The Combined Gas Law and Ideal Gas Behaviour | 组合气体定律与理想气体行为

📚 The Combined Gas Law and Ideal Gas Behaviour | 组合气体定律与理想气体行为

For Edexcel A Level Physics, the combined gas law is a central tool for linking pressure, volume and absolute temperature in a fixed mass of ideal gas. It combines Boyle’s law, Charles’s law and the pressure law into one relationship, and it prepares you for the ideal gas equation pV = nRT, kinetic theory and the behaviour of real gases.

在 Edexcel A Level 物理中，组合气体定律是联系一定质量理想气体的压强、体积和热力学温度的核心工具。它将波义耳定律、查理定律和压强定律合并为一个关系式，并为你后续学习理想气体方程 pV = nRT、分子动理论和真实气体的行为打下基础。

1. State Variables and Absolute Temperature | 状态参量与热力学温度

The macroscopic state of a fixed mass of gas is described by three measurable quantities: pressure p, volume V and thermodynamic temperature T. Pressure is measured in pascals (Pa), volume in cubic metres (m³) and temperature in kelvin (K). You must always convert Celsius temperatures to kelvin before using any gas law.

一定质量气体的宏观状态由三个可测量物理量描述：压强 p、体积 V 和热力学温度 T。压强的单位是帕斯卡（Pa），体积的单位是立方米（m³），温度的单位是开尔文（K）。在使用任何气体定律之前，必须先将摄氏温度换算为开尔文温度。

T(K) = θ(°C) + 273.15

Absolute zero, 0 K, is the lowest possible temperature. At this temperature, particles have minimum kinetic energy, and the pressure and volume of an ideal gas would theoretically reach zero. All gas law calculations are only valid when T is expressed in kelvin.

绝对零度 0 K 是最低可能温度。在此温度下，粒子具有最小动能，理想气体的压强和体积理论上会趋于零。所有气体定律的计算只有在 T 以开尔文表示时才成立。

2. Boyle’s Law | 波义耳定律

Boyle’s law states that for a fixed mass of gas at constant temperature, the pressure p is inversely proportional to the volume V. This can be written as pV = constant, or p₁V₁ = p₂V₂ for two sets of conditions.

波义耳定律指出：对于一定质量的气体，在温度保持不变时，压强 p 与体积 V 成反比。这可以写成 pV = 常数，或者对于两组状态写成 p₁V₁ = p₂V₂。

pV = constant (at constant T)

A graph of p against V gives a hyperbola, while a graph of p against 1/V gives a straight line through the origin. If the temperature is increased, the pV constant increases and the hyperbola moves further from the origin.

p 对 V 的图像是一条双曲线，而 p 对 1/V 的图像是一条过原点的直线。如果温度升高，pV 常数增大，双曲线会远离原点。

3. Charles’s Law | 查理定律

Charles’s law states that for a fixed mass of gas at constant pressure, the volume V is directly proportional to the absolute temperature T. This is expressed as V/T = constant, or V₁/T₁ = V₂/T₂.

查理定律指出：对于一定质量的气体，在压强保持不变时，体积 V 与热力学温度 T 成正比。这可以表示为 V/T = 常数，或者 V₁/T₁ = V₂/T₂。

V/T = constant (at constant p)

A graph of volume against temperature in kelvin is a straight line through the origin. If you plot volume against Celsius temperature, the line intercepts the temperature axis at -273.15 °C, which is absolute zero. This is a useful graphical demonstration of the Kelvin scale.

体积对开尔文温度的图像是一条过原点的直线。如果以摄氏温度为横轴作图，直线与温度轴的交点为 -273.15 °C，这就是绝对零度。这是开尔文温标的一个直观图像证明。

4. Pressure Law / Gay-Lussac’s Law | 压强定律（盖-吕萨克定律）

The pressure law, also called Gay-Lussac’s law, states that for a fixed mass of gas at constant volume, the pressure p is directly proportional to the absolute temperature T. This gives p/T = constant, or p₁/T₁ = p₂/T₂.

压强定律也称为盖-吕萨克定律，指出：对于一定质量的气体，在体积保持不变时，压强 p 与热力学温度 T 成正比。因此 p/T = 常数，或者 p₁/T₁ = p₂/T₂。

p/T = constant (at constant V)

This law explains why a sealed gas container can become dangerous when heated: the increase in temperature increases the average kinetic energy of particles, so they collide with the walls more frequently and with greater force, raising the pressure.

这一定律解释了为什么密封气体容器在受热时可能变得危险：温度升高使气体粒子的平均动能增大，粒子与器壁碰撞更频繁、更有力，从而使压强升高。

5. The Combined Gas Law | 组合气体定律

The three gas laws can be combined into a single relationship for a fixed mass of ideal gas. The combined gas law is pV/T = constant, so between two states we write p₁V₁/T₁ = p₂V₂/T₂. It is particularly useful when pressure, volume and temperature all change together.

三条气体定律可以合并为一定质量理想气体的单一关系式。组合气体定律为 pV/T = 常数，因此两个状态之间可以写成 p₁V₁/T₁ = p₂V₂/T₂。当压强、体积和温度同时变化时，这个公式尤其有用。

p₁V₁/T₁ = p₂V₂/T₂

To use the combined gas law correctly, the mass of gas and the number of moles must remain constant. All temperatures must be in kelvin, and pressure and volume units must be consistent on both sides of the equation.

正确使用组合气体定律的条件是气体的质量或物质的量必须保持不变。所有温度必须以开尔文为单位，压强和体积的单位在方程两边必须一致。

6. The Ideal Gas Equation | 理想气体状态方程

For n moles of an ideal gas, the combined gas law is generalised by the ideal gas equation pV = nRT, where R is the molar gas constant, R = 8.31 J mol⁻¹ K⁻¹. This equation connects the macroscopic measurable quantities with the amount of gas in moles.

对于 n 摩尔理想气体，组合气体定律推广为理想气体状态方程 pV = nRT，其中 R 是摩尔气体常数，R = 8.31 J mol⁻¹ K⁻¹。这个方程将宏观可测量与气体的物质的量联系起来。

pV = nRT

If you know the mass m of gas and its molar mass M, the number of moles is n = m/M, so pV = (m/M)RT. The density ρ = m/V can then be written as ρ = pM/(RT). This form is useful for comparing different gases under the same conditions.

如果已知气体的质量 m 和摩尔质量 M，则物质的量为 n = m/M，因此 pV = (m/M)RT。密度 ρ = m/V 可以写成 ρ = pM/(RT)。这种形式对于比较相同条件下的不同气体很有用。

7. Kinetic Theory Model | 分子动理论模型

The kinetic theory model explains gas pressure in terms of the motion of particles. Its main assumptions are:

分子动理论模型用粒子的运动来解释气体压强。其主要假设如下：

Gas consists of a large number of identical particles in random motion.
The volume of the particles is negligible compared with the volume of the container.
There are no intermolecular forces except during collisions.
All collisions between particles and with the container walls are perfectly elastic.
The duration of collisions is negligible compared with the time between collisions.
Newtonian mechanics applies to particle motion.

气体由大量相同的、做无规则运动的粒子组成。
粒子本身的体积与容器的体积相比可以忽略不计。
除碰撞瞬间外，粒子之间没有分子间作用力。
粒子之间以及粒子与器壁之间的所有碰撞都是完全弹性的。
碰撞持续的时间与两次碰撞之间的时间相比可以忽略不计。
粒子的运动遵循牛顿力学。

8. Pressure and RMS Speed | 压强与均方根速率

Kinetic theory leads to an expression for the pressure exerted by an ideal gas: pV = ⅓ N m c_rms², where N is the number of particles, m is the mass of one particle and c_rms is the root mean square speed of the particles.

分子动理论推导出理想气体压强公式：pV = ⅓ N m c_rms²，其中 N 是粒子总数，m 是单个粒子的质量，c_rms 是粒子的均方根速率。

pV = ⅓ N m c_rms²

The root mean square speed is a statistical average that reflects the spread of molecular speeds. It is related to the molar mass and temperature by c_rms = √(3RT/M), where M is the molar mass in kg mol⁻¹. Lighter molecules have higher rms speeds at the same temperature.

均方根速率是反映分子速率分布的统计平均值。它与摩尔质量和温度的关系为 c_rms = √(3RT/M)，其中 M 是摩尔质量，单位为 kg mol⁻¹。在相同温度下，较轻的分子具有更高的均方根速率。

9. Temperature and Average Kinetic Energy | 温度与平均平动动能

The average translational kinetic energy of a single particle in an ideal gas is directly proportional to the absolute temperature. It is given by E_k = ½ m c_rms² = (3/2) kT, where k is the Boltzmann constant, k = 1.38 × 10⁻²³ J K⁻¹.

理想气体中单个粒子的平均平动动能与热力学温度成正比。其表达式为 E_k = ½ m c_rms² = (3/2) kT，其中 k 是玻尔兹曼常数，k = 1.38 × 10⁻²³ J K⁻¹。

E_k = ½ m c_rms² = (3/2) kT

For a monatomic ideal gas, the total internal energy U is only the kinetic energy of the atoms, so U = (3/2) nRT. For diatomic gases, rotational energy also contributes, so the internal energy is larger at the same temperature. Temperature is therefore a measure of the average random kinetic energy of particles.

对于单原子理想气体，总内能 U 只是原子的动能，因此 U = (3/2) nRT。对于双原子气体，转动能也有贡献，因此相同温度下内能更大。因此温度是粒子平均无规则动能的量度。

10. Real Gases vs Ideal Gases | 真实气体与理想气体

Real gases obey the ideal gas equation only approximately. Deviations become significant at high pressure and low temperature, because real particles have a finite volume and exert intermolecular forces. Attractive forces reduce the pressure, while the finite size of particles makes the available volume smaller.

真实气体只能近似满足理想气体方程。在高压和低温下偏差变得显著，因为真实粒子具有有限体积并存在分子间作用力。吸引力使压强减小，而粒子的有限体积使可用的自由体积变小。

At low pressure and high temperature, real gases approach ideal behaviour because particles are far apart and their intermolecular forces are negligible. The concept of an ideal gas is a useful limiting model for many practical calculations.

在低压和高温下，真实气体接近理想行为，因为粒子相距很远，分子间作用力可以忽略不计。理想气体的概念是许多实际计算中非常有用的极限模型。

11. Worked Example | 例题精讲

A fixed mass of gas has a volume of 2.0 m³ at a pressure of 101 kPa and a temperature of 27 °C. The gas is then heated to 127 °C and its pressure increases to 150 kPa. Calculate the new volume of the gas.

一定质量的气体在压强为 101 kPa、温度为 27 °C 时体积为 2.0 m³。该气体随后被加热到 127 °C，压强升高到 150 kPa。计算气体的新体积。

First convert temperatures to kelvin: T₁ = 27 + 273 = 300 K; T₂ = 127 + 273 = 400 K. Using the combined gas law p₁V₁/T₁ = p₂V₂/T₂, rearrange to find V₂ = p₁V₁T₂/(T₁p₂).

首先将温度换算为开尔文：T₁ = 27 + 273 = 300 K；T₂ = 127 + 273 = 400 K。使用组合气体定律 p₁V₁/T₁ = p₂V₂/T₂，整理得 V₂ = p₁V₁T₂/(T₁p₂)。

V₂ = (101 kPa × 2.0 m³ × 400 K) / (300 K × 150 kPa) = 1.80 m³

The new volume is 1.80 m³. Notice that the pressure units cancel, so it is acceptable to use kPa in this ratio calculation. However, when using the ideal gas equation pV = nRT, pressure must be in pascals.

新的体积为 1.80 m³。注意压强单位在比值的计算中可以约去，因此此例中使用 kPa 是可以的。但在使用理想气体方程 pV = nRT 时，压强必须用帕斯卡。

12. Exam Tips and Common Pitfalls | 考试技巧与常见误区

In Edexcel A Level Physics exams, gas law questions often test unit conversion and the distinction between ideal and real gases. Common mistakes include using Celsius temperatures, mixing units and forgetting that the mass of gas must be constant.

在 Edexcel A Level 物理考试中，气体定律题目通常考查单位换算以及理想气体与真实气体的区别。常见错误包括使用摄氏温度、单位混用以及忘记气体质量必须保持不变。

Always convert temperature to kelvin before applying any gas law.
Use pascals for pressure when using R = 8.31 J mol⁻¹ K⁻¹.
Remember that the combined gas law applies only to a fixed mass or fixed number of moles.
Be able to sketch and interpret pV, p against 1/V and V against T graphs.
Distinguish between mean square speed, mean speed and root mean square speed.

应用任何气体定律前，始终将温度换算为开尔文。
使用 R = 8.31 J mol⁻¹ K⁻¹ 时，压强必须用帕斯卡。
记住组合气体定律只适用于固定质量或固定物质的量的气体。
能够绘制并解释 pV 图、p 对 1/V 图以及 V 对 T 图。
区分均方速率、平均速率和均方根速率。

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The Combined Gas Law and Ideal Gas Behaviour | 组合气体定律与理想气体行为