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ChE 359 Homework #4
Solutions prepared by Yandi Hu
1.
(25 points) We are going to look at gas bubbles expanding to fill all available
space.
a.
(10 points) Initially, the 10 indistinguishable gas bubbles are confined to a
space that only consists of 10 boxes, each of which is large enough to hold
only one gas bubble.
Calculate the number of ways
that the bubbles
can be partitioned among the 10 boxes, and use this information about the
multiplicity to calculate the entropy of the gas bubble system.
Since there
are 10 indistinguishable gas bubbles and 10 boxes that can hold one gas
bubble each, there is only one way that the bubbles can be partitioned
among the ten boxes.
We can also do this mathematically if we consider
that there are 10 indistinguishable boxes categorized into two types –
those with bubbles, and those without bubbles.
In this case, the
multiplicity is given by:
W
10
=
N
!
n
1
!
n
2
!
=
10!
10!0!
=
1
.
The reason we cannot
simply use BoseEinstein statistics is because the BoseEinstein model
as shown in class may have more than one bubble in a box, which is not
there
case
here.
The
multiplicity
of
the
system
is
thus:
S
10
=
k
ln
W
10
=
k
ln1
=
0
b.
(10 points) The space is then expanded to 50 boxes.
Calculate the number
of ways
that the bubbles can be partitioned among the 50 boxes.
Again,
we
just
use
the
same
formula
as
above.
W
50
=
N
!
n
1
!
n
2
!
=
50!
10!40!
=
10272278170
.
The multiplicity of the system is thus:
S
50
=
k
ln
W
50
=
k
ln10272278170
=
3.18
×
10
−
22
J
K
c.
(5 points) Justify why the entropy changes the way it does when the
available space is increased from 10 boxes to 50 boxes.
We see that the
entropy increases when the number of boxes is increased, due to an
increase in the number of possible arrangements of the gas bubbles in
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 Spring '10
 CynthiaLo
 Mole, Kinetics

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