Name: gas collisions: path length and time
Uploaded: 2025-05-22T09:05:59.000Z
Duration: 1 min 22 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.12

SC.912.P.12.10

Mean free path and Mean free time

考虑 N 个半径为 r 的气体分子在体积为 V 的圆柱形容器中以速度 v 随机运动。当一个分子与另一个分子发生碰撞时，它们中心之间的距离为 2r。
假设一个半径为 2r 的圆柱体，其轴线与分子的运动方向平行。当分子在短时间内运动并在圆柱体内发生碰撞时，可由此确定单位时间内的碰撞次数。
利用平均相对速度公式，可以确定单位时间内所有运动分子的碰撞次数。
该方程的倒数给出了碰撞之间的平均时间，称为平均自由时间。
同时，气体分子的平均自由程等于分子运动速度与两次碰撞之间的平均时间的乘积。通过代入相关参数，可计算出平均自由程，其值与单位体积内的分子数以及分子的横截面积成反比。
回顾理想气体状态方程并代入相关参数，即可获得该气体的宏观性质。

<ol>
	<li>Young， HD 和 Freedman， RA （2012）。大学物理与现代物理。加利福尼亚州旧金山:皮尔逊。页码: 630</li>
	<li>OpenStax 的。（2019）. 大学物理学第 1 卷。[Web 版本]。取自 <a href="https://openstax.org/books/college-physics-2e/pages/13-4-kinetic-theory-atomic-and-molecular-explanation-of-pressure-and-temperature>https://openstax.org/books/college-physics-2e/pages/13-4-kinetic-theory-atomic-and-molecular-explanation-of-pressure-and-temperature</a></li>
</ol>

平均自由路和平均自由时间

假设气缸中的气体分子。 当它们相互碰撞并改变速度和方向时，它们会以随机运动的方式移动。 碰撞之间所有路径长度的平均值称为“平均自由路径。” <img alt="Equation1" src="/files/ftp_upload/14168/14168_Equation_1_new.svg" style="text-align: center; height: 40px;"> 平均自由程与分子密度成反比，因为当体积内有更多分子时，它们相互碰撞的机会就更大，从而减少平均自由程。 此外，平均自由程与分子的直径成反比，因为如果它们是点质量，它们就永远不会碰撞。 因此，较大的分子与较短的平均自由程相关。 在恒定压力下，气体随温度升高而膨胀； 因此，分子之间的平均距离和平均自由程增加。 然而，当压力在恒定温度下增加时，气体会压缩，导致平均自由程减小。 平均自由程可以定义为平均速度与平均自由时间的乘积，其中平均自由时间是碰撞之间的平均时间。 假设摩尔质量为 39.9 g/mol 的氩原子——在 273 K 的温度和 1 atm 的压力下在圆柱体中随机移动。 取氩原子的半径为1.70 × 10-10m，确定氩原子的平均自由时间。 要解决该问题，首先要确定已知量和未知量，并将它们转换为 SI 单位。 其次，回想一下气体分子的 RMS 速度方程。 通过代入这些值，可按如下方式确定 RMS 速度： <img alt="Equation2" src="/files/ftp_upload/14168/14168_Equation_2_new.svg" style="text-align: center; height: 45px;"> 最后，回想一下平均空闲时间方程。 通过替换这些值，平均空闲时间可以确定如下： <img alt="Equation3" src="/files/ftp_upload/14168/14168_Equation_3_new.svg" style="text-align: center; height: 40px;">

考虑气缸中的气体分子。它们在相互碰撞时做无规则运动，并不断改变速度和方向。所有分子……的平均值

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

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 molecules have a shorter mean free path, while point masses would never collide.

Temperature and pressure also affect the mean free path. When temperature increases at constant pressure, a gas expands. The average distance between molecules increases, and the mean free path increases too. When pressure increases at constant temperature, the gas compresses. That reduces the mean free path.

The mean free path can also be written as the product of average speed and mean free time. Mean free time is the average time between collisions. A sample problem uses argon atoms with a molar mass of 39.9 g/mol at 273 K and 1 atm. The radius of an argon atom is taken as 1.70 × 10 -10 m. To find the mean free time, first identify the known and unknown values and convert them to SI units. Then use the RMS speed equation for gas molecules. After that, substitute the values into the mean free time equation to calculate the result.

Gas collisions: path length and time

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

Avogadro's Number and Gas Particle Motion

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...

molecular kinetic energy

Gas Pressure and Molecular Motion

Gas pressure comes from the motion of molecules in a container. The word gas came from a Flemish word meaning “chaos,” and J. B. van Helmont first used it to describe vapors. In a gas, molecules move in a continuous and random way. When a molecule hits a container wall, its velocity changes. The part of the velocity parallel to the wall stays the same. The part perpendicular to the wall reverses direction, but its size does not change. If the molecule’s velocity changes in the x-direction, its...

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

maxwell-boltzmann-distribution-problem-solving

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...

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 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

Learn how gas molecule collisions determine mean free path and mean free time, and see how density, size, temperature, and pressure affect them in argon.

Gas collisions: path length and time | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

Introduction to Scalars

Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, &#34;a class period lasts ...

Introduction to Vectors

To define some physical quantities, there is a need to specify both magnitude as well as direction. For example, when the U.S. Coast Guard dispatches a ...

vector-components-in-the-cartesian-coordinate-system

Vector Components in the Cartesian Coordinate System

Vectors are usually described in terms of their components in a coordinate system. Even in everyday life, we naturally invoke the concept of orthogonal ...

Vector Algebra: Graphical Method

Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the ...

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like ...

/files/media/science-education/science-education-thumbs/12613.jpg

Scalar Product (Dot Product)

The scalar multiplication of two vectors is known as the scalar or dot product. As the name indicates, the scalar product of two vectors results in a ...

Vector Product (Cross Product)

Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine of ...

Scalar and Vector Triple Products

Two vectors can be multiplied using a scalar product or a vector product. The resultant of a scalar product is scalar, while with vector products, the ...

/files/media/science-education/science-education-thumbs/14688.jpg

Gradient and Del Operator

In mathematics and physics, the gradient and del operator are fundamental concepts used to describe the behavior of functions and fields in space. The ...

/files/media/science-education/science-education-thumbs/14689.jpg

Divergence and Curl

The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as ...

/files/media/science-education/science-education-thumbs/14690.jpg

Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or ...

/files/media/science-education/science-education-thumbs/14692.jpg

Line, Surface, and Volume Integrals

A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a ...

/files/media/science-education/science-education-thumbs/14693.jpg

Distinguish between scalar and vector quantities and assess which should be used to describe an event.

SC.912.P.12.1

Position and Displacement

The position of an object defines its location relative to a convenient frame of reference at any particular time. A frame of reference is an arbitrary ...

Average Velocity

To calculate the other physical quantities in kinematics, we must introduce the time variable. The time variable allows us not only to state the position ...

Instantaneous Velocity - I

The average velocity during a time interval cannot tell us how fast or in what direction a particle is moving at any given time during the interval. To ...

Instantaneous Velocity - II

Instantaneous velocity is the quantity that measures how fast an object is moving along its path. In other words, the instantaneous velocity of an object ...

Average Acceleration

The importance of understanding acceleration spans our day-to-day experiences, as well as the vast reaches of outer space and the tiny world of subatomic ...

Instantaneous Acceleration

Acceleration is in the direction of the change in velocity, but it is not always in the direction of motion. When an object slows down, its acceleration ...

Kinematic Equations - I

When an object moves with constant acceleration, the velocity of the object changes at a constant rate throughout the motion. The kinematic equations of ...

Kinematic Equations - II

The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant ...

Kinematic Equations - III

The first two kinematic equations have time as a variable, but the third kinematic equation is independent of time. This equation expresses final velocity ...

Kinematic Equations: Problem Solving

When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the ...

Free-falling Bodies: Introduction

All objects, neglecting air resistance, fall with the same acceleration towards the Earth&#39;s center due to the force exerted by the Earth&#39;s ...

Free-falling Bodies: Example

An object falling without any air resistance under the influence of gravitational force is said to be in free-fall. For free-falling bodies, the ...

velocity-and-position-by-graphical-method

Velocity and Position by Graphical Method

Velocity and position can be calculated from the known function of acceleration as a function of time. The total area under the acceleration-time graph ...

Non-uniform Circular Motion

In&#160;uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius. However, not all ...

Velocity and Position by Integral Method

If acceleration as a function of time is known, then velocity and position functions can be derived using integral calculus. For constant acceleration, ...

Relative Velocity in One Dimension

The understanding of the concept of reference frames is essential to discuss relative motion in one or more dimensions. When we say that an object has a ...

Position and Displacement Vectors

To describe the motion of an object, one should first be able to describe its position (where it is at any particular time). More precisely, the position ...

Relative Velocity in Two Dimensions

Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to ...

average-and-instantaneous-velocity-vectors

Average and Instantaneous Velocity Vectors

To calculate other physical quantities in kinematics, the time variable must be introduced. The time variable not only allows us to state where an object ...

Acceleration Vectors

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, &#916;v, therefore the ...

Direction of Acceleration Vectors

Acceleration occurs when velocity changes in magnitude (an increase or decrease in speed), direction, or both. Although acceleration is in the direction ...

Projectile Motion

An object thrown in the air follows a parabolic path under the influence of Earth&#39;s gravitational force. The motion of such an object is called ...

/files/media/science-education/science-education-thumbs/12634.jpg

Projectile Motion: Equations

Projectile motion is commonly observed in our day-to-day life. For example, a basketball thrown by a player, an arrow shot from a bow, and kids jumping ...

Projectile Motion: Example

The theory of projectile motion is very useful for players of several sports to improve their performance. For example, a javelin thrower needs to throw ...

Uniform Circular Motion

Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller ...

Analyze the motion of an object in terms of its position, velocity, and acceleration (with respect to a frame of reference) as functions of time.

SC.912.P.12.2

Newton's First Law: Introduction

Motion draws our attention. Motion itself can be beautiful, causing us to marvel at the forces needed to create spectacular sights, such as that of a ...

Newton's First Law: Application

Experience suggests that an object at rest remains at rest if left alone, and that an object in motion tends to slow down and stop unless some effort is ...

Internal and External Forces

Newton&#39;s first law states that a net external force causes a change in motion. External forces act on an object or system, originating outside of the ...

Newton's Second Law

Newton&#39;s second law is closely related to his first law of motion. It mathematically gives the cause-and-effect relationship between force and changes ...

Newton's Third Law: Introduction

Whenever one body exerts a force on a second body, the first body experiences a force equal in magnitude and opposite in direction, to the force that it ...

Newton's Third Law: Examples

Newton&#39;s third law states that every action has an equal and opposite reaction. Consider a swimmer pushing off the side of a pool. They push against ...

Inertial Frames of Reference

Newton&#8217;s first law is usually considered to be a statement about reference frames. It provides a method for identifying a special type of reference ...

Non-inertial Frames of Reference

A reference frame accelerating or decelerating relative to an inertial frame is a non-inertial frame. To help understand this, consider what taking off in ...

first-law-particles-in-one-dimensional-equilibrium

First Law: Particles in One-dimensional Equilibrium

Newton&#39;s first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a ...

first-law-particles-in-two-dimensional-equilibrium

First Law: Particles in Two-dimensional Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic ...

Second Law: Motion under Same Force

Newton&#39;s laws can be applied to bodies at rest and bodies in motion. Newton&#39;s first law is applied to bodies in equilibrium, whereas the second ...

second-law-motion-under-same-acceleration

Second Law: Motion under Same Acceleration

Newton&#39;s second law of motion applies to bodies moving under the same acceleration. For example, when a baggage tractor pulls luggage carts, each cart ...

Interpret and apply Newton's three laws of motion.

SC.912.P.12.3

Gravitation

In the years before Newton, a general belief prevailed that different laws governed objects in the sky than objects on Earth. When Kepler wrote down the ...

/files/media/science-education/science-education-thumbs/14716.jpg

Newton's Law of Gravitation

Our everyday observation tells us that all objects close to the Earth naturally tend to fall to the ground. Early philosophers assumed that this downward ...

gravitation-between-spherically-symmetric-masses

Gravitation Between Spherically Symmetric Masses

The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, ...

/files/media/science-education/science-education-thumbs/14717.jpg

Gravity between Spherical Bodies

Newton&#39;s law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, ...

Describe how the gravitational force between two objects depends on their masses and the distance between them.

SC.912.P.12.4

Linear Momentum

The term momentum is used in various ways in everyday language, most of which are consistent with the precise scientific definition. Generally, momentum ...

Apply the law of conservation of linear momentum to interactions, such as collisions between objects.

SC.912.P.12.5

A. Motion can be measured and described qualitatively and quantitatively. Net forces create a change in motion. When objects travel at speeds comparable to the speed of light, Einstein's special theory of relativity applies.

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.

Gas collisions: path length and time

Description

Transcript

More videos by Standards