Combined Gas Law | 综合气体定律

📚 Combined Gas Law | 综合气体定律

The combined gas law is a fundamental principle in chemistry that integrates Boyle's Law, Charles's Law, and Gay-Lussac's Law into a single equation. It describes the relationship between pressure, volume, and temperature of a fixed amount of gas when the amount of gas remains constant. This article provides a clear, exam-focused revision guide for Edexcel A-Level Chemistry.

综合气体定律是化学中的基本定律，它将玻意耳定律、查理定律和盖-吕萨克定律整合为单一方程。它描述了在气体物质的量不变时，压强、体积和温度之间的关系。本文为 Edexcel A-Level 化学提供清晰、紧扣考点的复习指南。

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1. Definition of the Combined Gas Law | 综合气体定律的定义

The combined gas law states that for a fixed amount of gas, the ratio of the product of pressure and volume to the absolute temperature is a constant. Mathematically, it is expressed as:

综合气体定律指出：对于一定量的气体，压强与体积的乘积除以绝对温度所得的比值是一个常量。其数学表达式为：

P₁V₁ / T₁ = P₂V₂ / T₂

where P is pressure, V is volume, and T is the absolute temperature (in kelvins). This equation allows us to calculate changes in one variable when the other two change, provided the amount of gas does not change.

其中 P 是压强，V 是体积，T 是绝对温度（单位 K）。该方程允许我们在气体物质的量不变时，根据其中两个变量的变化计算第三个变量。

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2. Derivation from the Ideal Gas Equation | 从理想气体方程推导

re n is the amount of gas and R is the molar gas constant. For a fixed amount of gas, n and R are constant, so pV/T is also constant:

理想气体方程为 pV=nRT，其中 n 是气体物质的量，R 是摩尔气体常量。对于定量气体，n 和 R 均为常量，因此 pV/T 也为常量：

pV/T = nR = constant

Thus, for two different states of the same gas sample, we can write p₁V₁/T₁ = p₂V₂/T₂. This is exactly the combined gas law.

因此，对于同一样品气体的两个不同状态，我们可以写出 p₁V₁/T₁ = p₂V₂/T₂。这就是综合气体定律。

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3. Relationship with Individual Gas Laws | 与各分定律的关系

The combined gas law is a generalisation of three simpler laws:

综合气体定律是三个更简单定律的推广：

• Boyle’s Law (T constant): p₁V₁ = p₂V₂ — pressure inversely proportional to volume.
• Charles’s Law (p constant): V₁/T₁ = V₂/T₂ — volume directly proportional to absolute temperature.
• Gay-Lussac’s Law (V constant): p₁/T₁ = p₂/T₂ — pressure directly proportional to absolute temperature.

• 玻意耳定律（温度恒定）： p₁V₁ = p₂V₂ —— 压强与体积成反比。
• 查理定律（压强恒定）： V₁/T₁ = V₂/T₂ —— 体积与绝对温度成正比。
• 盖-吕萨克定律（体积恒定）： p₁/T₁ = p₂/T₂ —— 压强与绝对温度成正比。

When one condition is held constant, the combined gas law simplifies to the corresponding individual law. This is a frequent exam question asking you to identify which law applies.

当某一条件恒定时，综合气体定律就简化为对应的分定律。考试中常要求判断何时适用哪个定律。

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4. Standard Conditions and Units | 标准状况与单位

In A-Level chemistry, you must always convert temperature to kelvin (K). The conversion is:

在 A-Level 化学中，温度必须换算为开尔文（K）。换算关系为：

T(K) = T(°C) + 273.15

id. However, when using the ideal gas equation pV = nRT, you must use consistent SI units.

压强的单位可以是帕（Pa）、千帕（kPa）或标准大气压（atm）。体积的单位可以是 m³、dm³ 或 cm³。只要方程两边使用相同的单位，比值就保持不变。但在使用理想气体方程 pV=nRT 时，必须使用统一的国际单位制单位。

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5. Step-by-Step Problem Solving | 解题步骤

To solve combined gas law problems, follow these steps:

解决综合气体定律问题的步骤如下：

1. Identify the known and unknown variables (P, V, T).
2. Convert all temperatures to kelvins.
3. Ensure pressure and volume units are consistent on both sides.
4. Rearrange the equation P₁V₁/T₁ = P₂V₂/T₂ to solve for the unknown.
5. Substitute values and calculate, including units.

1. 确定已知量和未知量（P、V、T）。
2. 将所有温度换算为开尔文。
3. 确保方程两边的压强和体积单位一致。
4. 移项整理方程 P₁V₁/T₁ = P₂V₂/T₂，解出未知量。
5. 代入数值并计算，注意单位。

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6. Worked Example | 例题解析

Example: A gas occupies 250 cm³ at 20 °C and 100 kPa. What is its volume at 0 °C and 120 kPa?

例题：某气体在 20 °C、100 kPa 时体积为 250 cm³。在 0 °C、120 kPa 时，它的体积是多少？

Solution:

解答：

First, convert temperatures: T₁ = 20 + 273.15 = 293.15 K, T₂ = 0 + 273.15 = 273.15 K.

首先换算温度：T₁=20+273.15=293.15 K，T₂=0+273.15=273.15 K。

Use the combined gas law:

使用综合气体定律：

V₂ = P₁V₁T₂ / (P₂T₁)

Substituting the values:

代入数值：

V₂ = (100 kPa × 250 cm³ × 273.15 K) / (120 kPa × 293.15 K)

V₂ ≈ 194 cm³

The final volume is approximately 194 cm³. Always check that the units of kPa and cm³ cancel appropriately.

最终体积约为 194 cm³。务必检查 kPa 和 cm³ 单位是否正确约去。

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7. Common Mistakes | 常见错误

Students often lose marks in exams due to the following errors:

学生在考试中常因以下错误失分：

• Forgetting to convert Celsius to kelvin.
• Using different units for pressure or volume on the two sides of the equation.
• Confusing which variable is constant (e.g., using the combined law when n changes).
• Incorrectly rearranging the equation.

• 忘记将摄氏温度换算为开尔文温度。
• 方程两边使用不同的压强或体积单位。
• 混淆哪个变量恒定（例如在物质的量变化时仍使用综合定律）。
• 错误地移项。

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8. Application in Stoichiometry and Gas Density | 在化学计量和气体密度中的应用

. Additionally, gas density d can be related to the combined gas law: since d = m/V and pV ∝ T, for a fixed mass, d ∝ p/T.

综合气体定律在将气体体积从一组条件换算到另一组条件时至关重要。例如，在 Edexcel A-Level 化学中，你可能需要计算在室温常压（RTP）下反应产生的气体体积，但实际测量条件不同。此外，气体密度 d 也与综合气体定律相关：由于 d=m/V 且 pV∝T，对于固定质量的气体，d∝p/T。

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9. Connecting to the Ideal Gas Equation in Exams | 连接考试中的理想气体方程

t papers, a common question gives the same gas under different conditions and asks whether the combined gas law or the ideal gas equation is appropriate. The key is to check whether n (amount) is constant.

当气体物质的量发生变化时，不能直接使用综合气体定律，而必须使用完整的理想气体方程 pV=nRT。在 Edexcel 历年试卷中，常见题型是给出同一气体在不同条件下的数据，询问应使用综合气体定律还是理想气体方程。关键点是判断 n（物质的量）是否恒定。

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10. Revision Summary | 复习总结

The combined gas law is a powerful tool for solving gas problems with constant amount and variable P, V, and T. Always remember:

综合气体定律是解决物质的量恒定、P、V、T 变化的气体问题的有力工具。请牢记：

• Use P₁V₁/T₁ = P₂V₂/T₂ for fixed n.
• Convert all temperatures to kelvins.
• Keep units consistent.
• When n changes, return to pV = nRT.

• 当 n 恒定时，使用 P₁V₁/T₁ = P₂V₂/T₂。
• 将所有温度换算为开尔文。
• 保持单位一致。
• 当 n 变化时，回到 pV=nRT。

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