92PP - University Physics with Modern Physics

Question

CALCULATING LUNGS VALUE IN HUMANS. In humans, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli in the lungs. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical pair of lungs is about $\mathbf{480}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}$ and that the average volume of a single alveolus is $\mathbf{4}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{ }{\mathbf{\mu m}}^{\mathbf{3}}$. (The volume of a sphere is $\mathbf{V}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{\pi r}}^{\mathbf{3}}$ and the area of a sphere is $\mathbf{A}\mathbf{=}\mathbf{4}{\mathbf{\pi r}}^{\mathbf{2}}$).

 

1.92: What is total volume of the gas-exchanging region of the lungs? (a) $\mathbf{2000}\mathbf{ }{\mathbf{\mu m}}^{\mathbf{3}}$; (b) $\mathbf{2}\mathbf{ }{\mathbf{m}}^{\mathbf{3}}$; (c) $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{L}$; (d) $\mathbf{120}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{L}$.

Step-by-Step Solution

Answer

The answer is $2.0 \mathrm{L}$. So option (c) is the correct option. 

Step 1: Given data

Total numbers of alveoli, $\mathrm{N}=480\times {10}^6$ 

The average volume of alveoli. ${\mathrm{V}}_{\mathrm{alveoli}}=4.2\times {10}^6 {\mathrm{\mu m}}^3$  

Step 2: Introduction

Gas exchanging region of the lungs is called alveoli. The product of the total number of alveoli and the average volume of a single alveolus will give the total volume of the alveoli.

Step 3: Find the total volume of the gas-exchanging region of the lungs

Use the following formula to find the total volume of the gas-exchanging region,

 

${\mathrm{V}}_{\mathrm{T}}={\mathrm{NV}}_{\mathrm{alveoli}}$  

 

Substitute, $480\times {10}^6$ for N and $4.2\times {10}^6 {\mathrm{\mu m}}^3$ for ${\mathrm{V}}_{\mathrm{alveoli}}$ in the above equation, and you have

$\begin{array}{rcl}{\mathrm{V}}_{\mathrm{T}}& =& 480\times {10}^6\times 4.2\times {10}^6 {\mathrm{\mu m}}^3\\ & =& 2016\times {10}^{12} {\mathrm{\mu m}}^3\\ & =& 2.016\times {10}^{15} {\mathrm{\mu m}}^3\end{array}$  

Now, as known that,

$1 {\mathrm{\mu m}}^3={10}^{-15} \mathrm{L}$ 

 

Therefore, the total volume of all alveoli is,

$\begin{array}{rcl}{\mathrm{V}}_{\mathrm{T}}& =& 2.016\times {10}^{15 }{\mathrm{\mu m}}^3\times {10}^{-15} \mathrm{L}\\ & =& 2.016 \mathrm{L}\\ & \approx & 2 \mathrm{L} \end{array}$  

 

Hence, option (c) is correct.