6.021/Notes/2006-10-11

Nernst-Planck Equation: $$J_n = -z_nFD_n\frac{\partial c_n}{\partial x}-u_nz_n^2F^2c_n\frac{\partial\psi}{\partial x}$$

Continuity: $$\frac{\partial J_n}{\partial x} = -z_nF\frac{\partial c_n}{\partial t}$$

Poisson's Equation: $$\frac{\partial^2 \psi}{\partial x^2} = -\frac{1}{\epsilon}\sum_n z_nFc_n(x,t)$$

Flux through membranes

 * Assume membrane in steady state as before
 * concentrations of charge charge can't change so current is constant
 * Four inputs: voltage on inside and outside, concentration on inside and outside
 * $$J_n = G_n (V_m-V_n)$$
 * $$G_n = \frac{1}{\int_0^d{\frac{dx}{u_nz_n^2F^2c_n(x)}}}$$ (electrical conductivity)
 * always greater than zero, means transport will always go down electrochemical gradient (lose energy)
 * Not really constant (depends on concentration) but in real cells, will seldom see much change in concentrations so we will assume $$G_n$$ is constant.
 * $$V_m=\psi(0)-\psi(d)$$ (potential difference across membrane)
 * $$V_n=\frac{RT}{z_nF}{\rm ln}\frac{c^o_n}{c^i_n}$$ (Nernst equilibrium potential)
 * this constant is part of the model and not directly measurable (not physical)
 * is electrical representation of chemical phenomenon
 * But can indirectly measure this by changing $$V_m$$. The Nernst potential is the same potential that when applied externally to the membrane causes no current.
 * $$\frac{RT}{F}\approx 26mV$$ at room temperature
 * $$\frac{RT}{F}{\rm ln(10)}\approx 60mV$$, so can use $$V_n \approx \frac{60mV}{z_n}{\rm log}\frac{c^o_n}{c^i_n}$$