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

Si considerino le molecole di gas N con raggio r che si muovono casualmente con velocità v in un volume cilindrico V. Quando una molecola si scontra con un'altra molecola, la distanza tra i loro centri è di 2r.
Immaginate un cilindro con un raggio di 2r, con un asse parallelo alla velocità della molecola. Quando la molecola viaggia per un piccolo intervallo di tempo e si scontra all'interno del cilindro, è possibile determinare il numero di collisioni per unità di tempo.
Utilizzando l'equazione della velocità relativa media, è possibile determinare le collisioni per tutte le molecole in movimento per unità di tempo.
Il reciproco dell'equazione fornisce il tempo medio tra le collisioni, noto come tempo libero medio.
Nel frattempo, il cammino libero medio di una molecola di gas è il prodotto della velocità della molecola e del tempo medio tra le collisioni. Sostituendo i termini, è possibile determinare il percorso libero medio, che è inversamente proporzionale al numero di molecole per unità di volume e all'area della sezione trasversale della molecola.
Ricordando l'equazione del gas ideale e sostituendo i termini, si possono ottenere le proprietà macroscopiche del gas.

<ol>
	<li>Young, H.D e Freedman, R.A. (2012). Fisica universitaria con fisica moderna. San Francisco, CA: Pearson. Pagine 630</li>
	<li>OpenStax. (2019). Fisica universitaria Vol. 1. [Versione web]. Estratto da <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>

percorso libero medio e tempo libero medio

Consideriamo le molecole di gas in un cilindro. Si muovono in un moto casuale mentre collidono tra loro e cambiano velocità e direzione. La media di tutte le lunghezze dei percorsi tra le collisioni è nota come "cammino libero medio".

<img alt="Equation1" src="/files/ftp_upload/14168/14168_Equation_1_new.svg" style="text-align: center; height: 40px;" />

Il cammino libero medio varia inversamente con la densità delle molecole perché quando ci sono più molecole all'interno di un volume, hanno una maggiore probabilità di collidere tra loro, riducendo così il cammino libero medio. Inoltre, il cammino libero medio è inversamente correlato al diametro delle molecole perché se fossero masse puntiformi, non colliderebbero mai. Quindi, molecole più grandi sono associate a un cammino libero medio più breve.

Il gas si espande quando la temperatura aumenta a pressione costante; di conseguenza, la distanza media tra le molecole e il cammino libero medio aumenta. Tuttavia, quando la pressione viene aumentata a temperatura costante, il gas si comprime, portando a una diminuzione del cammino libero medio. Il cammino libero medio può essere definito come il prodotto della velocità media e del tempo libero medio, dove il tempo libero medio è il tempo medio tra le collisioni.

Consideriamo gli atomi di argon con una massa molare di 39,9 g/mol che si muovono casualmente in un cilindro a una temperatura di 273 K e una pressione di 1 atm. Prendendo il raggio di un atomo di argon pari a 1.70x10-10 m, determiniamo il tempo libero medio per gli atomi di argon.

Per risolvere il problema, identifica prima le grandezze note e sconosciute e convertile in unità del SI.

In secondo luogo, ricorda l'equazione della velocità RMS per le molecole di gas. Sostituendo i valori, la velocità RMS può essere determinata come segue:

<img alt="Equation2" src="/files/ftp_upload/14168/14168_Equation_2_new.svg" style="text-align: center; height: 45px;" />

Infine, ricorda l'equazione del tempo libero medio. Sostituendo i valori, il tempo libero medio può essere determinato come segue:

<img alt="Equation3" src="/files/ftp_upload/14168/14168_Equation_3_new.svg" style="text-align: center; height: 40px;" />

Percorso libero medio e tempo libero medio

Consideriamo le molecole di gas in un cilindro. Si muovono con un movimento casuale mentre si scontrano tra loro e cambiano velocità e direzione. La media di tutti ...

Education

Didattica

JoVE Core

Physics

Fisica

Unit 1

Unità 1

The Kinetic Theory of Gases

La teoria cinetica dei gas

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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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How Real Gases Deviate from Ideal Behavior

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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,...

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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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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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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.

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E. Chemical reaction rates change with conditions under which they occur. Chemical equilibrium is a dynamic state in which forward and reverse processes occur at the same rates.

Gas collisions: path length and time

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