📚 Year 10 CAIE Science: Quick Formula & Theorem Handbook | Year 10 CAIE 科学：公式定理速查手册
This concise reference guide brings together the key formulas, relationships and fundamental theorems that every Year 10 CAIE Science student needs at their fingertips. Covering Physics, Chemistry and Biology, the handbook is organised by topic to support quick revision and problem‑solving throughout the IGCSE course.
这本精简易查的手册汇集了每一位 Year 10 CAIE 科学学生都需要掌握的核心公式、关系式和基本定理。手册按学科主题编排，覆盖物理、化学与生物，旨在帮助学生在整个 IGCSE 学习过程中快速复习和高效解题。
1. Motion and Forces | 运动与力
Motion can be described using scalar and vector quantities. The following equations link displacement, velocity, acceleration and time for objects moving with uniform acceleration.
运动可以用标量和矢量来描述。以下公式将匀加速运动物体的位移、速度、加速度和时间联系起来。
Average speed (v) = total distance (s) ÷ time (t), or v = s / t. This applies when speed is constant or when calculating an overall average.
平均速度 (v) = 总路程 (s) ÷ 时间 (t)，即 v = s / t。该式用于速度恒定或计算整体平均值的情形。
For uniform acceleration, final velocity v = u + a t, where u is initial velocity, a is acceleration and t is time.
对于匀加速运动，末速度 v = u + a t，其中 u 为初速度，a 为加速度，t 为时间。
Distance travelled s = (u + v) t / 2, giving displacement from the average velocity.
运动距离 s = (u + v) t / 2，利用平均速度求得位移。
Alternatively, s = u t + ½ a t², which is useful when final velocity is unknown.
另可用 s = u t + ½ a t²，当末速度未知时尤为方便。
v² = u² + 2 a s links final speed directly with displacement and acceleration.
v² = u² + 2 a s 将末速度与位移和加速度直接关联。
Force (F) = mass (m) × acceleration (a), Newton’s second law. Weight W = m g, with g = 9.8 m/s² on Earth.
力 (F) = 质量 (m) × 加速度 (a)，即牛顿第二定律。重量 W = m g，地球上 g = 9.8 m/s²。
2. Energy, Work and Power | 能量、功与功率
Energy is conserved in all changes; it can be transferred, stored or dissipated. Work and power quantify these transfers.
能量在所有变化中守恒；它可被转移、储存或耗散。功和功率定量描述这些转移。
Work done W = F d, where F is the force in the direction of displacement d. Work is measured in joules (J).
做功 W = F d，其中 F 是位移方向上的力，d 为位移。功的单位是焦耳 (J)。
Kinetic energy KE = ½ m v², where m is mass and v is speed.
动能 KE = ½ m v²，其中 m 为质量，v 为速率。
Gravitational potential energy GPE = m g h, where h is the change in height.
重力势能 GPE = m g h，其中 h 是高度的变化量。
Power P = W / t (work done ÷ time) or P = ΔE / t. The unit is the watt (W).
功率 P = W / t（做功 ÷ 时间）或 P = ΔE / t。单位是瓦特 (W)。
Efficiency = (useful output energy ÷ total input energy) × 100%, or the equivalent power ratio.
效率 =（有用的输出能量 ÷ 总输入能量）× 100%，或采用相应的功率比值。
3. Thermal Physics | 热物理学
Heating and cooling involve energy transfers that can be calculated using specific heat capacity and latent heat, while the behaviour of gases is summarised by the gas laws.
加热和冷却涉及能量传递，可用比热容和潜热进行计算；气体的行为则由气体定律概括。
Thermal energy change Q = m c Δθ, where c is specific heat capacity and Δθ is the temperature change.
热量变化 Q = m c Δθ，式中 c 为比热容，Δθ 为温度变化。
Energy for change of state Q = m L, where L is the specific latent heat (of fusion or vaporisation).
物态变化所需能量 Q = m L，其中 L 是比潜热（熔化潜热或汽化潜热）。
For a fixed mass of gas at constant temperature, p₁ V₁ = p₂ V₂ (Boyle’s law).
对于一定质量的气体，温度不变时有 p₁ V₁ = p₂ V₂（玻意耳定律）。
At constant pressure, V₁ / T₁ = V₂ / T₂ where T is in kelvin (Charles’s law).
压强不变时，V₁ / T₁ = V₂ / T₂，温度 T 须用开尔文（查理定律）。
The combined gas law: p V / T = constant for a fixed mass of ideal gas.
理想气体联合公式：p V / T = 常数（适用于一定质量的理想气体）。
4. Waves | 波
Waves transfer energy without transferring matter. The wave equation links speed, frequency and wavelength for all types of wave.
波传递能量而不传递物质。波动方程将各种波的速度、频率和波长联系起来。
Wave speed v = f λ, where f is frequency in hertz (Hz) and λ is wavelength in metres.
波速 v = f λ，其中 f 为频率，单位赫兹 (Hz)；λ 为波长，单位米。
Period T = 1 / f, the time for one complete oscillation.
周期 T = 1 / f，即完成一次完整振动所需的时间。
Refractive index n = sin i / sin r (Snell’s law), where i is the angle of incidence and r is the angle of refraction. Also n = c / v_wave in the medium.
折射率 n = sin i / sin r（斯涅尔定律），其中 i 为入射角，r 为折射角。亦可用 n = c / v（介质中波速）。
Critical angle c satisfies sin c = 1 / n, provided light travels from the denser medium towards the boundary.
临界角 c 满足 sin c = 1 / n，前提是光从光密介质射向界面。
5. Electricity and Magnetism | 电学与磁学
Current, voltage and resistance are related by Ohm’s law. Power and energy in electric circuits are calculated using these quantities.
电流、电压和电阻由欧姆定律相关联。电路中的功率和能量可用这些量进行计算。
Ohm’s law: V = I R, where V is potential difference (V), I is current (A) and R is resistance (Ω).
欧姆定律：V = I R，式中 V 为电势差 (V)，I 为电流 (A)，R 为电阻 (Ω)。
Resistance R = ρ L / A, where ρ is resistivity, L length and A cross‑sectional area.
电阻 R = ρ L / A，ρ 为电阻率，L 为长度，A 为横截面积。
Power P = I V = I² R = V² / R. This gives the rate of energy transfer in a component.
功率 P = I V = I² R = V² / R。它表示元件中能量转换的速率。
Electrical energy E = I V t, where t is time in seconds.
电能 E = I V t，其中 t 为时间，单位秒。
In series circuits: current is the same everywhere, total resistance R_total = R₁ + R₂ + … and supply voltage is shared.
串联电路：各处电流相等，总电阻 R_total = R₁ + R₂ + …，电源电压在各电阻上分配。
In parallel circuits: voltage across each branch is the same, total current splits and 1/R_total = 1/R₁ + 1/R₂ + …
并联电路：各支路两端电压相等，总电流被分支，1/R_total = 1/R₁ + 1/R₂ + …
Transformer equation (for an ideal transformer): Vₚ / Vₛ = Nₚ / Nₛ, where p denotes primary and s secondary coil. Power input ≈ power output.
变压器公式（理想变压器）：Vₚ / Vₛ = Nₚ / Nₛ，下标 p 代表初级线圈，s 代表次级线圈。输入功率 ≈ 输出功率。
6. Atomic Physics | 原子物理
Radioactive decay is a random process. The half‑life and decay equations allow us to predict the activity of a sample over time.
放射性衰变是一种随机过程。半衰期和衰变方程可用于预测样品活度随时间的变化。
Activity A = λ N, where λ is the decay constant and N is the number of undecayed nuclei.
活度 A = λ N，λ 为衰变常量，N 为未衰变的原子核数目。
Number of undecayed nuclei after n half‑lives: N = N₀ (½)^n, and the same form applies to mass and count rate.
经过 n 个半衰期后未衰变核的数目：N = N₀ (½)^n，该形式同样适用于质量和计数率。
Background radiation must be subtracted from measured count rate to find the corrected count.
必须从测得的计数率中减去本底辐射，才能得到修正后的计数率。
Nuclear equations balance total mass number (top) and total atomic number (bottom): for alpha decay, mass number drops by 4 and atomic number by 2; for beta decay, mass number stays the same and atomic number increases by 1.
核反应方程要求质量数（左上标）和质子数（左下标）守恒：α衰变质量数减 4、质子数减 2；β⁻衰变质量数不变、质子数增 1。
7. Chemical Formulae and Equations | 化学式和方程式
A chemical formula shows the number and type of atoms in a substance. Balanced equations reflect the conservation of mass in a reaction.
化学式显示物质中原子的种类和数目。配平后的方程式体现了反应中质量守恒。
Relative formula mass (Mᵣ) is the sum of the relative atomic masses (Aᵣ) of all atoms in the formula. It has no units.
相对式量 (Mᵣ) 是化学式中所有原子相对原子质量 (Aᵣ) 的总和，无单位。
Empirical formula gives the simplest whole‑number ratio of atoms; molecular formula shows the actual number of each atom in a molecule.
实验式表示原子最简单整数比；分子式表示分子中各原子的实际数目。
Percentage mass of an element = (total mass of the element in the formula ÷ Mᵣ) × 100%.
某元素的质量分数 =（该元素在式中的总质量 ÷ Mᵣ）× 100%。
Reacting masses are calculated using mole ratios from the balanced equation: mass → moles → molar ratio → moles of target substance → mass.
根据配平方程中的摩尔比计算反应质量：质量 → 摩尔数 → 摩尔比 → 目标物摩尔数 → 质量。
8. Mole Concept and Stoichiometry | 摩尔概念与化学计量
The mole is the unit linking measurable masses to numbers of particles. The molar volume of a gas at room temperature and pressure (rtp) is a fixed value.
摩尔是连接可测量质量与粒子数目的单位。在常温常压 (rtp) 下，气体的摩尔体积是一个固定值。
Number of moles n = m / M, where m is mass in grams and M is molar mass in g/mol.
摩尔数 n = m / M，其中 m 为质量（克），M 为摩尔质量 (g/mol)。
For a gas at rtp (20 °C, 1 atm), n = V / 24, where V is volume in dm³ and 24 dm³ mol⁻¹ is the molar volume.
对于常温常压 (20 °C, 1 atm) 下的气体，n = V / 24，V 的单位是 dm³，摩尔体积为 24 dm³ mol⁻¹。
Concentration of a solution c = n / V, where V is volume of solution in dm³. Concentration may be expressed in mol/dm³ or g/dm³.
溶液浓度 c = n / V，V 为溶液体积 (dm³)。浓度可用 mol/dm³ 或 g/dm³ 表示。
Number of particles = n × Nₐ, where Nₐ (Avogadro’s constant) ≈ 6.02 × 10²³ mol⁻¹.
粒子数目 = n × Nₐ，Nₐ（阿伏伽德罗常数）约等于 6.02 × 10²³ mol⁻¹。
Percentage yield = (actual yield ÷ theoretical yield) × 100%.
产率 =（实际产量 ÷ 理论产量）× 100%。
9. Acids, Bases and Salts | 酸碱盐
Acids donate H⁺ ions in water; bases accept H⁺ or release OH⁻. The pH scale quantifies acidity, and neutralisation produces a salt and water.
酸在水中释放 H⁺ 离子；碱接受 H⁺ 或释放 OH⁻。pH 值量化了酸碱度，中和反应生成盐和水。
pH = –log₁₀[H⁺]; a low pH means high hydrogen ion concentration. Each unit change in pH represents a ten‑fold change in [H⁺].
pH = –log₁₀[H⁺]；pH 值低表示氢离子浓度高。pH 每变化 1，[H⁺] 改变 10 倍。
Common neutralisation: acid + metal oxide → salt + water; acid + metal hydroxide → salt + water.
常见中和反应：酸 + 金属氧化物 → 盐 + 水；酸 + 金属氢氧化物 → 盐 + 水。
Acid + metal carbonate → salt + water + carbon dioxide. This reaction is used to test for carbonate ions.
酸 + 金属碳酸盐 → 盐 + 水 + 二氧化碳。该反应用于检验碳酸根离子。
Titration formula: c₁ V₁ / n₁ = c₂ V₂ / n₂, where c is concentration, V is volume and n is the number of moles from the balanced equation.
滴定公式：c₁ V₁ / n₁ = c₂ V₂ / n₂，c 为浓度，V 为体积，n 为配平方程中的摩尔数。
10. Rates and Equilibrium | 速率与平衡
Reaction rate depends on concentration, temperature, surface area and catalysts. In reversible reactions, the equilibrium position shifts according to Le Chatelier’s principle.
反应速率受浓度、温度、表面积和催化剂影响。对于可逆反应，平衡位置依据勒夏特列原理发生移动。
Average rate = change in quantity (mass, volume) ÷ time. The instantaneous rate can be found from the gradient of a concentration‑time graph.
平均速率 = 某量（质量、体积）的变化 ÷ 时间。瞬时速率可由浓度–时间曲线的斜率求得。
For a gaseous system at equilibrium, increasing pressure shifts the position towards the side with fewer gas molecules.
对于达到平衡的气态体系，增大压强会使平衡向气体分子总数较少的方向移动。
Increasing temperature favours the endothermic direction; decreasing temperature favours the exothermic direction.
升高温度有利于吸热方向；降低温度有利于放热方向。
The equilibrium constant Kc = [products] / [reactants], raised to their stoichiometric coefficients. Its value changes only with temperature.
平衡常数 Kc = [生成物] / [反应物]，各浓度以其化学计量数为指数。Kc 的数值只随温度改变。
11. Organic Chemistry | 有机化学
Organic compounds are built on carbon skeletons. Homologous series share a general formula and graduated physical properties, while functional groups determine chemical reactivity.
有机化合物建立在碳骨架之上。同系物具有相同的通式与渐变的物理性质，而官能团决定其化学活性。
Alkanes: general formula CₙH₂ₙ₊₂. They undergo combustion and substitution reactions with halogens (in UV light).
烷烃通式：CₙH₂ₙ₊₂。可发生燃烧反应和在紫外光下与卤素的取代反应。
Alkenes: CₙH₂ₙ. They contain a C=C double bond and undergo addition reactions with hydrogen, halogens and steam.
烯烃通式：CₙH₂ₙ。含有 C=C 双键，能与氢气、卤素和水蒸气发生加成反应。
Alcohols: CₙH₂ₙ₊₁OH. They can be oxidised to carboxylic acids and burn with a clean flame.
醇通式：CₙH₂ₙ₊₁OH。可被氧化成羧酸，燃烧时火焰洁净。
Carboxylic acids: CₙH₂ₙ₊₁COOH. They are weak acids and react with alcohols to form esters (esterification).
羧酸通式：CₙH₂ₙ₊₁COOH。它们属于弱酸，可与醇发生酯化反应生成酯。
Polymers: addition polymerisation of alkenes produces long‑chain molecules; the repeating unit is derived from the monomer.
聚合物：烯烃经加聚反应生成长链分子；重复单元源自单体。
12. Biology: Key Equations and Laws | 生物学关键方程与定律
Biology relies on word equations for physiological processes and on genetic principles that predict inheritance patterns.
生物学通过文字方程式描述生理过程，并借助遗传学原理预测遗传模式。
Photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂, occurring in chloroplasts using light energy.
光合作用：6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂，在叶绿体中利用光能进行。
Aerobic respiration: C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O (+ ATP). This releases energy for cells.
有氧呼吸：C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O（+ ATP），为细胞释放能量。
Magnification = size of image ÷ actual size of specimen. This applies to drawings and microscope images.
放大倍数 = 图像尺寸 ÷ 标本实际尺寸。适用于绘图和显微镜图像。
Mendel’s Law of Segregation: each individual has two alleles for a gene; these separate during gamete formation so each gamete carries only one allele.
孟德尔分离定律：每个个体的一对等位基因在配子形成时彼此分离，每个配子只含一个等位基因。
Mendel’s Law of Independent Assortment: alleles of different genes are distributed to gametes independently of one another.
孟德尔自由组合定律：不同基因的等位基因在配子形成时独立分配。
Monohybrid cross ratios: for a heterozygous cross (Tt × Tt), the expected phenotype ratio is 3:1; for a test cross (Tt × tt), it is 1:1.
单基因杂交比例：杂合子杂交 (Tt × Tt) 预期表型比为 3:1；测交 (Tt × tt) 预期表型比为 1:1。
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