Another situation in which this concept of entropy offers insight is the Gibbs's paradox. 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.