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**13** 

*1Russia 2USA* 

**Statistical Thermodynamics of Material** 

This chapter outlines a theoretical framework for the microscopic approach to material transport in liquid mixtures, and applies that framework to binary one-phase systems. The material transport in this approach includes no hydrodynamic processes related to the macroscopic transfer of momenta. In analyzing the current state of thermodynamic theory, we indicate critically important refinements necessary to use non-equilibrium thermodynamics and statistical mechanics in the application to material transport in non-

**2. Thermodynamic theory of material transport in liquid mixtures: Role of the** 

The aim of this section is to outline the thermodynamic approach to material transport in mixtures of different components. The approach is based on the principle of local equilibrium, which assumes that thermodynamic principles hold in a small volume within a non-equilibrium system. Consequently, a small volume containing a macroscopic number of particles within a non-equilibrium system can be treated as an equilibrium system. A detailed discussion on this topic and references to earlier work are given by Gyarmati (1970). The conditions required for the validity of such a system are that both the temperature and molecular velocity of the particles change little over the scale of molecular length or mean free path (the latter change being small relative to the speed of sound). For a gas, these conditions are met with a temperature gradient below 104 K cm-1; for a liquid, where the heat conductivity is greater, the speed of sound higher and the mean free path is small, this condition for local equilibrium is more than fulfilled, provided the experimental

Thermodynamic expressions for material transport in liquids have been established based on equilibrium thermodynamics (Gibbs and Gibbs-Duhem equations), as well as on the principles of non-equilibrium thermodynamics (thermodynamic forces and fluxes). For a review of these models, see (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine,

**1. Introduction** 

isothermal mixtures.

1999; Haase, 1969).

**Gibbs-Duhem equation** 

temperature gradient is below *104 K cm-1*.

**Transport in Non-Isothermal Mixtures** 

Semen Semenov1 and Martin Schimpf2

*1Institute of Biochemical Physics RAS,* 

*2Boise State University, Boise* 

sediment micro and meio-benthic measures. *Ecological Indicators*, Vol. 6, pp. 525– 542, ISSN: 1470-160X.

