Question

A typical adult human lung contains about 300 million tiny cavities called alveoli. Estimate the average diameter of a single alveolus.

Short answer

The average diameter of a single alveolus is 0.234 mm.

Step-by-step solution

Step 1. Volume of a sphere

The volume of a sphere can be calculated using its radius. Both these values are directly related to each other. Hence, as the radius increases, the volume of the sphere also increases.

Step 2. Calculation of the volume of alveolus

Assume that the alveoli are spherical, and the volume of a typical human lung is about 2.05 liters.

Given that there are 300 million alveoli per lung.

So, the volume of a single alveolus is:

$\begin{array}{c}V=\left(\frac{2.05\mathrm{liters}}{\mathrm{lung}}\right)\left(\frac{1\mathrm{lung}}{3\times {10}^8\mathrm{alveoli}}\right)\\ =\left(6.833\times {10}^{-9}\frac{\mathrm{liters}}{\mathrm{alveolus}}\right)\left(\frac{{10}^6{\mathrm{mm}}^3}{1\mathrm{liter}}\right)\\ =6.833\times {10}^{-3}\frac{{\mathrm{mm}}^3}{\mathrm{alveolus}}\end{array}$

Step 3. Calculation of the radius and diameter of a single alveolus

The radius of a single alveolus can be calculated as:

$\begin{array}{c}V=\frac{4}{3}\pi r^3\\ \left(6.833\times {10}^{-3}{\mathrm{mm}}^3\right)=\frac{4}{3}\pi r^3\\ r=0.117\mathrm{mm}\end{array}$

The average diameter of a single alveolus can be calculated as:

$\begin{array}{c}d=2r\\ d=2\left(0.117\mathrm{mm}\right)\\ d=0.234\mathrm{mm}\end{array}$

Thus, the average diameter of a single alveolus is $0.234\mathrm{mm}$.