Name: maxwell-boltzmann gas speed ratio
Uploaded: 2023-09-18T17:11:17.000Z
Duration: 1 min 20 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.12

SC.912.P.12.10

maxwell-boltzmann-distribution-problem-solving

Maxwell-Boltzmann Distribution: Problem Solving

Considere um gás oxigênio em um recipiente com uma massa molar de 0,0320 kg/mol à temperatura ambiente. Determine a razão entre o número de moléculas com uma velocidade de cerca de 400 m/s e o número de moléculas com uma velocidade de cerca de 200 m/s, a velocidade quadrática média da raiz e a velocidade mais provável das moléculas de oxigênio à temperatura ambiente.
Para resolver o problema, identifique as quantidades conhecidas e desconhecidas.
Na forma diferencial, o número de moléculas próximas a uma velocidade é o produto da função de distribuição de probabilidade da velocidade vezes um pequeno intervalo da velocidade.
A proporção pode ser calculada dividindo o número de moléculas próximo às velocidades individuais.
Substitua os termos na função de distribuição de Maxwell-Boltzmann por moléculas de gás. A proporção do número de moléculas com velocidades individuais pode ser determinada.
Lembre-se da equação da velocidade quadrática média e da equação de velocidade mais provável para moléculas de gás. Substituindo os valores conhecidos, a velocidade quadrática média e a velocidade mais provável do gás podem ser determinadas.

<ol>
	<li>OpenStax. (2019). Física Universitária Vol. 2. [Versão da Web], páginas 94–95. Obtido de <a href="https://openstax.org/books/university-physics-volume-2/pages/2-4-distribution-of-molecular-speeds">https://openstax.org/books/university-physics-volume-2/pages/2-4-distribution-of-molecular-speeds</a></li>
</ol>

distribuição de maxwell-boltzmann: resolução de problemas

As moléculas individuais em um gás se movem em direções aleatórias, mas um gás contendo inúmeras moléculas tem uma distribuição previsível de velocidades moleculares, conhecida como distribuição de Maxwell-Boltzmann, f(v).

Esta função de distribuição f(v) é definida ao se afirmar que o número esperado N(v1,v2) de partículas com velocidades entre v1 e v2 é dado por

<img alt="Equation1" src="/files/ftp_upload/14167/14167_Equation_1.svg" style="text-align: center; height: 80px;" />

Desde que N seja adimensional e a unidade de f(v) seja segundos por metro, a equação pode ser convenientemente modificada para uma forma diferencial:

<img alt="Equation2" src="/files/ftp_upload/14167/14167_Equation_2.svg" style="text-align: center; height: 20px;" />

Considere uma amostra de gás nitrogênio em um cilindro com uma massa molar de 28,0 g/mol em uma temperatura ambiente de 27°C. Determine a razão entre o número de moléculas com uma velocidade muito próxima de 300 m/s para o número de moléculas com uma velocidade muito próxima de 100 m/s.

Para resolver o problema, examine a situação e identifique as quantidades conhecidas e desconhecidas.

Depois, converta todos os valores conhecidos para as unidades SI adequadas. Por exemplo, converta o peso molecular para quilogramas (kg) e a temperatura para kelvin (K). Lembre-se da função de distribuição para a equação de velocidade. Por último, substitua os valores conhecidos na equação para determinar a quantidade desconhecida.

<img alt="Equation3" src="/files/ftp_upload/14167/14167_Equation_3.svg" style="text-align: center; height: 280px;" />

Distribuição de Maxwell-Boltzmann: Resolução de Problemas

Moléculas individuais em um gás se movem em direções aleatórias, mas um gás contendo numerosas moléculas tem uma distribuição previsível de velocidades moleculares, que ...

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

maxwell-boltzmann gas speed ratio

The Maxwell-Boltzmann distribution describes how molecular speeds are spread in a gas. Even though individual molecules move in random directions, the whole gas has a predictable speed pattern. This distribution is written as f(v), where v is molecular speed.

The distribution function is used to find the expected number of particles with speeds between v1 and v2. Because N is dimensionless, f(v) has units of seconds per meter. The relationship can also be written in differential form for speed intervals that are very small.

A nitrogen gas sample in a cylinder gives a sample problem. The gas has a molar mass of 28.0 g/mol and a room temperature of 27 °C. The task is to find the ratio of molecules with speeds very close to 300 m/s to those with speeds very close to 100 m/s.

Solving the problem starts with identifying the known and unknown quantities. The next step is to convert the given values into SI units. That means changing the molar mass to kilograms and the temperature to kelvin. After that, the Maxwell-Boltzmann velocity equation is used, and the known values are substituted to calculate the ratio.

Maxwell-Boltzmann Gas Speed Ratio

equation of state

Gas Behavior in the Equation of State

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ideal gas equation

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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...

maxwell-boltzmann distribution: problem solving

The Maxwell-Boltzmann distribution describes how molecular speeds are spread in a gas. Even though individual molecules move in random directions, the whole gas has a predictable speed pattern. This distribution is written as f(v), where v is molecular speed. The distribution function is used to find the expected number of particles with speeds between v1 and v2. Because N is dimensionless, f(v) has units of seconds per meter. The relationship can also be written in differential form for speed...

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: 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...

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

Solve a Maxwell-Boltzmann distribution problem for nitrogen gas by converting units, using the velocity equation, and comparing molecule speed ratios.

Maxwell-Boltzmann Gas Speed Ratio | 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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Maxwell-Boltzmann Gas Speed Ratio

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