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

Рассмотрим газообразный кислород в контейнере с молярной массой 0,0320 кг/моль при комнатной температуре. Определите отношение числа молекул со скоростью около 400 м/с к числу молекул со скоростью около 200 м/с, среднеквадратичной скорости и наиболее вероятной скорости молекул кислорода при комнатной температуре.
Чтобы решить проблему, определите известные и неизвестные величины.
В дифференциальной форме число молекул, близких к скорости, является произведением функции распределения вероятностей скорости на небольшой интервал скорости.
Соотношение можно рассчитать, разделив количество молекул, близкое к отдельным скоростям.
Подставим члены в функцию распределения Максвелла-Больцмана для молекул газа. Можно определить соотношение количества молекул с индивидуальными скоростями.
Вспомним уравнение среднеквадратичной скорости и наиболее вероятное уравнение скорости для молекул газа. Подставляя известные значения, можно определить среднеквадратичную скорость и наиболее вероятную скорость газа.

<ol>
	<li>OpenStax. (2019). Университетская физика Том 2. [Веб-версия], стр. 94–95. Получено с <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>

Распределение Максвелла-Больцмана: решение задачи

Отдельные молекулы в газе движутся в случайных направлениях, но газ, содержащий множество молекул, имеет предсказуемое распределение молекулярных скоростей, которое называется распределением Максвелла-Больцмана, f(v).

Эта функция распределения f(v) определяется следующим образом: ожидаемое число N (v1,v2)  частиц со скоростями между v1  и v2  дано следующим выражением:

<img alt="Equation1" src="/files/ftp_upload/14167/14167_Equation_1.svg" style="text-align: center; height: 80px;" />

Так как N является безразмерной величиной, а единица измерения f(v)  - секунды на метр, уравнение удобно модифицировать в дифференциальную форму:

<img alt="Equation2" src="/files/ftp_upload/14167/14167_Equation_2.svg" style="text-align: center; height: 20px;" />

Рассмотрим образец азотного газа в цилиндре с молекулярной массой 28,0 г/моль при комнатной температуре 27 °C. Определите отношение числа молекул со скоростью, очень близкой к 300 м/с, к числу молекул со скоростью, очень близкой к 100 м/с.

Для решения задачи исследуйте ситуацию и определите известные и неизвестные величины.

Затем преобразуйте все известные значения в соответствующие единицы СИ. Например, конвертируйте молекулярную массу в килограммы (кг) и температуру в кельвины (К). Вспомните уравнение для функции распределения скорости. Наконец, подставьте известные значения в уравнение, чтобы найти неизвестную величину.

<img alt="Equation3" src="/files/ftp_upload/14167/14167_Equation_3.svg" style="text-align: center; height: 280px;" />

Отдельные молекулы в газе движутся в случайных направлениях, но газ, содержащий множество молекул, имеет предсказуемое распределение молекулярных скоростей, которые ...

Education

JoVE Core

Physics

Физика

Unit 1

Раздел 1

The Kinetic Theory of Gases

Кинетическая теория газов

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

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

The ideal gas equation links pressure, volume, temperature, and amount of gas. It describes a hypothetical gas and is built from four empirical gas laws: Boyle’s law, Charles’s law, Avogadro’s law, and Gay-Lussac’s law. When these proportionalities are combined, they give one constant called the universal gas constant. This gas constant has the same value for all real gases under certain conditions. In chemical calculations, that constant is used in the ideal gas equation. The equation works...

van der waals equation

How Real Gases Deviate from Ideal Behavior

Real gases often deviate from the ideal gas law, especially at low temperatures and high pressures. The van der Waals equation improves the ideal gas model by adding two corrections that account for these real-world effects. One correction accounts for attractive forces between molecules. These forces become stronger at higher densities and lower the measured pressure, so the equation adds a pressure term based on the square of the molar density and a positive constant a. The second...

pv-diagrams

Liquid-Gas Changes on pV Curves

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kinetic theory of an ideal gas

Avogadro's Number and Gas Particle Motion

Avogadro's number and gas particle motion help explain how gases behave at the particle level. A mole is the amount of a substance that contains as many molecules as there are atoms in exactly 12 grams of carbon-12. This idea came from Amedeo Avogadro, who proposed that equal volumes of gas at the same pressure and temperature contain equal numbers of molecules, no matter what the gas is. Avogadro's number is the number of molecules in one mole. Its value is 6.02 × 10^23 mol−1. This number...

molecular kinetic energy

Gas Pressure and Molecular Motion

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...

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

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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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Maxwell-Boltzmann Gas Speed Ratio

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