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 = CΔV to investigate the charge as follows. Use typical values for a cylindrical axon of cell wall thickness d = 1.1 ✕ 10⁻⁸ m, axon radius r = 2.0 ✕ 10¹ ?m, and cell-wall dielectric constant ? = 2.7.

A diagram shows a collection of positive and negative charges in and around an axon. The diagram is divided into three sections, one on top of the other.

The top section is labeled "External fluid". A row of positive charges labeled "Positive charge layer" lies along the bottom side of this section. Above the row of positive charges, there is an even mixture of positive and negative charges.

The middle section has thickness d and is labeled "Axon wall membrane". There are no charges in this section.

The bottom section is labeled "Internal fluid". A row of negative charges labeled "Negative charge layer" lies along the top side of this section. Below the row of negative charges, there is an even mixture of positive and negative charges that is less dense than the mixture of charges in the external fluid. The axon radius r is drawn from a point in the internal fluid's mixture of charges to the center of the axon wall membrane.

(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 ✕ 10⁻² V.)
C

How many K⁺ ions are on the outside of the axon assuming an initial potential difference of 7.0 ✕ 10⁻² 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 (Å²). An atom has a cross section of about 1 Å² (1 Å = 10⁻¹⁰ m). (Compare to normal atomic spacing of one atom every few Å.)
YesNo

(b) How much positive charge must flow through the cell membrane to reach the excited state of +3.0 ✕ 10⁻² V from the resting state of −7.0 ✕ 10⁻² 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 ✕ 10⁻² V, starting from the resting potential of −7.0 ✕ 10⁻² V? (Assume that no energy is required to first raise the potential to 0 V from the resting potential of −7.0 ✕ 10⁻² V.)
J

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 = CΔV to investigate the charge as follows. Use typical values for a cylindrical axon of cell wall thickness d = 1.1 ✕ 10⁻⁸ m, axon radius r = 2.0 ✕ 10¹ ?m, and cell-wall dielectric constant ? = 2.7.

A diagram shows a collection of positive and negative charges in and around an axon. The diagram is divided into three sections, one on top of the other.

The top section is labeled "External fluid". A row of positive charges labeled "Positive charge layer" lies along the bottom side of this section. Above the row of positive charges, there is an even mixture of positive and negative charges.

The middle section has thickness d and is labeled "Axon wall membrane". There are no charges in this section.

The bottom section is labeled "Internal fluid". A row of negative charges labeled "Negative charge layer" lies along the top side of this section. Below the row of negative charges, there is an even mixture of positive and negative charges that is less dense than the mixture of charges in the external fluid. The axon radius r is drawn from a point in the internal fluid's mixture of charges to the center of the axon wall membrane.

(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 ✕ 10⁻² V.)
C

How many K⁺ ions are on the outside of the axon assuming an initial potential difference of 7.0 ✕ 10⁻² 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 (Å²). An atom has a cross section of about 1 Å² (1 Å = 10⁻¹⁰ m). (Compare to normal atomic spacing of one atom every few Å.)

YesNo

(b) How much positive charge must flow through the cell membrane to reach the excited state of +3.0 ✕ 10⁻² V from the resting state of −7.0 ✕ 10⁻² 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 ✕ 10⁻² V, starting from the resting potential of −7.0 ✕ 10⁻² V? (Assume that no energy is required to first raise the potential to 0 V from the resting potential of −7.0 ✕ 10⁻² V.)
J

External fluid
Positive
charge
layer
Axon wall membrane
d
Negative
charge
layer
Internal fluid
+
- Axon radius = r
+

Accepted Answer

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