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

Rozważmy cząsteczki gazu N o promieniu r poruszające się losowo z prędkością v w cylindrycznej objętości V. Kiedy jedna cząsteczka zderza się z inną cząsteczką, odległość między ich środkami wynosi 2r.
Wyobraźmy sobie cylinder o promieniu 2r, z osią równoległą do prędkości cząsteczki. Gdy cząsteczka przemieszcza się przez krótki odstęp czasu i zderza się wewnątrz cylindra, można określić liczbę zderzeń w jednostce czasu.
Korzystając z równania średniej prędkości względnej, można wyznaczyć zderzenia wszystkich poruszających się cząsteczek w jednostce czasu.
Odwrotność równania daje średni czas między zderzeniami, znany jako średni czas wolny.
Tymczasem średnia swobodna ścieżka cząsteczki gazu jest iloczynem prędkości cząsteczki i średniego czasu między zderzeniami. Podstawiając te warunki, można określić średnią drogę swobodną, która jest odwrotnie proporcjonalna do liczby cząsteczek na jednostkę objętości i pola przekroju poprzecznego cząsteczki.
Przywołując równanie gazu idealnego i podstawiając wyrazy, można uzyskać makroskopowe właściwości gazu.

<ol>
	<li>Young, H.D i Freedman, R.A. (2012). Fizyka uniwersytecka z fizyką współczesną. San Francisco, Kalifornia: Pearson. Strony 630</li>
	<li>Protokół OpenStax. (2019). Fizyka uniwersytecka Vol. 1. [Wersja internetowa]. Pobrane z <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>

Średnia wolna droga i średni czas wolny

Rozważ cząsteczki gazu w cylindrze. Poruszają się losowo, zderzając się ze sobą i zmieniając prędkość i kierunek. Średnia wszystkich długości ścieżek pomiędzy kolizjami nazywana jest „średnią swobodną ścieżką”.

<img alt="Równanie1" src="/files/ftp_upload/14168/14168_Equation_1_new.svg" style="text-align: center; height: 40px;" />

Średnia droga swobodna zmienia się odwrotnie wraz z gęstością cząsteczek, ponieważ gdy w danej objętości znajduje się więcej cząsteczek, istnieje większe prawdopodobieństwo, że zderzą się one ze sobą, zmniejszając w ten sposób średnią drogę swobodną. Dodatkowo średnia droga swobodna jest odwrotnie proporcjonalna do średnicy cząsteczek, ponieważ gdyby były masami punktowymi, nigdy by się nie zderzyły. Zatem większe cząsteczki są powiązane z krótszą średnią ścieżką swobodną.

Gaz rozszerza się, gdy temperatura wzrasta pod stałym ciśnieniem; w ten sposób wzrasta średnia odległość między cząsteczkami i średnia droga swobodna. Jednakże, gdy ciśnienie wzrasta w stałej temperaturze, gaz ulega kompresji, co prowadzi do zmniejszenia średniej swobodnej drogi. Średnią drogę swobodną można zdefiniować jako iloczyn średniej prędkości i średniego czasu swobodnego, gdzie średni czas swobodny to średni czas pomiędzy kolizjami.

Rozważmy atomy argonu o masie molowej 39,9 g/mol, poruszające się losowo w cylindrze o temperaturze 273 K i pod ciśnieniem 1 atm. Przyjmując promień atomu argonu wynoszący 1,70″ 10-10m, określ średni czas wolny dla atomów argonu.

Aby rozwiązać problem, najpierw zidentyfikuj znane i nieznane wielkości i przelicz je na jednostki SI.

Po drugie, przypomnij sobie równanie prędkości RMS dla cząsteczek gazu. Zastępując wartości, prędkość RMS można określić w następujący sposób:

<img alt="Równanie2" src="/files/ftp_upload/14168/14168_Equation_2_new.svg" style="text-align: center; height: 45px;" />

Na koniec przypomnijmy sobie równanie średniego czasu wolnego. Podstawiając wartości, średni czas wolny można wyznaczyć w następujący sposób:

<img alt="Równanie3" src="/files/ftp_upload/14168/14168_Equation_3_new.svg" style="text-align: center; height: 40px;" />

Weźmy pod uwagę cząsteczki gazu w cylindrze. Poruszają się losowym ruchem, gdy zderzają się ze sobą i zmieniają prędkość oraz kierunek. Średnia ze wszystkich ...

Education

JoVE Core

Physics

Fizyka

Unit 1

Jednostka 1

The Kinetic Theory of Gases

Kinetyczna teoria gazów

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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

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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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maxwell-boltzmann-distribution-problem-solving

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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,...

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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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Gas collisions: path length and time

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