Counting distinguishable arrangements
Notes on General Chemistry
http://quantum.bu.edu/notes/GeneralChemistry/CountingDistinguishableArrangements.pdf
Last updated Tuesday, March 14, 2006 22:15:4605:00
Copyright © 2006 Dan Dill ([email protected])
Department of Chemistry, Boston University, Boston MA 02215
Spontaneous change proceeds in the direction that increases the number of distinguishable
arrangements. We consider two different kinds of arrangements. The first is the ways molecules can
be distributed in space. The second is the ways energy can be distributed among molecules.
à
Molecules distributed in space
The number of distinguishable ways to distribute
n
molecules of one type and
m
molecules of
another type is
W
molecules
H
n
,
m
L
=
H
n
+
m
L
!
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅ
n
!
m
!
.
For example, the number of ways of arranging 3 ink molecules and 3 water molecules is
H
3
+
3
L
!
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅ
3
!
3
!
=
6
μ
5
μ
4
μ
3
!
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅ
3
!
H
3
μ
2
μ
1
L
=
6
μ
5
μ
4
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅ
3
μ
2
μ
1
=
2
μ
5
μ
2
=
20.
Here is a way to derive
W
molecules
H
n
,
m
L
. The number of arrangements of
n
+
m
molecules, ignoring
whether the arrangements are distinguishable is
H
n
+
m
L
!
, since the first molecule can be in any of
n
+
m
places, the second can be in any of the remaining
n
+
m

1 places, and so on to the last
molecule which can be in the single remaining space. Now, the number of arrangements
H
n
+
m
L
!
must be the product of (1) the number of unique arrangements, (2) the number of ways
n
!
that a
particular arrangement of the
n
molecules can arise, and (3) the number of ways
m
!
that a particular
arrangement of the
m
molecules can arise:
W
molecules
H
n
,
m
L
μ
n
!μ
m
!=
H
n
+
m
L
!
.
Solving this expression for
W
molecules
H
n
,
m
L
, we get the expression above. Note that in this expression
n
!μ
m
!
is the number of ways that a particular one of the
W
molecules
H
n
,
m
L
distinguishable
arrangements can occur.
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View Full DocumentQuestions
How many distinguishable ways can
n
different objects be arranged:
n
,
n
2
,
n
n
or
n
!
?
Answer:
n
!
How many indistinguishable ways can
w
identical objects be arranged:
w
,
w
2
,
w
n
or
w
!
?
Answer:
w
!
How many ways can
w
different objects and
i
different objects (a total of
w
+
i
different
objects) be arranged:
w
+
i
,
H
w
+
i
L
2
,
H
w
+
i
L
w
+
i
, or
H
w
+
i
L
!
?
Answer:
H
w
+
i
L
!
What is true about the number,
W
p
H
w
,
i
L
, of distinquishable ways can
w
identical objects
on one kind and
i
identical objects of another kind (a total of
w
+
i
different objects) be
arranged:
W
p
H
w
,
i
L
w
!
i
!=
H
w
+
i
L
!
,
W
p
H
w
,
i
L
=
w
!
i
!
, or
W
p
H
w
,
i
L
=
H
w
+
i
L
?
Answer:
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