We define the volume density and basing on the Euler’s and Lagrange theorems derive the volume continuity equation. These
fundamental formulae define the volume fixed frame of reference in the multicomponent, solid and liquid solutions. The volume velocity is a unique frame of reference for all processes namely, the mass diffusion, charge transport, heat flow, etc. No basic changes are obligatory in the foundations of linear irreversible thermodynamics except recognizing the need to add volume density to the usual list of extensive physical properties undergoing transport in every continuum. The volume fixed frame of reference allows the use of the Navier-Lamé equation of mechanics of solids. Proposed modifications of Navier-Lamé and energy conservation equations are self-consistent with the literature for solid-phase continua dating back to the classical experiments of Kirkendall and their interpretation by Darken. We do show that when Darken constraints are used the general formulae reduce to the Darken expression
