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

Si consideri un ossigeno gassoso in un contenitore avente una massa molare di 0,0320 kg/mol a temperatura ambiente. Determinare il rapporto tra il numero di molecole con una velocità di circa 400 m/s e il numero di molecole con una velocità di circa 200 m/s, la velocità quadratica media e la velocità più probabile delle molecole di ossigeno a temperatura ambiente.
Per risolvere il problema, identificare le quantità note e sconosciute.
In forma differenziale, il numero di molecole vicine a una velocità è il prodotto della funzione di distribuzione di probabilità della velocità moltiplicata per un piccolo intervallo della velocità.
Il rapporto può essere calcolato dividendo il numero di molecole vicino alle singole velocità.
Sostituire i termini nella funzione di distribuzione di Maxwell-Boltzmann per le molecole di gas. È possibile determinare il rapporto tra il numero di molecole e le singole velocità.
Ricordiamo l'equazione della velocità quadratica media e l'equazione della velocità più probabile per le molecole di gas. Sostituendo i valori noti, è possibile determinare la velocità quadratica media e la velocità più probabile del gas.

<ol>
	<li>OpenStax. (2019). Fisica universitaria Vol. 2. [Versione web], pagine 94-95. Estratto da <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>

distribuzione di maxwell-boltzmann: risolvere i problemi

Molecole individuali in un gas si muovono in direzioni casuali, ma un gas contenente numerose molecole ha una distribuzione prevedibile delle velocità molecolari, nota come distribuzione di Maxwell-Boltzmann, f(v).

Questa funzione di distribuzione f(v) è definita affermando che il numero atteso N (v1,v2) di particelle con velocità comprese tra v1 e v2 è dato da:

<img alt="Equation1" src="/files/ftp_upload/14167/14167_Equation_1.svg" style="text-align: center; height: 80px;" />

Poiché N è adimensionale e l'unità di f(v) è secondi per metro, l'equazione può essere comodamente modificata in una forma differenziale:

<img alt="Equation2" src="/files/ftp_upload/14167/14167_Equation_2.svg" style="text-align: center; height: 20px;" />

Considera un campione di gas azoto in un cilindro con una massa molare di 28,0 g/mol a una temperatura ambiente di 27 °C. Determina il rapporto tra il numero di molecole con una velocità molto vicina a 300 m/s e il numero di molecole con una velocità molto vicina a 100 m/s.

Per risolvere il problema, esamina la situazione e identifica le grandezze note e sconosciute.

In secondo luogo, converte tutti i valori noti nelle unità del Sistema Internazionale adeguate. Ad esempio, converti il peso molecolare in chilogrammi (kg) e la temperatura in kelvin (K). Ricorda l'equazione della funzione di distribuzione per la velocità. Infine, sostituisci i valori noti nell'equazione per determinare la grandezza sconosciuta.



<img alt="Equation3" src="/files/ftp_upload/14167/14167_Equation_3.svg" style="text-align: center; height: 280px;" />

Distribuzione di Maxwell-Boltzmann: risolvere i problemi

Le singole molecole di un gas si muovono in direzioni casuali, ma un gas contenente numerose molecole ha una distribuzione prevedibile delle velocità molecolari, che ...

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

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...

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van der waals equation

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distribution of molecular speeds

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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

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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

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

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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.

Maxwell-Boltzmann Gas Speed Ratio

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