Only a tiny fraction of the diffusible ions move across a cell membrane in establishing a Nernst potential, so there is no detectable concentration change. Consider a typical cell with a volume of 10^-8 cm³, a surface area (A) of 10^-6 cm², and a membrane thickness (l) of 10^-6 cm. Suppose that [K⁺] = 155 mM inside the cell and [K⁺] = 4 mM outside the cell and that the observed Nernst potential across the cell wall is 0.085 V. The membrane acts as a charge-storing device called a capacitor, with a capacitance, C, given by

where ε₀ is the dielectric constant of a vacuum and the product ε₀ε is the dielectric constant of the membrane, having a typical value of 3 x 8.854 x 10^-12 C² N^-1 m^-2 for a biological membrane. The SI unit of capacitance is the farad, 1 F = 1 coulomb per volt = 1 C V^-1 = 1 x C² N^-1 m^-1.

(a) Determine the capacitance of the membrane for the typical cell described.

(b) What is the net charge required to maintain the observed membrane potential?

(c) How many K⁺ ions must flow through the cell membrane to produce the membrane potential?

(d) How many K⁺ ions are in the typical cell?

(e) Show that the fraction of the intracellular K⁺ ions transferred through the cell membrane to produce the membrane potential is so small that it does not change [K⁺] within the cell.

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