The case of the missing entropy

The free expansion of an ideal gas into a vacuum is commonly used to illustrate a spontaneous process; however, what is often overlooked for this process is that there is no change in temperature (inherently isothermal), no exchange of heat into or out of the system (adiabatic), and no work done on or by the system. That is, for the expansion of an ideal gas into a vacuum, there are no changes in the energetic states of the particles in the system and, thus, no change in the energetic entropy, Delta S (E) = 0. But, if Delta S (E) = 0, then why is the process spontaneous? The answer is because the number of positional microstates increases. This paper describes the "missing entropy" originating from the distribution of particles in a system, as measured by Delta S (D). A positive Delta S (D) is the only change associated with processes that are inherently isothermal, adiabatic, and include no external work. The entropy of the distribution of particles, S (D), was not included in Boltzmann's distribution of energy in particles, S (E), which is used to explain the second law of thermodynamics. In Gibbs development of statistical thermodynamics, Gibbs included both particle distribution and energy distribution in his ensembles but only included energy distribution in deriving the macroscopic laws of thermodynamics. Consequently, thermodynamics cannot explain why spontaneous processes occur in inherently isothermal systems where heat and work are zero, (Q = W = 0) and Delta U = Delta S (E) = 0. Examples of such processes include free expansion of an ideal gas, mixing of ideal gases, diffusion of an ideal solute, mixing of ideal solutes, osmosis with ideal solutions, and free discharge of a concentration battery with ideal solutions. This paper uses a combinatorial model of positional microstates to develop a statistical description of the entropy of the distribution of particles, S (D), and applies the model to calculate Delta S (D) for these six examples of inherently isothermal processes in ideal systems. Delta S (D) has implications for correctly understanding processes in systems that do no work on the surroundings, for understanding colligative properties, and for understanding biological evolution and other information-driven, inherently isothermal systems.