Name: gas collisions: path length and time
Uploaded: 2025-05-22T09:05:59.000Z
Duration: 1 min 22 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.12

SC.912.P.12.10

Mean free path and Mean free time

Considere N moléculas de gás com raio r movendo-se aleatoriamente com velocidade v em um volume cilíndrico V. Quando uma molécula colide com outra molécula, a distância entre seus centros é 2r.
Imagine um cilindro com raio de 2r, com um eixo paralelo à velocidade da molécula. Quando a molécula viaja por um pequeno intervalo de tempo e colide dentro do cilindro, o número de colisões por unidade de tempo pode ser determinado.
Usando a equação da velocidade relativa média, as colisões para todas as moléculas em movimento por unidade de tempo podem ser determinadas.
O recíproco da equação fornece o tempo médio entre colisões, conhecido como tempo livre médio.
Enquanto isso, o caminho livre médio de uma molécula de gás é o produto da velocidade da molécula e o tempo médio entre as colisões. Ao substituir os termos, o caminho livre médio pode ser determinado, que é inversamente proporcional ao número de moléculas por unidade de volume e à área da seção transversal da molécula.
Lembrando a equação do gás ideal e substituindo os termos, as propriedades macroscópicas do gás podem ser obtidas.

<ol>
	<li>Young, H.D. e Freedman, R.A. (2012). Física Universitária com Física Moderna. São Francisco, CA: Pearson. Pág. 630</li>
	<li>OpenStax. (2019). Física Universitária Vol. 1. [Versão Web]. Obtido de <a href="https://openstax.org/books/college-physics-2e/pages/13-4-kinetic-theory-atomic-and-molecular-explanation-of-pressure-and-temperature">https://openstax.org/books/college-physics-2e/pages/13-4-kinetic-theory-atomic-and-molecular-explanation-of-pressure-and-temperature</a></li>
</ol>

caminho livre médio e tempo livre médio

Considere as moléculas de gás em um cilindro. Elas se movem em um movimento aleatório conforme colidem umas com as outras e mudam de velocidade e direção. A média de todos os comprimentos dos caminhos entre as colisões é conhecida como "caminho livre médio";

<img alt="Equation1" src="/files/ftp_upload/14168/14168_Equation_1_new.svg" style="text-align: center; height: 40px;" />

O caminho livre médio varia inversamente com a densidade das moléculas, porque quando há mais moléculas dentro de um volume, elas têm uma maior chance de colidirem umas com as outras, reduzindo assim o caminho livre médio. Além disso, o caminho livre médio está inversamente relacionado com o diâmetro das moléculas, porque se elas fossem massas pontuais, elas nunca colidiriam. Assim, moléculas maiores estão associadas a um caminho livre médio mais curto.

O gás se expande quando a temperatura aumenta sob pressão constante; assim, a distância média entre as moléculas e o caminho livre médio aumentam. No entanto, quando a pressão é aumentada a uma temperatura constante, o gás se comprime, levando a uma diminuição no caminho livre médio. O caminho livre médio pode ser definido como o produto da velocidade média e do tempo livre médio, onde o tempo livre médio é o tempo médio entre as colisões.

Considere átomos de argônio com uma massa molar de 39,9 g/mol se movendo aleatoriamente em um cilindro a uma temperatura de 273 K e uma pressão de 1 atm. Levando em conta que o raio de um átomo de argônio é de 1,70 × 10-10 m, determine o tempo livre médio para os átomos de argônio.

Para resolver o problema, primeiro identifique as quantidades conhecidas e desconhecidas, e converta-as para unidades do SI.

Depois, lembre-se da equação de velocidade RMS para moléculas de gás. Substituindo os valores, a velocidade RMS pode ser determinada da seguinte forma:

<img alt="Equation2" src="/files/ftp_upload/14168/14168_Equation_2_new.svg" style="text-align: center; height: 45px;" />

Por último, lembre-se da equação de tempo livre médio. Substituindo os valores, o tempo livre médio pode ser determinado da seguinte forma:

<img alt="Equation3" src="/files/ftp_upload/14168/14168_Equation_3_new.svg" style="text-align: center; height: 40px;" />

Caminho livre médio e Tempo livre médio

Considere as moléculas de gás em um cilindro. Eles se movem em um movimento aleatório à medida que colidem uns com os outros e mudam de velocidade e direção. A média de todos ...

Education

JoVE Core

Physics

Física

Unit 1

Unidade 1

The Kinetic Theory of Gases

A Teoria Cinética dos Gases

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

gas collisions: path length and time

Gas collisions in a cylinder help explain the mean free path and mean free time. Gas molecules move in random motion. They collide with each other and change speed and direction. The mean free path is the average path length between collisions.

The mean free path depends on how many molecules are packed into a volume. It varies inversely with density. When more molecules are present, collisions happen more often, so the mean free path becomes shorter. It also depends on molecule size. Larger molecules have a shorter mean free path, while point masses would never collide.

Temperature and pressure also affect the mean free path. When temperature increases at constant pressure, a gas expands. The average distance between molecules increases, and the mean free path increases too. When pressure increases at constant temperature, the gas compresses. That reduces the mean free path.

The mean free path can also be written as the product of average speed and mean free time. Mean free time is the average time between collisions. A sample problem uses argon atoms with a molar mass of 39.9 g/mol at 273 K and 1 atm. The radius of an argon atom is taken as 1.70 × 10 -10 m. To find the mean free time, first identify the known and unknown values and convert them to SI units. Then use the RMS speed equation for gas molecules. After that, substitute the values into the mean free time equation to calculate the result.

Gas collisions: path length and time

equation of state

Gas Behavior in the Equation of State

The equation of state describes how a gas’s pressure, volume, temperature, and number of moles are related. These macroscopic properties help define the condition, or state, of the matter in a thermodynamics system. The relationship between these quantities can be written as a simple mathematical expression. In some cases, it is too complex to express in a formula. Even then, the link between the quantities still exists. When an equation of state cannot be written mathematically, scientists...

ideal gas equation

Gas Laws Behind the Ideal Equation

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van der waals equation

How Real Gases Deviate from Ideal Behavior

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Gas pressure comes from the motion of molecules in a container. The word gas came from a Flemish word meaning “chaos,” and J. B. van Helmont first used it to describe vapors. In a gas, molecules move in a continuous and random way. When a molecule hits a container wall, its velocity changes. The part of the velocity parallel to the wall stays the same. The part perpendicular to the wall reverses direction, but its size does not change. If the molecule’s velocity changes in the x-direction, its...

distribution of molecular speeds

Molecular Speed Curves and Vapor Pressure

Molecular speed curves show how the speeds of gas particles are spread out at a given temperature. For a gas with many molecules, this spread is predictable even though each molecule moves randomly in direction and speed. The pattern is called the Maxwell-Boltzmann distribution. Liquids have a similar speed distribution, but it is not exactly the same as in gases. This difference helps explain boiling and vapor pressure. A molecule must reach at least a certain speed, v a, to escape from the...

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maxwell-boltzmann-distribution-problem-solving

Maxwell-Boltzmann Gas Speed Ratio

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phase diagram

Water Phase Changes on a Pressure Chart

Water phase changes can be mapped on a pressure and temperature chart called a phase diagram. These plots show which phase a substance is in at different conditions. They also reveal where phase transitions happen, because the boundary lines mark the temperatures and pressures where two phases coexist. The water phase diagram shows how pressure affects boiling. As pressure rises, the boiling temperature increases gradually until it reaches 374 °C at 218 atm. A pressure cooker works this way.

mean free path and mean free time

Gas collisions in a cylinder help explain the mean free path and mean free time. Gas molecules move in random motion. They collide with each other and change speed and direction. The mean free path is the average path length between collisions. The mean free path depends on how many molecules are packed into a volume. It varies inversely with density. When more molecules are present, collisions happen more often, so the mean free path becomes shorter. It also depends on molecule size. Larger...

heat capacity: problem-solving

Heat Capacity of Gases by Degrees of Freedom

Heat capacity of gases depends on the gas type and the motion of its molecules. It is the amount of heat energy needed to raise the temperature of a unit mass of gas by one degree Celsius. This thermodynamic property is important in science and industry. To solve heat capacity problems, first identify the gas. A gas's molecular structure affects its heat capacity because the molecules have different degrees of freedom. These values can be found from tables or from empirical equations. Next,...

dalton's law of partial pressure

Gas Mixtures and Their Partial Pressures

Gas mixtures have a total pressure that comes from the partial pressures of each gas. A partial pressure is the pressure one gas would make if it filled the whole container by itself. This idea helps explain how gases behave when they are mixed together. When two or more gases share a container, their molecules move randomly and collide with one another. If the gases are at the same temperature, they reach thermal equilibrium. Their molecules then have the same average kinetic energy. In that...

escape velocities of gases

Why Lighter Gases Escape Earth First

Earth’s escape velocity is the speed an object needs to leave gravity behind. Near the top of the atmosphere, about 100 km above Earth, that speed is 11.1 km/s. Gas molecules can escape only if their speeds become high enough to reach that limit. The chance of escape depends on gas temperature and molecular speed. According to kinetic theory, the average kinetic energy of gas molecules is proportional to temperature. As temperature rises, the rms speed, or root-mean-square speed, also rises.

key terms and concepts

Learn how gas molecule collisions determine mean free path and mean free time, and see how density, size, temperature, and pressure affect them in argon.

Gas collisions: path length and time | JoVE K12

physics

JoVE Core - Fisica apresenta a teoria e a aplicação de tópicos fundamentais da física, incluindo cinemática, dinâmica, mecânica dos fluidos, estática, ondas e acústica, com vídeos animados concisos e de fácil compreensão. Além disso, os vídeos de cientistas em ação demonstram a aplicação de conceitos relacionados em experimentos de pesquisa clássica e original realizados nos laboratórios de todo o mundo hoje em dia.

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A. Motion can be measured and described qualitatively and quantitatively. Net forces create a change in motion. When objects travel at speeds comparable to the speed of light, Einstein's special theory of relativity applies.

B. Momentum is conserved under well-defined conditions. A change in momentum occurs when a net force is applied to an object over a time interval.

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Gas collisions: path length and time

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