Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Mean Free Path

平均自由程

The mean free path or average distance between collisions for a gas molecule may be estimated from kinetic theory. Serway's approach is a good visualization - if the molecules have diameter d, then the effective cross-section for collision can be modeled by

气体分子的平均自由程或碰撞间的平均距离可通过气体动理论估计。Serway的方法是一种良好的可视化方式——如果分子直径为d，那么碰撞的有效横截面积可以建模为

using a circle of diameter 2d to represent a molecule's effective collision area while treating the "target" molecules as point masses. In time t, the circle would sweep out the volume shown and the number of collisions can be estimated from the number of gas molecules that were in that volume.

用直径为2d的圆来表示分子的有效碰撞面积，同时将‘目标’分子视为质点。在时间t内，该圆会扫过所示的体积，碰撞次数可以从处于该体积内的气体分子数量进行估算。

The mean free path could then be taken as the length of the path divided by the number of collisions.

平均自由程可取为路径长度除以碰撞次数。

The problem with this expression is that the average molecular velocity is used, but the target molecules are also moving. The frequency of collisions depends upon the average relative velocity of the randomly moving molecules.

这个表达式的问题在于使用了平均分子速度，但目标分子也在运动。碰撞频率取决于随机运动分子的平均相对速度。
Refinement of mean free path
自由路径的细化
Calculation
计算
Frequency of collision
碰撞频率
 Index

Gas law concepts

Kinetic theory concepts

Reference: Rohlf
Ch. 2
索引 气体定律概念 气体动理论概念 参考：Rohlf 第2章
 
HyperPhysics***** Thermodynamics
HyperPhysics ***** 热力学
R Nave
 Go Back
返回





Refinement of Mean Free Path

均自由路径的细化

The intuitive development of the mean free path expression suffers from a significant flaw - it assumes that the "target" molecules are at rest when in fact they have a high average velocity. What is needed is the average relative velocity, and the calculation of that velocity from the molecular speed distribution yields the result

平均自由程的直观发展存在一个显著缺陷——它假设“目标”分子是静止的，但实际上它们具有较高的平均速度。所需的是相对平均速度，从分子速度分布计算出该速度即可得到结果。
Show
显示

which revises the expression for the effective volume swept out in time t

这修正了在时间t内有效扫过的体积的表达式

The resulting mean free path is

The number of molecules per unit volume can be determined from Avogadro's number and the ideal gas law, leading to

单位体积中的分子数可通过阿伏伽德罗常数和理想气体定律确定，从而

所得到的平均自由程是

It should be noted that this expression for the mean free path of molecules treats them as hard spheres, whereas real molecules are not. For noble gases, the collisions are probably close to being perfectly elastic, so the hard sphere approximation is probably a good one. But real molecules may have a dipole moment and have significant electrical interaction as they approach each other. This has been approached by using an electrical potential for the molecules to refine the calculation, and also by using the measured viscosity of the gas as a parameter to refine the estimate of the mean free path of molecules in real gases.

Mean free path related to viscosity
与粘度相关的平均自由程
应注意的是，这个表达式将分子视为硬球，而实际上分子并非如此。对于惰性气体，碰撞可能接近完全弹性，因此硬球近似可能是合理的。但真实分子可能具有偶极矩，并在相互接近时有显著的电相互作用。为此，人们通过使用分子间的电势来改进计算，同时也利用气体的粘度作为参数来改进对真实气体中分子平均自由程的估计。

Reference:

参考文献：

Wiki: mean free path

平均自由程

Wiki: viscosity

维基：粘度

Gas viscosity

Calculation

计算
 Index

Kinetic theory concepts
索引 气体动理论概念
 
HyperPhysics***** Thermodynamics
HyperPhysics ***** 热力学
R Nave
 Go Back
返回





Mean Free Path Calculation

平均自由程计算

The mean free path equation depends upon the temperature and pressure as well as the molecular diameter.

平均自由程方程还取决于温度和压力以及分子直径。

For pressure P0 = mmHg =inHg =kPa

对于压力 P 0 = mmHg = inHg = kPa
中文译文中的待填/计算数值依次对应：1：pmm 2：pin 3：pk。实际数值以上方原输入框为准。

and temperature T=K = C =F,

以及温度 T = K = C = F，
中文译文中的待填/计算数值依次对应：1：t 2：tc 3：tf。实际数值以上方原输入框为准。

Molecules of diameter x 10-10 meters (angstroms)

直径为10^-10米的分子（埃)
中文译文中的待填/计算数值依次对应：1：md。实际数值以上方原输入框为准。

should have a mean free path of
应有平均自由程为
 = x 10^m

which is times the molecular diameter

这是分子直径的 times
中文译文中的待填/计算数值依次对应：1：dm。实际数值以上方原输入框为准。

and times the average molecular separation of x 10^m.

乘以平均分子间距的10^m。
中文译文中的待填/计算数值依次对应：1：sm 2：sepb 3：sepp。实际数值以上方原输入框为准。

The values for pressure, temperature, and molecular diameter may be changed above to recalculate the mean free path. A nominal molecular diameter of 0.3nm = 3 x 10-10 m will give you a reasonable approximation. The pressure required for a given mean free path can be calculated by changing the value of the mean free path λ above. The calculation is not designed to allow any other changes (their values will be replaced unchanged by the calculation.)

压力、温度和分子直径的值可能被修改以重新计算平均自由程。一个名义上的分子直径为0.3纳米（即3×10-10米）将提供一个合理的近似值。为了计算给定平均自由程所需的压强，可以修改上方的平均自由程λ的值。该计算未设计为允许其他任何更改（它们的值将被计算过程不变地替换）。

From the mean free path and the average velocity, the mean time between collisions and the frequency of collisions can be calculated.

从平均自由程和平均速度可以计算出碰撞之间的平均时间及碰撞频率。
Example of mean free path compared to molecular separation.
平均自由程与分子间距的比较示例
Relation of mean free path to viscosity
平均自由程与粘度的关系
 Index

Kinetic theory concepts
索引 气体动理论概念
 
HyperPhysics***** Thermodynamics
HyperPhysics ***** 热力学
R Nave
 Go Back
返回






Mean Free Path Perspective

平均自由程视角

You may be surprised by the length of the mean free path compared to the average molecular separation in an ideal gas. An atomic size of 0.3 nm was assumed to calculate the other distances.

你可能会惊讶于理想气体中平均自由程与平均分子间距相比的长度。假设原子尺寸为0.3纳米来计算其他距离。

Mean Free Path Calculation
平均自由程计算
Frequency of collision
碰撞频率
 Index

Gas law concepts

Kinetic theory concepts
索引 气体定律概念 气体动理论概念
 
HyperPhysics***** Thermodynamics
HyperPhysics ***** 热力学
R Nave
 Go Back
返回





Average Relative Velocity

平均相对速度

In order to calculate the mean free path for a molecule of a gas, it is necessary to assess the average relative velocity of the molecules involved rather than just the average velocity of any given molecule. The relative velocity of any two molecules can be expressed in terms of their vector velocities.

为了计算气体中分子的平均自由程，必须评估涉及分子的平均相对速度，而不仅仅是单个分子的平均速度。任意两个分子的相对速度可以表示为它们的矢量速度。

The magnitude of the relative velocity can be expressed as the square root of the scalar product of the velocity with itself.

相对速度的大小可以表示为速度与自身的标量积的平方根。

This expression can be expanded as follows.

可以展开为如下形式

Taking the average of the terms leads to

对各项取平均值得到

Since the same average velocity would be associated with each molecule, this becomes

由于每个分子都会具有相同的平均速度，这便成为
Mean Free Path Calculation
平均自由程计算
Frequency of collisions
碰撞频率
 Index

Gas law concepts

Kinetic theory concepts
索引 气体定律概念 气体动理论概念
 
HyperPhysics***** Thermodynamics
HyperPhysics ***** 热力学
R Nave
 Go Back
返回





气体粘度
中文译文中的待填/计算数值依次对应：1：mfb 2：mfp。实际数值以上方原输入框为准。