[SOLVED] Phys 362 Statistical and Thermal Physics: Homework 7

In a free expansion a gas that is not in thermal contact with its surroundings is allowed to
expand into a vacuum region.a) Is any work done on this gas during a free expansion? Explain your answer.
b) Noting that the gas is thermally isolated, what is that change in energy during a free
expansion?c) Suppose that the gas is an ideal gas. During a free expansion does its temperature
increase or decrease? Explain your answer.d) Suppose that the gas is a van der Waals gas. During a free expansion does its temperature increase or decrease? Explain your answer.To make these more precise recall the identity
!z
y

x
=
!z
x

y
!x
y

z
.e) Use this identity and another identity (we will demonstrate this in class later)
!E
V

T
= T
!P
T

V
P
to show that
!T
V

E
= 1
cV
#
T
!P
T

V
P
$
.
f) Use the previous result to confirm whether a free expansion of an ideal gas results in a
temperature change. Repeat this for a van der Waals gas.Consider various samples of the same gas whose particle number densities = N/V are fixed.
Recall that an extensive variable is one which is proportional to the number of particles in
the system.a) Consider various samples of an ideal gas, all at the same pressure. Rewrite the pressure
and energy equations of state rather than volume. Which of the following are extensive:
V,T,E? Explain your answer.b) Consider various samples of a van der Waals gas, all at the same pressure. Rewrite the
pressure and energy equations of state rather than volume. Which of the following
are extensive: V,T,E? Explain your answer.Consider a system that consists of two subsystems, labeled A and B, each of whose volume
is fixed and whose particle number is fixed.Suppose that the two subsystems interact and
undergo an infinitesimal process in which energy flows from one subsystem to another. Using
only i)
dS = dSA + dSB
and ii) the definition of the thermodynamic temperature,
T =
! S
E
1
V,N
show that the second law of thermodynamics implies that energy must flow from the higher
to the lower temperature subsystem as they approach equilibrium.Consider a system that consists of two subsystems, labeled A and B, whose particle number
is fixed. Suppose that the two subsystems are at the same temperature, interact and undergo
an infinitesimal process in which the total volume of the pair of the systems is fixed.a) Provide a cartoon diagram, indicating the systems and how they might be insulated
from each other and the environment, which illustrates such a process.
b) Using only i)
dS = dSA + dSB
and ii) the definition of the thermodynamic pressure,
P =
! S
V

E,N
show that the second law of thermodynamics implies that volume of the higher pressure
system must increase as they approach equilibrium. Is it possible that energy is transferred in this process? Does this matter for the conclusion about the volume change?