Mass and Energy: Einstein's E = mc² | 质量与能量：爱因斯坦的 E = mc²

📚 Mass and Energy: Einstein’s E = mc² | 质量与能量：爱因斯坦的 E = mc²

Albert Einstein’s mass-energy equivalence formula, E = mc², stands as one of the most profound and iconic relationships in all of physics. In the A-Level CIE Physics syllabus, this principle forms the cornerstone of nuclear physics, explaining how tiny changes in mass can release enormous quantities of energy in nuclear reactions. This article explores the meaning, derivation context, and applications of this revolutionary equation.

阿尔伯特·爱因斯坦的质能等价公式 E = mc² 是整个物理学中最深刻、最具标志性的关系之一。在 CIE A-Level 物理考纲中，这一原理构成了核物理的基石，解释了核反应中微小质量变化如何释放出巨大的能量。本文将深入探讨这一革命性方程的含义、推导背景及其应用。

1. The Meaning of E = mc² | E = mc² 的含义

The equation E = mc² states that energy (E) and mass (m) are interchangeable — they are two different manifestations of the same underlying quantity. The constant c (the speed of light in a vacuum, approximately 3.0 × 10⁸ m s⁻¹) acts as the conversion factor. Because c² is an enormous number (9 × 10¹⁶ m² s⁻²), even a tiny amount of mass corresponds to a colossal amount of energy.

方程 E = mc² 表明能量（E）和质量（m）是可以相互转化的——它们是同一基本量的两种不同表现形式。常数 c（真空中的光速，约为 3.0 × 10⁸ m s⁻¹）充当转换因子。由于 c² 是一个巨大的数值（9 × 10¹⁶ m² s⁻²），即使极微小的质量也对应着巨大的能量。

The equation tells us that mass is not conserved on its own in nuclear reactions; rather, the total mass-energy of an isolated system is conserved. When a system loses mass, that mass is converted into energy, and conversely, energy can be converted into mass.

该方程告诉我们，在核反应中质量本身并不守恒；相反，孤立系统的总质能是守恒的。当系统失去质量时，这部分质量转化为能量；反之，能量也可以转化为质量。

2. The Equivalence Principle | 等价原理

The equivalence principle asserts that mass is a form of energy. This means that any object with mass has intrinsic “rest energy” simply by virtue of existing. For example, a 1 kg mass has a rest energy of E = (1)(3 × 10⁸)² = 9 × 10¹⁶ J, which is enough to supply a typical household for millions of years.

等价原理断言质量是能量的一种形式。这意味着任何具有质量的物体仅仅因其存在就具有内在的”静能量”。例如，1 kg 的质量具有 E = (1)(3 × 10⁸)² = 9 × 10¹⁶ J 的静能量，足以供一个普通家庭使用数百万年。

In classical mechanics, energy and mass were treated as completely separate quantities. Einstein’s insight unified them into a single conservation law. This equivalence applies universally — to particles, nuclei, and macroscopic objects alike.

在经典力学中，能量和质量被视为完全独立的量。爱因斯坦的洞见将它们统一到同一条守恒定律之中。这种等价关系普遍适用——无论是粒子、原子核还是宏观物体。

3. Mass Defect | 质量亏损

When protons and neutrons (collectively called nucleons) combine to form a nucleus, the mass of the resulting nucleus is always less than the sum of the masses of its individual nucleons. This difference is called the mass defect (Δm). This “missing” mass has been converted into the energy that binds the nucleus together.

当质子和中子（统称为核子）结合形成原子核时，所得原子核的质量总是小于其各核子质量之和。这个差值称为质量亏损（Δm）。这部分”缺失”的质量已转化为将原子核束缚在一起的结合能。

Δm = Z mₚ + N mₙ − m_nucleus

Where Z is the number of protons, N is the number of neutrons, mₚ is the proton mass (1.67262 × 10⁻²⁷ kg), mₙ is the neutron mass (1.67493 × 10⁻²⁷ kg), and m_nucleus is the measured mass of the nucleus. The mass defect is always positive — nuclei are always lighter than their constituent parts.

其中 Z 是质子数，N 是中子数，mₚ 是质子质量（1.67262 × 10⁻²⁷ kg），mₙ 是中子质量（1.67493 × 10⁻²⁷ kg），m_nucleus 是原子核的实测质量。质量亏损始终为正——原子核总是比其组成粒子的质量之和小。

4. Binding Energy | 结合能

The binding energy of a nucleus is the energy required to completely separate a nucleus into its individual protons and neutrons. According to E = mc², the binding energy (E_B) is given by:

原子核的结合能是将原子核完全拆分成独立的质子和中子所需的能量。根据 E = mc²，结合能（E_B）由下式给出：

E_B = Δm c²

For example, the mass defect for a helium-4 nucleus (²He₄) is approximately 0.0304 u. Converting to kilograms and multiplying by c² gives a binding energy of about 28.3 MeV. This is the energy that holds the two protons and two neutrons together in the alpha particle.

例如，氦-4 原子核（²He₄）的质量亏损约为 0.0304 u。换算为千克并乘以 c²，得到约 28.3 MeV 的结合能。这就是将两个质子和两个中子束缚在 α 粒子中的能量。

Binding energy per nucleon is the binding energy divided by the mass number A. This quantity tells us about nuclear stability — higher binding energy per nucleon means a more stable nucleus. Iron-56 (⁵⁶Fe₂₆) has the highest binding energy per nucleon (about 8.8 MeV), making it the most stable nucleus.

平均结合能（每个核子的结合能）是结合能除以质量数 A。该量反映了原子核的稳定性——平均结合能越高，原子核越稳定。铁-56（⁵⁶Fe₂₆）具有最高的平均结合能（约 8.8 MeV），因此是最稳定的原子核。

5. Binding Energy Curve | 结合能曲线

The graph of binding energy per nucleon against nucleon number A provides crucial insights into nuclear stability and energy release. For light nuclei, the binding energy per nucleon rises sharply with A, peaking around iron (A ≈ 56). Beyond iron, it gradually decreases.

平均结合能随核子数 A 变化的曲线为核稳定性和能量释放提供了重要的见解。对于轻核，平均结合能随 A 急剧上升，在铁（A ≈ 56）附近达到峰值。在铁之后，它逐渐下降。

Nucleus	Mass Number A	Binding Energy per Nucleon (MeV)
Hydrogen (¹H₁)	1	0
Helium (⁴He₂)	4	7.07
Carbon (¹²C₆)	12	7.68
Iron (⁵⁶Fe₂₆)	56	8.80
Uranium (²³⁸U₉₂)	238	7.57

This curve explains why energy is released in both nuclear fusion (combining light nuclei) and nuclear fission (splitting heavy nuclei). In both processes, the products are closer to iron on the curve, meaning they have higher binding energy per nucleon, and the excess energy is released.

这条曲线解释了为什么核聚变（轻核结合）和核裂变（重核分裂）都会释放能量。在这两个过程中，产物在曲线上都更接近铁的位置，即具有更高的平均结合能，多余的能量便以动能或其他形式释放出来。

6. Nuclear Fission | 核裂变

Nuclear fission occurs when a heavy nucleus, such as uranium-235 (²³⁵U₉₂) or plutonium-239, absorbs a neutron and splits into two smaller nuclei, typically releasing 2 or 3 additional neutrons and a large amount of energy. The total mass of the fission products and neutrons is less than the mass of the original nucleus and neutron.

核裂变发生在重核（如铀-235（²³⁵U₉₂）或钚-239）吸收一个中子后分裂成两个较小的原子核，通常释放 2 或 3 个额外中子和大量能量。裂变产物和中子的总质量小于原始原子核和吸收中子的质量。

A typical fission reaction of uranium-235 can be represented as:

铀-235 的典型裂变反应可表示为：

²³⁵U₉₂ + ¹n₀ → ¹⁴¹Ba₅₆ + ⁹²Kr₃₆ + 3¹n₀ + Energy

The mass difference in this reaction is approximately 0.2 u, which corresponds to about 200 MeV of energy released per fission event. This energy appears as kinetic energy of the fission products, kinetic energy of the neutrons, and gamma radiation.

该反应中的质量差约为 0.2 u，每次裂变事件释放约 200 MeV 的能量。这些能量表现为裂变产物的动能、中子的动能以及 γ 辐射。

7. Nuclear Fusion | 核聚变

Nuclear fusion is the process by which two light nuclei combine to form a heavier nucleus. This is the process that powers the Sun and other stars. For example, in the proton-proton chain, hydrogen nuclei fuse to form helium, releasing enormous energy in the process.

核聚变是两个轻核结合形成一个较重原子核的过程。这是太阳和其他恒星的能量来源。例如，在质子-质子链反应中，氢核融合形成氦，在此过程中释放出巨大的能量。

A simple fusion reaction studied at A-Level is the deuterium-tritium reaction:

A-Level 中常考的一个简单聚变反应是氘-氚反应：

²H₁ + ³H₁ → ⁴He₂ + ¹n₀ + 17.6 MeV

The mass of the products is slightly less than the mass of the reactants. The mass defect of approximately 0.0188 u corresponds to 17.6 MeV of energy. Per kilogram of fuel, fusion releases far more energy than fission, which is why it is the ultimate goal of fusion energy research.

产物的质量略小于反应物的质量。约 0.0188 u 的质量亏损对应 17.6 MeV 的能量。每千克燃料，聚变释放的能量远多于裂变，这也是聚变能研究成为终极目标的原因。

8. Calculations Using E = Δmc² | 利用 E = Δmc² 计算

At CIE A-Level, you must be able to perform calculations involving mass-energy equivalence. The key steps are:

在 CIE A-Level 考试中，你必须能够进行涉及质能等价的计算。关键步骤如下：

Determine the mass defect Δm in atomic mass units (u) by comparing the mass of the products with the mass of the reactants
Convert the mass defect from u to kilograms using 1 u = 1.661 × 10⁻²⁷ kg
Apply E = Δm c² using c = 3.0 × 10⁸ m s⁻¹
Convert the energy to electronvolts (eV) using 1 eV = 1.6 × 10⁻¹⁹ J, or use 1 u = 931.5 MeV/c²

用原子质量单位（u）确定质量亏损 Δm：比较产物质量与反应物质量。

将质量亏损从 u 转换为千克：利用 1 u = 1.661 × 10⁻²⁷ kg。

应用 E = Δm c²，其中 c = 3.0 × 10⁸ m s⁻¹。

将能量转换为电子伏特（eV）：利用 1 eV = 1.6 × 10⁻¹⁹ J，或直接使用 1 u = 931.5 MeV/c²。

Worked Example | 例题：

A neutron is absorbed by a uranium-235 nucleus, producing barium-141, krypton-92, and 3 neutrons. The masses are: m(U-235) = 235.044 u, m(Ba-141) = 140.914 u, m(Kr-92) = 91.926 u, m(n) = 1.009 u. Calculate the energy released.

一个中子被铀-235 原子核吸收，产生钡-141、氪-92 和 3 个中子。质量分别为：m(U-235) = 235.044 u，m(Ba-141) = 140.914 u，m(Kr-92) = 91.926 u，m(n) = 1.009 u。计算释放的能量。

Total mass of reactants = 235.044 + 1.009 = 236.053 u

Total mass of products = 140.914 + 91.926 + 3(1.009) = 235.867 u

Mass defect Δm = 236.053 − 235.867 = 0.186 u

In kilograms: Δm = 0.186 × 1.661 × 10⁻²⁷ = 3.089 × 10⁻²⁸ kg

Energy released: E = (3.089 × 10⁻²⁸)(3.0 × 10⁸)² = 2.78 × 10⁻¹¹ J ≈ 173 MeV

反应物总质量 = 235.044 + 1.009 = 236.053 u

产物总质量 = 140.914 + 91.926 + 3(1.009) = 235.867 u

质量亏损 Δm = 236.053 − 235.867 = 0.186 u

质量（千克）：Δm = 0.186 × 1.661 × 10⁻²⁷ = 3.089 × 10⁻²⁸ kg

释放能量：E = (3.089 × 10⁻²⁸)(3.0 × 10⁸)² = 2.78 × 10⁻¹¹ J ≈ 173 MeV

9. Units and Conversions | 单位与换算

Mastering unit conversions is essential for success in exam questions on mass-energy equivalence. The atomic mass unit (u) is convenient because it is defined so that one atom of carbon-12 has a mass of exactly 12 u.

掌握单位换算是回答质能等价考题的关键。原子质量单位（u）使用起来很方便，因为它被定义为碳-12 一个原子的质量恰为 12 u。

Quantity	Conversion
1 atomic mass unit	1 u = 1.661 × 10⁻²⁷ kg
Energy equivalent of 1 u	1 u = 931.5 MeV/c²
1 electronvolt	1 eV = 1.6 × 10⁻¹⁹ J
1 mega-electronvolt	1 MeV = 1.6 × 10⁻¹³ J
Speed of light	c = 3.0 × 10⁸ m s⁻¹

When using the equation E = Δm c², always ensure that mass is in kilograms, energy in joules, and speed in metres per second. Alternatively, if mass is given in u, you may convert directly to energy in MeV using the factor 931.5 MeV/u, provided you focus only on the mass difference.

使用 E = Δm c² 时，务必确保质量以千克为单位、能量以焦耳为单位、速度以米每秒为单位。另外，如果质量以 u 表示，也可以直接用 931.5 MeV/u 的换算因子将质量亏损转换为能量（以 MeV 为单位）。

10. Applications: Nuclear Power and Beyond | 应用：核能及其他领域

Mass-energy equivalence underpins several major technologies and natural phenomena. In nuclear power stations, the controlled fission of uranium or plutonium releases thermal energy from mass defect, which heats water to produce steam that drives turbines to generate electricity. A typical reactor converts roughly 1 gram of mass into energy per day of operation.

质能等价关系支撑着多项重大技术和自然现象。在核电站中，铀或钚的可控裂变从质量亏损中释放热能，加热水产生蒸汽，驱动涡轮发电。一个典型反应堆每天运行大约将 1 克质量转化为能量。

In medicine, positron emission tomography (PET) scans rely on the annihilation of an electron and a positron. When a positron meets an electron, both particles completely disappear and their combined mass (2 × 9.11 × 10⁻³¹ kg) is converted entirely into gamma photons, each carrying 0.511 MeV of energy. These gamma rays are detected to produce diagnostic images.

在医学领域，正电子发射断层扫描（PET）依赖于电子和正电子的湮灭。当正电子与电子相遇时，两个粒子完全消失，它们的总质量（2 × 9.11 × 10⁻³¹ kg）完全转化为 γ 光子，每个光子携带 0.511 MeV 的能量。通过探测这些 γ 射线来产生诊断图像。

In astrophysics, the Sun converts about 4 million tonnes of mass into energy every second through fusion. This energy sustains life on Earth and drives our climate system. Without the mass-energy equivalence of E = mc², the Sun’s energy output would be impossible to explain.

在天体物理学中，太阳每秒钟通过聚变将约 400 万吨质量转化为能量。这种能量维持着地球上的生命并驱动着我们的气候系统。没有 E = mc² 所揭示的质能等价关系，太阳的能量输出将无法解释。

11. Common Exam Mistakes | 常见考试误区

Students frequently lose marks on mass-energy questions for several avoidable reasons. Being aware of these pitfalls will help you perform better in your CIE examinations.

学生在质能题目上失分往往源于几个可以避免的原因。了解这些陷阱将帮助你在 CIE 考试中取得更好的成绩。

Unit errors: Forgetting to convert u to kg before substituting into E = mc², or mixing up J and eV / MeV units in the final answer
Using c instead of c²: Many students forget to square the speed of light, leading to answers that are wrong by a factor of 3 × 10⁸
Mass of reactants and products: Forgetting to include the masses of all neutrons released in fission, or omitting the absorbed neutron from the reactant side
Sign conventions: Confusing mass defect with mass gain, or stating binding energy with the wrong sign
Significant figures: Using insufficient significant figures in intermediate steps, leading to inaccurate final answers

单位错误：在代入 E = mc² 之前忘记将 u 转换为 kg，或者在最终答案中混淆 J 和 eV/MeV。

使用 c 而不是 c²：许多学生忘记将光速平方，导致答案差了 3 × 10⁸ 倍。

反应物与产物质量：忘记计入裂变释放的所有中子质量，或在反应物一侧遗漏被吸收的中子。

符号约定：混淆质量亏损与质量增加，或将结合能的符号写错。

有效数字：中间步骤有效数字不足，导致最终答案不准确。

12. Summary and Key Takeaways | 总结与核心要点

Mass and energy are two facets of the same physical reality, connected by Einstein’s famous equation E = mc². The immense magnitude of c² means that even miniscule mass changes produce enormous energy changes. In nuclear physics, this principle manifests through mass defect, which measures the mass difference between a nucleus and its constituent nucleons. This mass defect converts directly into binding energy.

质量与能量是同一物理实在的两个方面，由爱因斯坦著名方程 E = mc² 联系。c² 的巨大数值意味着即使极其微小的质量变化也能产生巨大的能量变化。在核物理中，这一原理通过质量亏损体现——质量亏损衡量原子核与其组成核子之间的质量差异。这种质量亏损直接转化为结合能。

For the CIE A-Level examination, you must be comfortable with: calculating mass defect in u and kg, converting between mass and energy using both E = Δm c² and the 931.5 MeV/u conversion factor, explaining the binding energy curve and its implications for fission and fusion, and performing step-by-step calculations for nuclear reactions. Master these skills, and you will handle any mass-energy question with confidence.

对于 CIE A-Level 考试，你必须熟练掌握：以 u 和 kg 计算质量亏损、使用 E = Δm c² 和 931.5 MeV/u 两种转换因子在质量与能量之间换算、解释结合能曲线及其对裂变和聚变的含义、以及逐步完成核反应的能量计算。掌握这些技能，你就能自信地应对任何质能相关题目。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mass and Energy: Einstein’s E = mc² | 质量与能量：爱因斯坦的 E = mc²