Musings on Entropy — Anomalous Entropies

There are three cases of interest.

First, there is the case of entropy of mixing. This case encapsulates many of the key issues associated with entropy but also has a standard interpretation. The standard interpretation glosses over these issues. Both will be investigated.

Second, is the case of non-mixing, which is generally referred to as Gibbs Paradox. This is the situation wherein an identical noble gas exists in two containers, separated only by a membrane. The membrane is removed. does the entropy increase or remain the same?

Third, is the case of special of distributions of states, which lead to unusual values for the entropy. These occur outside of equilibrium conditions. The question is do they reveal something interesting about entropy.

Let’s examine these cases in reverse order.

Non-equilibrium Conditions and Anomalous Entropies

Normally, a high temperature leads to a high entropy, as all states are occupied per some distribution. More states are occupied since the temperature is high. Similarly, a low temperature leads to a low entropy, with few states being occupied.

• High Temperature → High Entropy
• Low Temperature → Low Entropy

However, a sample can be selected to only include particles in specific states; For instance, given a gas at equilibrium in a container at a given STP, imagine selecting only those particles that have a given velocity or energy, perhaps by opening a small opening b/w the container and another neighboring container, or by using a membrane that is transparent to particles of a given energy or higher. Such a membrane would only allow particles with an energy above some threshold to pass through it. The net result would be a 2nd container filled with particles of a given energy, with no particles below that energy.

This process could be repeated a sufficient number of times with similarly prepared containers to ensure that the final container would contain the same number of particles as the original container but only with particles above some given temperature.

The temperature of this new container would be the same as the prior container but the entropy would be commensurately lower.

• High Temperature → Lower Entropy

The entropy would be lower since all the states below the energy threshold would not be populated.

This preparation could be repeated at any temperature eg for any range of states. The process would then ‘select a slice’ of particles from the source container that have the exact range of velocity states desired and move these to a separate container. The result is the same. The initial entropy of the source container is always higher than the entropy of the final container. The entropy of the universe has of course gone up since energy was required to cause the particles to separate. But the special preparation has produced something unusual — the specially prepared container has a temperature equal to or lower than the original container. And the specially prepared container has a lower entropy. Both attributes are established by design.

• Initial Entropy >Final Entropy
• Initial Temperature ≥ Final Temperature

The preparation procedure has lead to the anomalous situation of high a temperature within a given range and a lower entropy .

The selection process can be repeated until the final container contains the highest possible energy particles only and, ultimately, if it only contains particles in a single state, an entropy of zero.

This is akin to the start of the universe.

Entropy, work and reversible processes

How does entropy relate to work? Check out Isothermal Processes and Lost work

How does entropy figure into reversible and irreversible processes? Check out Musings on Entropy — Part III — Reversible and Irreversible Processes