📚 A-Level Physics Unit 5 Insert Jan 20: Key Concepts Explained | A-Level 物理 Unit 5 插入材料 (2020年1月) 概念解析
This article breaks down the essential equations, constants and physical ideas that appear in the A-Level Physics Unit 5 Data and Formulae Insert for the January 2020 examination. Mastering these concepts will give you the confidence to handle questions on thermal physics, nuclear processes and astrophysics.
本文详细梳理了2020年1月A-Level物理Unit 5考试插入材料中出现的核心方程、常数与物理思想。彻底掌握这些概念，能让你从容应对热力学、核过程以及天体物理的相关考题。
1. Ideal Gas Laws and Kinetic Theory | 理想气体定律与分子运动论
The insert provides two forms of the ideal gas equation: pV = nRT, which uses the number of moles n, and pV = NkT, which uses the number of molecules N. The Boltzmann constant k is given as 1.38 × 10⁻²³ J K⁻¹ and the Avogadro constant Nₐ as 6.02 × 10²³ mol⁻¹, linking the molecular and molar pictures.
插入材料中给出了理想气体方程的两种形式：利用摩尔数 n 的 pV = nRT 和利用分子数 N 的 pV = NkT。玻尔兹曼常数 k 取为 1.38 × 10⁻²³ J K⁻¹，阿伏伽德罗常数 Nₐ 为 6.02 × 10²³ mol⁻¹，将微观分子图像与宏观物质的量联系起来。
The formula for the mean translational kinetic energy of a single molecule, ½ m⟨c²⟩ = (3/2) kT, tells us that temperature is a direct measure of the average random kinetic energy of particles. For a mole of gas the total internal kinetic energy becomes (3/2) RT.
单个分子的平均平动动能公式 ½ m⟨c²⟩ = (3/2) kT 向我们表明，温度是粒子平均无规则动能的直接量度。对于一摩尔气体，总的内部动能就变为 (3/2) RT。
2. Internal Energy and the First Law of Thermodynamics | 内能与热力学第一定律
The insert lists the first law as ΔU = Q + W (or sometimes ΔU = Q − W depending on sign convention). In the sign convention used here, W is the work done on the system, so when a gas is compressed, W is positive and internal energy increases.
插入材料中热力学第一定律写为 ΔU = Q + W（根据符号约定也有 ΔU = Q − W 的形式）。在这里使用的符号约定中，W 代表外界对系统所做的功，因此气体被压缩时 W 为正，内能增加。
For a gas expanding at constant pressure, the work done is W = −pΔV (with the sign adjusted to the chosen convention). You can combine this with the ideal gas law to calculate energy transfers in isobaric changes. The insert also provides the specific heat capacity equations Q = mcΔθ and latent heat Q = mL, both essential for calorimetry and phase-change calculations.
对于恒压膨胀的气体，做功可表示为 W = −pΔV（符号根据约定调整）。你可以将此与理想气体定律结合，计算等压变化中的能量转移。插入材料还提供了比热容方程 Q = mcΔθ 和潜热方程 Q = mL，这两者对于量热学和相变计算必不可少。
3. Radioactive Decay and Half-Life | 放射性衰变与半衰期
The exponential decay law is given as N = N₀ e⁻λt, where λ is the decay constant. The activity A follows the same pattern: A = λN, and its decay is A = A₀ e⁻λt. The insert shows that activity is measured in becquerels (Bq), where 1 Bq = 1 decay per second.
插入材料中指数衰变规律写为 N = N₀ e⁻λt，λ 是衰变常数。活度 A 遵循相同的规律：A = λN，且它的衰减为 A = A₀ e⁻λt。插入材料表明活度的单位是贝克勒尔（Bq），1 Bq 等于每秒一次衰变。
The relationship between half-life T½ and decay constant, T½ = ln2 / λ, is one of the most frequently used expressions. Make sure you can derive it from N = N₀/2 and solve for time. Also useful is the number of undecayed nuclei remaining after an integer number of half-lives: N = N₀ (1/2)ⁿ, which follows directly from the exponential law.
半衰期 T½ 与衰变常数的关系 T½ = ln2 / λ 是最常用的表达式之一。务必掌握如何由 N = N₀/2 推导出它。同样有用的是经过整数个半衰期后剩余未衰变原子核的数量：N = N₀ (1/2)ⁿ，它直接由指数规律导出。
4. Mass-Energy Equivalence and Binding Energy | 质能等价与结合能
The insert reminds you of Einstein’s famous equation E = mc², usually applied with energy in joules, mass in kilograms and c = 3.00 × 10⁸ m s⁻¹. For nuclear energies it is often more convenient to use the unified atomic mass unit: 1 u = 931.5 MeV of energy equivalent.
插入材料提醒你牢记爱因斯坦著名的方程 E = mc²，通常使用时能量以焦耳为单位，质量以千克为单位，c = 3.00 × 10⁸ m s⁻¹。在核能计算中，使用原子质量单位往往更方便：1 u = 931.5 MeV 的能量当量。
The binding energy of a nucleus is the energy required to separate it into its individual nucleons. It is found from the mass defect Δm: binding energy = Δm c². The average binding energy per nucleon, binding energy / A, is a guide to stability; nuclei with values around 8.7 MeV per nucleon are the most stable.
原子核的结合能是将它拆散成单个核子所需的能量。它由质量亏损 Δm 求出：结合能 = Δm c²。平均每个核子的结合能，即结合能除以 A，是稳定性的指标；每个核子结合能约 8.7 MeV 的原子核最为稳定。
5. Nuclear Size and Density | 原子核的大小与密度
The insert gives the nuclear radius formula R = R₀ A^(1/3), with R₀ ≈ 1.2 fm (1 fm = 1 × 10⁻¹⁵ m). This shows that nuclear volume is proportional to the mass number A, implying that all nuclei have roughly the same density, about 2.3 × 10¹⁷ kg m⁻³.
插入材料给出了原子核半径公式 R = R₀ A^(1/3)，其中 R₀ ≈ 1.2 fm（1 fm = 1 × 10⁻¹⁵ m）。这表明原子核体积与质量数 A 成正比，意味着所有原子核的密度大致相同，约为 2.3 × 10¹⁷ kg m⁻³。
When answering questions, you may need to estimate the density of nuclear matter using ρ = mass / volume = (A × u) / (4/3 π R³). Substituting R = R₀ A^(1/3) causes A to cancel, confirming that nuclear density is independent of A. This constant density is a key piece of evidence for the strong nuclear force having a short range.
答题时你可能需要利用 ρ = 质量 / 体积 = (A × u) / (4/3 π R³) 估算核物质密度。代入 R = R₀ A^(1/3) 后 A 会消去，证实核密度与 A 无关。这种恒定的密度是强核力具有短程性的重要证据。
6. Stellar Luminosity and Black-Body Radiation | 恒星光度与黑体辐射
The insert provides the Stefan–Boltzmann law for a black-body radiator: L = σAT⁴, where L is luminosity, A is surface area, T is absolute temperature and σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. For a spherical star, L = 4πR²σT⁴, allowing you to calculate the star’s radius or effective temperature from its observed luminosity.
插入材料提供了黑体辐射的斯特藩–玻尔兹曼定律：L = σAT⁴，其中 L 为光度，A 为表面积，T 为绝对温度，σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴。对于球形恒星，有 L = 4πR²σT⁴，这让你能够从观测光度推算出恒星的半径或有效温度。
Black-body radiation concepts also help explain why stars appear different colours. A star’s peak wavelength shifts with temperature, and the insert includes Wien’s displacement law: λmax T = 2.898 × 10⁻³ m K. Hotter stars emit more blue light, while cooler stars appear red.
黑体辐射概念还有助于解释恒星为何呈现不同颜色。恒星的峰值波长随温度移动，插入材料中包含了 维恩位移定律：λmax T = 2.898 × 10⁻³ m K。温度更高的恒星发出更多蓝光，而温度较低的恒星则呈现红色。
7. Using Wien’s Law and Stefan–Boltzmann Together | 维恩定律与斯特藩-玻尔兹曼定律的联合应用
Many Unit 5 questions ask you to combine Wien’s law with the Stefan–Boltzmann law. For example, if you know the peak wavelength of a star, you can find its surface temperature using λmax T = constant, and then insert that temperature into L = 4πR²σT⁴ to find the radius or luminosity.
许多Unit 5试题要求你联合运用维恩定律和斯特藩–玻尔兹曼定律。例如，若已知一颗恒星的峰值波长，你可以通过 λmax T = 常数求出其表面温度，然后将该温度代入 L = 4πR²σT⁴ 计算半径或光度。
You should also be able to compare two stars. Writing L₁/L₂ = (R₁/R₂)² (T₁/T₂)⁴ lets you find relative sizes or temperatures without needing absolute values. This ratio technique saves time and reduces calculator errors.
你还应掌握如何比较两颗恒星。写出 L₁/L₂ = (R₁/R₂)² (T₁/T₂)⁴ 后，无需知道绝对值就能求出相对大小或温度。这一比值技巧能节省时间并减少计算器错误。
8. Hubble’s Law and the Expanding Universe | 哈勃定律与宇宙膨胀
The insert states Hubble’s law as v = H₀ d, where v is the recessional velocity of a galaxy, d is its distance from us and H₀ is the Hubble constant. The value of H₀ is given as approximately 2.2 × 10⁻¹⁸ s⁻¹ (or about 68 km s⁻¹ Mpc⁻¹ in astronomical units). This linear relationship is the cornerstone of evidence for an expanding universe.
插入材料中哈勃定律写为 v = H₀ d，其中 v 是星系的退行速度，d 是该星系与我们的距离，H₀ 是哈勃常数。H₀ 的值约为 2.2 × 10⁻¹⁸ s⁻¹（或天文单位下约 68 km s⁻¹ Mpc⁻¹）。这一线性关系是宇宙正在膨胀的关键证据。
The age of the Universe can be estimated from t ≈ 1 / H₀, assuming a constant expansion rate. This gives a rough value of 4.4 × 10¹⁷ s, or about 14 billion years. You may also need to link Hubble’s law with the cosmological redshift z: for low speeds, z ≈ v / c, so v ≈ zc and zc ≈ H₀ d.
假设膨胀速率恒定，宇宙年龄可以由 t ≈ 1 / H₀ 估算，得到约 4.4 × 10¹⁷ 秒，即大约 140 亿年。你可能还需要将哈勃定律与宇宙学红移 z 联系起来：低速下 z ≈ v / c，因此 v ≈ zc 且 zc ≈ H₀ d。
9. Interpreting Insert Constants and Unit Conversions | 插入材料常数的解读与单位换算
The insert lists a range of physical constants, such as the Planck constant h = 6.63 × 10⁻³⁴ J s, the speed of light c = 3.00 × 10⁸ m s⁻¹, and the electron charge e = 1.60 × 10⁻¹⁹ C. Always check that the units you use match those implied by the constants, especially when converting between joules and electronvolts (1 eV = 1.60 × 10⁻¹⁹ J).
插入材料列出了许多物理常数，如普朗克常数 h = 6.63 × 10⁻³⁴ J s，光速 c = 3.00 × 10⁸ m s⁻¹，以及电子电荷 e = 1.60 × 10⁻¹⁹ C。务必检查你所用的单位是否与常数所暗示的单位一致，特别是在焦耳和电子伏特之间转换时（1 eV = 1.60 × 10⁻¹⁹ J）。
For nuclear calculations, the mass of the electron, proton and neutron are provided in both kilograms and atomic mass units. Remember to use consistent units when calculating mass defect: either work entirely in u and convert to MeV using 931.5 MeV/u, or convert all masses to kg and use E = mc² with c in m s⁻¹.
对于核计算，插入材料以千克和原子质量单位两种形式给出了电子、质子和中子的质量。计算质量亏损时切记单位要统一：要么全部使用 u 并利用 931.5 MeV/u 转换为能量，要么将所有质量转换为 kg 并代入 E = mc²，其中 c 以 m s⁻¹ 为单位。
10. Practical Tips for Using the Insert in the Exam | 考试中使用插入材料的实用技巧
During the exam, the insert is your quick-reference tool. Don’t waste time memorising every constant — instead, practise locating them rapidly. Highlight the equations you find difficult and annotate a sample insert at home so that you can navigate it instinctively under timed conditions.
在考试中，插入材料就是你的快速参考工具。不要把时间浪费在记忆每一个常数上，而是要练习快速定位它们。把难记的方程高亮标注，在家里的样卷上做批注，这样你在限时条件下就能本能地找到所需信息。
Check the units of each formula before plugging numbers in. For example, when using pV = nRT, if p is in Pa and V in m³, R must be 8.31 J K⁻¹ mol⁻¹; if p is in kPa and V in dm³, you may need to adjust R or convert units. The insert usually gives R in J K⁻¹ mol⁻¹, so working in SI units is the safest strategy.
代入数据前先核对每个公式的单位。例如，使用 pV = nRT 时，若 p 以 Pa 为单位、V 以 m³ 为单位，则 R 必须取 8.31 J K⁻¹ mol⁻¹；若 p 以 kPa、V 以 dm³ 为单位，你可能需要调整 R 或转换单位。插入材料中 R 通常以 J K⁻¹ mol⁻¹ 给出，因此使用国际单位制是最稳妥的策略。
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