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Thermal

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Thermal & Kinetic Lecture 14 Permutations, Combinations, and Entropy Overview Permutations and combinations Interacting atoms and energy transfer – PowerPoint PPT presentation

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Title: Thermal

1
Thermal Kinetic Lecture 14 Permutations,
Combinations, and Entropy
Overview
Permutations and combinations
Interacting atoms and energy transfer
Distribution of energy at thermal equilibrium
2
Last time
Diffusion The Einstein model of a
solid Oscillators and quanta Permutations and
combinations.
3
Energy distributions
Consider bringing two identical blocks
together. What is the most probable distribution
of energy amongst the two blocks?
Most probable distribution is intuitively that
where total thermal energy is shared equally
between the two blocks.
However, what is the probability that the first
block has more energy than the second or, indeed,
ends up with all the thermal energy? Need to
consider possible arrangements of energy quanta
4
Energy distributions
We have 3N independent simple harmonic
oscillators (where N is the total number of atoms
in the solid). Number of ways of distributing
quanta of energy amongst these oscillators? Say
we have 3 quanta of energy to distribute amongst
2 oscillators
5
3 energy quanta distributed between 2 oscillators
ANS Total of 4 possibilities (including
arrangement shown on previous slide)
Oscillator 1
Oscillator 2
6
Counting arrangements
Clearly, we are not going to count by hand every
arrangement of energy possible for 3N oscillators
in, e.g., a mole of solid (N 6 x 1023). Need
to consider permutations and combinations. A
permutation is an arrangement of a collection of
objects where the ordering of the arrangement is
important.
ANS There are 10 choices for the 1st CD, 9 for
the 2nd, and 8 for the 3rd. Hence, 720
different lists.
The number of permutations of r objects selected
from a set of n distinct objects is denoted by
nPr where nPr n! / (n - r)!
7
Counting arrangements
?
A CD club member is asked to pick 3 CDs from a
list of 10 CDs. How many different choices are
possible?
ANS We need to divide the previous 720
arrangements by the total number of different
possible permutations of 3 choices (e.g. ABC
BAC CAB). This is 3! permutations. Hence, 120
choices are possible.
8
Counting arrangements
?
Take a collection of 10 pool (billiard) balls, 6
of which are yellow and 4 of which are red. How
many different arrangements of the coloured balls
are possible (eg RRYYYYYYRR)?
9
Counting arrangements
Returning to the distribution of energy quanta
amongst a collection of oscillators, we need to
establish a formula for the number of possible
arrangements.
Consider the case of 3 quanta of energy
distributed between 2 oscillators as before.
Well adopt the same representation as Chabay and
Sherwood, p. 348.
Thus, we have 4 objects arranged in a certain
sequence. We need N - 1 vertical bars to separate
N oscillators.
10
Counting arrangements
11
Counting arrangements
Number of ways to arrange q quanta of energy
amongst N 1D oscillators
12
How many ways can four quanta of energy be
arranged amongst four oscillators?

13
Counting arrangements
14
Microstates and macrostates
Each of the 35 different distributions of energy
is a microstate (i.e. an individual microscopic
configuration of the system, as shown above).
The 35 different microstates all correspond to
the same macrostate - in this case the
macrostate is that the total energy of the system
is 4
FUNDAMENTAL ASSUMPTION OF STATISTICAL
MECHANICS Each microstate corresponding to a
given macrostate is equally probable.
15
Microstates and macrostates poker hands
16
Whats the probability of being dealt the hand of
cards in the order shown?

1/2112
1/8192
1/2,256,781
1/311,875,200

17
Microstates and macrostates poker hands
18
As compared to the junk hand of cards, the
probability of being dealt the Royal Flush is

Higher
Lower
Exactly the same
Dont know

19
Microstates and macrostates poker hands
ANS This is 1 possibility out of a total of
52!/47! possibilities.
Youll have to bear with me again for the
answer......
20
Two interacting atoms six 1D oscillators
Remember, were still trying to find out why the
total thermal energy is shared equally between
the two blocks.
21
Are many ways are there of distributing four
quanta of energy amongst six 1D oscillators?

126
1260
1,024,256
None of these

22
If four quanta of energy are given to one atom or
the other, how many ways are there of
distributing the energy quanta?

23
Two interacting atoms six 1D oscillators
ANS 4 quanta distributed amongst 3 oscillators
15 ways x 2 30 ways
24
If three quanta are given to one atom, and one
quantum to the other, how many ways are there of
distributing the energy?

25
Two interacting atoms six 1D oscillators
?
If three quanta are given to one atom, and one
quantum to the other how many ways are there of
distributing the energy?
ANS 60 ways 3 quanta distributed amongst 3
oscillators on atom 1 10 ways. 1 quantum
distributed amongst 3 oscillators on atom 2 3
ways. However, could also have three quanta on
atom 2, one quantum on atom 1
ANS 36 ways. (i.e. 126 30 60, but make sure
you can get the same result by counting the
states as above.)
26
Two interacting atoms six 1D oscillators
For two interacting atoms it is most probable
that the thermal energy is shared equally.

We can look at this result in two ways
if frequent observations of the two atom system
are made, in 29 of the observations - i.e. 36
out of 126 the energy will be split evenly
for 100 identical two atom systems, at any given
instant 29 will have the thermal energy split
evenly between the two atoms.

27
Increasing the number of atoms
For two atoms, 29 of the time the system will
adopt a state where the energy is shared equally.
So, although this is the most probable
distribution, it happens lt ? of the time. It is
almost as likely (24) to find all the energy on
one atom or the other. What happens as we add
more atoms?
ANS 6.35 x 108
ANS 1
28
Increasing the number of atoms
No. of arrangements increases VERY quickly for
small changes in numbers of oscillators. For
300 oscillators (100 atoms) there are 1.7 x
1096 ways of distributing 100 quanta of
energy. 1 mole of any material contains 6 x
1023 atoms.
Is it possible that all the energy could be
concentrated on 1 atom? Are we ever likely to
see this happen?
Yes!
No!
29
Module Questionnaire
30
The aims and objectives were made clear