Physics 425, Spring 2007
Review III
II.6. Free Expansion and Throttling Process
a. Must understand that in a free expansion, i.e., a thermally insulated expansion of a
gas into a vacuum, the internal energy of the gas is conserved.
b. Must know why a free expansion cannot cool an idea gas.
c. Must be able to calculate the temperature in the final state of a real gas
undergoing a free expansion from the initial state.
d. Must know what is a throttling process: a steady gas flow passing through a
porous plug separating two chambers maintained in constant but different
pressures.
e. Must know why a throttling process cannot cool an ideal gas. Must know why the
throttling process in more efficient in cooling the real gas than the free expansion.
f. Must understand that the enthalpy H = E + PV is conserved in a throttling process.
#T
TV
1
($ % ) , the
g. Must know the meaning of Joule-Kevin coefficient µ " ) H =
#P
CP
T
inversion curve, and in what regime of the T-P diagram, the gas will be cooled or
heated through a throttling process.
h. Must know that free expansion and throttling !
process of a gas are IRreversible
processes even though they are adiabatic (see II.7.h), as they are not quasi-static
processes.
II.7 Heat Engines and Refrigerators
a. Must know what is the Kelvin postulate and understand why the “second kind
perfect engine” cannot be made according to the second law.
b. Must know what is a Carnot engine, and how it works (i.e., the Carnot cycle) so as
not to violate the second law.
c. Must be able to calculate work W and heat Q, using the first law, equation of state,
and adiabatic relations (if applicable), in a given engine cycle consisting of, e.g.,
quasi-static isothermal or adiabatic or isobaric expansions and compressions of an
ideal gas (review II.4 and HW problems Reif 5.22, 5.26 and Stowe 14-7, 14-8.)
Note: be very careful with the signs of W and Q in your calculation/definition.
Note: sometimes you can simplify your calculations knowing and using the fact
that the change in a state variable, like the internal energy, only depends on the
initial and final states and independent of the path of evolution.
d. Must be able to calculate the efficiency of an engine and the Carnot efficiency
W T1 $ T 2
"=
#
% "c .
Q1
!
T1
e. Must know what is the Clausius postulate and why a “perfect refrigerator” cannot
be made.
f. Must know how a refrigerator works so as not to violate the second law. Must be
able to calculate heat and work in a given refrigerator cycle. Note: an ideal
refrigerator cycle is a reversed Carnot (engine) cycle!
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g. Understand how the second law (entropy increase law) determines whether a
thermodynamic process is reversible or not; know of examples of reversible and
irreversible processes.
h. Must understand the relationship between adiabatic process and reversibility:
dQ
.
dS =
T
III Introduction to Statistical Physics
III.1 Probability distribution
a. Understand the probability Pr of a particle (of a gas system in thermal equilibrium)
in a given quantum state with energy Er.
!
e"E r / kT
.
# re"E r / kT
b. Must know the Boltzmann factor: Pr =
c. Must be able to calculate Pr for a given quantum state, and the ratio of
probabilities, sometimes equivalent to the ratio of numbers of particles, of
different quantum states !(review Ex.4 about ground and excitation states and
excitation rate).
d. Must be able to calculate the mean properties of a particle system with the
"
probability distribution function (Boltzmann factor): y =
# r y re"E r / kT
(review Ex.5
# re"E r / kT
and HW problems Reif 6.1, 6.6, 6.7, Stowe 16-9 about how to calculate the mean
energy and specific heat of a system).
!
III. 2 Equipartion and specific heats
a. Must know what is the equipartion theorem and be able to use it in the classical
limit. Understand how it is derived with probability distribution function.
b. Must understand why equipartition theorem breaks down when quantum
mechanics effects have to be taken into account. Must know how to derive the
condition for the classical limit: ε << kT., for example, for quantized translational
kinetic
"=
!
energy " =
L2 l(l + 1)h 2
,
=
2I
2I
p2
h2
=
n2
2m 2mL2
(in
one
dimension),
1
2
and vibrational energy " = kx rel 2 +
1
1
prel 2 = (n + )h#
2m
2
energy
(for a 1D
harmonic !oscillator). Note that the excitation temperature for the vibrational
degrees of freedom is 103 K for many materials.
c. Must be able to explain the discrepancy
in specific heats of di-atomic gas and
!
solids between classical calculations and laboratory measurements, with the
quantum mechanics consideration.
d. Must
understand
the
Einstein
model
for
solids
(Reif
7.7):
"
% $ (2
1
1
e$ E /T
E = 3N a h# ( + h# / kT
), cV = 3R' E *
& T ) e$ E /T " 1
2 e
"1
(
!
rotational
)
2
. Must understand the meaning of
Einstein temperature θE . Must know the temperature dependence and the high and
low temperature limits of mean energy and specific heats of solids according to the
Einstein model.
e. Know of Maxwellian velocity and speed distributions of gas particles. Know the
concepts of mean speed, most probable speed, and root-mean-square speed.
2
III.3 Paramagnetism
a. Must understand how the intrinsic magnetism of a material is generated and the
definition of magnetic moment along the z-axis, or the direction of the external
ge
ge
ge
magnetic field: µ z =
Lz , µ z =
Sz , or µ z =
J z for particles considering their
2m
2m
2m
orbital, spin, or total angular momentum.
eh
= 9.3" 10#24 J/T . Must understand why
b. Must know the Bohr magneton: µ B =
2me
!
magnetism of a material is usually most contributed by electron spin/orbit
motions.
c. Must know how to calculate
the magnetic energy of orbit/spin particles in an
!
external field: " = #µ z B . Understand that when the magnetic moment (or angular
momentum) is parallel to the external magnetic field, the magnetic energy is
lowest (most negative).
d. Must be
! able to calculate the mean magnetic moment and magnetization using the
probability distribution, or the Boltzmann factor, for simple cases. Must know the
magnetization at very high and very low temperatures – the Curie’s law
"
M =N µ =C
B
T
and magnetic saturation (review Ex 3, 4 and HW problems Reif 6.2,
6.3 and Stowe 20-9).
!
III.4 Quantum gas
a. Know of the difference between classical statistics and quantum statistics.
b. Must know the definition of occupation number: mean number of particles at a
_
given quantum state: n s =
" n s n se#n s ($ s # µ )/ kT
" n s e#n s ($ s # µ )/ kT
.
c. Must know the partition function Z s = " se#n s ($ s # µ )/ kT and how to calculate the
"
occupation number from the partition function: n s = "kT
!
1 #Z s
.
Z s #$ s
d. Must know what are Fermions
! and what are Bosons. Must know what values ns
can take for Fermions and Bosons. Know how to calculate the occupation number
for systems of Fermions and Bosons (review
HW problem Reif 9.1, Stowe 24-12).
!
e. Must know the Fermi-Dirac (FD) distribution and Bose-Einstein (BE) distribution:
"
ns =
1
e
(# s " µ )/ kT
±1
.
"
f. Must know the Maxwell-Boltzmann (MB) distribution n s = N
!
" V %1/ 3
classical limit $ '
#N &
>>
h
3mkT
e"# s / kT
at the
$ se"# s / kT
. Understand what conditions meet the classical
! MB distributions.
limit and what’s the relationship between FD, BE, and
g. Must know
the photon statistics. Qualitatively understand blackbody radiation and
!
be able to make simple calculations using Planck’s function and StefanBoltzmann’s law (review HW problem Reif 9.12):
3
2hc 2
1
J m-2s -1A -1sr -1 ,
5
hc / "kT
" e
#1
F (T ) = "T 4 J m2 s -1 , (" = 5.67 # 10$8 J m2 s -1 K -4 )
F (" ,T ) =
h. Must understand the concepts of “Fermi energy” and “Fermi temperature” of
2
!
h2 # 2 N &3
conduction
electrons
in
metals:
µ
=
% 3"
( ,
0
!
2m $
V'
i.
µ0
. Must understand the
k
1/ 3
#
N&
concept of “Fermi sphere” in the momentum space with its radius kF = % 3" 2 ( .
$
V'
1
Qualitatively understand! the physical picture of pressure integral P = npv (note: n
3
TF =
is the particle number density) in different gases and be
! able to make simple
estimates of gas pressure, radiation pressure, and electron degeneracy pressure for
classical particles, Bosons (photons), and Fermions !(electrons), respectively
(review Ex. 4 in III.4).
Mathematic preparation: you must be very familiar with simple calculations involving
exponential functions including some simple forms of sum or integration and
derivatives; you must be able to derive approximations of the exponential function
(such as by Taylor expansion) at high/low temperature limits.
------ End of Review III ------
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