Another situation in which this concept of entropy offers insight is the
Gibbs's paradox[9]. Gibbs's paradox arises from the way in
which the number of accessible states of a gas is calculated in classical
mechanics. Classically, a monatomic gas of N distinguishable molecules at
temperature T and volume V has 
accessible distinguishable
states:
where the 
is a constant which relates the phase
space volume to the number of states. This implies the following equation
for the entropy:
where k, 
, and 
are constants. We can ignore the additive
constant in the classical entropy, since it is independent of N, V, and T. This equation, however, leads to some undesired results. If one
begins with a gas of N molecules with temperature T and volume V the
gas has the entropy given above. If a partition is placed so that the volume
is divided into two sections of equal volume, the entropy of each side
becomes
so that the total entropy of the divided system is
which is 
lower than it was before the partition was
added. The entropy of the system has apparently been reduced by the addition
of the partition. This problematic result of the classical formulation is
usually avoided in by claiming that the molecules are indistinguishable.
This reduces the number of accessible states by N!, yielding an equation
for the entropy which eliminates the ``paradox.'' This modification is
justified by quantum mechanics; however, it is possible to resolve the
paradox in another, purely classical manner.
All of the above listed equations for entropy are accurate given any set of N molecules of known temperature in a known volume. In the equation for the
divided volume, however, we are not ``given'' 
molecules. Which of
the original N molecules is in which of the final two volumes is
undetermined by N, V, and T. If the ``entropy decrease'' is to be used
to do work, then this information must be gathered by the device doing the
work. Once this information is gathered, the device could replace the
partition with a special semipermeable membrane that is only permeable to
those molecules in the left hand side. The resulting osmotic pressure
difference can be used to extract work.
To determine which semipermeable membrane to use, the device must determine
which side of the partition each molecule is on. Each of the molecules can
be on either side, so the location of each molecule may be determined by a
binary number. The membrane building device must use one bit for each of the
N molecules, so the memory will have 
accessible states. The
entropy increase of the membrane builder is therefore 
, which is exactly enough to
counterbalance the decrease in entropy of the gas in the Gibbs's paradox.
Once information on which molecules are on which side is available, the
entropy expression for the case where we are ``given'' molecules
in each volume is accurate.