Question

Living cells "pump" singly ionized sodium ions, Na+, from the inside of the cell to the outside to maintain a membrane potential ?Vmembrane=Vin?Vout= -71mV It is calledpumping 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.

a) How much work must be done to move one sodium ion from the inside of the cell to the outside?

B )At rest, the human body uses energy at the rate of approximately 93W to maintain basic metabolic functions. It has been estimated that 24{\rm \\%} 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.

Answer 1

In order to answer this question, you need to know the equation for the potential energy gained U by a charge q moving through a electric potential difference of V:

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Part a) To find how much work needs to be done to move the singly ionized sodium atom through the pump, use the above equation, with q being 1.6x10^-19 C (a single unit of charge that comes from the atom being singly ionized), and 71x10^-3 V (the 71 mV) for the potential difference V:

So approximately 1.1x10^-20 J of work needs to be done to move a singly ionized sodium atom through the pump.

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Part b)

Step 1) To find out how many sodium pumps are operated by the bodies base metabolic needs, first find out how much energy is used to operate the pumps. Do that by taking the 24% (the percentage of power used to run the pumps) of the 93 J/s (the 93 W) that the body needs to run:

So all the pumps together use 22.32 J/s to run.

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Step 2) Set-up the equation governing the energy useage by the pumps, using that the total energy used by n number of pumps per second is P, with each pump using U amount of energy to run:

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Step 3) Rearrange that to solve for the number of pumps operated per second n:

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Step 4) Substitute in the 22.32 J/s found earlier for the total power P used by the pumps, and 1.136x10^-20 J for the energy needed to run a single pump U found in the first part:

So about 2x10²¹ sodium pumps are operating every second to maintain the bodies needs.