Question

2.30 Suppose that the radius of a typical cell is 10 mm and that inside it 106 ATP molecules are hydrolyzed each second. What is the power density of the cell in watts per cubic meter (1 W = 1 J s−1)? A computer battery delivers about 15 W and has a volume of 100 cm3. Which has the greater power density, the cell or the battery?

Answer 1

Answer and Explanation: One ATP release approximately 30.5kJ/mol of energy. Using Avogadro's constant, it is known one mol corresponds to $6.02.10^{23}$ molecules, so 106 ATP molecules releases:

E = $30.5.10^{3}*\frac{106}{6.02.10^{23}}$

E = $537.04.10^{-20}$J

A cell is approximately a sphere. Volume of a sphere is given by

V = $\frac{4}{3}.\pi.r^{3}$

Changing radius from mm to m:

r = $10^{-2}$m

Then, volume is

V = $\frac{4}{3}.\pi.(10^{-2})^{3}$

V = $4.2.10^{-6}$ m³

Power density is given by

$power=\frac{energy}{volume}$

$power=\frac{537.04.10^{-20}}{4.2.10^{-6}}$

$power=128.10^{-14}$

power = 12.8.$10^{-15}$ W/m³

Power density for a the cell is 12.8.$10^{-15}$ W/m³.

Power density for battery is

100 cm³ = $10^{-4}$m³

power = $\frac{15}{10^{-4}}$

power = $15.10^{4}$W/m³

Comparing cell and battery, battery has a greater power density.