📚 The Combined Gas Law and Ideal Gas Equation | 组合气体定律与理想气体方程
For Edexcel A-Level Physics, the ideal gas laws form a core part of Topic 9: Thermodynamics. This article connects Boyle’s law, Charles’s law, the pressure law and the combined gas law, then builds up to the ideal gas equation pV = nRT and the kinetic theory model of gases.
在爱德思 A-Level 物理中，理想气体定律是第 9 单元“热力学”的核心内容。本文将联系波义耳定律、查理定律、压力定律和组合气体定律，进而推导理想气体状态方程 pV=nRT，并介绍气体的分子动能理论模型。
1. The Ideal Gas Model | 理想气体模型
An ideal gas is a theoretical model used to simplify the behaviour of real gases. In this model, particles are point masses, so their own volume is negligible compared with the volume of the container. There are no forces between particles, except during perfectly elastic collisions. Particles move rapidly and randomly, obeying Newton’s laws. The model is most accurate for real gases at low pressure and high temperature.
理想气体是一种用来简化真实气体行为的理论模型。在该模型中，粒子被看作质点，因此其自身体积相对容器体积可以忽略。粒子之间除完全弹性碰撞外不存在作用力。粒子快速、随机地运动，并服从牛顿运动定律。该模型在低压、高温下对真实气体的描述最为准确。
2. Boyle’s Law | 波义耳定律
For a fixed mass of gas at constant temperature, pressure p is inversely proportional to volume V. This gives p ∝ 1/V, or pV = constant. For a change between two states, p₁V₁ = p₂V₂. A graph of p against V is a rectangular hyperbola, while a graph of p against 1/V is a straight line through the origin.
对于温度恒定的固定质量气体，压强 p 与体积 V 成反比。因此 p ∝ 1/V，或写作 pV = 常数。对于两个状态之间的变化，有 p₁V₁ = p₂V₂。p-V 图像是一条等轴双曲线，而 p-1/V 图像是一条过原点的直线。
p₁V₁ = p₂V₂
3. Charles’s Law | 查理定律
For a fixed mass of gas at constant pressure, volume V is directly proportional to absolute temperature T in kelvin. Therefore V/T = constant, or V₁/T₁ = V₂/T₂. Extrapolating the V–T graph for any gas gives zero volume at –273.15 °C, which is absolute zero.
对于压强恒定的固定质量气体，体积 V 与热力学温度 T（单位为开尔文）成正比。因此 V/T = 常数，或 V₁/T₁ = V₂/T₂。将任何气体的 V-T 图像外推，都会在 –273.15 °C 处得到零体积，这一温度就是绝对零度。
V₁/T₁ = V₂/T₂
4. The Pressure Law | 压力定律
For a fixed mass of gas at constant volume, pressure p is directly proportional to absolute temperature T. This is expressed as p/T = constant, or p₁/T₁ = p₂/T₂. Since temperature must be in kelvin, doubling the absolute temperature doubles the pressure if the volume is fixed.
对于体积恒定的固定质量气体，压强 p 与热力学温度 T 成正比。这可表示为 p/T = 常数，或 p₁/T₁ = p₂/T₂。由于温度必须使用开尔文温标，在体积不变时，热力学温度翻倍则压强也随之翻倍。
p₁/T₁ = p₂/T₂
5. The Combined Gas Law | 组合气体定律
Combining Boyle’s law, Charles’s law and the pressure law gives pV/T = constant for a fixed mass of gas. Therefore, between two states, p₁V₁/T₁ = p₂V₂/T₂. This equation allows you to find one unknown variable when pressure, volume and temperature all change together. Remember that temperatures must be absolute temperatures in kelvin.
综合波义耳定律、查理定律和压力定律，可得到固定质量气体的关系式 pV/T = 常数。因此，在两个状态之间，有 p₁V₁/T₁ = p₂V₂/T₂。当压强、体积和温度同时变化时，可利用该方程求出未知量。请记住，温度必须使用开尔文温标。
p₁V₁/T₁ = p₂V₂/T₂
6. The Ideal Gas Equation | 理想气体状态方程
The ideal gas equation pV = nRT is more general because it includes the amount of gas n in moles. R is the molar gas constant, 8.31 J mol⁻¹ K⁻¹. Pressure p must be in pascals Pa, volume V in cubic metres m³, and temperature T in kelvin. Rearranging allows calculation of n, p, V, or T for both fixed and changing amounts of gas.
理想气体状态方程 pV = nRT 更具普遍性，因为它包含气体的物质的量 n（单位为摩尔）。R 是摩尔气体常数，其值为 8.31 J mol⁻¹ K⁻¹。压强 p 必须以帕斯卡 Pa 为单位，体积 V 以立方米 m³ 为单位，温度 T 以开尔文为单位。通过变形可以计算 n、p、V 或 T，适用于质量不变或变化的气体系统。
pV = nRT
For example, if 2.0 mol of an ideal gas is held at 300 K in 0.0499 m³, then p = nRT/V = (2.0 × 8.31 × 300) / 0.0499 ≈ 1.0 × 10⁵ Pa.
例如，将 2.0 mol 理想气体置于 300 K、体积为 0.0499 m³ 的容器中，则 p = nRT/V = (2.0 × 8.31 × 300) / 0.0499 ≈ 1.0 × 10⁵ Pa。
7. Molar Gas Volume | 摩尔气体体积
At room temperature and pressure (RTP, about 20 °C and 101 kPa), one mole of an ideal gas occupies approximately 24.0 dm³. At standard temperature and pressure (STP, 0 °C and 101 kPa), one mole occupies about 22.4 dm³. These conversions are useful for moving between moles and gas volumes.
在常温常压（RTP，约 20 °C、101 kPa）下，1 摩尔理想气体约占 24.0 dm³。在标准状况（STP，0 °C、101 kPa）下，1 摩尔气体约占 22.4 dm³。这些换算关系常用于气体物质的量与体积之间的相互转换。
| Condition / 条件 | Temperature / 温度 | Molar volume / 摩尔体积 |
|---|---|---|
| RTP / 常温常压 | 20 °C, 101 kPa | 24.0 dm³ mol⁻¹ |
| STP / 标准状况 | 0 °C, 101 kPa | 22.4 dm³ mol⁻¹ |
8. Kinetic Theory and rms Speed | 动能理论与方均根速率
Kinetic theory links macroscopic pressure to microscopic particle motion. Pressure arises from collisions of gas particles with the container walls. For N particles of mass m in a box of volume V
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