Consider an ideal gas of N particles in a container with a volume V.
A partition separates the container into two sections with volumes
and 
, respectively, such that 
. Also, there
are 
particles in the volume and 
particles in the
volume . It is assumed that the number density is the same throughout
the system
If the partition is now removed, what should happen to
the total entropy? Since the particles are identical, the total
entropy should not increase as the partition is removed because
the two states cannot be differentiated due to the indistinguishability of
the particles. Let us analyze this thought experiment using
the classical expression entropy derived above (i.e., we leave off
the 
term).
The entropies 
and 
before the partition is removed are
and the total entropy is 
.
After the partition is removed, the total entropy is
Thus, the difference 
is
This contradicts our predicted result that 
. Therefore, the
classical expression must not be quite right.
Let us now restore the . Using the Stirling approximation
, the entropy can be written as
which is known as the Sackur-Tetrode equation. Using this expression for the entropy, the difference now becomes
However, since the density 
is
constant, the terms appearing in the log are all 1 and, therefore, vanish.
Hence, the change in entropy, as expected. Thus, it seems that the 1/N! term
is absolutely necessary to resolve the paradox. This means that only
a correct quantum mechanical treatment of the ideal gas gives rise to
a consistent entropy.
