Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Problem 75

Chapter 17, Problem 75

Estimate how many molecules of air are in each 2.0-L breath you inhale that were also in the last breath Galileo took. Assume the atmosphere is about 10 km high and of constant density. What other assumptions did you make?

Name: Physics for Scientists and Engineers
Author: Giancoli Douglas
ISBN: 9780137488179

Verified step by step guidance

Step 1: Calculate the total volume of Earth's atmosphere. Assume the atmosphere is a cylinder with a height of 10 km and a radius equal to Earth's radius (approximately 6371 km). Use the formula for the volume of a cylinder:

V = π r^{2} h

, where

r

is Earth's radius and

h

is the height of the atmosphere.

Step 2: Determine the total number of air molecules in the atmosphere. Use the ideal gas law

PV = nRT

to find the number of moles of air,

n

. Then multiply by Avogadro's number (

6.022 \times 10^{23}

) to convert moles to molecules. Assume standard atmospheric pressure and temperature for simplicity.

Step 3: Calculate the fraction of the atmosphere contained in a single breath. Divide the volume of a single breath (2.0 L or

2.0 \times 10^{-3}

cubic meters) by the total volume of the atmosphere calculated in Step 1.

Step 4: Estimate the number of molecules in a single breath. Multiply the fraction obtained in Step 3 by the total number of molecules in the atmosphere calculated in Step 2.

Step 5: Consider the assumptions made in this calculation. These include: (1) the atmosphere has a uniform density, (2) the ideal gas law applies uniformly throughout the atmosphere, (3) the mixing of air molecules is perfect and random over time, and (4) the Earth's radius and atmospheric height are approximations.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for estimating the number of air molecules in a given volume, as it allows us to calculate the number of moles of gas in a specific volume at a given temperature and pressure.

Molecular Density

Molecular density refers to the number of molecules per unit volume of a substance. In the context of air, understanding the average molecular density at sea level helps estimate how many molecules are present in a specific volume, such as a 2.0-L breath. This concept is crucial for determining how many of those molecules could have been present in the atmosphere during Galileo's time.

Atmospheric Mixing

Atmospheric mixing describes how air molecules are distributed throughout the atmosphere due to turbulence and convection. This concept is important for the question as it implies that the air inhaled by Galileo and the current breather is likely to contain a mix of molecules from different times, making it plausible to estimate the overlap of air molecules between breaths across centuries.

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