[SOLVED] Phys 362 Statistical and Thermal Physics: Homework 4

The work done on any gas can be computed via
W =
!
P dV.
a) There is only one type of process where it is legitimate to compute the work via
W = P
!
dV.Describe what conditions this process must satisfy for this to give the correct expression
for work. What curve would represent this process on a P V diagram?b) For ideal gases P = N kT /V and thus
W =
! Vf
Vi
NkT
V dV.
It may appear that this gives
W = NkT ! Vf
Vi
1
V
dV = NkT ln Vf
Vi
#
.But there is also only one type of process for which this is true. What condition must T
satisfy during the process for this to be true? What curve would represent this process
on a P V diagram?c) During an isobaric process, is T independent of V ? Would it be true for this process
that
W = NkT ! Vf
Vi
1
V dV = NkT ln Vf
Vi
#
?
Explain your answer.In general one must be careful when calculating work. One must ask: Does P (or T) depend
on V during the process? If it does, then the process of integration will be more complicated
that extracting a constant and integrating with respect to V as done in the cases above.An ideal gas which undergoes the following three-stage process, with initial volume Vi and
pressure Pi. First it expands to double its initial volume while the pressure remains constant.Then the volume is held constant and the pressure increases to three times the original
pressure. Then it returns to its initial state via a process which can be represented as a
straight line on a P V diagram. Sketch the entire process on a P V diagram. Determine an
expression for the work done on the gas in for the entire process.A monoatomic ideal gas undergoes a cyclical process, starting at an initial volume V1 and
initial pressure P1 and eventually ending at the same pressure and volume. During the first
stage of the process, the pressure and volume are related by
P = aV 2
where a is a constant with units of Pa/m6.During this process the volume is reduced to V1/2.
During the second and third stages either the pressure or else the volume remains constant.At the end of the second stage the gas reaches a pressure of P1.
a) Sketch the process on a P V diagram.
b) Determine the work done during each stage of the process.
c) Determine the change in internal energy during each stage of the process.
d) Determine the heat that leaves or enters the gas during each stage of the process.e) Over the entire cycle, does the gas do work on its surroundings or is work done on the
gas?
f) Consider the second stage. As the gas evolves during this stage is heat constantly
supplied or constantly removed? Or is heat supplied during some portions of this stage
and removed during other portions of this stage? Repeat this for the third stage.g) Suppose that the only cost to running the gas through this cycle is the heat added (the
heat removed is just lost). Determine what fraction of the heat added over the entire
cycle is converted to work.A monoatomic van der Waals gas satisfies the equation of state,

P + a
N2
V 2
#
(V N b) = NkT
and has thermal energy given by
E = 3
2
NkT a
N2
V
2
where a and b are constants that depend on the particular gas. Suppose that the gas undergoes
a constant volume process in which the pressure doubles. Determine expressions (in terms
of the initial pressure and volume) for the work done, the change in internal energy and the
heat supplied during this process.