Document 2378283

Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Tutorial 5 – Boltzmann Distribution & Helmholtz Free Energy
(Canonical Ensemble)
1.
The zipper problem
A zipper has N links; each link has a state in which it is closed with energy 0 and a state in
which it is open with energy  . We require, however, that the zipper can only unzip from
one side (say, the right), and that the m'th link may only open if and only if all of the links to
its right are already open. The system is kept at temperature  .
a. Find the partition function. Calculate from it the free energy and the entropy
b. Find the average number of open links at the limit of   
c. This model is a very simplified model of the unwinding of two-stranded DNA
molecules. For normal body temperature, what is the percent of open links of a DNA
coil containing approximately 3x109 links (take ε=5x10-27J)?
d. Bonus. Assume that each open link can have p states, all with energy  . Find the
partition function, and calculate the temperature in which the system changes its
properties dramatically (this phenomenon is referred to as a phase transition). What
is that change?
2.
Ideal gas in a box
a. Find the partition function
b. What is nq ? What is the thermal wavelength T ? What is their significance?
3.
Gas in gravity
Consider N point-like particles in a box of volume V  Lx Ly ( z2  z1 ) , located in a
gravitational field gzˆ .
a. Calculate the partition function of the system
b. Calculate the internal energy of the system
c. Calculate the force acting on the upper and lower walls of the box. What is the net
force?