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Note 4.
Entropy II: Molecular basis of Entropy
4.1 The molecular basis of Entropy
We have discussed some mathematical aspects of entropy, but without a microscopic
interpretation, entropy is often a vague and abstract concept. I will now give a
microscopic understanding of entropy. This will also (hopefully) help to shed light on the
nature of heat as well.
Let’s start by defining a new quantity
. We define
as the ln (natural log)
of how
many different arrangement a system could have. For example, consider flipping a coin.
When it lands, it can be either heads or tails. Thus, there are 2 different arrangements and
2
ln
1
coin
What if we had 2 coins? How many arrangements would be possible then? Each coin
would have 2 possibilities, so there would be 2
2
ways (both heads; 1st head, 2nd tails; 1st
tails, 2nd heads; both tails). In general, if we have
N
coins, then there are 2
N
different
rearrangements. This means that
2
ln
2
ln
N
N
coin
N
On thing we immediately notice is that
coin
coin
N
N
1
Thus,
is extensive!
Next, let’s consider something closer to the thermodynamic systems we have been
discussing. How many ways are there to rearrange the
N
particles of an ideal gas in a
volume
V
? Let’s say that each particle has a volume
b
. For simplicity, let’s imagine that
the box is cubical and that are gas particles are little cubes as well. Then, we ask, how
many ways can we arrange a cube of volume
b
in a cubical box of volume
V
? One way
to answer this question is to fill the large box with particles. The box of volume
V
can
hold
V
/
b
particles. What this means is that there are
V
/
b
different places we can put a
single particle. Thus, the
for a single gas particle is
)
ln(
1
b
V
particle
Now, what if we had
N
ideal gas particles? Each particle could be in any of the
V
/
b
locations (note that since these particles are ideal, they can be in the same place possibly.
Remember that molecules in ideal gas do not see each other!). Thus, there are
N
b
V
different ways to arrange these particles. This leads to

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