📚 Particles of a Gas | 气体粒子
In A-Level Physics, understanding the behaviour of gas particles is fundamental to thermodynamics and kinetic theory. A gas consists of a vast number of particles in constant, random motion, and their collective behaviour explains macroscopic properties such as pressure, temperature, and volume. This article explores the kinetic model of an ideal gas, the derivation of the pressure equation, the link between temperature and kinetic energy, the distribution of molecular speeds, and deviations from ideal behaviour.
在A-Level物理中，理解气体粒子的行为是热力学和动理论的基础。气体由大量不断做无规则运动的粒子组成，它们的集体行为解释了压强、温度和体积等宏观性质。本文探讨理想气体的动理论模型、压强方程的推导、温度与动能的联系、分子速率的分布以及偏离理想行为的情况。
1. The Nature of Gas Particles and the Ideal Gas Model | 气体粒子的本质与理想气体模型
Gas particles are atoms or molecules in constant, rapid, random motion. They have negligible volume compared with the empty space between them, and they exert almost no forces on one another except during brief collisions. This microscopic picture explains why a gas expands to fill its container completely.
气体粒子是不断做快速、无规则运动的原子或分子。与它们之间的空隙相比，粒子本身的体积可忽略不计；除短暂碰撞瞬间外，粒子之间几乎没有作用力。这一微观图像解释了气体为什么会完全充满其容器。
To make quantitative predictions, physicists use the ideal gas model. In this model, particles are treated as point masses that interact only through perfectly elastic collisions. Although no real gas is perfectly ideal, many gases at low pressure and high temperature follow ideal behaviour very closely, making the model extremely useful.
为进行定量预测，物理学家使用理想气体模型。在该模型中，粒子被视为质点，仅通过完全弹性碰撞发生相互作用。尽管没有真实气体是完全理想的，但许多气体在低压和高温下非常接近理想行为，使得该模型极为有用。
2. Assumptions of Kinetic Theory | 气体动理论的基本假设
To develop the kinetic theory of gases, we make several key simplifying assumptions about the particles:
为建立气体动理论，我们对粒子作出以下几条关键的简化假设：
- The gas consists of a very large number of identical, tiny particles (atoms or molecules).
- 气体由数量极多的相同微小粒子（原子或分子）组成。
- The volume of the particles themselves is negligible compared with the total volume of the gas.
- 粒子本身的体积与气体总体积相比可以忽略。
- There are no forces between particles except during collisions.
- 除碰撞瞬间外，粒子之间无相互作用力。
- All collisions between particles and with the walls of the container are perfectly elastic (kinetic energy is conserved).
- 所有碰撞（粒子之间以及粒子与容器壁之间）都是完全弹性碰撞（动能守恒）。
- Particles obey Newton's laws of motion.
- 粒子遵循牛顿运动定律。
These assumptions hold best for monatomic gases at low pressure and high temperature, where interactions are minimal.
这些假设在低压和高温的单原子气体中最接近实际情况，因为此时粒子间相互作用极弱。
3. How Gas Pressure Arises | 气体压强如何产生
The pressure exerted by a gas on the walls of its container is the result of countless collisions of particles with the walls. Each collision transfers momentum to the wall, and the average force per unit area over time gives the pressure.
气体对容器壁施加的压强，是大量粒子与壁面碰撞的结果。每次碰撞都将动量传递给壁面，一段时间内单位面积上的平均作用力即为压强。
When a particle of mass m hits a wall perpendicularly and bounces back with the same speed, its momentum changes by 2mv. This change creates an impulse on the wall. With billions of impacts per second, the wall experiences a steady, macroscopic force.
当一个质量为m的粒子垂直撞击壁面并以相同速率弹回时，其动量变化为2mv。这一变化对壁面产生冲量。每秒发生数以亿计的碰撞，壁面便受到持续稳定的宏观力。
4. Deriving the Pressure Equation | 推导气体压强方程
We consider a single particle moving with velocity components vx, vy, vz inside a cubic box of side L. A collision with the wall perpendicular to the x‑axis reverses the x‑component of velocity, giving a momentum change of 2mvx.
我们考虑一个粒子在边长为L的立方盒中运动，其速度分量为vx、vy、vz。与垂直于x轴的壁面碰撞会反向改变x方向速度分量，动量变化为2mvx。
The time between collisions with that wall is 2L/vx. The average force on that wall due to one particle is therefore mvx²/L. Summing over all N particles and averaging over all directions, we use the relationship ⟨v²⟩ = ⟨vx²⟩ + ⟨vy²⟩ + ⟨vz²⟩ = 3⟨vx²⟩. The final result is:
与同一壁面两次碰撞的时间间隔为2L/vx。因此一个粒子对该壁面的平均作用力为mvx²/L。对所有N个粒子求和并对所有方向取平均，利用 ⟨v²⟩ = ⟨vx²⟩ + ⟨vy²⟩ + ⟨vz²⟩ = 3⟨vx²⟩，最终得到：
pV = ⅓ Nm⟨c²⟩
where ⟨c²⟩ is the mean square speed of the particles. Since Nm/V is the density ρ, we can also write p = ⅓ ρ⟨c²⟩.
其中⟨c²⟩表示粒子方均速率的平均值。由于Nm/V即密度ρ，方程也可写作p = ⅓ ρ⟨c²⟩。
5. Connecting Pressure, Volume, and Temperature | 连接压强、体积和温度
From experiments, an ideal gas obeys pV = nRT, where n is the number of moles and R is the molar gas constant. Alternatively, using the number of particles N and Boltzmann's constant k, we have pV = NkT.
通过实验可知，理想气体满足pV = nRT，其中n为摩尔数，R为摩尔气体常数。也可用粒子数N和玻尔兹曼常量k表示为pV = NkT。
Equating the kinetic theory expression pV = ⅓ Nm⟨c²⟩ with pV = NkT we obtain:
½ m⟨c²⟩ = (3/2) kT
This is a profound result: the average translational kinetic energy of a gas particle is directly proportional to the absolute temperature. Temperature is thus a measure of the microscopic kinetic energy.
这是一个深刻的结果：气体粒子的平均平动动能与绝对温度成正比。因此温度是微观动能的量度。
6. Kinetic Energy and Absolute Temperature | 动能与绝对温度
The equation ½ m⟨c²⟩ hows that if the temperature becomes zero kelvin, the average kinetic energy would be zero – al motion. This defines the physical meaning of absolute zero and underpins the Kelvin scale.
方程 ½ m⟨c²⟩ = (3/2) kT 表明，如果温度达到零开尔文，平均动能将为零——粒子不再有任何平动。这定义了绝对零度的物理意义，也是开氏温标的基础。
Note that for a mixture of different gases at the same temperature, heavier molecules have a smaller mean square speed, so the average kinetic energy remains the same for all species. This equality leads to important concepts such as diffusion and effusion rates.
需要注意的是，对于处于同一温度下的混合气体，质量较大的分子方均速率较小，但所有种类粒子的平均动能相同。这种相等关系引出了扩散和隙流速率等重要概念。
7. The Maxwell–Boltzmann Speed Distribution | 麦克斯韦–玻尔兹曼速率分布
Not all particles in a gas move at the same speed. The distribution of molecular speeds is described by the Maxwell–Boltzmann distribution. The curve starts at zero, rises to a peak at the most probable speed vmp, and then tails off at higher speeds.
气体中并非所有粒子都以相同速率运动。分子速率的分布由麦克斯韦–玻尔兹曼分布描述。曲线从零开始上升，在最概然速率vmp处达到峰值，然后向高速端拖尾。
Key features of the distribution:
- The area under the curve represents the total number of particles.
- 曲线下面积代表粒子总数。
- The peak shifts to higher speeds and flattens when temperature increases.
- 温度升高时，峰向较高速度方向移动且曲线变得更平坦。
- Lighter gases (smaller molar mass) have wider distributions with higher average speeds at the same temperature.
- 相同温度下，较轻的气体（摩尔质量较小）分布更宽，平均速率更大。
Three characteristic speeds are often compared: most probable speed vmp < mean speed ⟨v⟩ < root‑mean‑square speed vrms. Their ratios are approximately 1 : 1.13 : 1.22.
常比较三种特征速率：最概然速率vmp < 平均速率⟨v⟩ < 方均根速率vrms。它们之比约为 1 : 1.13 : 1.22。
8. Understanding Root-Mean-Square (rms) Speed | 理解方均根速率
The root-mean-square speed crms is defined as the square root of the mean square speed: crms = √⟨c²⟩ om the kinetic theory, it can be expressed directly in terms of temperature and molar mass M:
crms = √3RT/M
方均根速率crms定义为平均平方速率的平方根：crms = √⟨c²⟩。根据动理论，它可直接用温度和摩尔质量M表示为：
crms = √3RT/M
032 kg mol⁻¹) it is roughly 480 m s⁻¹. This equation is frequently tested in calculations.
例如，在300 K时，氦原子（M = 4.0 × 10⁻³ kg mol⁻¹）的方均根速率约为 1370 m s⁻¹，而氧分子（M = 0.032 kg mol⁻¹）约为 480 m s⁻¹。此公式经常在计算题中考查。
9. Internal Energy of an Ideal Gas | 理想气体的内能
For a monatomic ideal gas, the only form of energy is the translational kinetic energy of the particles. The total internal energy U is therefore:
U = (3/2) nRT
对于单原子理想气体，其唯一能量形式是粒子的平动动能，因此总内能U为：
U = (3/2) nRT
The internal energy depends only on temperature and not on pressure or volume. This is a consequence of the absence of intermolecular forces in the ideal gas model. For diatomic or polyatomic gases, rotational and vibrational energies contribute additional terms, but the temperature proportionality remains fundamental.
内能仅取决于温度，与压强或体积无关。这是理想气体模型中不存在分子间作用力的结果。对于双原子或多原子气体，转动能和振动能会额外贡献项，但与温度成正比的特性仍是基本规律。
10. When Gases Behave Non-Ideally | 气体偏离理想行为时
Real gases deviate from the ideal gas law, especially at high pressure and low temperature. Under these conditions, two neglected assumptions become significant: particles have a finite volume, and intermolecular forces (van der Waals forces) cannot be ignored.
真实气体会偏离理想气体定律，尤其在高压和低温下。此时，两个被忽略的假设变得重要：粒子具有有限体积，且分子间作用力（范德瓦耳斯力）不可忽略。
| Condition | Ideal Gas Behaviour | Real Gas Behaviour |
|---|---|---|
| Low pressure / high temperature | pV = nRT closely followed | Approaches ideal behaviour |
广告 / Sponsored Cambridge International AS and A Level Physics Coursebook … 物理 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥584.0
更多咨询请联系16621398022（同微信） 备考资讯每日更新，关注不错过 考纲变化 · 科目分析 · 学习方法  在微信里长按二维码识别关注 或截图后用「扫一扫 → 相册」选择本图 公众号：tutorhao |