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Chapter 18: Problem 60

(III) During an action potential, Na\(^+\) ions move into the cell at a rate of about 3 \(\times\) 10\(^{-7} \mathrm{mol/m}^2 \cdot\) s. How much power must be produced by the "active Na\(^+\) pumping" system to produce this flow against a +30-mV potential difference? Assume that the axon is 10 cm long and 20 \(\mu\)m in diameter.

Short Answer

Expert verified
The power required is approximately 5.46 脳 10^{-10} W.

Step by step solution

01

Calculate the area of the axon

The axon is modeled as a cylinder. Use the formula for the lateral surface area of a cylinder, which is \( A = 2\pi rL \), where \( r \) is the radius and \( L \) is the length of the axon. Here, \( r = 20/2 = 10 \mu m = 10 \times 10^{-6} \) meters and \( L = 0.1 \) meters. Calculate \( A \): \[ A = 2\pi (10 \times 10^{-6}) (0.1) = 2\pi \times 10^{-6} \text{ m}^2 \] \[ \approx 6.283 \times 10^{-7} \text{ m}^2 \]
02

Calculate the total moles of Na鈦� moving

Determine the total moles of \( \text{Na}^+ \) ions per second using the given rate and the surface area calculated in Step 1. The rate is given as \( 3 \times 10^{-7} \text{ mol/m}^2 \cdot s \). So total moles per second is: \[ \text{Total moles} = (3 \times 10^{-7}) \times (6.283 \times 10^{-7}) \text{ mol/sec} \] \[ = 1.885 \times 10^{-13} \text{ mol/sec} \]
03

Calculate the work done per mole of Na鈦�

Use the formula for work associated with moving charge against a potential difference: \( W = nFE \), where \( n \) is the number of moles, \( F \) is the Faraday's constant (approximately \( 96485 \) C/mol), and \( E = 30 \times 10^{-3} \) V is the potential difference. Calculate \( W \) for one mole: \[ W = (1 \text{ mol}) \times (96485 \text{ C/mol}) \times (30 \times 10^{-3} \text{ V}) \] \[ = 2894.55 \text{ J/mol} \]
04

Calculate power required for Na鈦� movement

Power is defined as energy per unit time. Using the work done per mole (from Step 3) and the total moles per second (from Step 2), calculate the power: \[ \text{Power} = 1.885 \times 10^{-13} \text{ mol/sec} \times 2894.55 \text{ J/mol} \] \[ \approx 5.46 \times 10^{-10} \text{ W} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Na+ ions
In the context of an action potential, Na\(^+\) ions are crucial players. These ions are charged particles, specifically sodium ions, and they play a vital role in transmitting electrical signals along neurons.
When an action potential occurs, sodium channels in a neuron's membrane open, allowing Na\(^+\) ions to flow into the cell.
This influx of Na\(^+\) ions causes the interior of the cell to become more positive relative to the outside, triggering the nerve signal to propagate. It is this influx that fundamentally changes the electrical state of the nerve cell.
  • Na\(^+\) ions actively participate in depolarizing the neuron.
  • They move across the cell membrane very quickly during an action potential.
  • To maintain the proper balance of Na\(^+\) inside and outside the cell, neurons have specialized "pumping" systems to restore order after the action potential passes.
Understanding the flow and regulation of Na\(^+\) ions is key to comprehending how nerve impulses travel through our bodies.
Potential Difference
Potential difference, also known as voltage, is a vital concept in the study of action potentials. It refers to the difference in electric potential between two points. In neurons, it is the difference in charge across the neuron's membrane.
During an action potential, a +30 mV potential difference is created as Na\(^+\) ions enter the neuron.
This potential difference is essential because it provides the energy necessary to transmit the nerve impulse by altering the charge of the cell.
  • The potential difference is crucial for the opening and closing of ion channels in the nerve cell membrane.
  • It represents the driving force behind the movement of ions like Na\(^+\) across the cell membrane.
  • A sudden change in this potential difference is what generates and transmits the electrical signals in neurons.
By understanding potential difference, you can grasp how electrical signals are initiated and carried along nerves.
Axon
An axon is a long, slender projection of a neuron that transmits electrical impulses away from the neuron's cell body. This is the key pathway for signals to travel over relatively long distances in the body.
In the example problem, the axon's cylindrical shape is important because it affects the rate of ion movement and the overall power required to push ions against a potential difference.
As ions move into and out of the axon, they change its electrical potential, enabling nerve signals to propagate.
  • The axon can be quite long, as it is the main transmission line of the nervous system.
  • It has specialized structures to support its elongated structure and efficient signal transmission.
  • When considering power calculations, the axon's dimension鈥攊ts length and diameter鈥攑lay a crucial role.
Understanding how the axon's structural features contribute to ion movement and signal transmission can help explain the complex processes of neuronal communication.

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Most popular questions from this chapter

(II) How many kWh of energy does a 550-W toaster use in the morning if it is in operation for a total of 5.0 min? At a cost of 9.0 cents/kWh, estimate how much this would add to your monthly electric energy bill if you made toast four mornings per week.

(II) An electric device draws 5.60 A at 240 V. (\(a\)) If the voltage drops by 15\(\%\), what will be the current, assuming nothing else changes? (\(b\)) If the resistance of the device were reduced by 15\(\%\), what current would be drawn at 240 V?

An air conditioner draws 18 A at 220-V ac. The connecting cord is copper wire with a diameter of 1.628 mm. (\(a\)) How much power does the air conditioner draw? (\(b\)) If the length of the cord (containing two wires) is 3.5 m, how much power is dissipated in the wiring? (\(c\)) If no. 12 wire, with a diameter of 2.053 mm, was used instead, how much power would be dissipated in the wiring? (\(d\)) Assuming that the air conditioner is run 12 h per day, how much money per month (30 days) would be saved by using no. 12 wire? Assume that the cost of electricity is 12 cents per kWh.

(III) A 10.0-m length of wire consists of 5.0 m of copper followed by 5.0 m of aluminum, both of diameter 1.4 mm. A voltage difference of 95 mV is placed across the composite wire. (\(a\)) What is the total resistance (sum) of the two wires? (\(b\)) What is the current through the wire? (\(c\)) What are the voltages across the aluminum part and across the copper part?

(II) A power station delivers 750 kW of power at 12,000 V to a factory through wires with total resistance 3.0 \(\Omega\). How much less power is wasted if the electricity is delivered at 50,000 V rather than 12,000 V?
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