Thermodynamics and Statistical Mechanics I - Home Exercise 9

Thermodynamics and Statistical Mechanics I - Home Exercise 9
1. Using the temperature pressure ensemble to solve ideal gas
(a) Write the integral of the partition function for a gas at pressure P and temperature
R∞
T (Particle number is N ) using the integral IN −1 = V N −1 e−βP V dV . Hint: dont
0
forget to divide by something when you move from a sum to an integral.
(b) Show that IN −1 ∼ IN −2 .
(c) What is the partition function (solve the integral from (a))?
(d) Find the average volume hV i. Compare with ideal gas law.
(e) Find the volume fluctuations. Show that the relative volume fluctuations scale as
∼
√1
N
2. Ideal Gas and a Piston Consider an ideal gas of N particles with mass m, confined
to a cylindrical container with cross-section A. The container is sealed by a piston
of mass M at a height h. Neglect direct gravitational effects on the gas (what is the
condition on the mass in order for this assumption to be valid?)
(a) Calculate the canonical partition function of the gas.
(b) Calculate the partition function of the entire system. Note that
R∞
0
xN e−x dx = N !.
You may loosely choose the normalization factor for the partition function (does
it matter for this problem?)
(c) Calculate the average enthalpy.
(d) Calculate the average height hhi of the piston .
(e) Calculate the fluctuations in the height of the piston.
3. Isentropic relations of ideal gas
(a) Show that the differential changes for an ideal gas in an isentropic process satisfy
the following:
dp
p
+ γ dV
= 0,
V
dτ
τ
+ (γ − 1) dV
= 0,
V
dp
p
+
γ dτ
1−γ τ
= 0, where γ =
These relations apply even in the molecules have internal degrees of freedom.
1
Cp
CV
.
∂p
(b) The isentropic and isothermal bulk moduli are defined as Bσ = −V ∂V
and
σ
∂p
Bτ = −V ∂V
. Show that for an ideal gas Bσ = γBτ , Bτ = p. The velocity
τ
p
of sound in a gas is given by c = Bσ/ρ; there is very little heat transfer in a
sound wave. For an ideal gas of molecules of mass M we have p =
p
c = γτ/M . Here ρ is the mass density.
2
ρτ /M ,
so that