📚 Mass Defect and Nuclear Binding Energy: Worked Example with Δm = 0.00641 × 10⁻²⁷ kg | 质量亏损与核结合能：以 Δm = 0.00641 × 10⁻²⁷ kg 为例
In AQA A-Level Physics, one of the most important ideas in nuclear physics is that mass can be converted into energy. When protons and neutrons join together to form a nucleus, the total mass of the nucleus is slightly smaller than the total mass of the individual nucleons. This mass difference is called the mass defect, and it is directly related to the binding energy of the nucleus. In this revision article, we will explore this idea step by step, using a specific numerical example: Δm = 0.00641 × 10⁻²⁷ kg.
在 AQA A-Level 物理中，核物理最重要的概念之一就是质量可以转化为能量。当质子和中子结合形成原子核时，原子核的总质量会略小于所有独立核子的质量之和。这个质量差值称为“质量亏损”，它与原子核的结合能直接相关。在本篇复习文章中，我们将一步步探讨这个概念，并使用一个具体的数值例子：Δm = 0.00641 × 10⁻²⁷ kg。
1. What Is Mass Defect? | 什么是质量亏损？
The mass defect is defined as the difference between the total mass of the separate nucleons and the mass of the bound nucleus. When nucleons are bound together by the strong nuclear force, some of their mass is released as energy according to Einstein’s equation. This missing mass is not lost; it is converted into the energy that holds the nucleus together.
质量亏损的定义是：独立的核子总质量与束缚态原子核质量之间的差值。当核子通过强核力结合在一起时，部分质量会依照爱因斯坦方程以能量的形式释放。这部分“消失”的质量并没有丢失，而是转化为将原子核维系在一起的能量。
For a nucleus with Z protons and N neutrons, we can write:
对于一个含 Z 个质子和 N 个中子的原子核，可以写成：
Δm = Z mₚ + N mₙ − M_nucleus
where mₚ is the proton mass, mₙ is the neutron mass, and M_nucleus is the measured mass of the nucleus.
其中 mₚ 是质子质量，mₙ 是中子质量，M_nucleus 是原子核的实测质量。
2. Binding Energy | 结合能
The binding energy of a nucleus is the minimum energy required to separate the nucleus completely into its individual protons and neutrons. It can also be understood as the energy released when the nucleus is formed from its separate nucleons. A larger binding energy means the nucleus is more stable.
原子核的结合能是将原子核完全拆分成独立质子和中子所需的最小能量。也可以理解为：当原子核由独立核子形成时释放出的能量。结合能越大，原子核越稳定。
Using the mass–energy equivalence, the binding energy E is:
根据质能等价关系，结合能 E 为：
E = Δm c²
Here, c is the speed of light in a vacuum, c = 3.00 × 10⁸ m s⁻¹.
其中 c 是真空中的光速，c = 3.00 × 10⁸ m s⁻¹。
3. Einstein’s Mass–Energy Equation | 爱因斯坦质能方程
Einstein’s equation E = mc² is the bridge between mass and energy in nuclear physics. Even a very small mass defect corresponds to a very large amount of energy, because c² is an enormous number.
爱因斯坦方程 E = mc² 是核物理中连接质量与能量的桥梁。即使非常微小的质量亏损，也对应着巨大的能量，因为 c² 是一个巨大的数值。
For example, if the mass defect is Δm = 0.00641 × 10⁻²⁷ kg, the energy released is calculated as follows:
例如，若质量亏损为 Δm = 0.00641 × 10⁻²⁷ kg，释放的能量计算如下：
ΔE = Δm c² = (6.41 × 10⁻³⁰ kg) × (3.00 × 10⁸ m s⁻¹)²
ΔE = 5.77 × 10⁻¹³ J
This result is in joules, the SI unit of energy. In nuclear physics, it is often more convenient to use mega-electronvolts (MeV), so we need a unit conversion.
这个结果以焦耳为单位，即能量的国际单位。在核物理中，使用兆电子伏特（MeV）通常更方便，因此我们需要进行单位换算。
4. Converting Joules to MeV | 将焦耳换算为兆电子伏特
The electronvolt is defined as the energy gained by an electron when it moves through a potential difference of one volt. The conversion factor is:
电子伏特的定义是：电子在通过 1 伏特电势差时所获得的能量。换算关系为：
1 eV = 1.60 × 10⁻¹⁹ J
Therefore:
因此：
1 MeV = 1.60 × 10⁻¹³ J
For our worked example:
对于我们的例题：
ΔE = 5.77 × 10⁻¹³ J ÷ (1.60 × 10⁻¹³ J MeV⁻¹)
ΔE = 3.60 MeV
So the binding energy of this nucleus is 3.60 MeV. Notice how directly the given mass defect leads to an answer in MeV using this conversion.
因此，该原子核的结合能为 3.60 MeV。注意，已知质量亏损通过这个换算可以非常直接地得出以 MeV 为单位的答案。
5. Binding Energy per Nucleon | 每核子结合能
The binding energy per nucleon is a measure of nuclear stability. It is calculated by dividing the total binding energy by the number of nucleons A:
每核子结合能是衡量原子核稳定性的指标。计算方法是将总结合能除以核子数 A：
Binding energy per nucleon = E / A
If we assume the nucleus in our example has A = 4 nucleons, then:
如果我们假设该原子核有 A = 4 个核子，则：
Binding energy per nucleon = 3.60 MeV ÷ 4 = 0.900 MeV
This relatively small value suggests a comparatively weak binding per nucleon. Most stable nuclei have binding energies per nucleon in the range of 7–9 MeV, with iron-56 being one of the most stable nuclei.
这个相对较小的数值表明每个核子的平均结合能较弱。大多数稳定原子核的每核子结合能约为 7–9 MeV，其中铁-56 是最稳定的原子核之一。
6. Nuclear Stability and the Mass Defect | 原子核稳定性与质量亏损
Generally, a greater mass defect means more energy has been released during formation, so the nucleus is more tightly bound. However, the mass defect alone does not tell the whole story; we must consider the binding energy per nucleon for comparison between different nuclei.
一般而言，质量亏损越大，意味着形成过程中释放的能量越多，原子核结合越紧密。然而，仅看质量亏损并不能了解全貌；比较不同原子核时，我们必须考虑每核子结合能。
Here is a short comparison of common quantities in nuclear calculations:
以下是在核计算中常见物理量的一张小对比表：
| Quantity | Symbol | Value |
| Speed of light | c | 3.00 × 10⁸ m s⁻¹ |
| Proton mass | mₚ | 1.673 × 10⁻²⁷ kg |
| Neutron mass | mₙ | 1.675 × 10⁻²⁷ kg |
| Atomic mass unit | u | 1.661 × 10⁻²⁷ kg |
| Energy equivalent of 1 u | — | 931.5 MeV |
7. Fusion and Fission | 核聚变与核裂变
Mass defect explains why fusion and fission release energy. In nuclear fusion, light nuclei combine to form a heavier nucleus. If the product nucleus has a higher binding energy per nucleon than the reactants, energy is released. In nuclear fission, a heavy nucleus splits into lighter nuclei that have higher binding energy per nucleon, again releasing energy.
质量亏损解释了为什么核聚变和核裂变会释放能量。在核聚变中，轻原子核结合成较重的原子核。如果产物原子核的每核子结合能高于反应物，就会释放能量。在核裂变中，重原子核分裂成较轻的原子核，而这些较轻的原子核每核子结合能更高，也会释放能量。
In both cases, the total mass of the products is less than the total mass of the original particles. The missing mass appears as kinetic energy of the products or as electromagnetic radiation.
在这两种过程中，产物总质量都小于原始粒子的总质量。亏损的质量以产物动能或电磁辐射的形式出现。
8. Why Use Both kg and u? | 为什么同时使用 kg 和 u？
In AQA examinations, mass values may be given in kilograms or in atomic mass units (u). You must be comfortable converting between them. The key conversions are:
在 AQA 考试中，质量数值可能以千克或原子质量单位（u）给出。你必须熟练掌握它们之间的换算。关键换算如下：
- 1 u = 1.661 × 10⁻²⁷ kg
- 1 u = 931.5 MeV/c²
- 1 MeV = 1.60 × 10⁻¹³ J
If a mass defect is given in u, multiply by 931.5 to obtain the binding energy in MeV directly. For example, a mass defect of 0.00386 u corresponds to:
如果质量亏损以 u 为单位，直接乘以 931.5 即可得到以 MeV 为单位的结合能。例如，质量亏损 0.00386 u 对应：
E = 0.00386 × 931.5 = 3.60 MeV
This is the same energy we obtained earlier from the mass defect in kilograms. Both approaches are acceptable in exam answers, as long as you show your working clearly.
这与我们之前用千克质量亏损计算得到的能量相同。在考试答案中，两种方法都可以接受，只要步骤清晰即可。
9. AQA Exam Tips | AQA 考试提示
When answering mass defect and binding energy questions, follow these suggestions:
作答质量亏损与结合能题目时，建议遵循以下几点：
- Write down the formula E = Δm c² explicitly before substitution.
- Use standard units: convert mass to kg and energy to J before further conversion to MeV.
- Remember that c² = 9.00 × 10¹⁶ m² s⁻².
- Pay attention to significant figures; usually match the data given in the question.
- State whether the answer is the total binding energy or the binding energy per nucleon.
在代入数据之前，明确写出公式 E = Δm c²。
使用标准单位：先将质量换算为 kg，能量换算为 J，再进一步换算为 MeV。
记住 c² = 9.00 × 10¹⁶ m² s⁻²。
注意有效数字；通常与题目中给出的数据保持一致。
说明答案是总结合能还是每核子结合能。
10. Quick Practice Question | 快速练习题
A nucleus with mass number A = 4 has a mass defect of 0.00641 × 10⁻²⁷ kg. Calculate the binding energy per nucleon in MeV.
一个质量数 A = 4 的原子核，其质量亏损为 0.00641 × 10⁻²⁷ kg。计算其每核子结合能，单位为 MeV。
Worked solution:
解答过程：
ΔE = Δm c² = (6.41 × 10⁻³⁰)(9.00 × 10¹⁶) = 5.77 × 10⁻¹³ J
ΔE = 5.77 × 10⁻¹³ ÷ (1.60 × 10⁻¹³) = 3.61 MeV
Binding energy per nucleon = 3.61 ÷ 4 = 0.902 MeV
So the nucleus has a binding energy of about 3.61 MeV total, or 0.902 MeV per nucleon.
因此，该原子核的总结合能约为 3.61 MeV，即每核子结合能约为 0.902 MeV。
Understanding mass defect and binding energy is essential for AQA A-Level Physics. Once you can confidently convert between kilograms, joules, and mega-electronvolts, you will be able to solve a wide range of nuclear physics problems.
理解质量亏损与结合能是 AQA A-Level 物理的关键。一旦你能够熟练地在千克、焦耳和兆电子伏特之间进行换算，就能解决各类核物理问题。
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