Description
Problem 1:
In 1906, Berthelot proposed the following equation of state as an improvement of the ideal
gas equation of state:
P +
a
kT v2
(v − v0) = kT. (1)
Here, a and v0 are substance specific constants (they have to be determined for each
substance) and v is equal to V/N, with N being the number of atoms or molecules in the
substance.
Let us consider a gas that satisfies Berthelot’s equation of state as well as the
additional condition that the internal energy U approaches 5N kT/2 in the v → ∞ limit.
(a) Determine the Helmholtz free energy A (your answer will depend on an undetermined
constant).
(b) Determine the specific heat CV at constant volume,
CV =
∂U
∂T !
V
, (2)
in terms of T and v.
Problem 2:
In Problem 1, we determined the pressure and internal energy of a substance governed by
Berthelot’s equation of state, subject to the condition U → 5N kT/2 for v → ∞.
(a) What are the units of a and v0 that enter into Berthelot’s equation of state? Use your
result to rewrite the pressure and internal energy per particle as dimensionless quantities
(denote the dimensionless pressure by P˜ and the dimensionless internal energy per particle
by ˜).
Note: In the process, you will have to define a dimensionless volume ˜v and a dimensionless
temperature ˜τ (or more precisely, Boltzmann constant times temperature).
(b) Determine the critical dimensionless temperature ˜τc, which is obtained by enforcing
∂P˜
∂v˜
!
T
= 0 (3)
and
∂
2P˜
∂v˜
2
!
T
= 0. (4)
Explain how Eq. (3) relates to the isothermal compressibility. Explain whether or not
∂P˜
∂v˜
!
T
> 0 (5)
1
and
∂P˜
∂v˜
!
T
< 0 (6)
are physical.
(c) Plot isotherms in a P˜ versus ˜v diagram. Include the equation of state for ˜τ = ˜τc, ˜τ < τ˜c,
and ˜τ > τ˜c. Do you encounter negative P˜? Does this bother you?
Problem 3:
A gas consisting of N identical classical particles is confined inside a cylinder of length
L = a − b and with volume V = πR2L, i.e., the particles’ coordinates are restricted by
b < z < a and x
2 + y
2 < R2
.
The particles do not interact with each other and the motion
of a single particle is governed by the Hamiltonian H = ~p
2/(2m) + Kz, where K is a
constant. Let us assume that the gas is in equilibrium.
(a) Calculate the partition function of the gas.
(b) Determine the pressure Pa of the gas on the wall at z = a.
(c) Determine the pressure Pb of the gas on the wall at z = b.
(d) Compare the pressures Pa and Pb in the limit KL kT. Explain.
Problem 4:
This problem investigates the properties of the van der Waals equation of state,
(v − b)
P +
a
v
2
= kT, (7)
where a and b are substance specific constants and v = V/N.
(a) Show that the van der Waals equation of state reduces to the ideal gas equation of
state in the ultralow density limit and derive the leading-order correction to the ideal gas
equation of state in the low density limit.
(b) For neon, the following holds:
b −
a
kT → 1.3 × 10−5 m3
mol for T → ∞ (8)
and
b −
a
kT → 0 for T → 125 K. (9)
Using these relations, determine a and b for neon.
(c) Determine the critical volume vc and critical temperature Tc for neon from the conditions
∂P
∂v !
T
= 0 (10)
and
∂
2P
∂v2
!
T
= 0. (11)
What is the critical pressure? Compare your results with the experimentally determined
values of Tc = 44.5 K and Pc = 26.9 atm.
