Assume a length of axon membrane of about 0.10 m is excited by an action potenti

Question

Assume a length of axon membrane of about 0.10 m is excited by an action potential (length excited = nerve speed × pulse duration = 50.0 m/s ×0.0020 s = 0.10 m). In the resting state, the outer surface of the axon wall is charged positively with K+ ions and the inner wall has an equal and opposite charge of negative organic ions, as shown in the figure below. Model the axon as a parallel-plate capacitor and take C = 0A/d and Q = CVto investigate the charge as follows. Use typical values for a cylindrical axon of cell wall thickness d = 1.1 108 m, axon radius r = 1.6 101m, and cell-wall dielectric constant = 2.4.

(a) Calculate the positive charge on the outside of a 0.10-m piece of axon when it is not conducting an electric pulse. (Assume an initial potential difference of 7.0 102 V.)
C

How many K+ ions are on the outside of the axon assuming an initial potential difference of 7.0 102 V?
K+ ions

Is this a large charge per unit area? Hint: Calculate the charge per unit area in terms of electronic charge e per angstrom squared (2). An atom has a cross section of about 1 2 (1 = 1010 m). (Compare to normal atomic spacing of one atom every few .)

Yes/No    

(b) How much positive charge must flow through the cell membrane to reach the excited state of +3.0 102 V from the resting state of 7.0 102 V?
C

How many sodium ions (Na+) is this?
Na+ ions

(c) If it takes 2.0 ms for the Na+ ions to enter the axon, what is the average current in the axon wall in this process?
A

(d) How much energy does it take to raise the potential of the inner axon wall to +3.0 102 V, starting from the resting potential of7.0 102 V? (Assume that no energy is required to first raise the potential to 0 V from the resting potential of 7.0 102 V.)
J

Explanation / Answer

part a:

capacitance=k*epsilon*area/distance

=2.4*8.85*10^(-12)*pi*(16*10^(-6))^2/(1.1*10^(-8))

=1.5529*10^(-12) F

then charge =capacitance*voltage difference

=1.5529*10^(-12)*7*10^(-2)=1.0871*10^(-13) C

charge on each K+ ion=1.6*10^(-19) C

then number of K+ ion=1.0871*10^(-13)/(1.6*10^(-19))=6.7944*10^5

charge per unit area

=1.0871*10^(-13)/(pi*(16*10^(-6))^2)=1.3517*10^(-4) C/m^2

=8.4481*10^(-6) e/A^2

part b:

positive charge required=potential difference*capacitance

=(0.03-(-0.07))*1.5529*10^(-12)=1.5529*10^(-13) C

part c:

average current=charge/time

=1.5529*10^(-13) C/(0.002)=7.7645*10^(-11) A
part d:

energy required=charge*potential difference

=1.5529*10^(-13)*(0.03-(-0.07))=1.5529*10^(-14) J

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