Ideal gas in microcanonical ensemble'

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Lecture 6

• Ideal gas in microcanonical ensemble.
• Entropy.
• Sackur-Tetrode formula.
• De Broglie wavelength.
• Chemical potential.
• Ideal gas in canonical ensemble.
• Entropy of a system in a canonical ensemble.
• Free Energy.
• Maxwell Velocity Distribution.
• Principle of equipartition of energy.
• Heat capacity.
• Ideal gas in the grand canonical ensemble

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Ideal gas in microcanonical ensemble
Let us consider first the ideal gas in
microcanonical ensemble. In the microcanonical
ensemble for N non-interacting point particles of
mass M confined in the volume V with total energy
in ?E at E we must calculate
where the momentum space integral is to evaluated
subject to the constraint that
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by the construction of the ensemble. The
accessible volume in momentum space is that of a
shell of thickness (?E)(M/2E)1/2 on a hypersphere
of radius (2ME)1/2. If the result were sensitive
to the value of ?E employed, we would have
difficulty in deciding on a value ?E.
Fortunately we can prove that for a system of
large numbers of particles the value of ln ?? is
not sensitive to the value of ?E , we may even
replace ?E by entire range from 0 to E.
The proof now follows. We write
(6.3)
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for the volume of a ?-dimensional sphere of
radius R. The volume of a shell of thickness s at
the surface of this hypersphere is
(6.4)
or, by the definition of the exponential function
,
Therefore if v is large enough so that s?gtgtR, Vs
is practically the volume V(R) of the whole
sphere. If ?1023 as for a macroscopic system,
the requirement sgtgtR/1023 may be satisfied
without any practical imprecision in the
specification of the energy of the microcanonical
ensemble.
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We can replace now the constraint (6.2) by the
relaxed condition
because for any reasonable (not too thin) shell
the volume of the shell is essentially equal to
the volume of the entire hypersphere. In other
words, we want to evaluate the volume of the
3?-dimensional sphere of radius R(2ME)1/2. We
can evaluate the volume V? of the hypersphere by
the following argument Consider the integral
We may also write
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where R?-1S? denotes the surface area of the
?-dimensional sphere. On comparison of the two
results (6.7) and (6.8) we find
(6.7)
(6.8)
so that the volume of the sphere is
In the 3-dimensional case, V34?R3/3 and surface
are a34?R2. In the two-dimensional case, we
obtain V2?R2 and surface are a 22?R and in the
one-dimensional case V12R and surface are a 12.
We can write now that the phase space
and using the Stirling approximation to evaluate
factorial
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where in the expression for V? we have to put
?3N and R(2ME)1/2 .
It turns out that if the N particles are
identical we must not count as different
conditions, which differ only by interchange of
identical particles in phase space. We have to
overestimate the volume of phase space by a
factor which is N! under classical conditions.
Taking this factor into account e as the base of
natural logarithms
(6.14)
From (3.13) we have
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so that
in agreement with the elementary result for the
internal energy of a perfect monatomic gas. We
can consider (6.16) as establishing the
connection between ? and T. Further
whence
Using (6.16) and Sk?, we have the famous
Sackur-Tetrode formula for the entropy of an
ideal gas
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Thermal De Broglie Wavelength
We note that (2?MkT)1/2 has the character of an
average thermal momentum of a molecule. We define
as the thermal de Broglie wavelength associated
with a molecule. Than
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showing that the entropy is determined
essentially by the ratio of the volume per
particle to the volume ?3 associated with the de
Broglie wavelength.
The chemical potential of a perfect gas will be
as following
or
or using (6.17) we can write that
where p is a pressure and f(?) is a function of
the temperature alone.
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Ideal gas in canonical ensemble.
Let us derive the Sackur-Tetrode formula using
the canonical ensemble. Well start first with
partition function. For an ideal gas involving N
independent identical spineless particles the
partition function can be written as follows
Now
whence
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Now the integral may be evaluated using the
definite integral
giving for each integral (2?MkT)1/2. Thus using
the definition of the de Broglie wavelength ?.
We can calculate now the free energy using (4.41)
and taking into account that lnN!?Nln N.
whence other properties may be obtained from the
relations
and
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which is identically to E. We could also have
observed that
by the definition of ltEgt.
From (6.30) and (6.31) we have the Sackur-Tetrode
equation
in agreement with (6.19).
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Entropy of a system in a canonical ensemble
Let Es be the sth energy eigenvalue of a system,
and let
be the probability according to the canonical
ensemble that the system will be found in the
state s. Let us show that the entropy may be
expressed in the convenient and instructive form
We have from the above results
whence
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so that
and
Now we may write (6.36) as, using (6.35)
which is identical to (6.40).
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Let us consider the significance of the two terms
on the right of (6.41)
so that
in agreement with the definition of F.
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Maxwell Velocity Distribution
Let us apply the canonical ensemble equally to
macroscopic and to atom subsystems. Applied to a
single atom of mass M in volume V, we have for
?(E) (the occupancy probability of a unit volume
of phase space at energy E ) from (4.36)
as the probability of finding the atom in the
momentum range dpx, dpy, dpz at px, py, pz. We
see that
is the probability of finding the atom in the
velocity range dvx, dvy, dvz at vx, vy, vz .
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It remains to evaluate eF/kT. From (4.41) we have
and for a single atom (N1) we have from (6.29)
the result
Thus the probability
This result is known as the Maxwell distribution
of velocities.
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It may be noted that the steps following (6.41)
are devoted to normalizing the distribution to a
single atom. It would be just as easy to do this
by writing the result of the canonical
distribution as
and determine the normalization constant C by
integration ?w(v)dv1.
The probability P(v)dv that the atom will have
its speed in dv is
The probability P(E)dE that the atom will have
its energy in dE is
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Principle of equipartition of energy
Let us consider one of the variables in the
energy, say variable pj. A situation of
particular importance occurs when pj is an
additive quadratic term in the energy
We can then calculate the mean energy associated
with the variable pj
as the other terms in the exponent may be
canceled in the numerator and denominator. To
evaluate (6.51) we require the definite integrals
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in general
Therefore we have
In classical problem the mean energy associated
with each variable which contributes a quadratic
term to the energy has the value in thermal
equilibrium. The result is known as the principle
of equipartition of energy it depends
specifically on the quadratic assumption.
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For a free atom the Hamiltonian is
the total of three quadratic terms. The thermal
energy is therefore
for N atoms, and the heat capacity at constant
volume is
From this we have he Dulong-Petit result that the
heat capacity (at high temperature) of one mole
of monatomic solid is 3R, where R is the gas
constant. At low temperature the quantum energy
h? may exceed kT, and the problem requires
special treatment, leading to the Einstein and
Debye theories of the heat capacities of solids.
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Ideal gas in the grand canonical ensemble
We have for the grand partition function
We have already evaluated the quantity
Thus
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so that
or
Further
which is the ideal gas low. When we calculate
we get the Sackur-Tetrode formula as derived
previously.
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• Problems (Maxwell distribution)
• Show that the root mean square velocity of a
  Maxwellian gas at constant volume and temperature
  is (3kT/M)1/2
• (2) Show that the most probable speed of a
  Maxwellian gas at constant volume and temperature
  is (2kT/M)1/2
• (3) Show that the mean speed of a Maxwellian gas
  at constant volume and temperature is (8kT/?M)1/2.