📚 Combined Gas Law | 组合气体定律
The combined gas law is a fundamental relationship in thermal physics that unifies Boyle's law, Charles's law, and Gay‑Lussac's law into a single equation. It describes how the pressure, volume, and absolute temperature of a fixed mass of an ideal gas are interrelated. For any A‑Level Edexcel physics student, understanding this law is essential for tackling problems involving changing gas conditions and for building a bridge toward the ideal gas equation.
组合气体定律是热物理中一个基本的关系式，它将波义耳定律、查理定律和盖‑吕萨克定律统一为一个方程。它描述了固定质量理想气体的压强、体积和绝对温度之间如何相互关联。对于任何 Edexcel A‑Level 物理学生而言，理解该定律对于解决涉及气体状态变化的问题以及为掌握理想气体状态方程铺平道路都至关重要。
1. What is the Combined Gas Law? | 什么是组合气体定律？
tant. Written as an equation, it is (p × V) / T = constant, or more usefully, p₁V₁ / T₁ = p₂V₂ / T₂, where subscripts 1 and 2 refer to two different sets of conditions. All temperatures must be measured in kelvin for the relationship to hold.
组合气体定律指出，对于固定质量的理想气体，压强与体积的乘积除以热力学温度保持不变。写成等式为 (p × V) / T = 常数，或者更实用的形式为 p₁V₁ / T₁ = p₂V₂ / T₂，其中下标 1 和 2 代表两组不同的状态。所有温度都必须用开尔文表示，该关系式才能成立。
This law is powerful because it allows you to calculate how a gas will respond when two variables change simultaneously, something none of the three individual laws can handle on their own. For example, if you heat a gas in a sealed syringe and simultaneously allow the volume to change, both p and V can alter, and the combined law will link the before and after states.
该定律之所以强大，是因为它能让你计算气体在两个变量同时改变时的响应，而三个独立定律中没有一个能独自处理这种情况。例如，如果你在密封的注射器中加热气体并同时允许体积变化，那么压强和体积都会改变，组合定律就能将前后状态联系起来。
2. The Three Individual Gas Laws | 三个独立的气体定律
Boyle's law emperature, the volume of a fixed mass of gas is inversely proportional to its pressure: p ∝ ere T is held constant. This law explains why a balloon expands when the external pressure decreases.
波义耳定律指出，在温度不变的条件下，固定质量气体的体积与其压强成反比：p ∝ 1/V。等式形式为 p₁V₁ = p₂V₂，其中温度 T 保持不变。这一定律解释了为什么气球在外界压力降低时会膨胀。
Charles's law describes the direct proportionality between volume and absolute temperature at constant pressure: V ∝ T. It is expressed as V₁ / T₁ = V₂ / T₂, with p constant. Hot‑ is principle: heating the air increases its volume, reducing density so the balloon rises.
查理定律描述的是在压强不变的情况下，体积与热力学温度成正比：V ∝ T。其表达式为 V₁ / T₁ = V₂ / T₂，其中 p 保持不变。热气球就体现了这一原理：加热空气使体积增大，密度减小，气球得以升空。
Gay‑Lussac's law n called the pressure law) relates pressure and absolute temperature at constant volume: p ∝ T, giving p₁ / T₁ = p₂ / T₂. This is why aerosol cans carry warnings not to heat them — a rise in temperature leads to a dangerous increase in pressure when volume is fixed.
盖‑吕萨克定律（常称为压强定律）将恒定体积下的压强和热力学温度联系起来：p ∝ T，即 p₁ / T₁ = p₂ / T₂。这就是气雾罐上警告不可加热的原因——当体积固定时，温度升高会导致危险的压强增大。
3. Deriving the Combined Gas Law | 组合气体定律的推导
to state 2 (p₂, V₂, T₂) via an intermediate state. First, change the temperature from T₁ to T₂ while keeping volume constant; the pressure becomes p′ = p₁ × (T₂ / T₁). Second, change the volume from V₁ to V₂ while keeping temperature constant at T₂; applying Boyle's law gives p₂ = p′ × (V₁ / V₂).
当我们允许所有三个变量都发生变化时，组合气体定律就会自然产生。假设一个固定质量的气体通过一个中间状态从状态 1 (p₁, V₁, T₁) 过渡到状态 2 (p₂, V₂, T₂)。首先，在体积不变的情况下将温度从 T₁ 变为 T₂；压强变为 p′ = p₁ × (T₂ / T₁)。其次，在温度保持为 T₂ 不变的条件下将体积从 V₁ 变为 V₂；应用波义耳定律得到 p₂ = p′ × (V₁ / V₂)。
s that the ratio pV/T is invariant for a given amount of gas, provided it behaves ideally.
将第一步的 p′ 代入第二步，得到 p₂ = p₁ × (T₂ / T₁) × (V₁ / V₂)。重新整理后，就得到 p₁V₁ / T₁ = p₂V₂ / T₂。这表明，对于一定量的理想气体，pV/T 的比值是一个不变量。
4. The Equation and Its Meaning | 方程及含义
be in pascals, atmospheres, or any consistent unit, provided both sides use the same. Volume may be in m³, dm³, or litres, again consistently. Temperature must always be in kelvin (K). If a gas starts at 27 °C, you must convert to 300 K before substituting.
计算中使用的标准形式是 p₁V₁ / T₁ = p₂V₂ / T₂。压强单位可用帕斯卡、大气压或任何一致的单位，只要等式两边使用相同单位即可。体积可用 m³、dm³ 或升，同样需要统一。温度必须始终使用开尔文 (K)。如果气体的初始温度是 27 °C，必须转换为 300 K 后才能代入。
The equation reveals an intuitive truth: if you increase the temperature of a gas while letting it expand freely, the pressure might not rise as much as it would in a rigid container. Conversely, if you compress a gas, it heats up unless heat is removed. The law quantifies these everyday observations.
这个方程揭示了一个直观的事实：如果你在让气体自由膨胀的同时升高温度，其压强可能不会像在刚性容器中那样大幅上升。相反，如果你压缩气体，除非移走热量，否则它会升温。该定律将日常观察到的这些现象定量化。
5. Conditions for Application | 应用条件
For the combined gas law to be valid, several conditions must be met. The mass of gas must remain fixed — no gas escapes or enters the system. The gas must behave as an ideal gas, meaning intermolecular forces are negligible and the volume of the gas particles themselves is insignificant compared with the container volume.
要使组合气体定律成立，必须满足几个条件。气体的质量必须保持不变——没有气体逸出或进入系统。气体必须表现为理想气体，这意味着分子间作用力可以忽略不计，并且气体粒子本身的体积与容器体积相比微不足道。
In A‑Level problems, you typically assume ideal behaviour unless told otherwise. Additionally, the law applies to closed systems. If a vessel leaks or a piston moves in such a way that gas escapes, the fixed‑mass assumption breaks down and the law cannot be directly used.
在 A‑Level 问题中，除非特别说明，通常假设气体为理想气体。此外，该定律适用于封闭系统。如果容器泄漏或活塞移动导致气体逸出，固定质量的假设就不成立了，因而不能直接使用该定律。
6. Using the Combined Gas Law with Moles | 与摩尔一起使用
tended into the full ideal gas equation: pV = nRT, where R is the molar gas constant (8.31 J mol⁻¹ able when dealing with chemical amounts of gas, for example in calculating the volume of 1 mol of gas at room temperature and pressure.
当已知气体的摩尔数 n 时，组合气体定律可以扩展为完整的理想气体状态方程：pV = nRT，其中 R 是摩尔气体常数 (8.31 J mol⁻¹ K⁻¹)。由 pV/T = 常数出发，引入摩尔标度就得到 pV/T = nR，即 pV = nRT。在处理化学量的气体时，例如计算 1 mol 气体在室温和常压下的体积，这一形式是不可或缺的。
w the mass of a gas and its molecular mass, you can first find n, then use pV = nRT to solve for an unknown variable.
Edexcel 的考题经常会要求你通过阿伏伽德罗常数的概念，将组合气体定律与理想气体状态方程联系起来。如果你知道一种气体的质量及其分子量，就可以先求出 n，然后用 pV = nRT 解出未知量。
7. Worked Example 1 | 典型例题 1
A balloon contains 2.5 × 10⁻³ m³ of helium at 20 °C and 101 kPa. It is taken outdoors where the temperature drops to 5 °C and the pressure rises to 105 kPa. Calculate the new volume of the balloon.
一个气球装有 2.5 × 10⁻³ m³ 的氦气，温度为 20 °C，压强为 101 kPa。将它带到室外，此时气温降至 5 °C 且压强升至 105 kPa。计算气球的新体积。
Solution: Convert temperatures to kelvin: T₁ = 20 + 273 = 293 K, T₂ = 5 + 273 = 278 05 × 10³) × V₂ / 278. Rearranging gives V₂ = (101 × 2.5 × 10⁻³ × 278) / (293 × 105) ≈ 2.28 × 10⁻³ lloon shrinks slightly because the temperature drop dominates the small pressure increase.
解答：将温度转换为开尔文：T₁ = 20 + 273 = 293 K，T₂ = 5 + 273 = 278 K。应用 p₁V₁ / T₁ = p₂V₂ / T₂ → (101 × 10³) × (2.5 × 10⁻³) / 293 = (105 × 10³) × V₂ / 278。整理得 V₂ = (101 × 2.5 × 10⁻³ × 278) / (293 × 105) ≈ 2.28 × 10⁻³ m³。气球的体积略微缩小，因为温度下降的影响超过了小幅压强增大的影响。
8. Worked Example 2 | 典型例题 2
An ideal gas occupies 500 cm³ at 100 kPa and 27 °C. If the gas is compressed to 200 cm³ while being heated to 127 °C, what is the final pressure?
一种理想气体在 100 kPa 和 27 °C 下占据 500 cm³ 的体积。若将该气体压缩至 200 cm³ 并同时加热至 127 °C，最终压强是多少？
Solution: T₁ = 300 K, T₂ = 400 K. Using p₁V₁/T₁ = p₂V₂/T₂, we get p₂ = p₁ × (V₁/V₂) × (T₂/T₁) = 100 kPa × (500/200) × (400/300) = 100 × 2.5 × 1.333 ≈ 333 pressure more than triples because both compression and heating act together to raise it.
解答：T₁ = 300 K，T₂ = 400 K。运用 p₁V₁/T₁ = p₂V₂/T₂，得到 p₂ = p₁ × (V₁/V₂) × (T₂/T₁) = 100 kPa × (500/200) × (400/300) = 100 × 2.5 × 1.333 ≈ 333 kPa。由于压缩和加热共同作用，压强增大为原来的三倍多。
9. Common Mistakes | 常见错误
The most frequent error is forgetting to convert Celsius temperatures to kelvin. Using 27 °C instead of 300 K leads to nonsensical results. Always add 273 to a Celsius reading unless the problem already gives values in K. Another mistake is using inconsistent pressure units, such as mixing kPa and Pa without conversion.
最常见的错误是忘记将摄氏温度转换为开尔文。使用 27 °C 而非 300 K 会导致荒谬的结果。除非题目已给出开尔文数值，否则总是将摄氏温度读数加上 273。另一个错误是使用的压强单位不一致，比如混合使用 kPa 和 Pa 而不进行换算。
Students also sometimes confuse the subscripts or misidentify which state is '1' and which is '2'. It helps to tabulate the initial and final values clearly before substituting into the equation. Finally, failing to check whether the fixed‑mass approximation is valid can cause errors when the system is actually open.
学生们有时还会搞混下标，或者分不清哪一个状态是“1”、哪一个状态是“2”。在代入方程之前，清晰地将初始值和最终值列表有助于避免此类错误。此外，当系统实际上是开放系统时，没有检验固定质量近似是否有效也会导致错误。
10. Real Gases vs Ideal Gases | 真实气体与理想气体
The combined gas law assumes perfect ideal‑gas behaviour. However, real gases deviate from this law at high pressures and low temperatures. Under high pressure, gas particles are forced close together, making their own volume non‑negligible. At low temperatures, intermolecular attractive forces become significant, reducing the pressure below the ideal prediction.
组合气体定律假设气体具有完美的理想行为。然而，真实气体在高压和低温下会偏离这一规律。在高压下，气体粒子被挤压得很近，因此它们自身的体积变得不可忽略。在低温下，分子间的吸引力变得显著，使得实际压强低于理想预测值。
To account for these deviations, the van der Waals equation introduces correction terms. Although the detail is beyond the scope of A‑Level Edexcel, you should recognise that the combined gas law is an excellent approximation for many everyday conditions, such as air at room temperature and atmospheric pressure.
为了处理这些偏差，范德瓦尔斯方程引入了修正项。尽管其详细内容超出了 Edexcel A‑Level 的范围，但你应该认识到，组合气体定律对于许多日常条件（例如室温和大气压下的空气）来说是一个很好的近似。
11. Exam Tips and Tricks | 考试技巧
When tackling Edexcel exam questions, start by listing the known quantities with symbols: p₁, V₁, T₁, p₂, V₂, T₂. Convert temperatures to kelvin immediately. Rearrange the equation to solve for the unknown before plugging in numbers — this minimises calculation slips. Leave units in the expression to ensure they cancel appropriately.
在解答 Edexcel 考试题目时，首先用符号列出已知量：p₁、V₁、T₁、p₂、V₂、T₂。立即将温度转换为开尔文。在代入数字之前先对方程进行整理以解出未知量——这能最大程度地减少计算失误。在表达式中保留单位，以确保它们能适当地消去。
proach via the combined gas law if you treat n as part of the constant. Pay close attention to wording: 'sealed container' implies fixed mass, 'rigid' means constant volume, and 'isothermal' means constant temperature.
如果题目给出了摩尔数，通常直接使用 pV = nRT 会更快，但你仍然可以通过组合气体定律来求解，只需将 n 视为常数的一部分。请密切留意措辞：“sealed container”（密封容器）意味着质量固定，“rigid”（刚性）意味着体积不变，而“isothermal”（等温）意味着温度恒定。
12. Summary | 总结
The combined gas law elegantly unifies Boyle's, Charles's, and Gay‑Lussac' al gas, provided temperature is in kelvin. Its extension to pV = nRT links thermal physics and the mole concept, making it one of the most versatile equations in A‑Level physics.
组合气体定律将波义耳定律、查理定律和盖‑吕萨克定律优雅地统一成了一个强大的工具，用以预测变化条件下的气体行为。对于固定质量的理想气体，只要温度以开尔文为单位，就有 p₁V₁/T₁ = p₂V₂/T₂。将其扩展为 pV = nRT 后，便将热物理与摩尔概念联系了起来，使其成为 A‑Level 物理中用途最广泛的方程之一。
By practising systematic conversions and learning to spot the hallmark signs of each sub‑law, you can confidently solve multi‑step problems involving gases. Remember that real gases only approximate this behaviour, but for most A‑Level contexts the ideal assumption is entirely reliable.
通过练习系统的单位转换并学会识别各个子定律的标志性特征，你就能自信地解答涉及气体的多步骤问题。请记住，真实气体仅是近似地遵循这一行为，但对绝大多数 A‑Level 考题而言，理想气体的假设是完全可靠的。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图