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‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
Exercise 4 – Canonical Ensemble and Ideal Gas
1.
Canonical crystal
Consider a crystal containing N=5x1023 atoms, which may be found in one of the following
states: the ground state with E0=0 and an exited state with E1=ε=4x10-20J.
a. In the beginning ¼ of the atoms were in the exited state. What is the temperature of
the crystal?
b. The crystal is placed in thermal contact with a reservoir at a temperature of 300K.
How much energy will flow between the two systems (and in which direction) until
they reach equilibrium (i.e. what is the change in the energy of the crystal)?
2.
Classical spins

Consider a system of N spins in a magnetic field H attached to a reservoir at temperature

 . Each spin has a magnetic moment m that can continuously rotate, pointing in any
direction (this is referred to as "classical spin").
a. What are the possible energies of a single spin?
b. What is the partition function of the system?
c. Find the average magnetization in the z direction. Express it using Langevin's
function: L( x )  coth( x) 
1
x
d. Expand this result for small x, and in addition find the limit of Lengevin's function for
x   . Use this information to plot a qualitative graph of the magnetization as a
function of x
e. Show that Curie's law applies at high temperatures
Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
3.
‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Two types of gas (1)
In a 3D box there are two types of ideal gases. The number of particles is N1 and N2 and the
temperature is T.
a. Find the partition function of the system
b. Find the free energy and show that it obeys Ftot 
# of systems

i 1
Fi . Why is that?
c. Show that the pressure also obeys the same rule
d. What is the entropy if the two gases have the same number of molecules and if the
molecules have the same mass? Compare to the case where the two gases are
identical
e. Calculate numerical values for the entropy of the two cases in (d) in units of Joules/K
for the two gases if the molar mass of each gas is 4 g/mol. Assume one mole of each
gas is contained within the cube which possesses a length of 15 cm for each side and
which is maintained at a temperature of 300 K