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

Continuity: [math]\displaystyle{ \frac{\partial J_n}{\partial x} = -z_nF\frac{\partial c_n}{\partial t} }[/math]

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

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