Reversibility and the second law of thermodynamics

The previous discussion casts the concept of reversible and irreversible processes in a new light. Consider an isolated system (no mass or energy exchange with its surrounding environment is allowed) with N ideal gas particles, energy E, in volume V/2 (Fig. 6.5). The system is confined in F/2 by a barrier wall and is at equilibrium. The entropy of this macroscopic state is 

If the wall is removed, the gas will expand to volume F. At the new equilibrium state, the entropy is 

This is a well-known result in classical thermodynamics and is in accord with the second law of thermodynamics, which postulates that macroscopic phenomena move in the direction of increasing entropy. This is true not because it is theoretically impossible for the process to be reversed, but because it is practically impossible to observe a process reversal for systems with many particles. In other words, there is a non-zero probability that the system may be observed with all particles confined in V/2, but in actuality this probability is so small that we cannot expect to observe a system spontaneously decreasing its entropy. 

On the other hand, for a system with, say, N = 3 particles, there is a measurable probability that the particles will be confined in half the volume (see Exercises). 

This discussion can be extended to NVT systems. Consider the process of free expansion for a system of N particles from volume V/2 to volume V. In this case assume that the entire system is in thermal contact with a heat bath of constant temperature. The change in the Helmholtz free energies between the two equilibrium states can be computed with the help of the partition function 

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