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 80

Chapter 17, Problem 80

From the known value of atmospheric pressure at the surface of the Earth, estimate the total number of air molecules in the Earth’s atmosphere.

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Start by recalling that atmospheric pressure at the Earth's surface is approximately 101,325 Pa (Pascals). This pressure is due to the weight of the air column above a unit area. Use the relationship between pressure, force, and area: \( P = \frac{F}{A} \), where \( F \) is the force due to the weight of the air column and \( A \) is the area.

The force \( F \) can be expressed as \( F = mg \), where \( m \) is the total mass of the atmosphere and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²). Substituting this into the pressure equation gives \( P = \frac{mg}{A} \). Rearrange to solve for \( m \): \( m = \frac{P \cdot A}{g} \).

Next, calculate the total surface area of the Earth, \( A \), using the formula for the surface area of a sphere: \( A = 4\pi R^2 \), where \( R \) is the radius of the Earth (approximately 6.371 \(\times\) 10^6 \) m. Substitute this value into the equation for \( m \).

To estimate the total number of air molecules, use the relationship between the mass of the atmosphere and the number of molecules. The molar mass of air is approximately 29 g/mol (or 0.029 kg/mol). The number of moles, \( n \), is given by \( n = \frac{m}{M} \), where \( M \) is the molar mass. Then, use Avogadro's number, \( N_A = 6.022 \times 10^{23} \) molecules/mol, to find the total number of molecules: \( N = n \cdot N_A \).

Combine all the expressions and substitute the known values (\( P = 101,325 \) Pa, \( g = 9.8 \) m/s², \( R = 6.371 \times 10^6 \) m, \( M = 0.029 \) kg/mol, and \( N_A = 6.022 \times 10^{23} \) molecules/mol) to calculate the total number of air molecules in the Earth's atmosphere. Ensure unit consistency throughout the calculation.

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Key Concepts

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

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of air above a given point, typically measured in pascals (Pa) or atmospheres (atm). At sea level, the average atmospheric pressure is approximately 101,325 Pa. This pressure is crucial for understanding how air density and the number of air molecules can be estimated, as it relates directly to the mass of air in a given volume.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physics and chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law allows us to calculate the number of air molecules by rearranging the equation to solve for n, given the atmospheric pressure and volume of the atmosphere.

Molar Volume

Molar volume is the volume occupied by one mole of a substance at standard temperature and pressure (STP), which is approximately 22.4 liters for an ideal gas. This concept is essential for converting between the number of moles of air and the total number of molecules, as Avogadro's number (approximately 6.022 x 10^23 molecules per mole) can be used to find the total number of molecules in the atmosphere once the number of moles is determined.

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