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A zipper has N links; each link has a state in which it is closed with energy zero and a state in which it is open with energy e. We require, however, that the zipper can only unzip from the left end, and that the link numbers can only open if all links to the left (1,2, S- 1) are alread open. (a) Show that the partition function can be summed in the form 1- exp(-(N+1)e/T] Z = 1- exp(-e/T) high and low temperature

A zipper has N links; each link has a state in which it is closed with energy zero and a state in which it is open with energy e. We require, however, that the zipper can only unzip from the left end, and that the link numbers can only open if all links to the left (1,2, S- 1) are alread open. (a) Show that the partition function can be summed in the form 1- exp(-(N+1)e/T] Z = 1- exp(-e/T) high and low temperature

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A zipper has N links; each link has a state in which it is closed with energy zero and a state in which it is open with energy e. We require, however, that the zipper can only unzip from the left end, and that the link numbers can only open if all links to the left (1,2, ... ,s - 1) are already open. (a) Show that the partition function can be summed in the form 1– exp[-(N+ 1)e/T] Z = 1– exp(-e/T) (b) Assume that each link has length l, find the length of the zipper as a function of temperature. Also show this in the high and low temperature limits (T e and T €). Hint: While solving this problem, you will need to sum a standard type of series. You may look up the general sum for such a series on google.
Transcribed Image Text:A zipper has N links; each link has a state in which it is closed with energy zero and a state in which it is open with energy e. We require, however, that the zipper can only unzip from the left end, and that the link numbers can only open if all links to the left (1,2, ... ,s - 1) are already open. (a) Show that the partition function can be summed in the form 1– exp[-(N+ 1)e/T] Z = 1– exp(-e/T) (b) Assume that each link has length l, find the length of the zipper as a function of temperature. Also show this in the high and low temperature limits (T e and T €). Hint: While solving this problem, you will need to sum a standard type of series. You may look up the general sum for such a series on google.
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