Question

The average person takes \(500 \mathrm{~mL}\) of air into the lungs with each normal inhalation, which corresponds to approximately \(1 \times 10^{22}\) molecules of air. Calculate the number of molecules of air inhaled by a person with a respiratory problem who takes in only \(350 \mathrm{~mL}\) of air with each breath. Assume constant pressure and temperature.

Short answer

The person inhales \( 7 \times 10^{21} \) molecules of air each breath.

Step-by-step solution

Understand the Relationship

Given that 500 mL of air contains approximately \( 1 \times 10^{22} \) molecules, establish the direct relationship between volume and the number of molecules.

Set Up the Proportion

A direct proportion can be used since the problem assumes constant pressure and temperature. This gives the proportion: \[ \frac{500 \text{ mL}}{1 \times 10^{22} \text{ molecules}} = \frac{350 \text{ mL}}{x \text{ molecules}} \]

Solve for the Unknown Number of Molecules

Cross-multiply to solve for \( x \): \[ 500 \text{ mL} \times x \text{ molecules} = 350 \text{ mL} \times 1 \times 10^{22} \text{ molecules} \]

Simplify the Equation

Isolate \( x \) by dividing both sides by 500 mL: \[ x = \frac{350 \text{ mL} \times 1 \times 10^{22} \text{ molecules}}{500 \text{ mL}} \]

Calculate the Result

Perform the calculation: \[ x = \frac{350}{500} \times 1 \times 10^{22} \] \[ x = 0.7 \times 1 \times 10^{22} \] \[ x = 7 \times 10^{21} \text{ molecules} \]

Key concepts

Direct Proportion

The direct proportion is a fundamental mathematical concept, especially used in the fields of chemistry and physics. It describes a relationship between two quantities where if one quantity increases, the other one also increases at a consistent rate, and vice versa. In our exercise, this concept helps us relate the volume of air to the number of molecules it contains. Here, a normal person inhales 500 mL of air containing approximately \(1 \times 10^{22}\) molecules. When the volume of air changes to 350 mL due to a respiratory problem, we can use direct proportion to find out the number of molecules in the smaller volume.
We set up a proportion to solve the problem:
\[ \frac{500 \text{ mL}}{1 \times 10^{22} \text{ molecules}} = \frac{350 \text{ mL}}{x \text{ molecules}} \]
This equation is essential because it helps us to solve for the unknown value (\(x\)), showing a direct relationship.

Gas Volume

Gas volume is a measure of the amount of space that a gas occupies. In respiratory physiology, understanding gas volume is crucial to assessing lung function. The volume of gas that a person breathes in or out can be measured in liters (L) or milliliters (mL). When the volume of inhaled air is reduced from 500 mL to 350 mL due to a respiratory issue, we need to determine how many air molecules the person is still inhaling.
Assuming constant temperature and pressure, gases follow the ideal gas law, meaning that the number of molecules is directly proportional to the volume. Therefore, by calculating the new molecular count at 350 mL, we can better understand the impact of decreased lung capacity.
The equation set up as follows:
\[ 500 \text{ mL} \times x \text{ molecules} = 350 \text{ mL} \times 1 \times 10^{22} \text{ molecules} \]

Respiratory Physiology

Respiratory physiology involves studying how the lungs and respiratory system work to introduce oxygen into the body and remove carbon dioxide. Each breath draws air into the lungs, allowing gas exchange to occur. A typical human breath draws about 500 mL of air, containing roughly \(1 \times 10^{22}\) molecules. However, various respiratory conditions can reduce lung capacity, decreasing the volume of air a person can inhale per breath.
In the assignment, a person with respiratory problems only inhales 350 mL of air per breath, significantly less than normal. By understanding this, we can calculate the reduced number of air molecules reaching the lungs using the direct proportion equation. This helps medical professionals gauge the severity of a person's respiratory issue and tailor their treatment accordingly.

Molecular Calculations

Molecular calculations in chemistry often involve determining the number of molecules or moles, especially when dealing with gases. When given certain conditions such as volume, temperature, and pressure, you can calculate the number of molecules present. For example, when 500 mL of air contains \(1 \times 10^{22}\) molecules, you can deduce the number of molecules at a different volume, such as 350 mL.
In our example, we calculate the number of molecules by setting up a proportion:
\[ \frac{500 \text{ mL}}{1 \times 10^{22} \text{ molecules}} = \frac{350 \text{ mL}}{x \text{ molecules}} \]
Cross-multiplying and solving for \(x\), we find:
\(x = \frac{350 \times 10^{22}}{500}\)
\(x = 0.7 \times 10^{22}\)
\(x = 7 \times 10^{21}\) molecules.
This ensures we understand how various volumes of air determine molecular counts, which can be applied in different scientific and medical calculations.