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 = - 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. 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 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.

Short answer

The amount the work must be done to move one sodium ion is $1.1\times {10}^{-20}J$

The numbers of sodium ions arerole="math" localid="1648139356523" $2.0\times {10}^{21}ions$

Step-by-step solution

Theory of  work done source

The equation of the potential energy is

U=q/V

where q is the charge and V is the potential difference.

$$\begin{gathered} V=70mV=70\times {10}^{-3}V \\ q=1.6\times {10}^{-19}C \end{gathered}$$

$W=U$

Applying all data in the above equation

The process to calculate the numbers of sodium ions 

Formula

Number of sodium ions= power required to pump sodium ions/ work done to move the sodium ions

$N=\frac{20J}{1.1\times {10}^{-20}J}=2.0\times {10}^{21}J$