📚 Masses of elements and selected isotopes | 元素及选定同位素的质量
ction of mass can be converted into energy, as described by Einstein’s famous equation E = mc². This article examines how we define, measure, and apply atomic and isotopic masses, and how mass differences unlock enormous energy in stars and nuclear power.
在IB物理中，理解元素及其同位素的质量对于探索原子结构、核反应以及聚变和裂变等过程释放的能量至关重要。核层面的质量概念与日常经验不同，因为极小一部分质量可以转化为能量，正如爱因斯坦著名的方程 E = mc² 所描述的那样。本文将探讨如何定义、测量和运用原子质量与同位素质量，以及质量差异如何在恒星和核能中释放巨大能量。
1. Atomic mass unit (u) | 原子质量单位 (u)
cale. One unified atomic mass unit is defined as exactly 1/12 of the mass of a free carbon‑ antity makes it practical to handle the minuscule masses of protons, neutrons, and atoms without writing many powers of ten.
原子质量单位（符号 u）是在原子和核层面表示质量的标准单位。一个统一原子质量单位被精确定义为一个处于静止基态的自由碳‑12 原子质量的 1/12。其以千克表示的值约为 1.660539 × 10⁻²⁷ kg。这一微小量值可以方便地处理质子、中子和原子的极小质量，而不必反复书写十的多次幂。
2. The unified atomic mass unit and carbon‑12 | 统一原子质量单位与碳‑12
Historically, the atomic mass unit was linked first to hydrogen and then to oxygen. Since 1961, the internationally adopted standard has been based on the isotope ¹²C. By assigning exactly 12 u to a single ¹²C atom, all other atomic masses become relative to this reference. This choice also simplifies mass‑spectrometric measurements because carbon forms numerous stable molecular ions that serve as calibration points.
历史上，原子质量单位曾先后与氢和氧挂钩。自1961年起，国际采用的标准基于同位素 ¹²C。将一个 ¹²C 原子的质量严格定为 12 u，其他所有原子质量均以此为参照。这一选择也简化了质谱测量，因为碳能形成多种稳定的分子离子，可作为校准点。
3. Relative atomic mass (Aᵣ) | 相对原子质量 (Aᵣ)
Relative atomic mass, Aᵣ, is a dimensionless quantity that compares the average mass of an atom of an element to 1/12 of the mass of a ¹²C atom. For a monoisotopic element, Aᵣ equals the relative isotopic mass of the single naturally occurring isotope. For elements with several isotopes, Aᵣ is a weighted average that accounts for the natural abundances of each isotope. Therefore, Aᵣ values found in periodic tables are rarely integers.
相对原子质量 Aᵣ 是一个无量纲量，它表示某元素一个原子的平均质量与一个 ¹²C 原子质量的 1/12 的比值。对于单同位素元素，Aᵣ 等于唯一天然同位素的相对同位素质量。对于含有多种同位素的元素，Aᵣ 是考虑了各同位素天然丰度的加权平均值。因此，周期表中查到的 Aᵣ 值很少是整数。
4. Isotopic masses and relative isotopic mass | 同位素质量与相对同位素质量
Each nuclide possesses a precise isotopic mass, usually expressed in atomic mass units. For example, a neutral ¹⁶O atom has a mass of about 15.9949 u, while ¹⁷O has a mass of 16.9991 u. The relative isotopic mass is the dimensionless ratio of that isotopic mass to 1/12 of the mass of ¹²C; for ¹²C this ratio is exactly 12, for ¹⁶O it is about 15.9949. Modern mass spectrometers can determine these values to many decimal places, enabling very precise calculations of binding energies.
每种核素都具有一个精确的同位素质量，通常以原子质量单位表示。例如，一个中性的 ¹⁶O 原子质量约为 15.9949 u，而 ¹⁷O 的质量为 16.9991 u。相对同位素质量是该同位素质量与 ¹²C 原子质量 1/12 的无量纲比值；对于 ¹²C 比值为精确的 12，对于 ¹⁶O 约为 15.9949。现代质谱仪可将这些数值测至小数点后许多位，从而实现极精确的结合能计算。
5. Mass spectrometry and isotopic abundance | 质谱法与同位素丰度
A mass spectrometer ionises a sample and accelerates the ions through an electric field before passing them through a magnetic field, which deflects the ions according to their mass‑to‑charge ratio. By recording the relative ion currents, the instrument produces a mass spectrum showing peaks at different mass numbers. The height of each peak indicates the relative abundance of that isotope. This data is essential for determining the average atomic mass of an element.
质谱仪将样品电离，通过电场加速离子，再使之进入磁场；磁场根据离子的质荷比使其偏转。通过记录相对离子流强度，仪器产生一张质谱图，显示不同质量数位置的峰。每个峰的高度表明该同位素的相对丰度。这些数据是确定元素平均原子质量所必需的。
6. Calculating average atomic mass | 计算平均原子质量
The average atomic mass (used to obtain Aᵣ) is calculated by multiplying each isotopic mass by its fractional abundance and summing the products. For chlorine, which consists of ³⁵Cl (75.77 % abundance, mass 34.9689 u) and ³⁷Cl (24.23 % abundance, mass 36.9659 u), the computation is:
平均原子质量（用于求得 Aᵣ）通过将每种同位素质量乘以其丰度分数，再求和而得出。氯由 ³⁵Cl（丰度 75.77 %，质量 34.9689 u）和 ³⁷Cl（丰度 24.23 %，质量 36.9659 u）组成，其计算式为：
Aᵣ = (0.7577 × 34.9689 u) + (0.2423 × 36.9659 u) ≈ 35.45
Thus, the relative atomic mass of chlorine is approximately 35.45, matching the periodic‑table value. This weighted‑average concept is fundamental in IB Physics whenever isotopic data are provided.
因此，氯的相对原子质量约为 35.45，与周期表中的数值相符。每当题目给出同位素数据时，这一加权平均概念就是IB物理中的基础。
7. Mass defect and binding energy | 质量亏损与结合能
the mass defect, Δm. For a nucleus containing Z protons and N neutrons, Δm = (Z·mₚ + N·mₙ) – Δm c², using consistent units (e.g., Δm in kg gives E in joules; Δm in u gives E in MeV).
当质子和中子结合成原子核时，测得的原子核质量总是小于其单个核子质量之和。这一差值称为质量亏损，Δm。对于含有 Z 个质子和 N 个中子的原子核，Δm = (Z·mₚ + N·mₙ) – m(核)。亏损的质量被转化为将核子束缚在一起的能量。结合能由 E = Δm c² 给出，计算时需使用一致的单位（例如 Δm 以 kg 为单位时 E 为焦耳；Δm 以 u 为单位时 E 为 MeV）。
a free proton is 1.007276 u, and the mass of a free neutron is 1.008665 u. For the helium‑4 nucleus (⁴He, composed of 2 protons and 2 neutrons), the sum of the free‑ 31882 u. The measured mass of the ⁴He nucleus is 4.001506 u. The mass defect is 4.031882 u – 4.001506 u = 0.030376 u, which is equivalent to a binding energy of about 28.3 MeV.
一个自由质子的质量为 1.007276 u，一个自由中子的质量为 1.008665 u。对于氦‑4 核（⁴He，由 2 个质子和 2 个中子组成），自由核子质量之和为 2×1.007276 u + 2×1.008665 u = 4.031882 u。氦‑4 核的实测质量为 4.001506 u。质量亏损为 4.031882 u – 4.001506 u = 0.030376 u，这等效于约 28.3 MeV 的结合能。
8. Einstein’s mass–energy equivalence | 爱因斯坦质能等价
an object’s rest mass m to an equivalent rest energy. In nuclear physics, this equation explains how mass defect becomes binding energy. A convenient conversion factor is that 1 u of mass is equivalent to approximately 931.494 MeV of energy:
爱因斯坦公式 E = m c² 将物体的静止质量 m 与等价的静止能量联系起来。在核物理中，该方程解释了质量亏损如何成为结合能。一个方便的换算因子是：1 u 质量相当于约 931.494 MeV 能量：
1 u × c² = 931.494 MeV
dents are expected to use this factor in calculations of nuclear energy changes without re‑deriving it each time.
推导使用 1 u = 1.660539 × 10⁻²⁷ kg，c = 2.997925 × 10⁸ m s⁻¹，以及 1 eV = 1.602176 × 10⁻¹⁹ J。学生应能直接运用该因子进行核能量变化的计算，而无需每次都重新推导。
9. Binding energy per nucleon | 每个核子的结合能
B/A against A rises sharply for light nuclei, reaches a broad maximum around A = 56 (iron‑56 region) at about 8.8 MeV per nucleon, and then gently declines toward heavy nuclei.
将原子核的总结合能除以其质量数 A，即得每个核子的结合能 (B/A)。该值反映了原子核的稳定性：每个核子的结合能越高，原子核越稳定。B/A 对 A 的曲线在轻核处急剧上升，在 A ≈ 56（铁‑56 区域）达到约 8.8 MeV 每个核子的宽缓极大值，之后向重核方向缓慢下降。
This trend explains why energy can be released both by fusing light nuclei (fusion) and by splitting very heavy nuclei (fission): in both processes the products lie closer to the peak of the curve, so the total binding energy increases and the corresponding mass decrease appears as kinetic energy of the products.
这一趋势解释了为何使轻核聚合（聚变）和使极重核分裂（裂变）都能释放能量：在这两种过程中，产物都更靠近曲线的峰值，因此总结合能增加，对应的质量减少表现为产物动能。
10. Nuclear reactions and mass changes | 核反应与质量变化
rom the total mass after the reaction. The energy released or absorbed, often called the Q‑value, is given by Q = (initial mass – ndothermic). Mass is taken from accurate tables of nuclide masses, usually expressed in u.
在任何核反应中，反应前的总质量与反应后的总质量都存在差异。所释放或吸收的能量常称为 Q 值，由 Q = (初始质量 – 最终质量) × c² 计算。Q 为正代表释放能量（放热）；Q 为负代表需提供能量（吸热）。质量数据取自精确的核素质量表，通常以 u 为单位。
Example: the deuterium– 16049 u), ⁴He (4.002603 u), n (1.008665 u). The mass difference Δm = (2.014102 + 3.016049) – (4.002603 + 1.008665) = 0.018883 u. Multiplying by 931.494 MeV/u gives ≈ 17.59 MeV.
示例：氘‑氚聚变反应 ²H + ³H → ⁴He + n 释放约 17.6 MeV。采用精确质量：²H (2.014102 u)、³H (3.016049 u)、⁴He (4.002603 u)、n (1.008665 u)。质量差 Δm = (2.014102 + 3.016049) – (4.002603 + 1.008665) = 0.018883 u。乘以 931.494 MeV/u 可得 ≈ 17.59 MeV。
11. Application: fusion, fission, and energy release | 应用：聚变、裂变与能量释放
In nuclear fission, a heavy nucleus such as ²³⁵U absorbs a neutron and splits into lighter fragments plus several neutrons, releasing about 200 MeV per event. A typical fission reaction is:
在核裂变中，重核如 ²³⁵U 吸收一个中子后分裂成较轻的碎片及几个中子，每次事件释放约 200 MeV。一个典型的裂变反应为：
²³⁵U + ¹n → ¹⁴¹Ba + ⁹²Kr + 3¹n
eus, releasing about 26.7 MeV per cycle. Both applications rely entirely on the relation between mass and energy expressed through E = Δm c².
左侧总质量大于右侧总质量，质量亏损表现为裂变产物和中子的动能。这就是核电站和原子弹的原理。另一方面，聚变驱动着太阳：四个质子最终聚变为一个 ⁴He 核，每个循环释放约 26.7 MeV。这两种应用都完全依赖于通过 E = Δm c² 表达的质量与能量关系。
12. Summary and key points | 总结与关键点
Understanding the masses of elements and isotopes integrates several core IB Physics concepts: the atomic mass unit defined via ¹²C, relative atomic and isotopic masses, experimental determination by mass spectrometry, mass defect, and Einstein’s mass–energy equivalence. The binding energy per nucleon curve explains nuclear stability and the energy released in fusion and fission reactions. Mass differences, though tiny in absolute terms, release enormous energy because c² is a very large factor. Mastery of these ideas enables you to calculate energy changes in nuclear processes and to appreciate the origin of elements in stellar nucleosynthesis.
理解元素及同位素的质量融合了若干IB物理的核心概念：通过 ¹²C 定义的原子质量单位、相对原子质量与相对同位素质量、质谱实验测定、质量亏损以及爱因斯坦质能等价。每个核子结合能曲线解释了核稳定性以及聚变和裂变反应释放的能量。尽管质量差异的绝对数值极小，因为 c² 是一个非常大的因子，却能释放巨大能量。掌握这些概念后，你便能计算核过程中的能量变化，并理解恒星核合成中元素的起源。
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