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

Considérons un gaz d’oxygène dans un récipient ayant une masse molaire de 0,0320 kg/mol à température ambiante. Déterminez le rapport entre le nombre de molécules ayant une vitesse d’environ 400 m/s et le nombre de molécules ayant une vitesse d’environ 200 m/s, la vitesse quadratique moyenne et la vitesse la plus probable des molécules d’oxygène à température ambiante.
Pour résoudre le problème, identifiez les quantités connues et inconnues.
Sous forme différentielle, le nombre de molécules proches d’une vitesse est le produit de la fonction de distribution de probabilité de la vitesse multipliée par un petit intervalle de la vitesse.
Le rapport peut être calculé en divisant le nombre de molécules à proximité des vitesses individuelles.
Substituez les termes de la fonction de distribution de Maxwell-Boltzmann pour les molécules de gaz. Le rapport du nombre de molécules avec des vitesses individuelles peut être déterminé.
Rappelez-vous l’équation de la vitesse quadratique moyenne et l’équation de vitesse la plus probable pour les molécules de gaz. En substituant les valeurs connues, la vitesse quadratique moyenne et la vitesse la plus probable du gaz peuvent être déterminées.

<ol>
	<li>OpenStax. (2019). Physique universitaire Vol. 2. [Version Web], pages 94-95. Récupéré de <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>

distribution de maxwell-boltzmann : résolution de problèmes

Les molécules individuelles dans un gaz se déplacent dans des directions aléatoires, mais un gaz contenant de nombreuses molécules a une distribution prévisible des vitesses moléculaires, connue sous le nom de distribution de Maxwell-Boltzmann, f(v).

Cette fonction de distribution f(v) est définie en disant que le nombre attendu N(v1,v2) de particules avec des vitesses entre v1 et v2 est donné par

<img alt="Equation1" src="/files/ftp_upload/14167/14167_Equation_1.svg" style="text-align: center; height: 80px;" />

Puisque N est sans dimension et l'unité de f(v) est en secondes par mètre, l'équation peut être commodément modifiée sous une forme différentielle :

<img alt="Equation2" src="/files/ftp_upload/14167/14167_Equation_2.svg" style="text-align: center; height: 20px;" />

Considérez un échantillon de gaz azote dans un cylindre avec une masse molaire de 28,0 g/mol à une température ambiante de 27 °C. Déterminez le rapport entre le nombre de molécules avec une vitesse très proche de 300 m/s et le nombre de molécules avec une vitesse très proche de 100 m/s.

Pour résoudre le problème, examinez la situation et identifiez les quantités connues et inconnues.

Ensuite, convertissez toutes les valeurs connues dans les unités SI appropriées. Par exemple, convertissez la masse molaire en kilogrammes (kg) et la température en kelvins (K). Rappelons la fonction de distribution pour l'équation de la vitesse. Enfin, substituez les valeurs connues dans l'équation pour déterminer la quantité inconnue.

<img alt="Equation3" src="/files/ftp_upload/14167/14167_Equation_3.svg" style="text-align: center; height: 280px;" />

Distribution de Maxwell-Boltzmann : Résolution de problèmes

Les molécules individuelles d’un gaz se déplacent dans des directions aléatoires, mais un gaz contenant de nombreuses molécules a une distribution prévisible des vitesses moléculaires, qui ...

Education

Enseignement

JoVE Core

Physics

Physique

Unit 1

Unité 1

The Kinetic Theory of Gases

La théorie cinétique des gaz

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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

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van der waals equation

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kinetic theory of an ideal gas

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molecular kinetic energy

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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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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.

Maxwell-Boltzmann Gas Speed Ratio

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