Chemical Potential and the Transfer of Matter

The most frequent thermodynamic situation of a biological system is that of an exchange of heat and matter whereas its volume is kept fixed. Let us therefore extend the considerations of the preceding sections particularly to this kind of exchange. We start by defining the chemical potential of the component i by 

Let us again consider two subsystems (1) and (2) which can mutually exchange energy U and chemical components but are isolated as a whole. Writing the entropies of systems (1), (2) as functions of and 

If just one pair of chemical potentials, say for i = 1, violates (3.34) whereas 

we can no longer conclude that each of the fluxes of components 1 and 2 follows the direction from higher to lower values of its chemical potential. Due to some kind of an unknown internal coupling it might happen now that a single component flows uphill from a lower towards a higher value of its chemical potential, at least for a limited time interval until in the equilibrium state all fluxes will eventually come to rest. 

A particularly important generalization in view of subsequent applications of thermodynamics is the inclusion of a transfer of electric charges between the subsystems. Assume that the components i = 1, 2,. .. could also be ions and let z. be the valency of the component i such that z. = +1, +2,. .. denotes cations and z. 

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