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Nernst-Planck Equation

When ions are in solution, there are three mechanisms for movement: brownian motion (thermal), ordered drift due to a potential (voltage) field, and diffusion, ordered drift down a concentration gradient.

Figure: A glass filled with high concentration Na on the left and low concentration Na on the right

Consider first, drift down a concentration gradient. In solution, each molecule is not stationary but is moving and the motion results in collisions with neighbors. In a region of high concentration, collisions are more likely than in regions of low concentration. Thus, at the interface between a high concentration and a low concentration, there will be a collision gradient, more collisions on the high concentration side than on the low concentration side (see figure D.1.1. This gradient results in a drift of carriers into the low concentration region, increasing its concentration. An equilibrium is reached when the concentration equilibrates and the frequency of collision is spatially uniform. The flux associated with the drift is

$\displaystyle j = -\vert Z\vert D\frac{d[C]}{dt}$ (D.1.1)

where Z is the valence of the charge carrier and D is the diffusion constant.

Figure: A + charge in solution attracted to the negative plate.

Charge carriers are accelerated by the electrical attraction of the carrier within the electric field. As the charge is attracted, things get in the way that result in collisions. After each collision, velocity is lost resulting and is slowly recovered due to the acceleration caused by the attraction of the charge carrier and the potential field. To descirbe this, we start with the force that an unit charge feels within an electric field:

$\displaystyle F = -qE = \frac{d(mv)}{dt} = \frac{mv_d}{\tau}$    

where F is the force, q is the unit charge and E is the electric field. Remember that the electric field, E = dV/dx, is simply the change in potential at a point. Now the attractive force will change the momemtum of a charge carrier either positively (acceleration) or negatively (deceleration). We assume that the drift velocity is $ v_d$ and $ \tau$ is the time between the collisions of the charge carrier and something. From this, we can write the drift velocity between collisions due to the field (we ignore the collision events - acceleration and deceleration) as

$\displaystyle v_d = -\frac{qE\tau}{m}$    

Now define the mobility of the charge, $ \mu $ as

$\displaystyle \mu = \frac{q\tau}{m}$    

so that the drift velocity is

$\displaystyle v_d = -\mu E$    

which simply states that in the presence of a spatially uniform electric field, the charge will move with a fixed velocity known as the drift velocity, that is proportional to the charge and inversely proportional to the mass of the charge.

Now the current density associated with the flow of charge within an electric field in a solution is:

$\displaystyle j = v_d\vert Z\vert F[C]$    

where Z is the valence of the charge carrier, F is the number of coulombs of charge per mole of ion, and [C] is the concentration of charge carriers. But the drift velocity is $ -\mu E$ so that the current density can be written as

$\displaystyle j = \vert Z\vert F[C]\mu E = \sigma E$   $\displaystyle \textrm{ Ohm's law}$    

where $ \sigma$ = $ \vert Z\vert F\mu [C]$ is the conductivity. Note that the conductivity is proportional to the concentration of the ion carrier and its charge. Now, we combine the diffusive and electrical components of the flux under equilibrium conditions and have

$\displaystyle -\vert Z\vert D\frac{d[C]}{dx} + \vert Z\vert F[C]\mu E = -D\frac{d[C]}{dx} + F[C]\mu\frac{dV}{dx} = 0$    

Now we integrate this across the interface (cell membrane) that separates the extracellular fluid from the cytoplasm and have

$\displaystyle \int{D\frac{d[C]}{dx}}$ $\displaystyle = \int{F[C]\mu\frac{dV}{dx}}$    
$\displaystyle \int{D\frac{d[C]}{[C]}}$ $\displaystyle = \int{F\mu dV}$    
$\displaystyle D[ln[C_{\textrm{out}}] - ln[C_{\textrm{in}}]]$ $\displaystyle = F\mu (V_{\textrm{out}} - V_{\textrm{in}})$    
$\displaystyle V_{\textrm{out}} - V_{\textrm{in}}$ $\displaystyle = \frac{D}{F\mu} ln\frac{[C_{\textrm{out}}]}{[C_{\textrm{in}}]}$    

Einstein showed, in a cute little derivation that

$\displaystyle D = \mu RT$    

where R is the gas constant (8.314 J/K mole at 27 C), and T is the absolute temperature. At 27 C, RT/F = 8.314 * 300 / 96487 = 25.8 mV at 27 C. so for a monovalent cation or anion, the transmembrane potential due to a single charge carrier is:

$\displaystyle V_{\textrm{membrane}}(mV) = \frac{RT}{F} \frac{[C_{\textrm{out}}]}{[C_{\textrm{in}}]} = 25.8 ln\frac{[C_{\textrm{out}}]}{[C_{\textrm{in}}]}$    

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Next: Cubic Nonlinear ODE Up: Derivations For the Curious Previous: Derivations For the Curious   Index

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Frank Starmer 2004-05-19