📚 3 Electrons in atoms (HL) | 原子中的电子（HL）
In IB Chemistry Higher Level, the study of electrons in atoms moves beyond the simple shell model. We need quantum mechanics to explain atomic line spectra, the arrangement of electrons in orbitals, and trends in ionisation energy. This topic deepens your understanding of the periodic table, chemical bonding, and the properties of elements. Here we will cover everything from the Bohr model to writing electron configurations and interpreting ionisation evidence.
在IB化学高等级中，对原子中电子的研究超越了简单的壳层模型。我们需要量子力学来解释原子线状光谱、电子在轨道中的排布以及电离能的趋势。本部分将加深你对元素周期表、化学键和元素性质的理解。我们将涵盖从玻尔模型到书写电子构型、解读电离证据等所有内容。
1. From classical to quantum: the need for a new model | 从经典到量子：新模型的必要性
Rutherford's nuclear model proposed that electrons orbit the nucleus like planets around the Sun. However, classical physics predicted that an accelerating electron would continuously lose energy and spiral into the nucleus, which does not happen. Moreover, the model could not explain why atoms emit only certain wavelengths of light—line spectra. These failures made a new quantum model essential.
卢瑟福的核式模型认为电子像行星绕太阳那样围绕原子核运转。然而，经典物理预言加速运动的电子会持续损失能量并螺旋坠入原子核，但这并未发生。此外，该模型无法解释为什么原子只发射特定波长的光——线状光谱。这些缺陷使得全新的量子模型成为必需。
2. Electromagnetic spectrum and line spectra | 电磁波谱与线状光谱
d of light c = 3.00 × 10⁸ m s⁻¹. The relationship between wavelength λ and frequency ν is:
所有电磁辐射都以光速 c = 3.00 × 10⁸ m s⁻¹ 传播。波长 λ 与频率 ν 之间的关系为：
c = λ ν
The energy of a photon is quantised and given by Planck's equation:
光子的能量是量子化的，由普朗克方程给出：
E = h ν = hc / λ
atoms are excited, they emit light only at specific wavelengths, producing a line spectrum. The hydrogen emission spectrum in the visible region shows four discrete lines, which was a crucial puzzle.
其中 h = 6.63 × 10⁻³⁴ J s。当气态原子被激发时，它们只在特定波长处发光，产生线状光谱。氢在可见光区的发射光谱显示出四条分立谱线，这是一个关键的谜题。
3. Bohr model of the hydrogen atom: success and limitations | 氢原子的玻尔模型：成就与局限
Niels Bohr proposed that electrons occupy fixed circular orbits with quantised energy levels. The energy of an electron in level n is given by:
尼尔斯·玻尔提出电子占据具有量子化能级的固定圆形轨道。能级 n 中电子的能量为：
Eₙ = –2.18 × 10⁻¹⁸ J / n²
Transitions between levels absorb or emit photons whose energy matches the difference. This explained the hydrogen line spectrum perfectly using the Rydberg formula:
能级之间的跃迁吸收或发射能量与能级差匹配的光子。这完美地解释了氢线状光谱，并用里德伯公式表示：
1/λ = R (1/n₁² – 1/n₂²)
where R is the Rydberg constant. However, the Bohr model failed for multi-electron atoms, could not explain the splitting of spectral lines in magnetic fields, and treated electrons as simple particles without wave nature.
其中 R 为里德伯常数。然而，玻尔模型在多电子原子中失效，无法解释光谱线在磁场中的分裂，并将电子视为无波动性的简单粒子。
4. Wave-particle duality and the Schrödinger model | 波粒二象性与薛定谔模型
here p is momentum. This wave nature means electrons do not follow precise orbits but exist in standing waves around the nucleus. Schrö loped a wave equation whose solutions are wavefunctions ψ; the square of the wavefunction, |ψ|², gives the probability density of finding an electron in a region of space.
德布罗意提出所有粒子都具有由 λ = h / p 给出的波长（p 为动量）。这种波动性意味着电子并不遵循精确的轨道，而是在原子核周围以驻波形式存在。薛定谔发展了一个波动方程，其解为波函数 ψ；波函数的平方 |ψ|² 给出了在空间某区域找到电子的概率密度。
Thus an orbital is defined as a region where there is a high probability (typically >90%) of finding an electron. The modern quantum model discards the concept of fixed orbits and uses probability distributions to describe electron behaviour.
因此，轨道被定义为找到电子的概率很高（通常 >90%）的区域。现代量子模型摒弃了固定轨道的概念，使用概率分布来描述电子行为。
5. Quantum numbers: n, l, mₗ, mₛ | 量子数：n、l、mₗ、mₛ
Every electron in an atom is described by a set of four quantum numbers that arise from the Schrödinger equation:
原子中的每个电子都由一组四个量子数来描述，这些量子数源自薛定谔方程：
- Principal quantum number, n: a positive integer (1,2,3...), determines the main energy level and size of the orbital.
主量子数 n：正整数（1,2,3...），决定主能级和轨道的大小。 - Angular momentum quantum number, l: for a given n, l = 0,1,2,..., n–1. It defines the sub-shell shape: l=0 → s, l=1 → p, l=2 → d, l=3 → f.
角量子数 l：对于给定的 n，l = 0,1,2,..., n–1。它定义亚层形状：l=0 → s, l=1 → p, l=2 → d, l=3 → f。 - Magnetic quantum number, mₗ: for a given l, mₗ = –l, –(l–1), ..., 0, ..., +(l–1), +l. It specifies the orientation of the orbital in space.
磁量子数 mₗ：对于给定的 l，mₗ = –l, –(l–1), ..., 0, ..., +(l–1), +l。它规定轨道在空间中的取向。 - Spin magnetic quantum number, mₛ: has only two possible values, +½ and –½, representing the two spin states of an electron.
自旋磁量子数 mₛ：只有两个可能值 +½ 和 –½，代表电子的两种自旋状态。
These quantum numbers completely describe the state of an electron and satisfy the Pauli exclusion principle (see section 8).
这些量子数完整地描述了电子的状态，并满足泡利不相容原理（见第8节）。
6. Shapes and orientations of atomic orbitals (s, p, d) | 原子轨道的形状与取向（s、p、d）
passing through the nucleus. There are three p orbitals per energy level (n≥2): pₓ, pᵧ, pz, mutually perpendicular.
轨道通过其边界曲面进行可视化。s 轨道对于任何 n 都是球形。p 轨道由沿轴的两个瓣组成，有一个通过原子核的节平面。每个能级（n≥2）有三个 p 轨道：pₓ、pᵧ、pz，相互垂直。
d orbitals (l=2) have more complex shapes; five d orbitals exist for n≥3. Four of them (dxy, dxz, dyz, dx²–y²) have four lobes in planes, while dz² has two lobes and a doughnut-shaped ring. The table summarises:
d 轨道（l=2）具有更复杂的形状；对于 n≥3 存在五个 d 轨道。其中四个（dxy、dxz、dyz、dx²–y²）在平面中有四个瓣，而 dz² 有两个瓣和一个环形圈。下表总结：
| Sub-shell | l | Number of orbitals | Maximum electrons |
|---|---|---|---|
| s | 0 | 1 | 2 |
| p | 1 | 3 | 6 |
| d | 2 | 5 | 10 |
| f | 3 | 7 | 14 |
The shapes and orientations are crucial for understanding covalent bonding and hybridisation in later topics.
这些形状和取向对于理解后续内容中的共价键和杂化至关重要。
7. Energy levels and sublevels; the Aufbau principle | 能级与亚层；构造原理
In a multi-electron atom, the energy of an orbital depends on both n and l. Due to shielding and penetration effects, the order of filling is not simply 1s, 2s, 2p, 3s, 3p, 3d... Instead, the 4s sublevel is slightly lower in energy than 3d. The Aufbau principle states that electrons occupy the lowest energy orbitals available. The typical sequence is:
在多电子原子中，轨道能量同时取决于 n 和 l。由于屏蔽和穿透效应，电子填充的顺序并非简单的 1s、2s、2p、3s、3p、3d... 实际上，4s 亚层的能量略低于 3d。构造原理指出电子先填充能量最低的可用轨道。典型顺序为：
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s ...
This sequence can be remembered using a diagonal diagram or by following the periodic table's s, p, d, f blocks. Understanding this order is essential for writing correct electron configurations.
这个顺序可以用对角线图记忆，或者按元素周期表的 s、p、d、f 区来把握。理解这一顺序对于写出正确的电子构型至关重要。
8. Hund's rule and Pauli exclusion principle | 洪特规则与泡利不相容原理
No two electrons in an atom can have the same set of four quantum numbers (Pauli exclusion principle). This means an orbital can hold at most two electrons, and they must have opposite spins (↑↓).
原子中不能有两个电子具有完全相同的一组四个量子数（泡利不相容原理）。这意味着一个轨道最多容纳两个电子，并且它们必须自旋相反（↑↓）。
Hund' spins before pairing begins. For example, in a nitrogen atom (Z=7) the 2p³ arrangement is ↑ ↑ ↑, not ↑↓ ↑ or ↓ ↑ ↑ with a pair. This minimises electron-electron repulsion and gives lower energy.
洪特最大多重度规则指出，在填充简并轨道（能量相同的轨道，如三个 2p 轨道）时，电子在配对之前先以平行自旋单独占据每个轨道。例如，氮原子（Z=7）的 2p³ 排布为 ↑ ↑ ↑，而不是带有配对的 ↑↓ ↑ 或 ↓ ↑ ↑。这使电子间排斥最小，能量更低。
9. Writing electron configurations for atoms and ions (up to Z=36) | 书写原子和离子的电子构型（至Z=36）
6): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶. A shorthand notation uses the preceding noble gas core: [Ar] 4s² 3d⁶. For ions, electrons are added or removed according to the same energy order, but for transition metals, the 4s electrons are removed before 3d electrons when forming cations.
电子构型可以用完整符号书写，例如 Fe（Z=26）：1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶。简写形式使用前一周期的稀有气体原子实：[Ar] 4s² 3d⁶。对于离子，电子按照相同的能量顺序增加或移除，但对于过渡金属，形成阳离子时 4s 电子先于 3d 电子失去。
, electrons simply fill the next available orbital: O²⁻ (Z=8) becomes [He] 2s² 2p⁶ = [Ne].
因此，Fe²⁺ 为 [Ar] 3d⁶，Fe³⁺ 为 [Ar] 3d⁵。对于阴离子，电子直接填充下一个可用轨道：O²⁻（Z=8）变为 [He] 2s² 2p⁶ = [Ne]。
10. Exceptions to the Aufbau principle: Cr and Cu | 构造原理的例外：铬和铜
ed configuration of Cr is [Ar] 4s² 3d⁴, but the actual configuration is [Ar] 4s¹ 3d⁵. Similarly, the expected Cu configuration is [Ar] 4s² 3d⁹, but the real configuration is [Ar] 4s¹ 3d¹⁰.
构造原理有两个重要的例外：铬（Z=24）和铜（Z=29）。Cr 的预期构型为 [Ar] 4s² 3d⁴，但实际为 [Ar] 4s¹ 3d⁵。同样，Cu 的预期为 [Ar] 4s² 3d⁹，但实际为 [Ar] 4s¹ 3d¹⁰。
These exceptions arise because a half-filled d sub-shell (d⁵) and a completely filled d sub-shell (d¹⁰) confer extra stability. The energy gain achieved by promoting one 4s electron to the 3d sub-level outweighs the cost of the promotion.
这些例外产生的原因是半充满（d⁵）和全充满（d¹⁰）d 亚层具有额外的稳定性。将一个 4s 电子激发到 3d 亚层所获得的能量增益超过了激发所需的代价。
11. Ionisation energy: trends and evidence for sub-shells | 电离能：趋势与亚层存在的证据
sation energy generally increases due to increasing nuclear charge and similar shielding, making the outer electron harder to remove. Down a group, ionisation energy decreases because the outer electron is further from the nucleus and more shielded.
第一电离能是指从一摩尔气态原子中移去一摩尔电子所需的能量：X(g) → X⁺(g) + e⁻。在同一周期中，电离能通常递增，因为核电荷增加而屏蔽效应相似，使得外层电子更难移除。在同一族中，电离能递减，因为外层电子离核更远，受到的屏蔽更强。
However, there are small drops, for example from Mg (1s²2s²2p⁶3s²) to Al (3p¹) and from P (3p³) to S (3p⁴). These drops provide clear evidence for sub-shells: Al's electron is in a higher-energy 3p orbital, so less energy is needed. In S, the paired electron in a 3p orbital experiences repulsion, making it easier to remove.
然而，也存在小幅下降，例如从 Mg（3s²）到 Al（3p¹）以及从 P（3p³）到 S（3p⁴）。这些下降为亚层的存在提供了明确证据：Al 的外层电子处于能量更高的 3p 轨道，因此所需能量较少；在 S 中，3p 轨道中配对的电子受到排斥，使其更容易被移去。
Successive ionisation energies show large jumps when electrons are removed from a new inner shell, confirming the shell structure. For instance, removing the 5th electron from Si requires a huge jump because the 2p core electrons are much closer to the nucleus.
逐级电离能在移除新的内壳层电子时会显示大幅跃升，证实了壳层结构。例如，从 Si 中移去第五个电子需要巨大的能量跃升，因为 2p 核心电子离核近得多。
12. Electron configuration and chemical properties | 电子构型与化学性质
The electron configuration determines an element's chemical behaviour. Elements in the same group have similar outer-shell configurations, leading to similar reactivity and bonding patterns. For example, all alkali metals have ns¹ and readily lose one electron to form +1 ions.
电子构型决定了元素的化学行为。同一族元素具有相似的外层构型，从而产生相似的反应性和成键模式。例如，所有碱金属都具有 ns¹ 构型，容易失去一个电子形成 +1 离子。
Transition metals exhibit variable oxidation states because they can lose different numbers of 4s and 3d electrons. Magnetic properties also follow from unpaired electrons: paramagnetism occurs in species with unpaired electrons (e.g., O₂, Fe³⁺), while diamagnetism occurs when all electrons are paired.
过渡金属表现出可变的氧化态，因为它们可以失去不同数量的 4s 和 3d 电子。磁性也源自未成对电子：具有未成对电子的物种表现出顺磁性（如 O₂、Fe³⁺），而所有电子都配对时则表现出抗磁性。
Thus, mastering electron configurations enables you to predict oxidation states, magnetic behaviour, and the type of ions an element is likely to form—key skills for the IB examination.
因此，掌握电子构型使你能够预测氧化态、磁性行为以及元素可能形成的离子类型——这是 IB 考试的关键技能。
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