Problem 47 - Physics

Question

A young male adult takes in about \(5.0 \times 10^{-4} \mathrm{~m}^{3}\) of fresh air during a normal breath. Fresh air contains approximately \(21 \%\) oxygen. Assuming that the pressure in the lungs is \(1.0 \times 10^{5} \mathrm{~Pa}\) and air is an ideal gas at a temperature of \(310 \mathrm{~K}\), find the number of oxygen molecules in a normal breath.

Step-by-Step Solution

Answer

Approximately \(2.47 \times 10^{21}\) oxygen molecules are in a normal breath.

Step 1: Calculate Moles of Air

First, calculate the volume of air in the breath using the given volume and find the number of moles using the ideal gas law. The ideal gas law is: \[ PV = nRT \] Where:- \(P = 1.0 \times 10^5 \text{ Pa}\) (pressure)- \(V = 5.0 \times 10^{-4} \text{ m}^3\) (volume)- \(R = 8.314 \text{ J/(mol K)}\) (ideal gas constant)- \(T = 310 \text{ K}\) (temperature)Rearrange the formula to solve for moles, \(n\): \[ n = \frac{PV}{RT} = \frac{(1.0 \times 10^5) \times (5.0 \times 10^{-4})}{8.314 \times 310} \approx 0.0195 \text{ mol} \]

Step 2: Calculate Moles of Oxygen

Since fresh air is approximately 21% oxygen, calculate the number of moles of oxygen. Multiply the total moles of air by 0.21: \[ n_{\text{O}_2} = 0.0195 \times 0.21 = 0.004095 \text{ mol} \]

Step 3: Calculate Number of Oxygen Molecules

Convert the moles of oxygen to molecules using Avogadro's number. Avogadro's number is \(6.022 \times 10^{23} \text{ molecules/mol}\). Multiply the moles of oxygen by Avogadro's number: \[ N = 0.004095 \times 6.022 \times 10^{23} \approx 2.47 \times 10^{21} \text{ molecules} \]

Key Concepts

Oxygen Molecules

Oxygen molecules in the air are essential for human respiration. Each molecule consists of two oxygen atoms bonded together, represented as \( \text{O}_2 \). When you breathe in, your lungs fill with fresh air, which contains about 21% of these oxygen molecules. This allows the oxygen to pass into your bloodstream and fuel your body's functions.
Knowing the number of oxygen molecules in a breath is vital for understanding respiratory processes. We often use the ideal gas law to find this in scientific terms.

Avogadro's Number

Avogadro's number is a key concept in chemistry. It tells us how many molecules or atoms are in one mole of a substance. The value, \( 6.022 \times 10^{23} \), helps convert moles to molecules, making calculations between the macroscopic and microscopic scales possible.
For instance, to find out how many molecules of oxygen are in a breath, we use Avogadro's number to convert moles of oxygen into actual molecules. This allows us to understand the sheer number of molecules involved in everyday processes like breathing.

Moles of Gas

In chemistry, a mole is a counting unit that helps measure quantities of substances. The ideal gas law, \( PV = nRT \), allows us to find the moles of gas given the pressure (\( P \)), volume (\( V \)), temperature (\( T \)), and the ideal gas constant (\( R \)).
For a given volume of air, calculating moles involves rearranging the ideal gas law formula to \( n = \frac{PV}{RT} \). This step is crucial to determining how much of each component, like oxygen, is in a sample of air.

Pressure and Temperature

Pressure and temperature play pivotal roles in the behavior of gases. According to the ideal gas law, they are directly involved in determining the volume and number of moles of a gas.
Pressure is the force that the gas molecules exert against the walls of their container, measured in pascals (Pa). Temperature affects how fast these molecules move, measured in Kelvin (K).

• An increase in temperature typically increases the volume of the gas if the pressure remains constant.
• Under higher pressure, the gas molecules are compressed into a smaller volume.

Understanding these concepts helps in applying the ideal gas law to real-world scenarios, like inhaling oxygen during a breath at a specific temperature and pressure.