Storage Elements Generalized Capacitances

As a generalized capacitance we simply define any extensive quantity E. of the system which may change its value with time t E. = E. (t). In most cases, the E. will be the mole numbers of chemical species, E. =, or the electric charge E. = Q. 

The time variation of E. is due to fluxes I. in the system I. = dE./dt which defines a capacitive flux I. . Frequently, it is more convenient to define a flux not by the time change of E. but by that of its volume density or concentration c. = 

The reason for calling a time-dependent concentration a capacitance has already been explained in Section 2.3 in context with the relaxation behaviour of the blackbox membrane model. Let us assume that the concentration c. of a molecule of kind i is given as a unique function of its chemical potential c. = c. (y. ). Taking the time derivative and making use of the associated reference direction we then obtain 

The right-hand side of (4.3) precisely has the form of the constitutive relation for an electrical capacitance. We thus call C. (y. ) as defined in (4.4) a material capacity. For ideal systems like dilute solutions we obtain by inserting (3.77) into (4.4) 

Not only (4.5) but even the assumption that c. is a unique function of its chemical potential y. are approximations which are valid only for special systems like ideal systems. In the general case, the concentration c. of molecules of kind i will depend on the chemical potentials y. of all other molecules j including i. 

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In the present section, we start our thermodynamic stability considerations by recalling and generalizing the definition of capacitances as introduced in Section 4.2. Quite generally, capacitances are the equilibrium parts of thermodynamic networks or, in other words, they are reversible storage elements for extensive quantities like energy U, volume V, and mole numbers Np 2 property of the reversibility of the storage process is expressed by Gibbs relation which in its version (3.65) for the volume densities s = S/V, u = U/V, c. = N. /V can be written as... 

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