Ch. 18 - Kinetic Theory of Gases

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 18 - Kinetic Theory of Gases

문제 46a

18장, 문제 46a

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

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.