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Assume a length of axon membrane of about 0.10 m is excited by an action potential (length excited $=$ nerve speed $\times$ pulse duration $=50.0 \mathrm{m} / \mathrm{s} \times 2.0 \times 10^{-3} \mathrm{s}=0.10$ $\mathrm{m}$ ). In the resting state, the outer surface of the axon wall is charged positively with $\mathrm{K}^{+}$ ions and the inner wall has an equal and opposite charge of negative organic ions, as shown in Figure $\mathrm{P} 18.43$ . Model the axon as a parallel-plate capacitor and take $C=\kappa \epsilon_{0} A / d$ and $Q=C \Delta V$ to investigate the charge as follows. Use typical values for a cylindrical axon of cell wall thickness $d=1.0 \times 10^{-8} \mathrm{m},$ axon radius $r=1.0 \times 10^{1} \mu \mathrm{m},$ and cell-wall dielectric constant $\kappa=3.0$ . (a) Calculate the positive charge on the outside of a 0.10-m piece of axon when it is not conducting an electric pulse. How many $\mathrm{K}^{+}$ ions are on the outside of the axon assuming an initial potential difference of $7.0 \times 10^{-2} \mathrm{V}$ ? Is this a large charge per unit area? Hint: Calculate the charge per unit area in terms of electronic charge $e$ per squared $\left({\mathrm{Å}}^{2}\right) .$ An atom has a cross section of about $1 {\mathrm{Å}}^{2}$ $(1 Å$ $=10^{-10} \mathrm{m} )$. (b) How much positive charge must flow through the cell membrane to reach the excited state of $+3.0 \times 10^{-2} \mathrm{V}$ from the resting state of $-7.0 \times 10^{-2} \mathrm{V}$ ? How many sodium ions (Na^ $^{+} )$ is this? (c) If it takes 2.0 $\mathrm{ms}$ for the $\mathrm{Na}^{+}$ ions to enter the axon, what is the average current in the axon wall in this process? (d) How much energy does it take to raise the potential of the inner axon wall to $+3.0 \times 10^{-2} \mathrm{V},$ starting from the resting potential of $-7.0 \times 10^{-2} \mathrm{V}$ ?

a. no answer available

b. no answer available

c. no answer available

d. w=7 \cdot 5 \times 10^{-12} \mathrm{J}

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University of Michigan - Ann Arbor

Numerade Educator

Hope College

McMaster University

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