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

Рассмотрим N молекул газа с радиусом r, произвольно движущихся со скоростью v в цилиндрическом объеме V. При столкновении одной молекулы с другой молекулой расстояние между их центрами составляет 2r.
Представьте себе цилиндр радиусом 2r, с осью, параллельной скорости молекулы. Когда молекула движется в течение короткого промежутка времени и сталкивается внутри цилиндра, можно определить количество столкновений в единицу времени.
Используя уравнение средней относительной скорости, можно определить столкновения для всех движущихся молекул в единицу времени.
Обратная величина уравнения дает среднее время между столкновениями, известное как среднее свободное время.
Между тем, средний свободный пробег молекулы газа является произведением скорости молекулы и среднего времени между столкновениями. Путем подстановки членов можно определить средний свободный пробег, который обратно пропорционален числу молекул на единицу объема и площади поперечного сечения молекулы.
Вспомнив уравнение идеального газа и подставив его слагаемые, можно получить макроскопические свойства газа.

<ol>
	<li>Янг, Х.Д. и Фридман, Р.А. (2012). Университетская физика с современной физикой. Сан-Франциско, Калифорния: Пирсон. С. 630</li>
	<li>OpenStax. (2019). Университетская физика Том 1. [Веб-версия]. Получено из <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>

Средняя длина свободного пути и Среднее свободное время

Рассмотрим молекулы газа в цилиндре. Они движутся в случайном движении, сталкиваясь друг с другом, меняют скорость и направление. Среднее всех длин пути между столкновениями известно как "средний свободный путь".

<img alt="Equation1" src="/files/ftp_upload/14168/14168_Equation_1_new.svg" style="text-align: center; height: 40px;" />

Средний свободный путь изменяется обратно пропорционально плотности молекул, потому что при большем количестве молекул внутри объёма у них больше шансов столкнуться друг с другом, что уменьшает средний свободный путь. Кроме того, средний свободный путь обратно связан с диаметром молекул, потому что если они были бы точечными массами, они никогда бы не сталкивались. Таким образом, большие молекулы ассоциируются с более коротким средним свободным путём.

Газ расширяется при увеличении температуры при постоянном давлении; таким образом, среднее расстояние между молекулами и средний свободный путь увеличиваются. Однако, когда давление увеличивается при постоянной температуре, газ сжимается, что приводит к уменьшению среднего свободного пути. Средний свободный путь может быть определён как произведение средней скорости и среднего свободного времени, где среднее свободное время - это среднее время между столкновениями.

Рассмотрим атомы аргона с молярной массой 39,9 г/моль, движущиеся случайным образом в цилиндре при температуре 273 К и давлении 1 атм. Используя радиус атома аргона, равный 1.70 × 10-10м,, определите среднее свободное время для атомов аргона.

Для решения задачи, сначала определите известные и неизвестные величины и приведите их к системе СИ.

Во-вторых, вспомните уравнение для средней скорости RMS для молекул газа. Подставив значения, средняя скорость может быть определена следующим образом:

<img alt="Equation2" src="/files/ftp_upload/14168/14168_Equation_2_new.svg" style="text-align: center; height: 45px;" />

Наконец, вспомните уравнение для среднего свободного времени. Подставив значения, среднее свободное время может быть определено следующим образом:

<img alt="Equation3" src="/files/ftp_upload/14168/14168_Equation_3_new.svg" style="text-align: center; height: 40px;" />

Рассмотрим молекулы газа в цилиндре. Они движутся в случайном движении, сталкиваясь друг с другом и меняя скорость и направление. Среднее из всех...

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

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...

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Gas Laws Behind the Ideal Equation

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How Real Gases Deviate from Ideal Behavior

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Water Phase Changes on a Pressure Chart

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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...

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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

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Gas collisions: path length and time

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