Concentrational Flux

Consider first the current due to the concentration gradient. which will be given by

In words, Equation (A-3) states that the current through the membrane of area A will be equal to the flux, &C, of the ion down the concentration gradient (number of ions per second per unit area of membrane) multiplied by Z (the valence of the ion) and F (Faraday's constant; 96,500 coulombs per mole of univalent ion). The factor ZF translates the flux of ions into flux of charge and hence into an electrical current. The flux 0C for a given ion (call the ion Y, for example) will depend on the concentration gradient of Y across the membrane (that is, [Y]in - [Y]out) and on the membrane permeability to Y, pY. Quantitatively, this relation is given by

Note that pY has units of velocity (cm/sec), in order for &C to have units of molecules/sec/cm2 (remember that [Y] has units of molecules/cm3). The permeability coefficient, pY, is in turn given by pY = DY/a (A-5)

where DY is the diffusion constant for Y within the membrane and a is the thickness of the membrane. DY can be expanded to yield

where u is the mobility of the ion within the membrane and RT (the gas constant times the absolute temperature) is the thermal energy available to drive ion movement. Substituting Equation (A-6) in (A-5) and the result in (A-4) yields

Equation (A-7) gives us the flux through a membrane of thickness a, but we would like a more general expression that gives us the flux through any arbitrary plane in the presence of a concentration gradient. To arrive at this expression, consider the situation diagrammed in Figure A-1, which shows a segment

Membrane

Figure A-1 Segment of membrane separating two compartments.

of membrane separating two compartments. The dimension perpendicular to the membrane is called x, and the membrane extends from 0 to a (thickness = a). In this situation, Equation (A-7) can be expressed in the form of an integral equation:

Here, C stands for the concentration of the ion; therefore, in reference to Figure A-1, Ca is [Y]in and C0 is [Y]out. Differentiating both sides ofEquation (A-8) yields

which can be arranged to give the more general form of Equation (A-7) that we desire:

de dx

Equation (A-10) can be substituted into Equation (A-3) to give us the ionic current due to the concentration gradient.

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