📚 Year 13 Cambridge Science: Quick Reference Handbook of Formulas and Theorems | Year 13 剑桥科学：公式定理速查手册
Welcome to the essential quick reference for Year 13 Cambridge Science students. This handbook compiles the most critical formulas and theorems across Physics, Chemistry, and Biology, enabling rapid review and reinforcement of key concepts before the A Level examinations.
欢迎使用这本针对 Year 13 剑桥科学学生的重要速查手册。本手册汇集了物理、化学和生物三科中最关键的公式和定理，帮助你在 A Level 考前快速回顾和巩固核心概念。
1. Kinematics Equations | 运动学公式
For motion with constant acceleration along a straight line, four fundamental equations relate initial velocity u, final velocity v, acceleration a, time t, and displacement s.
描述匀加速直线运动的四个基本方程联系初速度 u、末速度 v、加速度 a、时间 t 和位移 s。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
2. Force, Momentum and Newton’s Laws | 力、动量与牛顿定律
Newton’s second law states that the net force on an object is equal to the product of its mass and acceleration: Fnet tum: Δp = FΔt. In a closed system, total momentum is conserved: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
牛顿第二定律指出物体所受合力等于质量与加速度的乘积：Fnet = ma。动量 p 定义为 p = mv。冲量等于动量的变化：Δp = FΔt。在封闭系统中总动量守恒：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
3. Work, Energy and Power | 功、能量和功率
nergy near Earth’s surface: Eₚ = mgh. Power is the rate of doing work: P = W/t = Fv. Efficiency = (useful power output)/(total power input) × 100%.
恒力做功 W = Fd cosθ，其中 θ 是力与位移的夹角。动能：Eₖ = ½mv²。地表附近的重力势能：Eₚ = mgh。功率是做功的快慢：P = W/t = Fv。效率 = (有用输出功率)/(总输入功率) × 100%。
4. Waves and Optics | 波与光学
The critical angle for total internal reflection is found from sin C = 1/n. Young’s double‑slit fringe spacing: Δx = λL/d, where L is distance to screen and d is slit separation.
波速 v = fλ。折射率 n = c/v，c 为光速。斯涅尔定律：n₁ sinθ₁ = n₂ sinθ₂。全内反射的临界角满足 sin C = 1/n。杨氏双缝干涉的条纹间距：Δx = λL/d，其中 L 是到屏的距离，d 是缝间距。
5. Electric Circuits | 电路
Ohm’s law: V = IR. For resistors in series: Rtotal = R₁ + R₂ + ···. For resistors in parallel: 1/Rtotal entering a junction = ΣI leaving. Kirchhoff’s second law: Σ emf = Σ IR in any closed loop.
欧姆定律：V = IR。串联电阻：Rtotal = R₁ + R₂ + ···。并联电阻：1/Rtotal = 1/R₁ + 1/R₂ + ···。电功率可用 P = IV = I²R = V²/R 表示。基尔霍夫第一定律：流入节点的电流之和等于流出电流之和。基尔霍夫第二定律：在任一闭合回路中，电动势代数和等于各电阻上电压降的代数和。
6. Thermal Physics | 热物理
l kinetic energy of a molecule is (3/2)kT. Heat transfer without phase change: Q = mcΔθ. During phase change: Q = mL, where L is specific latent heat.
温度换算：T(K) = θ(°C) + 273.15。理想气体状态方程 pV = nRT，R = 8.31 J mol⁻¹ K⁻¹。根据分子运动论，分子的平均平动动能为 (3/2)kT。无相变时传热：Q = mcΔθ。相变时：Q = mL，L 为相变潜热。
7. Nuclear Physics | 核物理
Mass–energy equivalence is given by E = mc². The number of undecayed nuclei follows N = N₀ e–λt, where λ is the decay constant. Half‑life T½ λN, measured in becquerels (Bq). Binding energy per nucleon determines nuclear stability.
质能等价方程为 E = mc²。未衰变原子核的数量遵循 N = N₀ e–λt，λ 为衰变常数。半衰期 T½ = ln2 / λ。放射性活度 A = λN，单位是贝可 (Bq)。每个核子的结合能决定原子核的稳定性。
8. Mole Concept and Stoichiometry | 摩尔概念与化学计量
tration c = n / V. The ideal gas equation links pV = nRT. Percentage yield = (actual yield / theoretical yield) × 100%. Atom economy = (mass of desired product / total mass of reactants) × 100%.
物质的量 n = m / M（质量除以摩尔质量）。常温常压下，1 mol 气体体积约为 24 dm³。浓度 c = n / V。理想气体状态方程为 pV = nRT。产率 = (实际产量 / 理论产量) × 100%。原子利用率 = (目标产物质量 / 总反应物质量) × 100%。
9. Energetics and Thermodynamics in Chemistry | 化学热力学与能量学
Standard enthalpy change of reaction ΔH = ΣΔHf°(products) – ΣΔHf lpy change for a reaction is independent of the pathway. Gibbs free energy change: ΔG = ΔH – TΔS. At equilibrium, ΔG = –RT ln K. For bond enthalpy calculations, ΔH ≈ Σ(bond energies broken) – Σ(bond energies formed).
反应的标准焓变 ΔH = ΣΔHf°(生成物) – ΣΔHf°(反应物)。盖斯定律指出化学反应的焓变与途径无关。吉布斯自由能变：ΔG = ΔH – TΔS。平衡时 ΔG = –RT ln K。利用键焓估算 ΔH ≈ Σ(断裂键键能) – Σ(形成键键能)。
10. Chemical Equilibrium and Kinetics | 化学平衡与动力学
For a general reaction aA + bB ⇌ cC + dD, the equilibrium constant Kc = [C]c[D]d / [A]a[B]b the shift in equilibrium to counteract a change. Rate equations take the form rate = k[A]m[B]n are orders of reaction. The Arrhenius equation links rate constant and temperature: k = Ae–Ea/(RT) or in linear form ln k = ln A – Ea/(RT).
对于一般反应 aA + bB ⇌ cC + dD，平衡常数 Kc = [C]c[D]d / [A]a[B]b。勒夏特列原理预测平衡朝着减弱变化的方向移动。速率方程形如 rate = k[A]m[B]n，其中 m 和 n 为反应级数。阿伦尼乌斯方程关联速率常数与温度：k = Ae–Ea/(RT)，或其线性形式 ln k = ln A – Ea/(RT)。
11. Genetics and Probability | 遗传学与概率
ratio of 3 : 1. A dihybrid cross (unlinked genes) gives the ratio 9 : 3 : 3 : 1. The Hardy– + 2pq + q² = 1, where p and q represent dominant and recessive allele frequencies. The chi‑squared (χ²) test is χ² = Σ((O – E)² / E); degrees of freedom and critical values are used to test significance.
具有完全显性的单基因杂交产生 3 : 1 的表型比例。双基因杂交（非连锁）呈现 9 : 3 : 3 : 1 的比例。哈迪–温伯格定律描述等位基因和基因型频率：p + q = 1；p² + 2pq + q² = 1，其中 p 和 q 代表显性和隐性等位基因的频率。卡方 (χ²) 检验：χ² = Σ((O – E)² / E)；利用自由度和临界值判断显著性。
12. Ecology and Population Growth | 生态与种群增长
intrinsic rate of increase. Logistic growth incorporates carrying capacity K: dN/dt = rN(1 – N/K). The Lincoln index (mark– dividuals. Ecological efficiency = (energy at trophic level n+1 / energy at trophic level n) × 100%.
指数式种群增长模型为 dN/dt = rN，r 为内禀增长率。逻辑斯谛增长纳入环境容纳量 K：dN/dt = rN(1 – N/K)。林肯指数（标记重捕法）估算种群数量 N = (M × C) / R，其中 M 为初始标记个体数，C 为第二次捕获总数，R 为捕获的标记个体数。生态效率 = (下一营养级能量 / 上一营养级能量) × 100%。
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