Question

5. Consider a polymer consisting of N links (like links of a chain) of length L. The links are massless. One end of the polymer is fixed at height z = 0. The other end is attached to a mass m. Each link can pivot about its attachment with the previous link or the fixed end so that a link may carry the polymer "up" or "down" a distance L. For example, the figure shows a polymer containing four links, taking the polymer down, up, down, and down. The figure is schematic; all links are actually vertical. The mass at the end of the polymer is in a uniform gravitational field g, so it has a gravitational potential energy mgz. The polymer is in weak thermal contact with a reservoir at temperature T. For simplicity, suppose N is even and suppose the only energy in the system is the gravitational potential energy of the mass. (a) Without doing any calculations, what is the gravitational potential energy of the system when it is very cold (T « mgL)? Be sure and give a brief explanation. (b) Again, without any calculations, what is the gravitational potential energy of the system when it is very hot (T >> NmgL)? Again, a brief explanation is desired. (c) What is the partition function of the system? (d) What is the gravitational potential energy of the system at temperature T?

Answer 1

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mEO Spleted e(N-anmg -4 mgiT e (aem3th e8imcosh- ee Joravi tadtional potediol Engy a Ot a tempematine NLog (Cosh

Pagen NLogo(cosh( 2 SinCmg Mguh) Cos T tamh mgL