Living cells 'pump' singly ionized sodium ions, Na+, from the inside of the cell to the outside to maintain a membrane potential ΔV_membrane=V_in−V_out=−70 mV. It is called pumping because work must be done to move a positive ion from the negative inside of the cell to the positive outside, and it must go on continuously because sodium ions 'leak' back through the cell wall by diffusion. At rest, the human body uses energy at the rate of approximately 100 W to maintain basic metabolic functions. It has been estimated that 20% of this energy is used to operate the sodium pumps of the body. Estimate—to one significant figure—the number of sodium ions pumped per second.

검증된 단계별 안내

Determine the total power used by the sodium pumps. Since 20% of the body's total power (100 W) is used for sodium pumps, calculate this fraction: \( P_{\text{sodium}} = 0.2 \times 100 \text{ W} \).

Calculate the work done to pump a single sodium ion. The work is given by \( W = q \Delta V_{\text{membrane}} \), where \( q \) is the charge of a single sodium ion (\( q = 1.6 \times 10^{-19} \text{ C} \)) and \( \Delta V_{\text{membrane}} = -70 \text{ mV} \) (convert to volts: \( -70 \times 10^{-3} \text{ V} \)).

Find the number of sodium ions pumped per second. The power used by the sodium pumps is related to the work done per ion and the number of ions pumped per second: \( P_{\text{sodium}} = n \cdot W \), where \( n \) is the number of ions per second. Rearrange to solve for \( n \): \( n = \frac{P_{\text{sodium}}}{W} \).

Substitute the values for \( P_{\text{sodium}} \) and \( W \) into the equation for \( n \). Ensure all units are consistent (e.g., power in watts, work in joules).

Estimate \( n \) to one significant figure. This will give the approximate number of sodium ions pumped per second by the sodium pumps in the body.

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주요 개념

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Membrane Potential

Membrane potential refers to the voltage difference across a cell's membrane, resulting from the distribution of ions inside and outside the cell. In this context, a resting membrane potential of -70 mV indicates that the inside of the cell is more negative compared to the outside. This potential is crucial for various cellular processes, including the generation of action potentials in neurons and muscle cells.

Sodium-Potassium Pump

The sodium-potassium pump is a vital membrane protein that actively transports sodium ions (Na+) out of the cell and potassium ions (K+) into the cell, using ATP as an energy source. This pump is essential for maintaining the resting membrane potential and cellular homeostasis. By moving three sodium ions out for every two potassium ions moved in, it creates a concentration gradient that is critical for nerve impulse transmission and muscle contraction.

Energy Consumption in Cells

Cells consume energy to perform various functions, including maintaining ion gradients across their membranes. In the human body, approximately 20% of the total energy expenditure is dedicated to operating sodium pumps, which is significant given the body's resting energy use of about 100 W. Understanding this energy allocation helps in estimating the number of ions pumped, as it links the energy used to the work done in moving ions against their concentration gradients.