The accessibility of arbitrary boundary conditions in membrane transport

Abstract

Onsager's principle of least dissipation of energy applied to the transport of arbitrary solutions of electrolytes through a membrane involves the variation with respect totime andposition. The time variation yields the familiar expression of the entropy production being restricted to aninfinitesimal volume elementqdx. In order to go over to the entropy production in a finite volume elementqΔx, Hamilton's variation of the endpointx in thestationary state must be performed. The usual difficulties arising from this variational problem of order one in the concentration gradients can be overcome by a theorem of Caratheodory dealing with the problem of “arbitrary extremals”. Regarding that variational problems result inpotential representations it will be shown that two conditions must be satisfied in order to obtain such a representation: Theisotonicity and the restriction to ions ofequal valency. The potential function found in this way is identical with Planck's equation of the diffusion potential found already 1890. It represents thesingular solution of the Nernst differential equation. Thegeneral solution of this equation, however, presents no potential function. Thus, there exist in a membrane neighbouring states beingnot accessible along extremals. Thus, Onsager's principle of least dissipation of energy can be satisfied only in discrete cases.