📚 Year 12 CAIE Science: Quick-Reference Handbook for Formulas and Theorems | Year 12 CAIE 科学：公式定理速查手册
This handbook provides a concise, exam-focused summary of essential formulas and theorems for Year 12 CAIE Sciences, covering Physics, Chemistry and Biology at the AS Level. Each entry is paired with a clear explanation to support rapid recall and effective revision.
本手册为 Year 12 CAIE 科学课程提供简明、贴近考纲的必要公式与定理摘要，涵盖 AS 阶段的物理、化学与生物。每个条目均配有清晰说明，帮助快速回顾和高效备考。
1. Kinematics Equations | 运动学方程
For uniform acceleration in a straight line, four key equations link displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
对于匀加速直线运动，四个核心方程关联位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t)。
v = u + at
Equation (1): Final velocity equals initial velocity plus acceleration multiplied by time.
方程 (1)：末速度等于初速度加上加速度乘以时间。
s = ut + ½at²
Equation (2): Displacement when starting at initial velocity u and accelerating uniformly.
方程 (2)：以初速度 u 匀加速时的位移。
s = ½(u + v)t
Equation (3): Displacement equals the average velocity multiplied by time.
方程 (3)：位移等于平均速度乘以时间。
v² = u² + 2as
Equation (4): Links final velocity, initial velocity, acceleration and displacement without involving time.
方程 (4)：关联末速度、初速度、加速度和位移，不涉及时间。
2. Dynamics and Momentum | 动力学与动量
Newton's second law states that the net force acting on an object is equal to the product of its mass and acceleration. Momentum is the product of mass and velocity and is conserved in isolated systems.
牛顿第二定律指出，物体所受合外力等于其质量与加速度的乘积。动量是质量与速度的乘积，在孤立系统中守恒。
F = ma
Vector form of Newton's second law where F is the resultant force.
牛顿第二定律的矢量形式，F 为合外力。
p = mv
Momentum p is a vector quantity.
动量 p 是矢量。
FΔt = Δp
The impulse–momentum theorem: Impulse equals change in momentum.
冲量–动量定理：冲量等于动量的变化。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Conservation of linear momentum for a two-body collision; u₁, u₂ are initial velocities, v₁, v₂ are final velocities.
两个物体碰撞的动量守恒；u₁、u₂ 为初速度，v₁、v₂ 为末速度。
3. Work, Energy and Power | 功、能量与功率
Work done is the product of force and displacement in the direction of the force. Energy can be stored as kinetic or gravitational potential energy. Power is the rate of doing work.
功是力与沿力方向位移的乘积。能量可以储存为动能或重力势能。功率是做功的速率。
W = Fs cos θ
Work done by a constant force F acting at an angle θ to the displacement s.
恒力 F 与位移 s 成 θ 角时所做的功。
Eₖ = ½mv²
Kinetic energy of an object of mass m moving at speed v.
质量为 m、速度为 v 的物体的动能。
Eₚ = mgΔh
Change in gravitational potential energy near the Earth's surface.
近地表面的重力势能变化。
P = W / t = Fv
Power as work per unit time and as the product of force and velocity (for constant force parallel to velocity).
功率定义为单位时间做功，也可表示为力与速度的乘积（力与速度平行且恒定时）。
4. Electric Circuits | 电路
Ohm's law, resistance, and rules for combining resistors govern current electricity. Electrical power can be expressed in several useful forms.
欧姆定律、电阻以及电阻组合规则支配着电流。电功率常有多种实用表达形式。
V = IR
Ohm's law for a resistor at constant temperature; V is potential difference, I is current, R is resistance.
恒温下电阻的欧姆定律；V 为电势差，I 为电流，R 为电阻。
R = ρL / A
Resistance of a uniform wire of length L, cross-sectional area A and resistivity ρ.
长度为 L、横截面积为 A、电阻率为 ρ 的均匀导线的电阻。
Rₜₒₜₐₗ = R₁ + R₂ (series), 1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ (parallel)
Resistors in series add directly; for parallel, reciprocals sum to give the reciprocal of total resistance.
串联电阻直接相加；并联电阻的倒数之和等于总电阻的倒数。
P = IV = I²R = V² / R
Power dissipated in a resistor, derived from Ohm's law.
电阻消耗的功率，由欧姆定律导出。
5. Waves and Optics | 波与光学
The relationship between wave speed, frequency and wavelength is fundamental. For light, Snell's law and the critical angle describe refraction and total internal reflection.
波速、频率和波长之间的关系是基础。对于光，斯涅耳定律和临界角描述了折射与全内反射。
v = f λ
Wave speed equals frequency multiplied by wavelength.
波速等于频率乘以波长。
n₁ sin θ₁ = n₂ sin θ₂
Snell's law: n is refractive index, θ is the angle from the normal.
斯涅耳定律：n 为折射率，θ 为与法线的夹角。
n = c / vₘ
Absolute refractive index n is the ratio of the speed of light in vacuum c to the speed in the medium vₘ.
绝对折射率 n 是真空光速 c 与介质中光速 vₘ 之比。
sin c = 1 / n
Critical angle c for light travelling from a medium of refractive index n into vacuum (or air).
光从折射率为 n 的介质射向真空（或空气）时的临界角 c。
6. Thermal Physics | 热物理学
The ideal gas equation links pressure, volume, temperature and amount of gas. The first law of thermodynamics accounts for energy transfers as heat and work.
理想气体方程关联压强、体积、温度和气体量。热力学第一定律解释了热与功的能量传递。
pV = nRT
Ideal gas equation for n moles; R = 8.31 J mol⁻¹ K⁻¹.
n 摩尔理想气体方程；R = 8.31 J mol⁻¹ K⁻¹。
pV = NkT
Form using the number of molecules N and Boltzmann constant k.
使用分子数 N 和玻尔兹曼常数 k 的形式。
ΔU = Q + W
First law of thermodynamics (CAIE convention): increase in internal energy ΔU equals heat added to the system Q plus work done ON the system W.
热力学第一定律（CAIE 约定）：内能增量 ΔU 等于系统吸收的热量 Q 加上外界对系统做的功 W。
7. Atomic and Nuclear Physics | 原子与核物理
The photoelectric effect confirms the particle nature of light. Radioactive decay follows an exponential law characterised by the decay constant and half-life.
光电效应证实了光的粒子性。放射性衰变遵循以衰变常数和半衰期为特征的指数规律。
E = hf
Photon energy; h is Planck's constant, f is frequency.
光子能量；h 为普朗克常数，f 为频率。
hf = φ + Kₘₐₓ
Einstein's photoelectric equation: photon energy equals work function φ plus maximum kinetic energy of emitted electron.
爱因斯坦光电方程：光子能量等于逸出功 φ 加上逸出电子的最大动能。
A = -dN/dt = λN
Activity A is the rate of decay, proportional to the number of undecayed nuclei N; λ is the decay constant.
活度 A 为衰变速率，正比于未衰变核数目 N；λ 为衰变常数。
N = N₀ e⁻λt
Exponential decay law; N₀ is the initial number of nuclei.
指数衰变规律；N₀ 为初始核数。
T½ = ln2 / λ
Half-life T½ related to decay constant; approximately 0.693 / λ.
半衰期 T½ 与衰变常数的关系：约为 0.693 / λ。
8. Stoichiometry and the Mole | 化学计量与摩尔
Amount of substance (mole) links mass, molar mass, gas volumes and solution concentrations. These relationships are the quantitative backbone of chemistry.
物质的量（摩尔）将质量、摩尔质量、气体体积和溶液浓度联系起来，是化学定量计算的支柱。
n = m / M
Mole calculation: amount n equals mass m divided by molar mass M.
摩尔计算：物质的量 n 等于质量 m 除以摩尔质量 M。
n = V / Vₘ
t room temperature and pressure (rtp), Vₘ = 24.0 dm³ mol⁻¹ under CAIE standard conditions.
气体的摩尔数：V 为室温常压下的体积，CAIE 标准条件下 Vₘ = 24.0 dm³ mol⁻¹。
c = n / V
Concentration c (mol dm⁻³) is amount of solute divided by volume of solution.
浓度 c (mol dm⁻³) 为溶质物质的量除以溶液体积。
c₁V₁ = c₂V₂
Dilution relationship: the number of moles remains constant when a solution is diluted.
稀释关系：溶液稀释时物质的量不变。
9. Chemical Kinetics and Equilibrium | 化学动力学与平衡
The rate of a reaction often depends on reactant concentrations raised to some powers. At dynamic equilibrium, the ratio of product and reactant concentrations is constant at a given temperature.
反应速率通常取决于反应物浓度的某次幂。在动态平衡时，产物浓度与反应物浓度之比在给定温度下为常数。
rate = k [A]ᵐ [B]ⁿ
Rate equation: k is the rate constant, m and n are orders of reaction with respect to A and B.
速率方程：k 为速率常数，m 和 n 分别为对 A 和 B 的反应级数。
Kc = [C]ᶜ[D]ᵈ / [A]ᵃ [B]ᵇ
Equilibrium constant Kc for the general reaction aA + bB ⇌ cC + dD; concentrations are equilibrium values in mol dm⁻³.
一般反应 aA + bB ⇌ cC + dD 的平衡常数 Kc，浓度为平衡时的值（mol dm⁻³）。
10. Biological Molecules and Enzyme Kinetics | 生物分子与酶动力学
Key biochemical tests identify macromolecules. The Michaelis–Menten model describes enzyme-catalysed reaction rates, highlighting the significance of Vmax and Km.
关键生化测试用于鉴定大分子。米氏模型描述了酶催化反应速率，突出 Vmax 和 Km 的重要性。
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