Boyle's Law | 玻意耳定律

📚 Boyle’s Law | 玻意耳定律

Boyle’s law is one of the fundamental gas laws studied at A-Level. It links the pressure and volume of a fixed mass of gas under isothermal conditions and forms a key part of the ideal gas model.

玻意耳定律是 A-Level 学习中基础的气体定律之一。它将一定质量气体在等温条件下的压强与体积联系起来，是理想气体模型的重要组成部分。

1. Definition and Statement | 定义与表述

Boyle’s law states that for a fixed mass of an ideal gas at constant temperature, the pressure p exerted by the gas is inversely proportional to its volume V.

玻意耳定律指出，对于一定质量的理想气体，在温度保持不变的条件下，气体产生的压强 p 与其体积 V 成反比。

This relationship can be written as:

这一关系可以表示为：

p ∝ 1/V

Equivalently, the product of pressure and volume remains constant as long as the temperature and mass do not change.

等价地，只要温度和质量不变，压强与体积的乘积就保持恒定。

pV = constant

2. Formula and Useful Forms | 公式与常用形式

For calculations between two equilibrium states of the same gas at constant temperature, Boyle’s law is usually written as:

对于同一气体在恒温下的两个平衡状态之间的计算，玻意耳定律通常写成：

p₁V₁ = p₂V₂

Here p₁ and V₁ are the initial pressure and volume, while p₂ and V₂ are the final pressure and volume. The pressure must be absolute pressure, not gauge pressure.

其中 p₁ 和 V₁ 是初始压强和体积，p₂ 和 V₂ 是末态压强和体积。压强必须使用绝对压强，而不是表压。

The law assumes three conditions: the mass of gas is fixed, the temperature remains constant, and the gas behaves ideally, meaning the pressure is low enough for intermolecular forces and molecular size to be negligible.

该定律有三个适用条件：气体质量固定、温度保持恒定、气体行为接近理想气体，即压强足够低，使分子间作用力和分子自身大小可以忽略。

3. Experimental Verification | 实验验证

A common experiment uses a sealed syringe or a glass tube connected to a pressure sensor. A fixed mass of dry air is trapped, and the volume can be changed by moving a piston or adjusting a mercury column.

常见实验使用密封注射器或连接压强传感器的玻璃管。将一定质量的干燥空气封存，通过移动活塞或调节水银柱来改变体积。

The volume is changed slowly in small steps. After each change, the gas is allowed to return to room temperature before the pressure is recorded. This keeps the process approximately isothermal.

缓慢地以小步改变体积。每次改变后，让气体恢复到室温，再记录压强。这样可使过程近似等温。

If the product pV is calculated for each pair of readings, it should remain roughly constant. A more sensitive test is to plot p against 1/V and check for a straight line through the origin.

如果计算每组数据的 pV 乘积，它应大致保持恒定。更灵敏的检验方法是绘制 p 对 1/V 的图像，并检查是否得到一条过原点的直线。

4. The p–V Graph | p–V 图像

For an ideal gas obeying Boyle’s law, a graph of pressure p against volume V is a rectangular hyperbola. This curve shows that as volume increases, pressure decreases, but the curve never touches the axes.

对于遵循玻意耳定律的理想气体，压强 p 对体积 V 的图像是一条等轴双曲线。该曲线表明，随着体积增大，压强减小，但曲线永远不会接触坐标轴。

Along a single isothermal curve, the temperature is constant. Different curves on the same axes represent different temperatures; for a fixed mass of gas, a curve further from the origin corresponds to a higher temperature.

在同一条等温曲线上，温度保持不变。同一坐标系中的不同曲线代表不同温度；对一定质量的气体，离原点越远的曲线对应越高的温度。

The product pV at any point on the curve has the same numerical value. If the gas deviates from ideal behaviour at very high pressures, the graph will depart from a perfect hyperbola.

曲线上任意一点的 pV 乘积具有相同的数值。如果气体在极高压强下偏离理想行为，图像将偏离完美的双曲线。

5. The p–1/V Graph | p–1/V 图像

Since p is inversely proportional to V, plotting pressure against the reciprocal of volume gives a straight line passing through the origin.

由于 p 与 V 成反比，因此绘制压强对体积倒数的图像会得到一条通过原点的直线。

The straight-line form is useful because it is easier to judge whether experimental points lie on a straight line than on a hyperbola. The gradient of the line equals the constant pV value for that fixed mass and temperature.

直线形式非常有用，因为判断实验点是否落在直线上比判断双曲线更容易。该直线的斜率等于该固定质量和温度下的 pV 常数值。

p = constant × 1/V

If the plotted line is not straight or does not pass through the origin, this indicates a systematic error, a temperature change, or a gas leak during the experiment.

如果绘制的直线不直或不过原点，说明实验中存在系统误差、温度变化或气体泄漏。

6. Molecular Explanation | 分子动理论解释

According to the kinetic theory of gases, gas pressure is caused by the collisions of gas molecules with the walls of the container. Each collision exerts a small force on the wall, and the total force per unit area is the pressure.

根据气体动理论，气体压强是由气体分子与容器壁碰撞产生的。每次碰撞对壁面施加一个微小的力，单位面积上的总力就是压强。

When the volume of a fixed mass of gas is reduced at constant temperature, the same number of molecules is confined to a smaller space. The number of collisions per second with the walls increases, so the pressure rises.

当一定质量的气体在恒温下体积减小时，相同数量的分子被限制在更小的空间中。每秒与壁面的碰撞次数增加，因此压强升高。

The average kinetic energy of the molecules does not change because the temperature is constant. Therefore, the increase in pressure arises from more frequent collisions, not from harder individual collisions.

由于温度不变，分子的平均动能不变。因此，压强升高源于碰撞频率增加，而不是单个碰撞变得更强。

This microscopic picture explains the inverse proportionality: halving the volume roughly doubles the collision frequency, so the pressure doubles.

这一微观图像解释了反比关系：体积减半大约使碰撞频率加倍，因此压强加倍。

7. Worked Example | 例题解析

A fixed mass of gas occupies a volume of 2.5 × 10⁻³ m³ at a pressure of 1.2 × 10⁵ Pa. The gas is compressed isothermally until its volume is 1.0 × 10⁻³ m³. Calculate the final pressure.

一定质量的气体在压强为 1.2 × 10⁵ Pa 时体积为 2.5 × 10⁻³ m³。该气体被等温压缩至体积为 1.0 × 10⁻³ m³。计算末态压强。

Since the temperature and mass are constant, Boyle’s law applies:

由于温度和质量不变，可应用玻意耳定律：

p₁V₁ = p₂V₂

Rearrange to make p₂ the subject:

整理公式，以 p₂ 为未知量：

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Boyle’s Law | 玻意耳定律