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 46b

18장, 문제 46b

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.

검증된 단계별 안내

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

λ = (k_B T) / (√2 π d² P)

, where

k_B

is the Boltzmann constant,

T

is the temperature in Kelvin,

d

is the diameter of the air molecules, and

P

is the pressure.

Rearrange the formula to solve for pressure

P

P = (k_B T) / (√2 π d² λ)

. Here,

λ

is given as the diameter of the air molecules,

d ≈ 3 × 10⁻¹⁰ \, \(\text{m}\)

Convert the temperature from Celsius to Kelvin using the formula:

T_K = T_C + 273.15

. For

T = 20 \, \(\degree\) \(\text{C}\)

T_K = 20 + 273.15 = 293.15 \, \(\text{K}\)

Substitute the known values into the rearranged formula:

P = (1.38 × 10⁻²³ \, \(\text{J/K}\) \(\times\) 293.15 \, \(\text{K}\)) / (√2 π (3 × 10⁻¹⁰ \, \(\text{m}\))² (3 × 10⁻¹⁰ \, \(\text{m}\)))

. Here,

k_B = 1.38 × 10⁻²³ \, \(\text{J/K}\)

Simplify the expression to calculate the pressure

P

. Ensure all units are consistent, and the final result will be in Pascals (Pa).

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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, as pressure increases, the mean free path decreases because molecules are closer together, leading to more frequent collisions.

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. Understanding how pressure affects molecular behavior is crucial for determining conditions like the mean free path.

Kinetic Theory of Gases

The kinetic theory of gases describes the behavior of gases in terms of the motion of their molecules. It explains how temperature, pressure, and volume are interrelated, and provides insights into concepts like mean free path. This theory helps predict how changes in conditions affect gas properties, including molecular spacing and collision rates.