Download Statistical - Jordan University of Science and Technology

Jordan University of Science and Tech.
Comprehensive exam
First Semester
Physics Department
Statistical Mechanics
2006/2007
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SOLVE THREE OUT FOUR. ALL QUESTIONS CARRY EQUAL MARKS.
GIVE CLEAR AND COMPLETE SOLUTION, EACH ON A SEPARATE SHEET.
Q1-a: Assume that at very low temperature, the molar heat capacity of copper is equal to
( 7x10-4 T ) J K-1 mole-1 , where T is the absolute temperature. Show that if
( 10-7 ) J of heat is added to a mole of copper, which is initially at the absolute
zero, the temperature of the copper rises to (0.069 K). Find the increase in the
entropy of the copper. The volume of the copper is kept constant. What is the
number of microstates accessible to the copper.
kB = 1.38 x10-23 J K-1 .
Q1-b- Consider a crystal which has N lattice points and the same number of interstitial
positions. Let ε be the energy necessary to remove an atom from a lattice site to an interstitial
position and let n be the number of atoms occupying interstitial sites in equilibrium.
a) what is the internal energy of the system.
b) What is the total energy. Give an asymptotic formula when n>> 1?
c) In equilibrium at temperature T, how many such defects are there in the solid. i.e what
is n? assume n>>1.
Q2 -a- Assume the earth's atmosphere is pure nitrogen in thermodynamic equilibrium at
300K. Calculate the height above sea level at which the density of the atmosphere is
one half its sea level value.
Given that: g=10m/s2. R=8.31 J/k.mole. molecular weight of nitrogen is 28 g/mole.
Q2-b- The average kinetic energy of the hydrogen atoms in certain stellar atmosphere in
Kelvin is 1ev.
a) what is the temperature of the atmosphere in Kelvins?
b) What is the ratio of the number of atoms in the second exited state (n=3) to the
number in the ground state (n=1).
Q3- a- The partition function of a system is given by:
Z = exp(aVT 4 )
Where a is a constant, V is the volume and T is the temp. of the gas. Calculate:
The internal energy, the entropy and the equation of state of the system.
Q3 -b- In free electron model you are given the number density of the electrons n and Fermi
energy εF of a non- interacting electron gas at T=0.
Find the isothermal compressibility κ = β-1 where β is the isothermal Bulk modulus .
Recall that β = -V(dP/dV)T
Q4- N weakly coupled particles obeying Maxwell- Boltzmann statistics may each exist in one
of the 3 non-degenerate energy levels of energies -E, 0, +E. The system is in contact with
a thermal reservoir at temperature T.
a- what is the entropy of the system at T=0.
b- what is the maximum possible entropy of the system.
c- what is the minimum possible energy of the system.
d- what is the partition function of the system.
e- what is the most probable energy of the system.
f- if C(T) is the heat capacity of the system, what is the value of ∫C(T) dT/T