Air entering the lungs ends up in tiny sacs called alveoli.From the alveoli, the oxygen diffuses into the blood. The average radius of the alveoli is 0.0050 cm and the air inside contains 14 per cent oxygen. Assuming that the pressure in the alveoli is 1.0 atm and the temperature is `37^(@)C`, calculate the number of oxygen molecules in one of the alveoli.

Options

A. `6xx10^(13)`
B. `10^(24)`
C. `1.7xx10^(22)`
D. `1.7xx10^(12)`

Explanation

To calculate the number of oxygen molecules in one of the alveoli, we can follow these steps:

### Step 1: Calculate the Volume of the Alveolus
The alveolus can be approximated as a sphere. The formula for the volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

Where:
- \( r \) is the radius of the alveolus.

Given:
- Radius \( r = 0.0050 \, \text{cm} = 5.0 \times 10^{-3} \, \text{cm} \)

Substituting the value into the formula:

\[ V = \frac{4}{3} \pi (5.0 \times 10^{-3})^3 \]

Calculating this gives:

\[ V \approx \frac{4}{3} \times 3.14 \times (1.25 \times 10^{-7}) \approx 5.23 \times 10^{-7} \, \text{cm}^3 \]

Since \( 1 \, \text{cm}^3 = 1 \, \text{mL} \), we can also express this volume as:

\[ V \approx 5.23 \times 10^{-7} \, \text{mL} \]

### Step 2: Convert Volume to Liters
To convert the volume from mL to liters:

\[ V = 5.23 \times 10^{-7} \, \text{mL} \times 10^{-3} = 5.23 \times 10^{-10} \, \text{L} \]

### Step 3: Calculate the Number of Moles of Air
Using the ideal gas law, we can calculate the number of moles \( n \):

\[ n = \frac{PV}{RT} \]

Where:
- \( P = 1.0 \, \text{atm} \)
- \( V = 5.23 \times 10^{-10} \, \text{L} \)
- \( R = 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \)
- \( T = 37 \, \text{°C} = 310 \, \text{K} \) (convert to Kelvin by adding 273)

Substituting the values:

\[ n = \frac{(1.0)(5.23 \times 10^{-10})}{(0.0821)(310)} \]

Calculating this gives:

\[ n \approx \frac{5.23 \times 10^{-10}}{25.453} \approx 2.06 \times 10^{-11} \, \text{mol} \]

### Step 4: Calculate the Number of Moles of Oxygen
Since the air contains 14% oxygen, the number of moles of oxygen \( n_{O_2} \) is:

\[ n_{O_2} = 0.14 \times n \]

Substituting the value of \( n \):

\[ n_{O_2} = 0.14 \times 2.06 \times 10^{-11} \approx 2.89 \times 10^{-12} \, \text{mol} \]

### Step 5: Calculate the Number of Oxygen Molecules
To find the number of molecules, we multiply the number of moles of oxygen by Avogadro's number \( N_A \):

\[ N = n_{O_2} \times N_A \]

Where \( N_A = 6.022 \times 10^{23} \, \text{molecules/mol} \).

Substituting the values:

\[ N = (2.89 \times 10^{-12}) \times (6.022 \times 10^{23}) \approx 1.74 \times 10^{12} \, \text{molecules} \]

### Final Answer
The number of oxygen molecules in one of the alveoli is approximately \( 1.74 \times 10^{12} \).

---