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Chapter 14: Problem 15

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 that air is an ideal gas at a temperature of \(310 \mathrm{K}\), find the number of oxygen molecules in a normal breath.

Short Answer

Expert verified
About \(2.45 \times 10^{21}\) oxygen molecules are inhaled per breath.

Step by step solution

01

Understand the Given Data

We have been given the following information:- Volume of air per breath: \( V = 5.0 \times 10^{-4} \text{ m}^3 \).- Percentage of oxygen in the air: \( 21\% \).- Pressure in the lungs: \( P = 1.0 \times 10^5 \text{ Pa} \).- Temperature of the air: \( T = 310 \text{ K} \).We are required to find the number of oxygen molecules inhaled with each breath.
02

Use the Ideal Gas Law

The ideal gas law is given by: \( PV = nRT \), where:- \( P \) is the pressure,- \( V \) is the volume,- \( n \) is the number of moles,- \( R = 8.314 \text{ J/(mol K)} \) is the ideal gas constant,- \( T \) is the temperature in Kelvin.Rearrange to solve for \( n \): \[ n = \frac{PV}{RT} \]
03

Calculate Total Moles of Air

Insert the given values into the equation:\[ n = \frac{(1.0 \times 10^5 \text{ Pa})(5.0 \times 10^{-4} \text{ m}^3)}{(8.314 \text{ J/(mol K)})(310 \text{ K})} \]Calculate:\[ n = \frac{(5.0 \times 10^1)}{(2.57734)} \]\[ n \approx 0.0194 \text{ moles of air} \]
04

Determine Moles of Oxygen

Since oxygen makes up 21% of fresh air by volume, the moles of oxygen can be calculated as:\[ n_{\text{oxygen}} = 0.21 \times 0.0194 \text{ moles} \]\[ n_{\text{oxygen}} \approx 0.004074 \text{ moles of oxygen} \]
05

Convert Moles of Oxygen to Molecules

Use Avogadro's number to convert moles to molecules. Avogadro's number is \( 6.022 \times 10^{23} \text{ molecules/mol} \).\[ \text{Number of oxygen molecules} = 0.004074 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mol} \]Calculate:\[ \text{Number of oxygen molecules} \approx 2.45 \times 10^{21} \text{ molecules} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moles of Gas
The concept of moles is essential in understanding chemical quantities. A mole is a standard scientific unit that is used to measure large quantities of tiny entities such as atoms, molecules, or other specified particles. In the context of gases, the number of moles ( ext{n}) is directly calculated from the ideal gas law:\[PV = nRT\]where:\
  • \(P\) is the pressure of the gas in Pascals.
  • \(V\) is the volume of the gas in cubic meters.
  • \(R\) is the ideal gas constant, approximately 8.314 J/(mol K).
  • \(T\) is the temperature in Kelvin.
By rearranging the formula to solve for \(n\), we can find the moles of gas:\[n = \frac{PV}{RT}\]This equation shows how changing any of these parameters affects the number of moles. In the exercise given, the pressure of the air in the lungs, the volume of air per breath, and the temperature provided allow us to accurately calculate the total moles of air inhaled with each breath.
Oxygen Percentage
The percentage of oxygen in the air is a vital factor in calculating various properties of inhaled air. Earth's atmosphere is made up of several gases, among which oxygen constitutes approximately 21% of the volume, although this percentage can differ slightly based on different conditions and locations. For instance, in the context of a normal breath, you can utilize this oxygen percentage to find out the moles of oxygen from the total moles of air.Considering the given scenario, where there are 0.0194 moles of air in a breath, the moles of oxygen can be estimated as:\[n_{\text{oxygen}} = 0.21 \times n_{\text{total}}\]where \(n_{\text{total}}\) is the total moles of air calculated. In this way, the percentage of oxygen plays a crucial role in determining how many moles specifically belong to oxygen at that breath volume.
Avogadro's Number
Avogadro's number, defined as \(6.022 \times 10^{23}\), is a fundamental constant in chemistry. It represents the number of atoms or molecules present in one mole of a substance. This concept is crucial for converting between the microscopic scale of molecules and the macroscopic scale we observe in everyday life.In the context of ideal gases, once we have determined the moles of a particular gas, such as oxygen in a breath, we can calculate the number of molecules using Avogadro's number. For the oxygen calculation given in the exercise: \[\text{Number of Oxygen Molecules} = n_{\text{oxygen}} \times 6.022 \times 10^{23} \text{ molecules/mol}\]This allows us to transition from a chemical quantity familiar in laboratory measurements (moles) to a more tangible idea of the sheer number of molecules, helping us understand chemical phenomena at a deeper level.

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Most popular questions from this chapter

A Goodyear blimp typically contains \(5400 \mathrm{m}^{3}\) of helium (He) at an absolute pressure of \(1.1 \times 10^{5}\) Pa. The temperature of the helium is 280 K. What is the mass (in kg) of the helium in the blimp?

When you push down on the handle of a bicycle pump, a piston in the pump cylinder compresses the air inside the cylinder. When the pressure in the cylinder is greater than the pressure inside the inner tube to which the pump is attached, air begins to flow from the pump to the inner tube. As a biker slowly begins to push down the handle of a bicycle pump, the pressure inside the cylinder is \(1.0 \times 10^{5} \mathrm{Pa}\), and the piston in the pump is \(0.55 \mathrm{m}\) above the bottom of the cylinder. The pressure inside the inner tube is \(2.4 \times 10^{5} \mathrm{Pa}\). How far down must the biker push the handle before air begins to flow from the pump to the inner tube? Ignore the air in the hose connecting the pump to the inner tube, and assume that the temperature of the air in the pump cylinder does not change.

Manufacturers of headache remedies routinely claim that their own brands are more potent pain relievers than the competing brands. Their way of making the comparison is to compare the number of molecules in the standard dosage. Tylenol uses \(325 \mathrm{mg}\) of acetaminophen \(\left(\mathrm{C}_{8} \mathrm{H}_{9} \mathrm{NO}_{2}\right)\) as the standard dose, whereas Advil uses \(2.00 \times 10^{2} \mathrm{mg}\) of ibuprofen \(\left(\mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2}\right) .\) Find the number of molecules of pain reliever in the standard doses of (a) Tylenol and (b) Advil.

Very fine smoke particles are suspended in air. The translational rms speed of a smoke particle is \(2.8 \times 10^{-3} \mathrm{m} / \mathrm{s},\) and the temperature is \(301 \mathrm{K}\). Find the mass of a particle.

Initially, the translational rms speed of a molecule of an ideal gas is \(463 \mathrm{m} / \mathrm{s}\). The pressure and volume of this gas are kept constant, while the number of molecules is doubled. What is the final translational rms speed of the molecules?
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