# 6.021/Notes/2006-10-11

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

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

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

## 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
• $\displaystyle{ J_n = G_n (V_m-V_n) }$
• $\displaystyle{ G_n = \frac{1}{\int_0^d{\frac{dx}{u_nz_n^2F^2c_n(x)}}} }$ (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 $\displaystyle{ G_n }$ is constant.
• $\displaystyle{ V_m=\psi(0)-\psi(d) }$ (potential difference across membrane)
• $\displaystyle{ V_n=\frac{RT}{z_nF}{\rm ln}\frac{c^o_n}{c^i_n} }$ (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 $\displaystyle{ V_m }$. The Nernst potential is the same potential that when applied externally to the membrane causes no current.
• $\displaystyle{ \frac{RT}{F}\approx 26mV }$ at room temperature
• $\displaystyle{ \frac{RT}{F}{\rm ln(10)}\approx 60mV }$, so can use $\displaystyle{ V_n \approx \frac{60mV}{z_n}{\rm log}\frac{c^o_n}{c^i_n} }$