Lecture 27 Overview

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Lecture 27 Overview
Final May 8, SEC 117
3 hours (4-7 PM), 6 problems (mostly Chapters 6,7)

• Boltzmann Statistics, Maxwell speed distribution
• Fermi-Dirac distribution, Degenerate Fermi gas
• Bose-Einstein distribution, BEC
• Blackbody radiation

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Problem 1 (partition function, average energy)
The neutral carbon atom has a 9-fold degenerate
ground level and a 5-fold degenerate excited
level at an energy 0.82 eV above the ground
level. Spectroscopic measurements of a certain
star show that 10 of the neutral carbon atoms
are in the excited level, and that the population
of higher levels is negligible. Assuming thermal
equilibrium, find the temperature.
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Problem (final 2005, partition function)

• Consider a particle with five microstates with
  energies 0, ?, ?, ?, and 2? ( ? 1 eV ) in
  equilibrium with a reservoir at temperature T
  0.5 eV.
• Find the partition function of the particle.
• Find the average energy of the particle.
• What is the average energy of 10 such particles?

the average energy of a single particle
the same result youd get from this
the average energy of N 10 such particles
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Problem 2006 (partition function, average energy)
Consider a system of N particles with only
3 possible energy levels separated by ? (let the
ground state energy be 0). The system occupies a
fixed volume V and is in thermal equilibrium with
a reservoir at temperature T. Ignore interactions
between particles and assume that Boltzmann
statistics applies. (a) (2) What is the
partition function for a single particle in the
system? (b) (5) What is the average energy per
particle? (c) (5) What is probability that the
2? level is occupied in the high temperature
limit, kBT gtgt ?? Explain your answer on
physical grounds. (d) (5) What is the average
energy per particle in the high temperature
limit, kBT gtgt ?? (e) (3) At what temperature is
the ground state 1.1 times as likely to be
occupied as the 2? level? (f) (25) Find the heat
capacity of the system, CV, analyze the low-T
(kBTltlt?) and high-T (kBT gtgt ?) limits, and sketch
CV as a function of T. Explain your answer on
physical grounds.
(a)
(b)
(c)
all 3 levels are populated with the same
probability
(d)
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Problem 2006 (partition function, average energy)
(e)
(f)
CV
Low T (?gtgt?)
high T (?ltlt?)
T
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Problem (Boltzmann distribution)

• A solid is placed in an external magnetic field B
  3 T. The solid contains weakly interacting
  paramagnetic atoms of spin ½ so that the energy
  of each atom is ? B, ? 9.310-23 J/T.
• Below what temperature must one cool the solid so
  that more than 75 percent of the atoms are
  polarized with their spins parallel to the
  external magnetic field?
• An absorption of the radio-frequency
  electromagnetic waves can induce transitions
  between these two energy levels if the frequency
  f satisfies he condition h f 2 ? B. The power
  absorbed is proportional to the difference in the
  number of atoms in these two energy states.
  Assume that the solid is in thermal equilibrium
  at ? B ltlt kBT. How does the absorbed power depend
  on the temperature?

(a)
(b)
The absorbed power is proportional to the
difference in the number of atoms in these two
energy states
The absorbed power is inversely proportional to
the temperature.
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Problem (Boltzmann distribution)
Consider an isothermic atmosphere at T300K in a
uniform gravitational field. Find the ratio of
the number of molecules in two layers one is 10
cm thick at the earths surface, and another one
is 1 km thick at a height of 100 km. The mass of
an air molecule m 510-26 kg, the acceleration
of free fall g10 m/s2.
- more air in the 10-cm-thick layer at the
earths surface
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Problem (Maxwell distr.)
Find the temperature at which the number of
molecules in an ideal Boltzmann gas with the
values of speed within the range v - vdv is a
maximum.
maximum
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Problem 2006 (maxwell-boltzmann)
(a) Find the temperature T at which the root mean
square thermal speed of a hydrogen molecule H2
exceeds its most probable speed by 400 m/s. (b)
The earths escape velocity (the velocity an
object must have at the sea level to escape the
earths gravitational field) is 7.9x103 m/s.
Compare this velocity with the root mean square
thermal velocity at 300K of (a) a nitrogen
molecule N2 and (b) a hydrogen molecule H2.
Explain why the earths atmosphere contains
nitrogen but not hydrogen.
Significant percentage of hydrogen molecules in
the tail of the Maxwell-Boltzmann distribution
can escape the gravitational field of the Earth.
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Problem (degenerate Fermi gas)
The density of mobile electrons in copper is
8.51028 m-3, the effective mass the mass of a
free electron. (a) Estimate the magnitude of the
thermal de Broglie wavelength for an electron at
room temperature. Can you apply Boltzmann
statistics to this system? Explain.
- Fermi distribution
(b) Calculate the Fermi energy for mobile
electrons in Cu. Is room temperature sufficiently
low to treat this system as degenerate electron
gas? Explain.
- strongly degenerate
(c) If the copper is heated to 1160K, what is the
average number of electrons in the state with
energy ?F 0.1 eV?
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Problem 2006 (electrons in Lithium)
Metallic Lithium has a Fermi temperature of
5.5x104 K and a Debye temperature of 400K. (a)
(7) Find the total density of electron gas in
Lithium. Assume that the effective mass of
electrons equals the free electron mass. (b) (10)
Find (approximately) the density of electrons in
Lithium that can carry current at 300K. (c) (8)
Find at what temperature the phonon and electron
contributions to the heat capacity become equal.
(a)
(b)
The phonon contribution to the heat capacity
(c)
The electron contribution to the heat capacity