Linear phenomenological laws

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Linear phenomenological laws

Description

Linear phenomenological laws are at the core of the thermodynamics of irreversible processes TIP, considered to apply near equilibrium but more generally in transport processes (e.g. Fick's law). In TIP, linearity is discussed as the dependence of generalized flows on generalized forces, I = -L·F, where L is expected to be constant (as a prerequisite for linearity) and must not be a function of the force F (affinity) for Onsager reciprocity to apply. This paradigm is challenged by the ergodynamic concept of fundamentally non-linear isomorphic flow-force relations and is replaced by the generalized isomorphic flow-pressure relations. Flows I [MU·s^-1] and forces F [J·MU^-1] are conjugated pairs, the product of which yields power, I·F = P [J·s^-1 = W]. Then vectoral and vectorial transport processes are inherently non-linear flow-force relationships, with L = u·c in continuous transport processes along a gradient (c is the local concentration), or L = u·α (α is the free activity in a discontinuous transport process across a semipermeable membrane) — formally not different from (isomorphic to) scalar chemical reactions.

Reference: Gnaiger 2020 BEC MitoPathways, Gnaiger 1989 Energy Transformations, MitoPedia: Ergodynamics

Communicated by Gnaiger Erich 2022-10-20

Linear laws in transport processes are not linear flow-force laws

In TIP, the linearity between 'generalized flows' and 'generalized forces' is emphasized to apply to transport processes:

• 'In contrast to the simple behavior of transport processes, the linear relations between rates and affinities are not always sufficient and in many cases it is necessary to take into account non-linear relations like (5.14).' - (p. 59 in Prigogine 1967 Interscience)
• 'Even in the validity range of the Gibbs formula the relations between chemical rates and affinities may be non-linear (see, i.e., (5.14)). This is in contrast with transport processes where we restrict ourselves to linear laws with respect to the forces. - (p. 94 in Prigogine 1967 Interscience)

Problems

1. 'Rates' or generalized fluxes (p. 40 in Prigogine 1967 Interscience) must be distinguished from flows.
2. An amazingly vague terminology is used in physical chemistry, where 'natural assumptions' are not expected, since such assumptions can be easily replaced by rigorous isomorphic analyses and are falsified by such anayses. 'It is quite natural to assume, at least close to equilibrium, that we have linear relations between the rates and the affinities. Such a scheme automatically includes empirical laws as Fourier's law for heat flow or Fick's law for diffusion.' - (p. 44/45 in Prigogine 1967 Interscience)
3. Fick's law for diffusion is a linear law of the form I = -u·ΔΠ for discontinuous transport across a semipermeable membrane. I [mol·s^-1] times Π [Pa = J·m^-3] does not yield power P [W = J·s^-1]. Fick's law, therefore, must not be considered as a linear phenomenological flow-force law, but instead a linear flow-pressure law, where the diffusion coefficient is D = u·RT and Π = RT·Δc. This yields the more familiar form of Fick's law as I = -D·Δc = -u·RT·Δc. The free activity α is pressure divided by force, i.e. pressure Π = RT·Δc [J·m^-3] divided by force F = RT·Δlnc [J·mol^-1]. Then α = Δc/Δlnc [mol·m^-3]. From Fick's law in the form I = -u·α·F it can be seen that the phenomenological coefficient combines the term mobility u related to the molecular details (particle size, membrane permeability) and free activity α, L = u·α. Then correctly P = I·F. In contrast, the ergodynamic pressure concept combines the free activity α and the force F which are interdependent, Π = α·F. Note: The symbols for isomorphic force F and pressure Π are simplified here in a 'short-hand' style and should be replaced by iconic symbols.

MitoPedia concepts: Ergodynamics