📚 Year 13 OCR Science: Quick Reference Handbook of Formulas and Theorems | Year 13 OCR 科学：公式定理速查手册
This quick reference handbook consolidates the essential formulas, equations, and core theorems required for OCR A Level Sciences at Year 13. Covering Physics, Chemistry, and Biology, it provides a bilingual revision tool to reinforce key concepts, ensuring that both English and Chinese terminology are clearly understood and readily applicable in problem-solving and examinations.
这本速查手册汇总了 Year 13 OCR A Level 科学课程中必备的公式、方程和核心定理。内容涵盖物理、化学和生物，以中英双语形式呈现，帮助巩固关键概念，确保学生能够清晰理解并在解题和考试中熟练运用相关术语。
1. Physics: Mechanics and Motion | 物理：力学与运动
The SUVAT equations predict motion with constant acceleration along a straight line. They form the basis for analysing projectile motion, free fall, and vehicle dynamics.
SUVAT 方程组适用于匀加速直线运动，是分析抛体运动、自由落体和车辆动力学的基础。
- v = u + a t (final velocity = initial velocity + acceleration × time)
- v = u + a t （末速度 = 初速度 + 加速度 × 时间）
- s = u t + ½ a t² (displacement = initial velocity × time + half acceleration × time squared)
- s = u t + ½ a t² （位移 = 初速度 × 时间 + 二分之一加速度 × 时间的平方）
- v² = u² + 2 a s
- v² = u² + 2 a s
- s = ½ (u + v) t
- s = ½ (u + v) t
Newton’s second law links resultant force, mass, and acceleration: F = m a. Momentum p = m v is conserved when no external forces act.
牛顿第二定律 F = m a 建立合力、质量和加速度的关系。动量 p = m v 在无外力作用时守恒。
The work–energy theorem states that net work done = change in kinetic energy: W = ½ m v² − ½ m u². Power is the rate of work: P = W / t = F v.
功–能定理：净功等于动能变化量，即 W = ½ m v² − ½ m u²。功率是做功的速率：P = W / t = F v。
2. Physics: Circular Motion and Gravitation | 物理：圆周运动与引力
For an object moving at constant speed v in a circle of radius r, the centripetal acceleration is a = v²/r = ω² r, and the centripetal force is F = m v²/r = m ω² r, where ω = θ/t = 2πf.
物体以恒定速率 v 在半径为 r 的圆周上运动时，向心加速度 a = v²/r = ω² r，向心力 F = m v²/r = m ω² r，其中角速度 ω = θ/t = 2πf。
Newton’s law of gravitation: F = −G M m / r². Gravitational field strength g = F/m = G M / r². Kepler’s third law relates orbital period T and radius: T² ∝ r³.
牛顿万有引力定律：F = −G M m / r²。引力场强度 g = F/m = G M / r²。开普勒第三定律给出轨道周期与半径的关系：T² ∝ r³。
| Quantity / 物理量 | Symbol / 符号 | Unit / 单位 |
|---|---|---|
| Gravitational constant | G | N m² kg⁻² |
| Angular velocity | ω | rad s⁻¹ |
| Centripetal acceleration | a | m s⁻² |
3. Physics: Electricity and Magnetism | 物理：电与磁
Ohm’s law: V = I R. Resistivity ρ links resistance to geometry: R = ρ L / A. Power dissipated: P = V I = I² R = V²/R.
欧姆定律：V = I R。电阻率 ρ 给出电阻与几何结构的关系：R = ρ L / A。功率耗散：P = V I = I² R = V²/R。
Kirchhoff’s first law: sum of currents entering a junction = sum leaving. Second law: sum of e.m.f. = sum of p.d. around a closed loop.
基尔霍夫第一定律：流入节点的电流总和等于流出电流总和。第二定律：闭合回路中电动势总和等于电势差总和。
For capacitors, charge Q = C V, energy stored E = ½ Q V = ½ C V² = ½ Q²/C. Time constant τ = R C for discharge V = V₀ e^(−t/RC).
电容器：电荷 Q = C V，储存能量 E = ½ Q V = ½ C V² = ½ Q²/C。RC 放电时间常数 τ = R C，电压 V = V₀ e^(−t/RC)。
Magnetic force on a moving charge: F = B Q v sin θ. Force on a current-carrying conductor: F = B I L sin θ. Faraday’s law: induced e.m.f. ε = −N dΦ/dt.
运动电荷受到的磁场力：F = B Q v sin θ。载流导体受到的力：F = B I L sin θ。法拉第定律：感应电动势 ε = −N dΦ/dt。
4. Physics: Waves and Quantum Phenomena | 物理：波与量子现象
Wave speed: v = f λ. For two-source interference, path difference = n λ for constructive, = (n + ½) λ for destructive. Diffraction grating: n λ = d sin θ.
波速：v = f λ。双源干涉中，加强条件为波程差 n λ，减弱条件为 (n + ½) λ。衍射光栅公式：n λ = d sin θ。
Photon energy: E = h f = h c / λ. Photoelectric effect: h f = Φ + ½ m v²_max. De Broglie wavelength: λ = h / p.
光子能量：E = h f = h c / λ。光电效应方程：h f = Φ + ½ m v²_max。德布罗意波长：λ = h / p。
Energy levels in atoms produce discrete line spectra. The hydrogen spectrum is described by 1/λ = R (1/n₁² − 1/n₂²) where R is the Rydberg constant.
原子能级产生分立线状光谱。氢原子光谱由公式 1/λ = R (1/n₁² − 1/n₂²) 描述，其中 R 为里德伯常数。
5. Chemistry: Physical Chemistry – Energetics and Kinetics | 化学：物理化学 – 能量学与动力学
The enthalpy change of reaction ΔH can be calculated using Hess’s law or bond enthalpies: ΔH = Σ (bond energies broken) − Σ (bond energies formed).
反应焓变 ΔH 可用赫斯定律或键焓计算：ΔH = Σ（断裂键的键能）− Σ（形成键的键能）。
Gibbs free energy: ΔG = ΔH − T ΔS. A feasible reaction occurs when ΔG < 0. For equilibrium, ΔG = 0 and ΔG° = −R T ln K.
吉布斯自由能：ΔG = ΔH − T ΔS。当 ΔG < 0 时反应可行。达到平衡时 ΔG = 0，且标准条件下 ΔG° = −R T ln K。
The rate equation: Rate = k [A]ᵐ [B]ⁿ. The order m,n is determined experimentally. The Arrhenius equation links rate constant k to temperature: k = A e^(−Eₐ/RT) or ln k = ln A − Eₐ/(R T).
速率方程：Rate = k [A]ᵐ [B]ⁿ，反应级数 m、n 由实验确定。阿伦尼乌斯方程给出速率常数 k 与温度的关系：k = A e^(−Eₐ/RT) 或 ln k = ln A − Eₐ/(R T)。
6. Chemistry: Equilibria and Acid–Base | 化学：平衡与酸碱
For a reversible reaction aA + bB ⇌ cC + dD, the equilibrium constant K_c = [C]^c [D]^d / ([A]^a [B]^b). For gases, K_p = (P_C^c P_D^d) / (P_A^a P_B^b).
对于可逆反应 aA + bB ⇌ cC + dD，平衡常数 K_c = [C]^c [D]^d / ([A]^a [B]^b)。对气体而言，K_p = (P_C^c P_D^d) / (P_A^a P_B^b)。
The pH scale: pH = −log₁₀[H⁺]; pOH = −log₁₀[OH⁻]; pK_w = pH + pOH = 14 at 25°C. For weak acids: K_a = [H⁺][A⁻]/[HA]; pK_a = −log₁₀ K_a.
pH 标度：pH = −log₁₀[H⁺]；pOH = −log₁₀[OH⁻]；25°C 时 pK_w = pH + pOH = 14。弱酸：K_a = [H⁺][A⁻]/[HA]；pK_a = −log₁₀ K_a。
Buffer solutions resist pH change; the Henderson–Hasselbalch equation is: pH = pK_a + log₁₀([A⁻]/[HA]).
缓冲溶液能抵抗 pH 变化；亨德森–哈塞尔巴尔赫方程：pH = pK_a + log₁₀([A⁻]/[HA])。
7. Chemistry: Organic Synthesis and Analysis | 化学：有机合成与分析
Infrared spectroscopy identifies functional groups by absorption bands. Characteristic frequencies: O–H (alcohols) ~3200–3550 cm⁻¹, C=O ~1680–1750 cm⁻¹.
红外光谱通过吸收峰识别官能团。特征频率：醇 O–H 约 3200–3550 cm⁻¹，羰基 C=O 约 1680–1750 cm⁻¹。
Mass spectrometry gives molecular ion peak M⁺ and fragmentation patterns. High-resolution MS gives exact molecular formula.
质谱提供分子离子峰 M⁺ 和碎片峰。高分辨质谱可确定精确的分子式。
NMR spectroscopy: number of signals indicates equivalent proton environments; splitting follows n+1 rule; chemical shift δ = (ν_sample − ν_ref)/ν_ref × 10⁶.
核磁共振谱：信号数反映等效质子环境；裂分遵循 n+1 规则；化学位移 δ = (ν_sample − ν_ref)/ν_ref × 10⁶。
Condensation polymers form with loss of small molecules (e.g., H₂O). Polyesters and polyamides are common examples.
缩合聚合反应在形成聚合物的同时脱去小分子（如 H₂O）。常见例子包括聚酯和聚酰胺。
8. Biology: Genetics and Evolution | 生物：遗传与进化
The Hardy–Weinberg principle describes allele frequencies in a non-evolving population: p + q = 1; p² + 2pq + q² = 1. Here p = frequency of dominant allele, q = recessive.
哈迪–温伯格定律描述非进化群体中的等位基因频率：p + q = 1；p² + 2pq + q² = 1，其中 p 为显性等位基因频率，q 为隐性等位基因频率。
Chi-squared test: χ² = Σ (O − E)² / E. Used to test goodness of fit or independence in genetics experiments. Degrees of freedom = number of categories − 1.
卡方检验：χ² = Σ (O − E)² / E，用于遗传学中检验适合度或独立性。自由度 = 类别数 − 1。
Spearman’s rank correlation coefficient r_s = 1 − (6 Σ d²) / (n(n²−1)), where d is the difference in ranks. It tests the strength of association between two variables.
斯皮尔曼等级相关系数 r_s = 1 − (6 Σ d²) / (n(n²−1))，其中 d 为等级差，用于检测两变量间的关联强度。
9. Biology: Photosynthesis and Respiration | 生物：光合作用与呼吸作用
Photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂. The light-dependent reactions produce ATP and reduced NADP; the Calvin cycle fixes CO₂ using ATP and NADPH.
光合作用：6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂。光反应产生 ATP 和还原型 NADP；卡尔文循环消耗 ATP 和 NADPH 固定 CO₂。
Aerobic respiration: C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + ~38 ATP. Glycolysis yields 2 ATP; Krebs cycle and oxidative phosphorylation account for the rest.
有氧呼吸：C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + 约 38 ATP。糖酵解产生 2 ATP，其余来自克雷布斯循环和氧化磷酸化。
Respiratory quotient RQ = CO₂ produced / O₂ consumed. RQ values indicate the substrate used: 1.0 for carbohydrates, ~0.7 for lipids, ~0.8 for proteins.
呼吸商 RQ = 产生的 CO₂量 / 消耗的 O₂量。RQ 值指示呼吸底物类型：碳水化合物约 1.0，脂质约 0.7，蛋白质约 0.8。
10. Biology: Ecology and Energy Transfer | 生物：生态与能量传递
Efficiency of energy transfer between trophic levels = (energy available after transfer / energy available before transfer) × 100%. Typically 5–20%.
营养级间能量传递效率 =（传递后可利用能量 / 传递前可利用能量）× 100%。通常为 5–20%。
Lincoln index for estimating population size: N = (n₁ × n₂) / m, where n₁ = number captured and marked in first sample, n₂ = number in second sample, m = number marked in second sample.
林肯指数估算种群数量：N = (n₁ × n₂) / m，其中 n₁ 为第一次捕获并标记的个体数，n₂ 为第二次捕获总数，m 为第二次捕获中带标记的个体数。
Net primary productivity NPP = GPP − R, where GPP is gross primary productivity and R is respiratory loss. Units: kJ m⁻² y⁻¹.
净初级生产力 NPP = GPP − R，其中 GPP 为总初级生产力，R 为呼吸消耗量。单位：kJ m⁻² y⁻¹。
11. Key Theorems and Laws Across Sciences | 科学中的关键定理与定律
Conservation laws (mass, energy, momentum, charge) are fundamental across physics and chemistry. Le Chatelier’s principle states that if a dynamic equilibrium is disturbed, the system adjusts to partially counteract the change.
守恒定律（质量、能量、动量、电荷）是物理和化学的基础。勒夏特列原理指出，当动态平衡受到扰动时，体系会向着减弱这种改变的方向移动。
In thermodynamics, the first law: ΔU = Q + W (internal energy change = heat added + work done on system). The second law introduces entropy: for a spontaneous process, ΔS_total ≥ 0.
热力学第一定律：ΔU = Q + W（内能变化 = 吸热 + 外界对系统做功）。第二定律引入熵的概念：自发过程的总熵变 ΔS_total ≥ 0。
The central dogma of molecular biology: DNA → RNA → protein. Transcription and translation are governed by the genetic code, which is degenerate and universal.
分子生物学中心法则：DNA → RNA → 蛋白质。转录和翻译受遗传密码控制，密码具有简并性和通用性。
12. Statistical Tests and Data Handling | 统计检验与数据处理
Standard deviation s = √[ Σ(x − x̄)² / (n−1) ]. Standard error of the mean SE = s / √n. Student’s t-test: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂) for unpaired data.
标准差 s = √[ Σ(x − x̄)² / (n−1) ]；标准误差 SE = s / √n。学生 t 检验（非配对）：t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)。
Correlation coefficient r ranges from −1 to +1. For critical values, p < 0.05 indicates a significant correlation at the 5% level.
相关系数 r 取值在 −1 到 +1 之间。当 p < 0.05 时，在 5% 显著性水平下，相关性具有统计学意义。
Uncertainty calculations: absolute uncertainty = ± half the smallest division; percentage uncertainty = (absolute / measured) × 100%. Combined uncertainties: if quantities added/subtracted, add absolute uncertainties; if multiplied/divided, add percentage uncertainties.
不确定度计算：绝对不确定度 = 最小分度的一半；百分不确定度 =（绝对/测量值）× 100%。不确定度合成：加减运算时绝对不确定度相加，乘除运算时百分不确定度相加。
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