Membrane Thermodynamics for Osmotic Phenomena

In this chapter, we briefly review the thermodynamic ensembles and associated energy functions using the seven thermodynamic variables. The energy E, the entropy S, and the system volume V are used to derive the temperature T and pressure P. The chemical potential μ is derived as the change of the system energy with respect to the number of matters N in the isobaric‐isothermal environment. A dilute solution is defined as a homogeneous mixture of solvent and inert solutes, where the total number and volume of solutes are much smaller than those of the solvent. Gibbs free energy of the dilute solution is used to rigorously derive the osmotic pressure by equilibrating chemical potentials of solutes and solvent. Nonequilibrium of the filtration systems is reviewed by introducing the irreversible thermodynamic model with Onsager’s reciprocal theorem. Direct applications of the irreversible thermodynamic model are currently limited due to the absence of the exact nonequilibrium statistical mechanics. We hope this chapter, containing a review of statistical mechanics, related to membrane separations and osmosis phenomena, helps researchers and especially graduate students, who seek an in‐depth understanding of membrane separation from the theoretical statistical physics as applied to chemical and environmental engineering