molecules into a onedimensional chain. Each molecule can align either its long axis (of length ##l_1## and energy ##E_1##) or short axis (of length ##l_2## and energy ##E_2##). Suppose that the chain is subject to tension ##\tau##. a) Calculate the number of different ways of arranging the polymer such that there are n molecules aligned by its long axis. b) Using the Gibbs canonical ensemble in 1-dimension find the average energy ##\langle E \rangle## and the average length of the chain ##\langle L \rangle##. [h2]Homework Equations[/h2] Gibbs canonical partition function $$\Xi=\sum_{V_S}\sum_{i}exp[-\beta (E_i+pV_S)]$$ Hint: You can take the sum in the partition function as $$\sum_{V_S}\sum_{i}=\sum_{\{ n \}}g(n)$$ where ##\{ n \}## denotes the states of the chain with ##n## molecules aligned by its long axis and ##g(n)## a degeneracy factor. [h2]The Attempt at a Solution[/h2] a) For the first problem, let's denote for simplicity a molecule in vertical position with ##0## and in horizontal position with ##1##. Since each molecule can only adopt two configurations ##(0,1)##, then ##N## molecules can adopt a total of ##2^N## possible configuration. Now if we want the number of ##n## molecules on state ##1##, it is the same to find the total number of ways of arranging n molecules in N spaces, that is: $$g(n)={N \choose n}$$ b) I'm really not sure how to proceed on this one and how to even include the tension ##\tau##. I began by calculating the Gibbs canonical partition function as: $$\Xi=\sum_{V_S}\sum_{i}exp[-\beta (E_i+pV_S)]$$ Using the hint, the sum can be simplified to: $$\Xi=\sum_{\{ n \}}g(n)exp[-\beta (E_n+pV_S)]=\sum_{\{ n \}}{N \choose n}exp[-\beta (E_n+pV_S)]$$ The energy ##E_n## for the state with ##n## molecules aligned along it's axis is ##n(\epsilon_2-\epsilon_1)=n\Delta\epsilon##. On plus, it is subject to a tension which I think should be ##\tau (l_2-l_1)=\tau\Delta l##. I'm not sure if I should include this tension in the energy, or replace ##pV_S## by ##\tau \Delta l##. After that, I have no idea on how to find the average energy and length. [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Advanced Physics Homework Help
One-dimensional polymer (Statistical Physics)
Back
Top