📚 Year 12 CAIE Science: Quick Reference Formula & Theorem Handbook | 12年级CAIE科学：公式定理速查手册
This compact revision guide brings together the most essential formulae, theorems, and principles from the Year 12 CAIE Science curriculum. It covers core topics in Physics, Chemistry, and Biology, giving you a quick way to recall what you need in problem-solving and exam preparation. Every expression is presented with clear English explanations immediately followed by their Chinese equivalents so you can study bilingually.
这本精简的复习手册汇集了12年级CAIE科学课程中最关键的公式、定理和原理，涵盖物理、化学、生物的核心内容，助你在解题和备考中快速回顾所需知识。每个表达式都配有清晰的英文说明，并紧接对应的中文解释，方便你进行双语学习。
1. Kinematics & Motion | 运动学与运动
Uniform acceleration in a straight line is described by the SUVAT equations. The four equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
匀加速直线运动由SUVAT方程组描述。四个方程联系了位移s、初速度u、末速度v、加速度a和时间t。
v = u + at
s = ut + ½at²
s = ½(u + v)t
v² = u² + 2as
The gradient of a displacement–time graph gives velocity, and the area under a velocity–time graph gives displacement. Acceleration is the gradient of a velocity–time graph.
位移-时间图的斜率表示速度，速度-时间图下的面积表示位移；加速度是速度-时间图的斜率。
2. Dynamics & Forces | 动力学与力
Newton’s laws of motion form the basis of dynamics. The resultant force on an object equals the rate of change of its momentum, leading to the familiar formula when mass is constant.
牛顿运动定律是动力学的基础。物体所受的合力等于其动量变化率，当质量恒定时就得到熟知的公式。
F = ma
Weight is the gravitational force on a mass: W = mg, where g ≈ 9.81 m s⁻² on Earth. Momentum p = mv is conserved in a closed system with no external forces.
重量是作用在物体上的引力：W = mg，地球表面 g ≈ 9.81 m s⁻²。在没有外力的封闭系统中，动量 p = mv 守恒。
For a spring obeying Hooke’s law, the extension is directly proportional to the applied force:
对于遵守胡克定律的弹簧，伸长量与施加的力成正比：
F = kΔx
The coefficient of friction μ relates frictional force to the normal reaction: f ≤ μR, with the equality holding at limiting friction.
摩擦系数 μ 联系摩擦力与法向反作用力：f ≤ μR，等号在极限摩擦时成立。
3. Work, Energy & Power | 功、能量和功率
Work done by a constant force is the product of the force and the displacement in the direction of the force.
恒力做的功等于力的大小与该力方向上位移的乘积。
W = F d cos θ
Kinetic energy and gravitational potential energy are the two principal mechanical energy stores.
动能和重力势能是两种主要的力学能量形式。
KE = ½mv²
GPE = mgh
Power is the rate of doing work or transferring energy. For an object moving at constant velocity against a force, power can be expressed using speed.
功率是做功或能量转移的速率。当物体以恒速克服阻力运动时，功率可用速度表达。
P = W/t = Fv
In a closed system, total mechanical energy is conserved provided only conservative forces do work.
在一个封闭系统中，若只有保守力做功，总机械能守恒。
4. Waves & Optics | 波与光学
The fundamental wave equation links speed, frequency, and wavelength for all types of progressive waves.
基本波动方程联系一切前进波的速度、频率和波长。
v = f λ
For light, the refractive index n of a medium is the ratio of the speed of light in vacuum to that in the medium. Snell’s law governs refraction at a boundary.
对于光波，介质的折射率 n 是真空光速与介质中光速之比。斯涅尔定律描述界面的折射。
n = c / v
n₁ sin θ₁ = n₂ sin θ₂
The critical angle for total internal reflection occurs when the refracted angle is 90°.
当折射角为90°时，产生全内反射的临界角满足：
sin θc = n₂ / n₁ (n₁ > n₂)
Diffraction gratings produce maxima at angles satisfying d sin θ = nλ, where d is the slit spacing and n is the order number.
衍射光栅产生亮纹的角度满足 d sin θ = nλ，其中 d 是狭缝间距，n 是级数。
5. Electricity | 电学
Ohm’s law states that the potential difference across a conductor is proportional to the current through it, provided temperature remains constant.
欧姆定律表明，在温度不变的条件下，导体两端的电势差与通过的电流成正比。
V = IR
Resistance arises from the ratio V/I, and for a uniform wire it depends on resistivity, length, and cross-sectional area.
电阻定义为 V/I，对于均匀导线，其值取决于电阻率、长度和截面积。
R = ρL / A
Electrical power can be expressed in three equivalent forms using Ohm’s law.
电功率可通过欧姆定律表达为三种等价形式。
P = IV = I²R = V²/R
For resistors in series and parallel, the equivalent resistance follows these rules.
串联和并联电阻的等效电阻分别遵循以下法则。
Series: Rtotal = R₁ + R₂ + …
Parallel: 1/Rtotal = 1/R₁ + 1/R₂ + …
6. Quantum & Nuclear Physics | 量子与核物理
The energy of a photon is directly proportional to its frequency, with Planck’s constant h as the proportionality constant.
光子的能量与其频率成正比，比例系数为普朗克常数 h。
E = hf
Einstein’s photoelectric equation relates photon energy to work function φ and maximum kinetic energy of emitted electrons.
爱因斯坦光电方程将光子能量、功函数 φ 与发射电子的最大动能联系起来。
hf = φ + ½mv²max
The de Broglie wavelength of a particle with momentum p shows wave–particle duality.
具有动量 p 的粒子其德布罗意波长体现了波粒二象性。
λ = h / p = h / mv
Radioactive decay follows an exponential law: N = N₀ e–λt, where λ is the decay constant and half-life t½ = ln 2/λ.
放射性衰变遵循指数规律：N = N₀ e–λt，衰变常数 λ，半衰期 t½ = ln 2/λ。
7. Chemical Calculations & Energetics | 化学计量与热化学
The mole is the central unit in chemistry. The amount of substance n links mass and molar mass, as well as gas volume at given conditions.
摩尔是化学的核心单位。物质的量 n 将质量、摩尔质量以及特定条件下的气体体积联系起来。
n = m / M
n = V / 24 dm³ mol⁻¹ (at RTP)
Concentration of a solution is expressed in mol dm⁻³.
溶液的浓度以 mol dm⁻³ 表示。
c = n / V
The ideal gas equation combines pressure, volume, temperature, and amount of gas.
理想气体状态方程综合了压力、体积、温度和气体物质的量。
pV = nRT (R = 8.31 J K⁻¹ mol⁻¹)
Enthalpy change is calculated from temperature change using q = mcΔT, then ΔH = –q / n (for exothermic). Hess’s law allows the manipulation of thermochemical cycles.
焓变通过量热公式 q = mcΔT 计算，然后 ΔH = –q / n（放热反应）。赫斯定律使得我们可以通过热化学循环进行间接计算。
ΔH = ΣΔHf°(products) – ΣΔHf°(reactants)
8. Chemical Equilibrium & Kinetics | 化学平衡与动力学
For a reversible reaction aA + bB ⇌ cC + dD, the equilibrium constant Kc is defined in terms of equilibrium concentrations.
对于可逆反应 aA + bB ⇌ cC + dD，平衡常数 Kc 根据平衡浓度定义。
Kc = [C]c[D]d / [A]a[B]b
Le Chatelier’s principle predicts how a system at equilibrium responds to changes in concentration, pressure, or temperature.
勒夏特列原理预测平衡系统对浓度、压力或温度变化作出的反应。
The rate of a reaction is often given by a rate law determined experimentally.
反应速率通常由实验确定的速率方程表示。
Rate = k[A]m[B]n
The Arrhenius equation relates the rate constant to temperature and activation energy.
阿伦尼乌斯方程将速率常数与温度和活化能联系起来。
k = A e–Ea/(RT)
A catalyst provides an alternative pathway with lower activation energy, thereby increasing the rate without being consumed.
催化剂提供了一条活化能较低的替代路径，从而加快反应速率而本身不消耗。
9. Biological Molecules & Enzymes | 生物分子与酶
Carbohydrates, lipids, and proteins are built from smaller monomer units. The general formula for many monosaccharides is C₆H₁₂O₆.
糖类、脂质和蛋白质由较小的单体构建而成。许多单糖的通式为 C₆H₁₂O₆。
For proteins, the primary structure is the sequence of amino acids linked by peptide bonds. Enzyme activity is described by the lock‑and‑key and induced‑fit models.
蛋白质的一级结构是由肽键连接的氨基酸序列。酶的作用可用锁钥模型和诱导契合模型描述。
The Michaelis–Menten equation quantifies the rate of an enzyme‑catalysed reaction in terms of substrate concentration.
米氏方程以底物浓度量化酶促反应的速率。
v = Vmax[S] / (Km + [S])
Benedict’s test and biuret test are key qualitative tests for reducing sugars and proteins, respectively.
班氏试剂检验还原糖，双缩脲试剂检验蛋白质，是重要的定性实验。
10. Cell Biology & Genetics | 细胞生物学与遗传
Magnification is calculated from the observed image size and the actual object size.
放大倍数由观测图像大小和物体实际大小计算。
Magnification = Image size / Actual size
The mitotic index gives the proportion of cells undergoing mitosis and is used to identify zones of growth.
有丝分裂指数表示正在进行有丝分裂的细胞比例，用于识别生长区。
Mitotic index = Number of cells in mitosis / Total number of cells
In population genetics, the Hardy–Weinberg principle predicts allele and genotype frequencies in a non‑evolving population.
在种群遗传学中，哈迪-温伯格原理预测非进化种群中的等位基因和基因型频率。
p + q = 1
p² + 2pq + q² = 1
The chi‑squared test (χ²) is used to compare observed and expected frequencies in genetic crosses.
卡方检验 (χ²) 用于比较遗传杂交中观测值与期望值的频率。
χ² = Σ ((O – E)² / E)
Standard deviation and standard error are used to assess the spread and reliability of biological data.
标准差和标准误差用于评估生物学数据的离散程度和可靠性。
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