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

考虑一个容器中的氧气，其摩尔质量为 0.0320 kg/mol，在室温下。确定速度约为 400 m/s 的分子数与速度约为 200 m/s 的分子数之比，以及氧气分子在室温下的方均根速度和最概然速度。
为了解决问题，需确定已知量和未知量。
在微分形式下，接近某一速度的分子数量等于该速度的概率分布函数与速度微小区间之乘积。
该比值可通过将接近各个速度的分子数相除来计算。
将气体分子的麦克斯韦-玻尔兹曼分布函数中的各项进行代入，即可确定具有不同速率的分子数之比。
回顾气体分子的均方根速率方程和最概然速率方程。通过代入已知数值，可确定该气体的均方根速率和最概然速率。

<ol>
	<li>OpenStax 的。（2019）. 大学物理学第 2 卷。[网络版]，第 94-95 页。取自 <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>

麦克斯韦-玻尔兹曼分布：解决问题

气体中的单个分子沿随机方向移动，但包含大量分子的气体具有可预测的分子速度分布，称为麦克斯韦-玻尔兹曼分布，f(v)。 此分布函数 f(v) 的定义是预期数 N (v1,v2) 速度在 v1 和 v2 之间的粒子由下式给出 <img alt="Equation1" src="/files/ftp_upload/14167/14167_Equation_1.svg" style="text-align: center; height: 80px;"> 由于 N 是无量纲的，并且 f(v) 的单位是秒/米，所以方程可以方便地修改为微分形式： <img alt="Equation2" src="/files/ftp_upload/14167/14167_Equation_2.svg" style="text-align: center; height: 20px;"> 假设在 27 °C 室温下在钢瓶中摩尔质量为 28.0 g/mol 的氮气样品。 如求速度非常接近 300 m/s 的分子数与速度非常接近 100 m/s 的分子数之比。 要解决问题，需检查情况并确定已知量和未知量。 其次，将所有已知值转换为适当的 SI 单位。 例如，将分子量转换为千克 (kg)，将温度转换为开尔文 (K)。 回想一下速度方程的分布函数。 最后，将已知值代入方程即可求出未知量。 <img alt="Equation3" src="/files/ftp_upload/14167/14167_Equation_3.svg" style="text-align: center; height: 280px;">

气体中的单个分子朝随机方向运动，但含有大量分子的气体具有可预测的分子速率分布，这种分布……

Education

教育

JoVE Core

Physics

物理学

Unit 1

第一单元

The Kinetic Theory of Gases

气体分子动理论

KEY TERMS AND CONCEPTS

关键术语与概念

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

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pv-diagrams

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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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dalton's law of partial pressure

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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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Maxwell-Boltzmann Gas Speed Ratio

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