2 Modelling a Gas | 2 气体建模

📚 2 Modelling a Gas | 2 气体建模

Modelling a gas is fundamental in thermal physics, bridging macroscopic observables like pressure, volume and temperature with the microscopic motion of countless molecules. The ideal gas model provides a simplified yet powerful framework that allows us to understand and predict the behaviour of real gases under a wide range of conditions. By exploring the assumptions, laws and kinetic theory that underpin this model, we build a foundation for thermodynamics, engine cycles and the study of matter itself.

气体建模是热物理的基础，它将压强、体积和温度等宏观可观测量与无数分子的微观运动联系起来。理想气体模型提供了一个简化但强大的框架，使我们能够理解并预测实际气体在多种条件下的行为。通过探索支撑该模型的假设、定律和分子动理论，我们为热力学、热机循环以及物质本身的研究打下了基础。

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1. The Ideal Gas Model | 理想气体模型

An ideal gas is a theoretical model in which the particles are point masses that occupy no volume and exert no intermolecular forces on one another, except during perfectly elastic collisions. All collisions between particles and with the container walls are assumed to conserve kinetic energy, and the motion of the particles follows Newton’s laws. This abstraction strips away complexities to reveal the direct relationships between pressure, volume and temperature.

理想气体是一种理论模型，其粒子被视为点质量，不占据体积，且除发生完全弹性碰撞外彼此之间没有分子间作用力。粒子之间以及粒子与容器壁之间的所有碰撞均假定动能守恒，粒子的运动遵循牛顿定律。这种抽象剥离了复杂性，从而揭示压强、体积和温度之间的直接关系。

No real gas behaves perfectly as an ideal gas, but many gases at low pressure and high temperature approximate the model very closely. The ideal gas concept allows us to define temperature and internal energy in a clear, microscopic way, which is essential before studying deviations.

没有任何真实气体能完全表现为理想气体，但许多气体在低压和高温下非常接近该模型。理想气体的概念使我们能够以清晰的微观方式定义温度和内能，这对于研究偏离情况至关重要。

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2. Macroscopic State Variables | 宏观状态参量

The state of a fixed mass of gas is described by three macroscopic variables: pressure p, volume V and thermodynamic temperature T. Pressure is the force exerted per unit area on the walls of the container, measured in pascals (Pa); volume is the space occupied by the gas, measured in cubic metres (m³); and temperature is a measure of the average kinetic energy of the particles, measured in kelvin (K).

一定质量气体的状态由三个宏观变量描述：压强 p、体积 V 和热力学温度 T。压强是施加在容器壁上单位面积的力，单位为帕斯卡（Pa）；体积是气体所占的空间，单位为立方米（m³）；温度是粒子平均动能的量度，单位为开尔文（K）。

Temperature must always be expressed in kelvin when using gas laws. The kelvin scale is an absolute scale where 0 K corresponds to the lowest possible energy state of a system. To convert from degrees Celsius to kelvin, use T (K) = θ (°C) + 273.15.

在使用气体定律时，温度必须始终以开尔文表示。开尔文温标是绝对温标，0 K 对应于系统可能的最低能量状态。将摄氏度转换为开尔文时，使用 T (K) = θ (°C) + 273.15。

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3. The Ideal Gas Equation | 理想气体状态方程

The relationship among pressure, volume and temperature for an ideal gas is summarised by the ideal gas equation. In its molar form, it is expressed as:

理想气体的压强、体积和温度之间的关系由理想气体状态方程概括。其摩尔形式表示为：

pV = nRT

Here n is the amount of substance in moles, R is the universal gas constant (8.31 J mol⁻¹ K⁻¹), and T is the absolute temperature. This equation implies that for a fixed amount of gas, the product pV/T remains constant. It unifies the empirical gas laws: Boyle’s law (p ∝ 1/V at constant T), Charles’s law (V ∝ T at constant p) and the pressure law (p ∝ T at constant V).

其中 n 是物质的量（摩尔），R 是通用气体常数（8.31 J mol⁻¹ K⁻¹），T 是绝对温度。该方程意味着对于一定量的气体，乘积 pV/T 保持恒定。它统一了经验气体定律：玻意耳定律（T 一定时 p ∝ 1/V）、查理定律（p 一定时 V ∝ T）和压强定律（V 一定时 p ∝ T）。

Another useful form replaces moles with the number of gas particles N: pV = NkT, where k is the Boltzmann constant (1.38 × 10⁻²³ J K⁻¹). This version directly connects macroscopic quantities to the number of molecules.

另一种有用的形式用气体粒子数 N 代替摩尔数：pV = NkT，其中 k 是玻尔兹曼常数（1.38 × 10⁻²³ J K⁻¹）。该形式直接将宏观量与分子数目联系起来。

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4. Moles, Molar Mass and Avogadro’s Number | 摩尔、摩尔质量与阿伏伽德罗常数

The mole is the SI unit for amount of substance. One mole contains exactly 6.022 × 10²³ elementary entities, a number known as Avogadro’s number N_A. For any substance, the mass of one mole is its molar mass M, measured in grams per mole (g mol⁻¹), which is numerically equal to the relative molecular mass.

摩尔是物质的量的国际单位。一摩尔恰好包含 6.022 × 10²³ 个基本单元，这个数称为阿伏伽德罗常数 N_A。对于任何物质，一摩尔的质量就是其摩尔质量 M，单位为克每摩尔（g mol⁻¹），数值上等于相对分子质量。

The number of moles n can be found from the total mass m_total and molar mass M using n = m_total / M. The total number of particles N is then N = n N_A. These relations are fundamental when switching between the macroscopic and particle descriptions of a gas.

摩尔数 n 可以根据总质量 m_total 和摩尔质量 M 用 n = m_total / M 求出。那么总粒子数 N 为 N = n N_A。这些关系式是在气体的宏观描述和粒子描述之间转换时的基础。

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5. Kinetic Theory Assumptions | 分子动理论假设

Kinetic theory links the microscopic motion of particles to macroscopic pressure and temperature. It rests on a set of simplifying assumptions that define the ideal gas at the particle level. The main assumptions are:

分子动理论将粒子的微观运动与宏观压强和温度联系起来。它建立在一组简化假设之上，这些假设在粒子层面上定义了理想气体。主要假设如下：

• The gas consists of a large number of identical, tiny particles in constant, random motion.

  气体由大量相同的微小粒子组成，它们处于持续的无规运动中。

• The volume of the particles is negligible compared with the total volume of the gas.

  与气体总体积相比，粒子本身的体积可以忽略不计。

• There are no intermolecular forces between particles, except during collisions.

  除碰撞瞬间外，粒子之间没有分子间作用力。

• Collisions between particles and with the container walls are perfectly elastic, so kinetic energy is conserved.

  粒子之间以及粒子与容器壁之间的碰撞是完全弹性的，因此动能守恒。

• The time spent in collisions is negligible compared with the time between collisions.

  碰撞所持续的时间与两次碰撞之间的时间相比可以忽略。

• Newtonian mechanics applies to the motion of the particles.

  粒子的运动遵循牛顿力学。

These assumptions are reasonable for monatomic gases at low pressure and high temperature. They fail when particle volume or intermolecular attractions become significant, leading to real gas behaviour.

这些假设对于低压、高温下的单原子气体是合理的。当粒子体积或分子间引力变得显著时，它们就不成立了，从而导致实际气体行为。

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6. Pressure from Molecular Collisions | 分子碰撞产生的压强

Using the kinetic theory assumptions, we can derive an expression for the pressure exerted by an ideal gas. Consider N particles of mass m moving with random velocities in a cubic container of side length L. By analysing the change in momentum when a particle collides elastically with a wall, and summing over all particles, we obtain:

利用分子动理论的假设，我们可以推导出理想气体施加的压强表达式。考虑 N 个质量为 m 的粒子在边长为 L 的立方体容器中无规运动。通过分析粒子与器壁弹性碰撞时的动量变化，并对所有粒子求和，我们得到：

p = ⅓ ρ <c²>

where ρ = Nm/V is the density of the gas and <c²> is the mean square speed of the particles. This equation reveals that pressure is proportional to the density of the gas and to the average of the squared speeds of the particles.

其中 ρ = Nm/V 是气体的密度，<c²> 是粒子方均速率。该方程表明压强与气体密度和粒子速率平方的平均值成正比。

An equivalent form is pV = ⅓ Nm<c²>. From here, combining with the ideal gas equation pV = NkT, we can immediately link the microscopic quantity <c²> to the absolute temperature T.

一个等价形式是 pV = ⅓ Nm<c²>。由此，结合理想气体状态方程 pV = NkT，我们可以立即将微观量 <c²> 与绝对温度 T 联系起来。

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7. Average Translational Kinetic Energy | 平均平动动能

By equating the kinetic theory pressure formula pV = ⅓ Nm<c²> with the ideal gas equation pV = NkT, we find:

将分子动理论的压强公式 pV = ⅓ Nm<c²> 与理想气体状态方程 pV = NkT 联立，我们得到：

½ m<c²> = (3/2) kT

The left‑hand side is the average translational kinetic energy <E_k> of a single particle. Thus, the absolute temperature of an ideal gas is directly proportional to the average random kinetic energy of its particles. This is a profound microscopic interpretation of temperature.

左边是单个粒子的平均平动动能 <E_k>。因此，理想气体的绝对温度与其粒子的平均无规动能成正比。这是对温度的深刻微观诠释。

For monatomic particles, all internal energy is translational. For polyatomic molecules, rotational and vibrational modes also contribute, but the translational part always obeys the same relation. The total internal energy U for N particles is therefore U = N × (3/2) kT = (3/2) nRT.

对于单原子粒子，所有内能都是平动动能。对于多原子分子，转动和振动模式也有贡献，但平动部分总是遵循相同的关系。因此 N 个粒子的总内能 U 为 U = N × (3/2) kT = (3/2) nRT。

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8. Root Mean Square Speed and Temperature | 方均根速率与温度

The root mean square (rms) speed c_rms is defined as the square root of the mean square speed: c_rms = √<c²>. From the relation ½ m<c²> = (3/2) kT, we can express c_rms in terms of temperature and particle mass:

方均根速率 c_rms 定义为方均速率的平方根：c_rms = √<c²>。由 ½ m<c²> = (3/2) kT，我们可以将 c_rms 用温度和粒子质量表示：

c_rms = √(3kT / m)

Because k = R/N_A and m = M/N_A, this can also be written as c_rms = √(3RT / M), where M is the molar mass. This shows that at a given temperature, lighter gas molecules move faster than heavier ones. For example, at room temperature, hydrogen molecules have an rms speed about four times that of oxygen molecules.

由于 k = R/N_A 且 m = M/N_A，该式也可写为 c_rms = √(3RT / M)，其中 M 是摩尔质量。这表明在给定温度下，较轻的气体分子比较重的分子运动得更快。例如，在室温下，氢分子的方均根速率约为氧分子的四倍。

The rms speed is a statistical measure; individual particles have a wide distribution of speeds, described by the Maxwell–Boltzmann distribution. However, c_rms provides a convenient single value to characterise particle speeds when calculating pressure or kinetic energy.

方均根速率是一种统计量度；单个粒子的速率分布范围很广，由麦克斯韦–玻尔兹曼分布描述。然而，在计算压强或动能时，c_rms 提供了一个方便的特征值。

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9. Internal Energy of an Ideal Gas | 理想气体的内能

The internal energy U of an ideal gas is the sum of the kinetic energies of all its particles, since there are no intermolecular potential energies in the model. For a monatomic ideal gas, it depends only on temperature and the number of particles:

理想气体的内能 U 是所有粒子动能之和，因为模型中没有分子间势能。对于单原子理想气体，它只取决于温度和粒子数目：

U = (3/2) nRT = (3/2) NkT

This result has two important consequences. First, a change in internal energy ΔU is proportional to the change in temperature ΔT, with ΔU = (3/2) nR ΔT. Second, during an isothermal process (constant T), the internal energy of an ideal gas does not change; any heat added goes entirely into work done by the gas.

这个结果有两个重要推论。第一，内能的变化 ΔU 与温度的变化 ΔT 成正比，即 ΔU = (3/2) nR ΔT。第二，在等温过程中（温度 T 恒定），理想气体的内能不发生变化；吸收的热量全部转化为气体对外做功。

For diatomic and polyatomic gases, the situation is more complex because rotational and vibrational energies contribute. The molar heat capacities differ, but the principle that U is a function of temperature alone remains valid for all ideal gases.

对于双原子和多原子气体，情况更为复杂，因为转动和振动能量也有贡献。摩尔热容有所不同，但内能仅仅是温度的函数这一原则对所有理想气体仍然成立。

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10. Real Gases and Deviations from Ideality | 实际气体与理想性的偏离

Real gases deviate from ideal behaviour when the assumptions of kinetic theory break down. Two main factors are responsible: the finite volume of gas molecules and the existence of attractive intermolecular forces. At high pressures, the volume occupied by the molecules themselves becomes significant compared with the container volume, so the available space is less than V. At low temperatures, molecules move slowly enough that attractive forces cause them to stick together during collisions, reducing the pressure compared with an ideal gas.

当分子动理论的假设不再成立时，实际气体就会偏离理想行为。两个主要因素是：气体分子具有有限的体积，以及分子间存在吸引力。在高压下，分子自身占据的体积相比于容器体积变得显著，因此可用空间小于 V。在低温下，分子运动足够缓慢，吸引力使它们在碰撞时黏在一起，从而导致压强低于理想气体的预期值。

The van der Waals equation modifies the ideal gas law to account for these effects: (p + a n²/V²)(V – nb) = nRT, where a and b are empirical constants. The term nb corrects for molecular volume, and a n²/V² corrects for intermolecular attractions. This model better describes the liquid–vapour transition and critical point behaviour.

范德华方程对理想气体定律进行了修正，以考虑这些效应：(p + a n²/V²)(V – nb) = nRT，其中 a 和 b 为经验常数。项 nb 修正了分子体积，而 a n²/V² 修正了分子间吸引力。该模型能更好地描述液–气相变和临界点行为。

A convenient way to visualise deviations is the compressibility factor Z = pV/nRT. For an ideal gas, Z = 1 at all conditions. For real gases, Z < 1 when attractive forces dominate (moderate pressures) and Z > 1 when repulsive forces due to finite volume dominate (very high pressures). Understanding these deviations is crucial in applications such as liquefaction and high‑pressure gas storage.

一种直观表示偏离的方法是压缩因子 Z = pV/nRT。对于理想气体，在任何条件下 Z = 1。对于实际气体，当吸引力占主导时（中等压强），Z < 1；当有限体积导致的排斥力占主导时（极高压力），Z > 1。理解这些偏离在液化与高压气体储存等应用中至关重要。

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11. The Maxwell–Boltzmann Distribution | 麦克斯韦–玻尔兹曼分布

The kinetic theory model gives us average speeds, but the actual speeds of particles in a gas are spread over a wide range. The Maxwell–Boltzmann distribution describes the probability of finding a particle with a given speed at a certain temperature. The distribution curve starts at zero, rises to a peak at the most probable speed, and then gradually tails off at high speeds.

分子动理论模型给出了平均速率，但气体中粒子的实际速率分布在一个很宽的范围内。麦克斯韦–玻尔兹曼分布描述了在某一温度下找到具有特定速率的粒子的概率。分布曲线从零开始，在最概然速率处达到峰值，然后在高速率端逐渐下降。

Key features include: the most probable speed c_mp = √(2kT/m), the average speed <c> = √(8kT/πm), and the rms speed c_rms = √(3kT/m). Note that c_mp < <c> < c_rms. As temperature increases, the entire distribution shifts to higher speeds and flattens, indicating a broader spread of kinetic energies.

主要特征包括：最概然速率 c_mp = √(2kT/m)，平均速率 <c> = √(8kT/πm)，以及方均根速率 c_rms = √(3kT/m)。注意 c_mp < <c> < c_rms。随着温度升高，整个分布向高速方向移动并变得平坦，表明动能分布更加分散。

The distribution also explains why a small fraction of particles have energies high enough to escape the Earth’s atmosphere, and why evaporation can occur at temperatures well below the boiling point.

该分布还解释了为什么有一小部分粒子具有足够高的能量逸出地球大气层，以及为什么蒸发可以在远低于沸点的温度下发生。

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12. Linking Kinetic Theory to Experimental Observations | 分子动理论与实验观测的联系

The power of the gas model lies in its ability to predict measurable quantities. Boyle’s law, Charles’s law and the pressure law all emerge naturally from the kinetic theory derivation of pV = ⅓ Nm<c²> combined with the temperature dependence of <c²>. The model also predicts that the pressure exerted by a gas is independent of the type of gas, as long as the temperature and number density are fixed — a fact confirmed by Avogadro’s law.

气体模型的威力在于其预测可测量量的能力。玻意耳定律、查理定律和压强定律都可以从分子动理论推导出的 pV = ⅓ Nm<c²> 以及 <c²> 对温度的依赖关系中自然得出。该模型还预测，只要温度和粒子数密度固定，气体施加的压强就与气体种类无关——这一事实被阿伏伽德罗定律所证实。

Experiments such as the determination of the molar volume of a gas at STP (22.4 dm³) and the measurement of specific heat capacities provide strong indirect evidence for the kinetic model. Brownian motion, in which pollen grains or smoke particles jitter randomly when suspended in a fluid, offers direct visual evidence of molecular motion and supports the statistical foundations of the theory.

诸如测定标准状况下气体的摩尔体积（22.4 dm³）和测量比热容等实验，为分子动模型提供了有力的间接证据。布朗运动，即花粉颗粒或烟雾颗粒悬浮在流体中时出现无规抖动，为分子运动提供了直接的视觉证据，并支持了该理论的统计基础。

Thus, modelling a gas is not an abstract mathematical exercise; it is a cornerstone of modern physics that connects the invisible particulate world with the everyday macroscopic one.

因此，气体建模并非抽象的数学练习；它是现代物理学的基石，将看不见的粒子世界与日常的宏观世界联系起来。

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