IB Physics: Core Formulas & Graph Analysis of Gas Laws | IB物理：气体定律核心公式与图像分析

📚 IB Physics: Core Formulas & Graph Analysis of Gas Laws | IB物理：气体定律核心公式与图像分析

The gas laws are fundamental to understanding the behaviour of ideal gases in thermodynamics. In IB Physics, you are expected to know not only the formulas but also their graphical representations, which often appear in Paper 1 and Paper 2 questions. This article reviews the essential formulas and provides a step-by-step guide to interpreting the associated graphs.

气体定律是热力学中理解理想气体行为的基础。在IB物理中，你不仅需要掌握公式，还要理解它们的图像表达，这些内容经常出现在卷一和卷二的考题中。本文回顾核心公式，并逐步指导你解读相关图像。

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1. The Ideal Gas Assumptions | 理想气体的假设

The ideal gas model simplifies reality by assuming molecules occupy negligible volume, have no intermolecular forces, and undergo perfectly elastic collisions. The average kinetic energy of the gas molecules is directly proportional to the absolute temperature.

理想气体模型简化了实际情况：假设分子体积可忽略、分子间无作用力、碰撞完全弹性。气体分子的平均动能与绝对温度成正比。

The ideal gas equation of state is:

PV = nRT

where P is pressure, V is volume, n is the number of moles, R is the universal gas constant (8.31 J mol⁻¹ K⁻¹), and T is the absolute temperature in kelvin.

其中 P 为压强，V 为体积，n 为物质的量，R 为普适气体常量（8.31 J mol⁻¹ K⁻¹），T 为开尔文绝对温度。

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2. Boyle’s Law | 玻意耳定律

At constant temperature (isothermal process), the pressure of a fixed amount of gas is inversely proportional to its volume. This is Boyle’s Law:

在温度恒定的条件下（等温过程），一定量气体的压强与体积成反比。这就是玻意耳定律：

P ∝ 1/V or P₁V₁ = P₂V₂

The P–V graph at a fixed temperature is a rectangular hyperbola. For higher temperatures, the curve shifts further from the origin because the product PV = nRT is larger.

在固定温度下，P–V 图是一条等轴双曲线。温度越高，曲线离原点越远，因为乘积 PV = nRT 更大。

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3. Charles’s Law | 查理定律

At constant pressure (isobaric process), the volume of a fixed amount of gas is directly proportional to its absolute temperature:

在压强恒定的条件下（等压过程），一定量气体的体积与绝对温度成正比：

V ∝ T or V₁/T₁ = V₂/T₂

The V–T graph at constant pressure is a straight line passing through the origin. The slope of this line is nR/P, so a lower pressure gives a steeper slope.

在等压下，V–T 图是一条通过原点的直线。其斜率为 nR/P，压强越低，斜率越大。

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4. Gay-Lussac’s Law | 盖-吕萨克定律

At constant volume (isochoric process), the pressure of a fixed amount of gas is directly proportional to its absolute temperature:

在体积恒定的条件下（等容过程），一定量气体的压强与绝对温度成正比：

P ∝ T or P₁/T₁ = P₂/T₂

The P–T graph at constant volume is again a straight line through the origin, with slope nR/V. A smaller volume gives a larger slope.

在等容条件下，P–T 图也是一条通过原点的直线，斜率为 nR/V。体积越小，斜率越大。

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5. Avogadro’s Law | 阿伏伽德罗定律

For a gas at constant temperature and pressure, the volume is directly proportional to the number of moles n. This law is built into the ideal gas equation, as V = (RT/P) n.

在温度和压强恒定时，气体的体积与物质的量 n 成正比。这一定律已包含在理想气体方程中，即 V = (RT/P)n。

It is rarely tested as a standalone graph, but you may need to compare two gases using equal volumes under identical conditions to deduce the same number of particles.

该定律极少单独出图像题，但你可能需要在相同条件下比较两种气体，利用等体积来推断相同的粒子数。

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6. The Ideal Gas Law in Combined Form | 理想气体定律的综合形式

When a gas changes from an initial state (P₁, V₁, T₁) to a final state (P₂, V₂, T₂), the combined gas law avoids using n explicitly:

当气体从初态（P₁, V₁, T₁）变化到末态（P₂, V₂, T₂）时，综合气体定律不需要明确写出 n：

P₁V₁/T₁ = P₂V₂/T₂

This equation is valid for a fixed mass of gas. Always convert Celsius temperatures to kelvin before substituting, because the proportionalities hold only for absolute temperature.

该方程对固定质量的气体成立。代入前务必把摄氏温度转换为开尔文，因为上述正比关系只对绝对温度成立。

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7. Graph Analysis: Isotherms | 图像分析：等温线

Isotherms are curves of constant temperature on a P–V diagram. Each curve follows P = nRT / V. For a given amount of gas, a curve further from the origin represents a higher temperature.

等温线是 P–V 图上温度恒定的曲线。每条曲线满足 P = nRT / V。对一定量的气体，离原点越远的曲线代表温度越高。

When solving problems, choose two points on the same isotherm and apply Boyle’s law. If the gas moves from one isotherm to another, use the combined gas law to relate all state variables.

解题时，在同一等温线上取两个点应用玻意耳定律。如果气体从一条等温线移动到另一条等温线，则使用综合气体定律联系所有状态变量。

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8. Graph Analysis: Isochores and Isobars | 图像分析：等容线与等压线

An isochoric (constant volume) process is shown as a straight line on a P–T graph through the origin. A higher density of lines means different fixed volumes; the line with the steeper slope corresponds to the smaller volume.

等容过程（体积恒定）在 P–T 图上表现为通过原点的直线。不同固定体积对应不同斜率，斜率越大的直线对应体积越小。

An isobaric (constant pressure) process is shown as a straight line on a V–T graph through the origin. The line with the steeper slope corresponds to the lower pressure.

等压过程（压强恒定）在 V–T 图上表现为通过原点的直线。斜率越大的直线对应压强越低。

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9. Work Done and Area under Curves | 做功与曲线下面积

On a P–V diagram, the work done by a gas during expansion is equal to the area under the curve between the initial and final volumes. If volume increases, the work done by the gas is positive. In an isothermal expansion, the work is W = nRT ln(V₂/V₁).

在 P–V 图上，气体膨胀过程中对外做的功等于曲线下从初态到末态的面积。如果体积增加，气体对外做功为正。等温膨胀时，功 W = nRT ln(V₂/V₁)。

For a cyclic process, the net work done is the area enclosed by the loop. Clockwise loops correspond to positive net work (heat engine), while counter-clockwise loops correspond to negative net work (refrigerator).

对循环过程，净功等于闭合路径所包围的面积。顺时针循环对应正净功（热机），逆时针循环对应负净功（制冷机）。

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10. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧

One common mistake is using Celsius temperature in gas law formulas. Always convert to kelvin by adding 273.15.

一个常见错误是在气体定律公式中使用摄氏温度。务必通过加上273.15转换为开尔文。

Another pitfall is confusing isothermal and isobaric graphs: remember that P–V curves are hyperbolas, while V–T and P–T lines are straight lines through the origin. Also, double-check whether a graph axis uses pressure or volume, as many questions deliberately swap axes.

另一个陷阱是混淆等温线与等压线：记住 P–V 曲线是双曲线，而 V–T 和 P–T 是过原点的直线。同时，仔细检查坐标轴是压强还是体积，很多题目会故意对调坐标轴。

In exam questions, read off values from the graph carefully, use the correct number of significant figures, and state the law or relationship you are applying before substituting numbers.

在考题中，仔细读取图像数值，使用正确的有效数字，并在代入数据前写明你使用的定律或关系式。

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11. Summary of Key Graphs | 关键图像总结

Law	Formula	Graph	Slope/Shape
Boyle’s Law	P₁V₁ = P₂V₂	P–V at constant T	Hyperbola
Charles’s Law	V₁/T₁ = V₂/T₂	V–T at constant P	Straight line through origin
Gay-Lussac’s Law	P₁/T₁ = P₂/T₂	P–T at constant V	Straight line through origin
Ideal Gas Law	PV = nRT	Various	Depends on fixed variable

By mastering these formulas and graph interpretations, you will be able to solve gas law problems quickly and accurately in the IB exam.

通过掌握这些公式和图像解读方法，你将能在IB考试中快速而准确地解决气体定律问题。

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