📚 The Kinetic Model of Gases | 气体的动力学模型
The kinetic model explains macroscopic gas properties such as pressure and temperature by considering the motion of large numbers of microscopic particles. For CIE A-Level Physics, you need to be able to state the model’s assumptions, derive pV = ⅓ N m ⟨c²⟩, and connect temperature to mean kinetic energy.
动力学模型通过大量微观粒子的运动来解释气体的宏观性质，例如压强和温度。在 CIE A-Level 物理中，你需要能陈述模型假设、推导 pV = ⅓ N m ⟨c²⟩，并把温度与分子平均动能联系起来。
1. What is the kinetic model? | 什么是动力学模型？
A gas consists of a very large number of tiny particles in continuous, random motion. The model treats these particles as point-like masses that obey Newtonian mechanics between collisions.
气体由大量微小的粒子组成，它们持续进行无规则运动。该模型把这些粒子视为点状质量，在碰撞之间遵循牛顿力学。
Macroscopic quantities such as pressure, temperature and internal energy emerge from the average behaviour of these particles. The symbol ⟨c²⟩ denotes the mean square speed of the molecules.
压强、温度和内能等宏观量来源于这些粒子的平均行为。符号 ⟨c²⟩ 表示分子的方均速率。
2. Core assumptions of an ideal gas | 理想气体的核心假设
The kinetic theory is built on a set of simplifying assumptions. You must be able to recall them precisely because many exam questions start from these statements.
动力学理论建立在一组简化假设之上。你必须能够准确复述这些假设，因为许多考题都从这些表述出发。
- The gas contains a very large number of identical molecules moving in random directions.
气体包含大量相同的分子，沿随机方向运动。 - The volume of the molecules is negligible compared with the volume of the container.
与容器体积相比，分子本身体积可以忽略。 - Collisions between molecules and with the walls are perfectly elastic.
分子之间以及分子与器壁的碰撞是完全弹性的。 - The time of a collision is negligible compared with the time between collisions.
碰撞持续时间与两次碰撞之间的时间相比可以忽略。 - There are no intermolecular forces except during collisions.
除碰撞瞬间外，分子之间不存在作用力。 - Newtonian mechanics applies to the motion of the molecules.
牛顿力学适用于分子的运动。
3. Pressure arises from collisions | 压强来源于分子碰撞
When a molecule collides with a wall, its momentum changes. By Newton’s second law, this change requires a force. The average force per unit area over many collisions is the gas pressure.
当分子与器壁碰撞时，其动量发生改变。根据牛顿第二定律，这种变化需要力。大量碰撞产生的单位面积平均力就是气体压强。
The pressure is steady because the number of collisions per second is huge, so random fluctuations average out to give a constant macroscopic value.
压强是稳定的，因为每秒碰撞次数极大，随机涨落被平均掉，从而给出恒定的宏观值。
4. Deriving the pressure equation | 推导压强方程
Consider N molecules in a cube of side L. Focus on one molecule moving with an x-component of velocity u. Its change in momentum when it hits a wall is 2 m u.
考虑边长为 L 的立方体中有 N 个分子。研究一个 x 方向分速度为 u 的分子。它撞击器壁时动量变化为 2 m u。
The time between successive hits on the same wall is 2 L / u, so the average force from one molecule is m u² / L.
同一器壁相邻两次碰撞的时间间隔为 2 L / u，因此单个分子产生的平均力为 m u² / L。
For a large number of molecules moving randomly, the mean value of u² is one third of the mean square speed ⟨c²⟩. The total force is therefore N m ⟨c²⟩ / (3 L).
对于大量无规则运动的分子，u² 的平均值是方均速率 ⟨c²⟩ 的三分之一。因此总力为 N m ⟨c²⟩ / (3 L)。
Dividing by the wall area L² gives pressure p = N m ⟨c²⟩ / (3 L³). Since L³ = V, we obtain the key equation.
除以器壁面积 L² 得到压强 p = N m ⟨c²⟩ / (3 L³)。因为 L³ = V，我们得到关键方程。
pV = ⅓ N m ⟨c²⟩
This equation links the macroscopic quantities p and V to the microscopic quantities N, m and ⟨c²⟩.
该方程把宏观量 p、V 与微观量 N、m 和 ⟨c²⟩ 联系起来。
5. Alternative form using density | 用密度表示的压强形式
Since density ρ = total mass / volume = N m / V, the pressure equation can be written as p = ⅓ ρ ⟨c²⟩.
因为密度 ρ = 总质量 / 体积 = N m / V，压强方程可写成 p = ⅓ ρ ⟨c²⟩。
This form is useful when the number of molecules is not known, but the density of the gas is known. It shows that pressure depends on density and mean square speed.
当分子数量未知但气体密度已知时，这种形式很有用。它表明压强取决于密度和方均速率。
6. Linking temperature to kinetic energy | 温度与动能的联系
Comparing pV = ⅓ N m ⟨c²⟩ with the ideal gas equation pV = N k T gives ⅓ N m ⟨c²⟩ = N k T, where k is the Boltzmann constant.
将 pV = ⅓ N m ⟨c²⟩ 与理想气体状态方程 pV = N k T 比较，得到 ⅓ N m ⟨c²⟩ = N k T，其中 k 为玻尔兹曼常数。
Rearranging shows that the mean translational kinetic energy of a molecule is proportional to the absolute temperature.
整理后表明分子的平均平动动能与热力学温度成正比。
(1/2) m ⟨c²⟩ = (3/2) k T
Temperature is therefore a measure of the average random kinetic energy of gas molecules. If the kelvin temperature doubles, the mean kinetic energy per molecule doubles.
因此温度是气体分子平均无规则动能的量度。如果开尔文温度加倍，每个分子的平均动能也加倍。
7. Root-mean-square speed | 方均根速率
The root-mean-square speed c_rms is the square root of the mean square speed: c_rms = √⟨c²⟩. It is a useful average speed for kinetic calculations.
方均根速率 c_rms 是方均速率的平方根：c_rms = √⟨c²⟩。它是动力学计算中常用的平均速率。
From (1/2) m c_rms² = (3/2) k T, we get c_rms = √(3 k T / m).
由 (1/2) m c_rms² = (3/2) k T，可得 c_rms = √(3 k T / m)。
For one mole, using m = M / N_A and k = R / N_A gives c_rms = √(3 R T / M), where M is the molar mass and R is the molar gas constant.
对于 1 mol，利用 m = M / N_A 和 k = R / N_A，可得 c_rms = √(3 R T / M)，其中 M 为摩尔质量，R 为摩尔气体常数。
c_rms = √(3 k T / m) = √(3 R T / M)
The rms speed increases with temperature and is smaller for heavier molecules at the same temperature.
方均根速率随温度升高而增大；在相同温度下，较重分子的方均根速率较小。
8. Internal energy of an ideal gas | 理想气体的内能
For a monatomic ideal gas, the only internal energy is the translational kinetic energy of the atoms. The total internal energy U is the sum of the kinetic energies of all N atoms.
对于单原子理想气体，内能只是原子的平动动能。总内能 U 是所有 N 个原子动能之和。
U = N × (1/2) m ⟨c²⟩. Substituting (1/2) m ⟨c²⟩ = (3/2) k T gives U = (3/2) N k T.
U = N × (1/2) m ⟨c²⟩。代入 (1/2) m ⟨c²⟩ = (3/2) k T，得 U = (3/2) N k T。
Using N = n N_A and k = R / N_A, this becomes U = (3/2) n R T.
利用 N = n N_A 和 k = R / N_A，可写成 U = (3/2) n R T。
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导