Although science knows from the beginning of scientific thinking that the material world around us comprises discrete objects (particles) that cannot be divided without limitations at a given abstraction level, most of the scientific laws were devised for a continuous material, such as the laws about pressure and current. Our laws about mass and charge (and so: on energy, impulse and so on), are about continuous concepts. Furthermore, science has created abstract concepts such as space and time, mass and charge, continuous and discrete views. Classical science considered only one such concept at a time. Thermodynamics was the first discipline which connected two of those separated attributes, the discrete and continuous views.
Thermodynamics could provide a framework for handling ions and determining their thermodynamical properties. However, as good thermodynamic textbooks (including the one on the thermodynamics of the membrane [97]) emphasize, thermodynamics derives its concepts for non-interacting particles, so one cannot expect its validity for ionic solutions [22]. Boltzmann assumed that, in the absence of long-range interaction between the particles, the sizes of cells in the phase space do not change. In addition, he required the presence of a vast number of particles in a homogeneous, isotropic, infinite volume, which is typically not the case in biology.