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

Consider N gas molecules with radius r moving randomly with speed v in a cylindrical volume V. When one molecule collides with another molecule, the distance between their centers is 2r.
Imagine a cylinder with a radius of 2r, with an axis parallel to the molecule&#39;s velocity. When the molecule travels for a small time interval and collides inside the cylinder, the number of collisions per unit time can be determined.
Using the average relative velocity equation, the collisions for all the moving molecules per unit time can be determined.
The reciprocal of the equation gives the average time between collisions, known as the mean free time.
Meanwhile, the mean free path of a gas molecule is the product of the molecule&#39;s speed and the average time between collisions. By substituting the terms, the mean free path can be determined, which is inversely proportional to the number of molecules per unit volume and the cross-sectional area of the molecule.
Recalling the ideal-gas equation and substituting the terms, the macroscopic properties of the gas can be obtained.

beteken een vrij pad en een gemiddelde vrije tijd

Beschouw de gasmoleculen in een cilinder. Ze bewegen in een willekeurige beweging terwijl ze met elkaar botsen en veranderen van snelheid en richting. Het gemiddelde van alle padlengtes tussen botsingen staat bekend als het "gemiddelde vrije pad." 
 <img alt="Equation1" src="/files/ftp_upload/14168/14168_Equation_1_new.svg" style="text-align: center; height: 40px;"> 

 Het gemiddelde vrije pad varieert omgekeerd evenredig met de dichtheid van de moleculen, want als er meer moleculen in een volume zitten, hebben ze een grotere kans om met elkaar te botsen, waardoor het gemiddelde vrije pad kleiner wordt. Bovendien is het gemiddelde vrije pad omgekeerd evenredig aan de diameter van de moleculen, omdat ze nooit zouden botsen als ze puntmassa's zouden zijn. Grotere moleculen zijn dus geassocieerd met een korter gemiddeld vrij pad. 

 Het gas zet uit als de temperatuur stijgt onder constante druk; dus neemt de gemiddelde afstand tussen moleculen en het gemiddelde vrije pad toe. Wanneer de druk echter bij een constante temperatuur wordt verhoogd, comprimeert het gas, wat leidt tot een afname van het gemiddelde vrije pad. Het gemiddelde vrije pad kan worden gedefinieerd als het product van de gemiddelde snelheid en de gemiddelde vrije tijd, waarbij de gemiddelde vrije tijd de gemiddelde tijd tussen botsingen is. 

 Beschouw argonatomen met een molaire massa van 39,9 g/mol, die zich willekeurig in een cilinder bewegen bij een temperatuur van 273 K en een druk van 1 atm. Neem de straal van een argonatoom als 1,70 × 10-10m, bepaal de gemiddelde vrije tijd voor argonatomen.

 Om het probleem op te lossen, identificeert u eerst de bekende en onbekende grootheden en converteert u deze naar SI-eenheden. 

 Ten tweede: denk eens aan de RMS-snelheidsvergelijking voor gasmoleculen. Door de waarden te vervangen, kan de RMS-snelheid als volgt worden bepaald: <img alt="Equation2" src="/files/ftp_upload/14168/14168_Equation_2_new.svg" style="text-align: center; height: 45px;"> 

 Herinner ten slotte de vergelijking van de gemiddelde vrije tijd. Door de waarden te vervangen, kan de gemiddelde vrije tijd als volgt worden bepaald: <img alt="Equation3" src="/files/ftp_upload/14168/14168_Equation_3_new.svg" style="text-align: center; height: 40px;">

Beteken een vrij pad en een gemiddelde vrije tijd

Consider the gas molecules in a cylinder. They move in a random motion as they collide with each other and change speed and direction. The average of all ...

Education

JoVE Core

Physics

Natuurkunde

Unit 1

Onderdeel 1

The Kinetic Theory of Gases

De kinetische gastheorie

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

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...

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...

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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...

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kinetic theory of an ideal gas

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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

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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...

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maxwell-boltzmann-distribution-problem-solving

Maxwell-Boltzmann Gas Speed Ratio

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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,...

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

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.

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.

C. The Law of Universal Gravitation states that gravitational forces act on all objects irrespective of their size and position.

D. Gases consist of great numbers of molecules moving in all directions. The behavior of gases can be modeled by the kinetic molecular theory.

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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