The Physical Meaning and Microscopic Mechanism of Internal Energy | 内能的物理意义与微观机制

📚 The Physical Meaning and Microscopic Mechanism of Internal Energy | 内能的物理意义与微观机制

In thermal physics, internal energy is one of the most fundamental ideas. It links the macroscopic temperature of a substance to the random motion and interactions of its microscopic particles.

在热学中，内能是最基本的概念之一。它将物质的宏观温度与微观粒子的无规则运动及其相互作用联系起来。

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1. Definition of Internal Energy | 内能的定义

Internal energy, usually given the symbol U, is the total energy stored inside a substance due to the kinetic energy of its particles and the potential energy arising from the forces between them. It does not include the kinetic energy of the substance as a whole or any external potential energy such as gravitational potential energy.

内能通常用符号 U 表示，是指物质内部储存的总能量，包括粒子动能以及粒子间相互作用力所产生的势能。它不包括物体整体运动的动能，也不包括重力势能等外部势能。

For an ideal gas, the particles do not exert forces on each other except during collisions, so the internal energy is purely the total random kinetic energy of the particles. For real solids, liquids and gases, intermolecular forces also contribute a potential-energy term.

对于理想气体，除碰撞瞬间外粒子间没有相互作用力，因此内能仅仅是粒子无规则运动的总动能。对于真实的固体、液体和气体，分子间力还会贡献势能项。

A common mistake is to confuse internal energy with heat or temperature. Heat is energy transferred because of a temperature difference, while temperature is a measure of the average kinetic energy of the particles, not the total internal energy.

常见的错误是将内能与热量或温度混为一谈。热量是因温度差而转移的能量，温度则是粒子平均动能的量度，而不是总内能。

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2. Microscopic Kinetic Energy and Temperature | 微观动能与温度

All particles in a substance are in constant random motion. In a gas, atoms or molecules move freely; in a liquid, they slide past each other; in a solid, they vibrate about fixed positions. The greater the random motion, the higher the temperature.

物质中的所有粒子都处于永不停息的无规则运动中。在气体中，原子或分子自由运动；在液体中，它们相互滑移；在固体中，它们在固定位置附近振动。无规则运动越剧烈，温度就越高。

The average kinetic energy of a particle is proportional to the absolute temperature T measured in kelvin:

粒子的平均动能与以开尔文为单位的绝对温度 T 成正比：

average kinetic energy ∝ T

For a monatomic ideal gas, the mean kinetic energy of one atom is given by:

对于单原子理想气体，一个原子的平均动能为：

½ m ⟨v²⟩ = 3/2 kT

where m is the mass of one atom, ⟨v²⟩ is the mean square speed, and k is the Boltzmann constant. This equation shows that temperature is a direct measure of the microscopic kinetic energy of the particles.

其中 m 是一个原子的质量，⟨v²⟩ 是均方速率，k 是玻尔兹曼常数。该方程表明，温度是粒子微观动能的直接量度。

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3. Intermolecular Potential Energy | 分子间势能

When molecules are close together, attractive and repulsive forces between them create potential energy. If the average separation of molecules changes, the potential energy changes even if the temperature stays the same.

当分子彼此靠近时，分子间的引力和斥力会产生势能。如果分子的平均间距改变，即使温度不变，势能也会改变。

During a phase change such as melting or boiling, the temperature remains constant, so the average kinetic energy of the particles does not change. However, energy is still absorbed or released because the particles move apart or come together, changing the intermolecular potential energy.

在熔化或沸腾等相变过程中，温度保持不变，因此粒子的平均动能不变。然而，由于粒子间距增大或减小，分子间势能发生变化，体系仍然会吸收或释放能量。

In an ideal gas, molecules are treated as point particles with negligible intermolecular forces, so the potential energy term is zero. This is why the internal energy of an ideal gas depends only on temperature.

在理想气体中，分子被视为没有体积的质点，分子间力可以忽略，因此势能项为零。这就是理想气体内能只取决于温度的原因。

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4. Degrees of Freedom and Equipartition | 自由度与能量均分

Each independent way a particle can store energy is called a degree of freedom. A monatomic gas atom has three translational degrees of freedom, corresponding to motion in the x, y and z directions.

粒子储存能量的每一种独立方式称为一个自由度。单原子气体原子有三个平动自由度，分别对应 x、y、z 三个方向的运动。

According to the principle of equipartition of energy, each degree of freedom contributes an average energy of ½ kT per particle. For one mole of gas, this contribution is ½ RT per degree of freedom.

根据能量均分定理，每个自由度对每个粒子贡献的平均能量为 ½ kT。对于一摩尔气体，每个自由度贡献 ½ RT。

For a gas with f degrees of freedom, the molar internal energy is:

对于自由度为 f 的气体，其摩尔内能为：

U = (f/2) nRT

where n is the number of moles and R is the molar gas constant.

其中 n 是摩尔数，R 是摩尔气体常量。

Type of gas	Degrees of freedom f	Molar internal energy
Monatomic, for example helium or argon	3	3/2 RT
Diatomic, for example oxygen or nitrogen	5 at ordinary temperatures	5/2 RT

At higher temperatures, diatomic molecules may also have vibrational degrees of freedom, increasing the value of f. In CIE A-Level questions, the usual assumption is f = 3 for monatomic and f = 5 for diatomic gases.

在较高温度下，双原子分子还可能具有振动自由度，使 f 值增大。在 CIE A-Level 试题中，通常假定单原子气体 f = 3，双原子气体 f = 5。

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5. The First Law of Thermodynamics | 热力学第一定律

The first law of thermodynamics states that the change in internal energy of a system is equal to the heat supplied to the system plus the work done on the system:

热力学第一定律指出，系统内能的变化等于系统吸收的热量加上外界对系统做的功：

ΔU = Q + W

In this sign convention, Q is positive when heat is added to the system, and W is positive when work is done on the system. If the system does work on its surroundings, W is negative.

在此符号约定中，Q 为正值表示系统吸热，W 为正值表示外界对系统做功。如果系统对外界做功，则 W 为负值。

Some textbooks use the alternative convention ΔU = Q − W, where W is the work done by the gas. Always read the question carefully and state the convention you are using.

有些教材使用另一种约定 ΔU = Q − W，其中 W 表示气体对外做功。做题时务必仔细阅读题目，并说明你所采用的符号约定。

The first law is a statement of conservation of energy. It shows that internal energy can be changed either by heating or by doing mechanical work, and that heat and work are equivalent forms of energy transfer.

热力学第一定律是能量守恒的表述。它表明内能可以通过加热或做功来改变，热量和功是能量转移的两种等效形式。

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6. Work Done and Internal Energy | 做功与内能

When a gas expands, its particles push against the surroundings, so the gas does positive work and its internal energy falls if no heat is supplied. When a gas is compressed, work is done on the gas and its internal energy rises.

当气体膨胀时，粒子推动外界，气体对外做正功。如果没有热量供给，气体的内能就会减少。当气体被压缩时，外界对气体做功，内能会增加。

For a gas at constant pressure, the work done by the gas during a volume change ΔV is:

在恒压条件下，气体体积变化 ΔV 过程中气体对外做的功为：

W = p ΔV

In an adiabatic process, no heat enters or leaves the system, so Q = 0. The first law then gives:

在绝热过程中，系统与外界没有热量交换，因此 Q = 0。此时热力学第一定律给出：

ΔU = W

For adiabatic compression, W is positive, so the internal energy and temperature of the gas increase. For adiabatic expansion, W is negative, so the gas cools.

对于绝热压缩，W 为正值，因此气体的内能和温度升高。对于绝热膨胀，W 为负值，因此气体温度降低。

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7. Heat Capacity and Internal Energy | 热容与内能

The molar heat capacity at constant volume, cᵥ, is the energy required to raise the temperature of one mole of a substance by one kelvin while the volume is kept constant. For an ideal gas, all this heat goes into increasing the internal energy:

定容摩尔热容 cᵥ 是在体积保持不变的条件下，使一摩尔物质温度升高一开尔文所需的能量。对于理想气体，这部分热量全部用于增加内能：

ΔU = n cᵥ ΔT

Comparing this with the equipartition result for an ideal gas gives:

将其与能量均分的结果比较，可得理想气体：

cᵥ = (f/2) R

For a monatomic ideal gas, f = 3, so cᵥ = 3/2 R. For a diatomic gas with f = 5, cᵥ = 5/2 R.

对于单原子理想气体，f = 3，因此 cᵥ = 3/2 R。对于 f = 5 的双原子气体，cᵥ = 5/2 R。

When a gas is heated at constant pressure, it must also do work against the external pressure, so more heat is needed for the same temperature rise. The molar heat capacity at constant pressure cₚ is therefore larger than cᵥ. For an ideal gas:

当气体在恒压下受热时，气体还必须反抗外界压强做功，因此同样升高一开尔文需要更多的热量。所以定压摩尔热容 cₚ 大于 cᵥ。对于理想气体：

cₚ − cᵥ = R

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8. Internal Energy During Phase Changes | 相变中的内能

During melting, boiling or sublimation, the temperature of a pure substance remains constant. The average kinetic energy of the particles therefore stays constant, but the internal energy still changes because the intermolecular potential energy changes.

在熔化、沸腾或升华过程中，纯物质的温度保持不变。因此粒子的平均动能不变，但内能仍然会改变，因为分子间势能发生了变化。

The energy absorbed or released during a phase change at constant temperature is called latent heat. In terms of internal energy:

相变过程中在恒定温度下吸收或释放的能量称为潜热。用内能表述为：

ΔU = mL

where m is the mass of the substance and L is the specific latent heat. This energy changes the separation of molecules, not their average speed.

其中 m 是物质质量，L 是比潜热。该能量改变的是分子间距，而不是分子的平均速率。

For example, when ice melts at 0 °C, water molecules become free to move as a liquid. The bonds between molecules are partially broken, so the potential energy increases while the temperature remains at 0 °C during the transition.

例如，当冰在 0 °C 熔化时，水分子可以自由移动成为液体。分子间键部分断裂，因此势能增大，而在相变过程中温度保持在 0 °C。

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9. Kinetic Theory and Internal Energy | 分子动理论与内能

The kinetic theory of gases relates macroscopic quantities such as pressure to the microscopic motion of molecules. For an ideal gas, the pressure is given by:

分子动理论将压强等宏观量与分子的微观运动联系起来。对于理想气体，压强由下式给出：

pV = ⅓ N m c²

where N is the number of molecules, m is the mass of one molecule, and c² is the mean square speed. The total translational kinetic energy of the gas is:

其中 N 是分子总数，m 是一个分子的质量，c² 是均方速率。气体的总平动动能为：

U = ½ N m c²

Combining these two equations gives:

将以上两式联立可得：

pV = 2/3 U

Using the ideal gas equation pV = nRT, we obtain U = 3/2 nRT for a monatomic gas. This confirms that the internal energy of an ideal gas depends only on its temperature and the number of moles.

利用理想气体状态方程 pV = nRT，可得单原子气体 U = 3/2 nRT。这证实了理想气体的内能只取决于温度与摩尔数。

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10. Exam Tips and Common Misconceptions | 考试要点与常见误区

Students often lose marks by confusing the sign conventions in the first law, or by using U = 3/2 nRT for all gases. Remember that the factor 3/2 applies only to monatomic ideal gases.

学生常因混淆热力学第一定律的符号约定，或对所有气体都使用 U = 3/2 nRT 而失分。请记住，3/2 这个系数只适用于单原子理想气体。

• Always check whether a process is isothermal, adiabatic, isobaric or isochoric before applying the first law.

  在应用热力学第一定律之前，先判断过程是等温、绝热、等压还是等容。

• For an ideal gas, a change in internal energy requires a change in temperature. No temperature change means no change in internal energy.

  对于理想气体，内能的变化必然伴随温度的变化。温度不变，内能就不变。

• During a phase change, temperature is constant but internal energy changes because potential energy changes.

  在相变过程中，温度不变，但内能因势能变化而改变。

• Use the correct units: T in kelvin, p in pascals, V in cubic metres, and n in moles.

  注意单位：T 用开尔文，p 用帕斯卡，V 用立方米，n 用摩尔。

In CIE A-Level questions, you may be asked to estimate the internal energy change of a gas, compare internal energies at the same temperature, or explain why internal energy is zero for an ideal gas at absolute zero. These questions test whether you understand the microscopic meaning of internal energy, not just the equation.

在 CIE A-Level 试题中，你可能会被要求估算气体内能的变化、比较相同温度下的内能，或解释为什么理想气体在绝对零度时内能为零。这些问题考查的是对内能微观意义的理解，而不只是公式的记忆。

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