FYS4130: Problem set 6
The Virial Theorem
Clausius assigned the name virial to the quantity denoted C, defined by
X
C=
pi · ri .
(1)
i
The theorem refers to a system of interacting particles with well-defined timeaverage kinetic energy hT i, and with bounded ri and vi over long time.
If we consider a particle of mass m and velocity v, that moves under the
action of a central force
F = − γM m/r2 r̂,
(2)
where M m is the mass of the object that generates the interaction. The
lineal momentum of the mass m is:
p = mv,
v = vθ θ̂ + vr r̂
(3)
where θ̂ and r̂ are unit vectors in polar coordinates. We can now define two
quantities; the vector quantity L = r × p and a scalar quantity A = p · r. For
the angular momentum L, it is known that for a central force parallel to r, its
time derivative is L̇ = 0. While for A we obtain:
Ȧ =
=
mv̇ · r + mv · ṙ
2
F · r + mv .
(4)
(5)
Since F = − ∂V
∂r r̂, with V = −γmM/r being the gravitational potential we get.
Ȧ = −V + 2T
(6)
Since the angular momentum is conserved, L = mvr is a constant and r and v
are bound in time. Taking the time average of Ȧ we obtain
Z
A − A0
1 τ dA
dt =
,
(7)
hȦi =
τ 0 dt
τ
which implies that hȦi → 0 for τ (A − A0 ), and thus
hV i = −2hT i.
(8)
1: Assume that a star is a sphere of radius R, and mass Ms . Its total gravitational potential energy is then
V =−
3GMs2
.
5R
(9)
Use the equipartition principle to find the mean kinetic energy of a single
atom in the interior of the sun, and use the virial theorem to estimate
the Sun interior temperature. The average mass of an atom in the sun is
m = Ms /N = 2.2 · 10−27 kg, and the mass of the sun is Ms = 2 · 1030 kg.
Compare your result with data from more accurate calculations.
1
2: Consider an ideal gas confined in a box of volume V . The force differential
on the gas molecules, defined by the pressure exerted by the wall of the
box in a differential area dA can be written as
dF = pdA n̂,
so the total force will be
(10)
Z
F =
P dA n̂
(11)
Use the virial theorem (in the form of Eq.(5), where mv 2 is twice the
kinetic energy) to find the ideal gas law
hpiV = N kT
(12)
Hint: Remember Gauss’s divergence theorem, and 5 · r = 3.
Virial Expansion
The equation of state of a real gas can be written in a form
B2
B3
B4
Pv
=1+
+ 2 + 3 + ···
RT
v
v
v
(13)
called the virial expansion, which is valid for dilute gases with molar density
1/v = n/V . Find the first virial coefficients B2 , B3 , B4 for the van der Waals
gas equation of state
a
P + 2 (v − b) = RT,
(14)
v
where a and b are constants.
Hint: Assume that b/v 1.
Equipartition Theorem
A molecule consisting of two different atoms can rotate in space about two axes
each having a moment of inertia I. The third axis, which joins the two atoms
is a symmetry axis, thus there is no rotation about that axis. In this problem
consider only the rotational motion of the molecule (neglect translational and
vibrational motions).
1: Use the equipartition theorem from classical statistical mechanics to find the
heat capacity of this molecule at temperature T .
Quantum mechanically the molecule has energy levels
Ej =
~2
j(j + 1)
2I
j = 0, 1, 2, . . .
(15)
2: Use quantum statistical mechanics to write down an expression in terms of a
sum for the canonical partition function for this molecule (do not attempt
to evaluate the sum). Give also general expressions (in terms of derivatives
etc.) for how to compute the average energy and heat capacity from the
canonical partition function.
2
3: Derive an expression for the temperature dependence of the molecule’s heat
capacity C(T ) at low temperatures β 1. In what range of temperatures
is your expression valid?
4: Use quantum statistical mechanics to derive an expression for the molecule’s
heat capacity C(T ) valid for high temperatures. Compare the result to
the one you got in 1.
Fluctuatons in the Grand Canonical Ensemble
1: Using the partition function for the grand canonical ensemble, prove that
the variation of number of particles can be expressed as
∂hN i
,
(16)
h∆N 2 i = hN 2 i − hN i2 =
∂α β
where β = 1/T , α = µ/T and hN i is the average number of particles.
2: Let hNi i be the average occupancy of a single level of a boson system. Show
that
h∆Ni2 i = h Ni i(1 + hNi i)
(17)
3: Show that for a fermion system we have
h∆Ni2 i = h Ni i(1 − hNi i)
(18)
4: Finally show that for a Maxwell-Boltzmann gas we have
h∆Ni2 i = h Ni i
3
(19)