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

실온에서 몰 질량이 0.0320kg/mol인 용기에 담긴 산소 가스를 생각해 보십시오. 약 400m/s 속도의 분자 수와 약 200m/s 속도의 분자 수의 비율, 제곱평균제곱근 속도 및 실온에서 산소 분자의 가장 가능성이 높은 속도를 결정합니다.
문제를 해결하려면 알려진 수량과 알려지지 않은 수량을 식별하십시오.
미분 형태에서, 속도에 가까운 분자의 수는 속도의 확률 분포 함수에 속도의 작은 간격을 곱한 곱입니다.
이 비율은 개별 속도에 가까운 분자 수를 나누어 계산할 수 있습니다.
Maxwell-Boltzmann 분포 함수의 항을 기체 분자에 대입합니다. 개별 속도를 가진 분자 수의 비율을 결정할 수 있습니다.
제곱평균제곱근(root-mean-square) 속도 방정식과 기체 분자에 대한 가장 가능성 있는 속도 방정식을 상기해 보십시오. 알려진 값을 대입하여 기체의 제곱근 평균 속도와 가장 가능성이 높은 속도를 결정할 수 있습니다.

<ol>
	<li>오픈스택스. (2019). 대학 물리학 Vol. 2. [웹 버전], 94–95쪽. 원본 주소 <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 분포: 문제 해결

기체 내의 개별 분자는 임의의 방향으로 움직이지만 수많은 분자를 포함하는 기체는 맥스웰-볼츠만 분포(Maxwell-Boltzmann distribution)로 알려진 예측 가능한 분자 속도 분포를 갖습니다. f(v ).

이 분포 함수 f(v)는 예상 숫자 N (v)으로 정의됩니다. em>1,v2) 속도가 v1 사이인 입자  및 v2 는 다음과 같이 제공됩니다.

<img alt="Equation1" src="/files/ftp_upload/14167/14167_Equation_1.svg" style="text-align: center; height: 80px;" />

N은 무차원이고 f(v)'의 단위는 미터당 초이므로 방정식은 다음과 같이 편리하게 사용할 수 있습니다. 차등 형식으로 수정됨:

<img alt="Equation2" src="/files/ftp_upload/14167/14167_Equation_2.svg" style="text-align: center; height: 20px;" />

상온 27°C에서 몰 질량이 28.0g/mol인 실린더에 들어 있는 질소 가스 샘플을 생각해 보세요. 300m/s에 매우 가까운 속도를 갖는 분자 수와 100m/s에 매우 가까운 속도를 갖는 분자 수의 비율을 구하십시오.

문제를 해결하려면 상황을 조사하고 알려진 수량과 알려지지 않은 수량을 식별하세요.

둘째, 알려진 모든 값을 적절한 SI 단위로 변환합니다. 예를 들어, 분자량을 킬로그램(kg)으로 변환하고 온도를 켈빈(K)으로 변환합니다. 속도 방정식의 분포 함수를 기억해 보세요. 마지막으로 알려진 값을 방정식에 대입하여 알려지지 않은 양을 결정합니다.

<img alt="Equation3" src="/files/ftp_upload/14167/14167_Equation_3.svg" style="text-align: center; height: 280px;" />

Maxwell-Boltzmann 분포: 문제 해결

기체의 개별 분자는 임의의 방향으로 움직이지만 수많은 분자를 포함하는 기체는 예측 가능한 분자 속도 분포를 가지고 있습니다.

Education

교육

JoVE Core

Physics

물리학

Unit 1

제1단원

The Kinetic Theory of Gases

기체 분자 운동론

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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van der waals equation

How Real Gases Deviate from Ideal Behavior

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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

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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

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Maxwell-Boltzmann Gas Speed Ratio

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