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

Stellen Sie sich ein Sauerstoffgas in einem Behälter vor, das bei Raumtemperatur eine Molmasse von 0,0320 kg/mol hat. Bestimmen Sie das Verhältnis der Anzahl der Moleküle mit einer Geschwindigkeit von etwa 400 m/s zur Anzahl der Moleküle mit einer Geschwindigkeit von etwa 200 m/s, der mittleren quadratischen Geschwindigkeit und der wahrscheinlichsten Geschwindigkeit von Sauerstoffmolekülen bei Raumtemperatur.
Um das Problem zu lösen, identifizieren Sie die bekannten und unbekannten Größen.
In der differentiellen Form ist die Anzahl der Moleküle nahe einer Geschwindigkeit das Produkt aus der Wahrscheinlichkeitsverteilungsfunktion der Geschwindigkeit multipliziert mit einem kleinen Intervall der Geschwindigkeit.
Das Verhältnis kann berechnet werden, indem die Anzahl der Moleküle in der Nähe der einzelnen Geschwindigkeiten dividiert wird.
Ersetzen Sie Gasmoleküle durch die Terme in der Maxwell-Boltzmann-Verteilungsfunktion. Das Verhältnis der Anzahl der Moleküle mit den einzelnen Geschwindigkeiten kann bestimmt werden.
Erinnern wir uns an die Gleichung für die mittlere quadratische Geschwindigkeit und die wahrscheinlichste Geschwindigkeitsgleichung für Gasmoleküle. Durch Ersetzen der bekannten Werte können die mittlere quadratische Geschwindigkeit und die wahrscheinlichste Geschwindigkeit des Gases bestimmt werden.

<ol>
	<li>OpenStax. (2019). Universitätsphysik Vol. 2. [Webversion], Seiten 94–95. Abgerufen von <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>

maxwell-boltzmann-verteilung: problemlösung

Einzelne Moleküle in einem Gas bewegen sich in zufällige Richtungen, aber ein Gas, das zahlreiche Moleküle enthält, hat eine vorhersagbare Verteilung der Molekülgewschwindigkeiten, die als Maxwell-Boltzmann-Verteilung f(v) bekannt ist.

Diese Verteilungsfunktion f(v) wird definiert, indem man sagt, dass die erwartete Anzahl N(v1,v2) der Teilchen mit Geschwindigkeiten zwischen v1 und v2 gegeben ist durch:

<img alt="Equation1" src="/files/ftp_upload/14167/14167_Equation_1.svg" style="text-align: center; height: 80px;" />

Da N dimensionslos ist und die Einheit von f(v) Sekunden pro Meter ist, kann die Gleichung bequem in eine differentielle Form umgewandelt werden:

<img alt="Equation2" src="/files/ftp_upload/14167/14167_Equation_2.svg" style="text-align: center; height: 20px;" />

Betrachten Sie eine Probe von Stickstoffgas in einem Zylinder mit einer molaren Masse von 28,0 g/mol bei einer Raumtemperatur von 27 °C. Bestimmen Sie das Verhältnis zwischen der Anzahl der Moleküle mit einer Geschwindigkeit von annähernd 300 m/s und der Anzahl der Moleküle mit einer Geschwindigkeit von annähernd 100 m/s.

Um das Problem zu lösen, betrachten Sie die Situation und identifizieren Sie bekannte und unbekannte Größen.

Als nächstes konvertieren Sie alle bekannten Werte in geeignete SI-Einheiten. Zum Beispiel wandeln Sie das Molekulargewicht in Kilogramm (kg) und die Temperatur in Kelvin (K) um. Erinnern Sie sich an die Verteilungsfunktion für die Geschwindigkeitsgleichung. Schließlich setzen Sie die bekannten Werte in die Gleichung ein, um die unbekannte Größe zu bestimmen.

<img alt="Equation3" src="/files/ftp_upload/14167/14167_Equation_3.svg" style="text-align: center; height: 280px;" />

Maxwell-Boltzmann-Verteilung: Problemlösung

Einzelne Moleküle in einem Gas bewegen sich in zufällige Richtungen, aber ein Gas, das zahlreiche Moleküle enthält, hat eine vorhersagbare Verteilung von Molekülgeschwindigkeiten, die ...

Education

Lehre

JoVE Core

Physics

Physik

Unit 1

Einheit 1

The Kinetic Theory of Gases

Die kinetische Gastheorie

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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

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...

van der waals equation

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...

pv-diagrams

Liquid-Gas Changes on pV Curves

pV curves show how gas pressure changes with volume. A pV diagram is a graph of pressure versus volume for the gas being studied. It helps describe how the substance behaves. When the gas acts like an ideal gas, the ideal gas equation gives the link between pressure and volume. A common pV curve is an isotherm, which is a line drawn at fixed temperature and fixed number of molecules. For an ideal gas, pressure times volume stays constant. That means volume goes down as pressure goes up. The...

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...

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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.

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.

Maxwell-Boltzmann Gas Speed Ratio

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