Ch. 18 - Kinetic Theory of Gases

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. 18 - Kinetic Theory of Gases

Problem 46a

Chapter 18, Problem 46a

At about what pressure would the mean free path of air molecules be 0.30 m? Assume T = 20° C.

Verified step by step guidance

Determine the formula for the mean free path (λ) of air molecules. The mean free path is given by:

λ = \frac{1}{4 π r_{2} n}

, where

r_{2}

is the molecular radius squared, and

n

is the number density of molecules.

Relate the number density of molecules

n

to the pressure

P

using the ideal gas law:

n = \frac{P}{k T}

, where

k

is the Boltzmann constant and

T

is the temperature in kelvins.

Substitute

n

from the ideal gas law into the mean free path formula:

λ = \frac{k}{4 π r_{2} P}

. Rearrange this equation to solve for pressure

P

P = \frac{k T}{4 π r_{2} λ}

Convert the given temperature from Celsius to kelvins using the formula:

T = T °_{C} + 273.15

. For

20 ° C

, this gives

T = 293.15

Substitute the known values into the rearranged formula for

P

k = 1.38 \times 10 ⁻ 23

J/K,

T = 293.15

r_{2} = 0.2 \times 10 ⁻ 9

m (assuming a typical molecular radius for air), and

λ = 0.30

m. Simplify to find the pressure.

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Key Concepts

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

Mean Free Path

The mean free path is the average distance a molecule travels between collisions with other molecules. It is influenced by the density of the gas and the size of the molecules. In gases, a longer mean free path indicates fewer collisions, which typically occurs at lower pressures.

Gas Pressure

Gas pressure is the force exerted by gas molecules colliding with the walls of a container per unit area. It is directly related to the number of molecules and their kinetic energy, which is influenced by temperature. Lowering the pressure increases the mean free path, as there are fewer molecules in a given volume.

Ideal Gas Law

The Ideal Gas Law, expressed as PV = nRT, relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the number of moles (n) and the universal gas constant (R). This law helps in calculating the conditions under which the mean free path can be determined, especially when considering changes in pressure and temperature.

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Related Practice

Textbook Question

At about what pressure would the mean free path of air molecules be equal to the diameter of air molecules, ≈ 3 x 10⁻¹⁰ m? Assume T = 20° C.

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Textbook Question

Water is in which phase when the pressure is 0.01 atm and the temperature is 90°C?

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Textbook Question

Below a certain threshold pressure, the air molecules (0.3-nm diameter) within a research vacuum chamber are in the “collision-free regime,” meaning that a particular air molecule is as likely to cross the container and collide with the opposite wall as it is to collide with another air molecule. Estimate the threshold pressure for a vacuum chamber of side 1.0 m at 20°C.

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Textbook Question

The escape speed from the Earth is 1.12 x 10⁴ m/s (Section 8–7). So a gas molecule traveling away from Earth near the outer boundary of the Earth’s atmosphere would, at this speed, be able to escape from the Earth’s gravitational field and be lost to the atmosphere. Can you explain why our atmosphere contains oxygen but not helium?

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Textbook Question

For what range of pressures and temperatures can CO₂ be a liquid? Refer to Fig. 18-6.

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Estimate the time needed for a glycine molecule (see Table 18–3) to diffuse a distance of 25μm in water at 20°C if its concentration varies over that distance from 1.00 mol/m³ to 0.50 mol/m³. Compare this “speed” to its rms (thermal) speed. The molecular mass of glycine is about 75 u.