📚 Year 12 CIE Science: Formula & Theorem Quick Reference Handbook | Year 12 CIE 科学：公式定理速查手册
This quick reference handbook brings together the essential formulas, theorems, and key expressions needed for Year 12 CIE Sciences, covering Physics, Chemistry, and Biology. Each entry is presented with a concise English explanation, the relevant formula in a clear central display, and a matching Chinese explanation, making it easy to revise on the go.
本速查手册汇集了 Year 12 CIE 科学（物理、化学、生物）中必备的公式、定理与核心表达式。每个条目都配有简洁的英文说明、居中展示的公式以及对应的中文解释，方便随时复习查阅。
1. Kinematics Formulas | 运动学公式
The SUVAT equations describe uniformly accelerated motion along a straight line. Each equation links four of the five variables: s (displacement), u (initial velocity), v (final velocity), a (acceleration), t (time).
v = u + at
SUVAT 方程描述匀加速直线运动。每个方程联系五个变量中的四个：s（位移）、u（初速度）、v（末速度）、a（加速度）、t（时间）。方程一：v = u + at，表示末速度等于初速度加上加速度与时间的乘积。
Equation without final velocity: s = ut + ½at².
s = ut + ½at²
不含末速度的方程：s = ut + ½at²，位移等于初速度乘以时间加上二分之一加速度乘以时间的平方。
Equation without time: v² = u² + 2as.
v² = u² + 2as
不含时间的方程：v² = u² + 2as，末速度的平方等于初速度的平方加上两倍加速度与位移的乘积。
The average velocity form is useful when acceleration is constant: s = ½(u + v)t.
s = ½(u + v)t
平均速度形式在加速度恒定时很有用：s = ½(u + v)t，位移等于初速度与末速度平均值乘以时间。
2. Dynamics & Momentum | 动力学与动量
Newton’s second law states that the net force on an object is equal to the rate of change of its momentum. For constant mass, it simplifies to F = ma.
F = ma
牛顿第二定律指出，物体所受合外力等于其动量变化率。对于质量不变的物体，简化为 F = ma，即力等于质量乘以加速度。
Momentum p is the product of mass and velocity. The principle of conservation of momentum holds for isolated systems.
p = mv
动量 p 是质量与速度的乘积。在孤立系统中动量守恒。
Impulse is the change in momentum and equals the average force multiplied by the time interval: Impulse = FΔt = Δp.
Impulse = FΔt = Δp
冲量等于动量的变化，也等于平均作用力与作用时间的乘积：冲量 = FΔt = Δp。
3. Work, Energy & Power | 功、能与功率
Work done by a constant force is the product of the force, displacement and the cosine of the angle between them.
W = Fd cosθ
恒力做的功等于力、位移以及两者夹角余弦的乘积：W = Fd cosθ。
Kinetic energy is the energy of motion. The work–energy theorem links net work to change in kinetic energy.
Eₖ = ½mv²
动能是物体由于运动而具有的能量。动能定理将合外力做的功与动能的变化联系起来：Eₖ = ½mv²。
Gravitational potential energy change near Earth’s surface: ΔEₚ = mgΔh.
ΔEₚ = mgΔh
近地球表面的重力势能变化：ΔEₚ = mgΔh，其中 g 为重力场强度。
Power is the rate of doing work or transferring energy.
P = W/t = Fv
功率是做功或能量转移的速率：P = W/t，对于恒力且速度方向与力同向时也可写作 P = Fv。
4. Materials & Hooke’s Law | 材料与胡克定律
Hooke’s law states that extension is proportional to applied force within the elastic limit. The spring constant k measures stiffness.
F = kx
胡克定律指出，在弹性限度内，弹簧的伸长量与施加的力成正比。劲度系数 k 描述弹簧的刚度：F = kx。
Stress, strain and Young modulus characterise material behaviour. Young modulus E = tensile stress / tensile strain.
E = σ/ε = (F/A) / (ΔL/L₀)
应力、应变和杨氏模量描述材料的力学行为。杨氏模量 E = 拉伸应力/拉伸应变 = (F/A)/(ΔL/L₀)。
Strain energy stored in a stretched wire or spring: Eₚ = ½Fx = ½kx².
Eₚ = ½Fx = ½kx²
拉伸的弹性势能：Eₚ = ½Fx = ½kx²，等于力–位移图线下的面积。
5. Waves | 波动
The wave equation links wave speed v, frequency f and wavelength λ. It applies to all periodic waves.
v = fλ
波动方程联结波速 v、频率 f 和波长 λ：v = fλ，适用于所有周期性波。
For refraction, Snell’s law relates angles of incidence and refraction to refractive indices. For a boundary with vacuum or air, n₁sinθ₁ = n₂sinθ₂.
n₁ sinθ₁ = n₂ sinθ₂
折射定律（斯涅耳定律）：n₁ sinθ₁ = n₂ sinθ₂。光从介质1进入介质2时，入射角与折射角的正弦之比等于折射率之比。
Critical angle for total internal reflection: sin C = n₂/n₁ (where n₁ > n₂). For glass to air, sin C = 1/n.
sin C = 1/n
全反射临界角：sin C = 1/n（光从折射率为 n 的介质射向空气时）。
6. Electricity | 电学
Ohm’s law states that current through a conductor is proportional to potential difference at constant temperature.
V = IR
欧姆定律：在恒定温度下，通过导体的电流与两端电势差成正比，即 V = IR。
Electrical power can be expressed in several forms, all equivalent for resistive components.
P = IV = I²R = V²/R
电功率的三种等效形式：P = IV、P = I²R、P = V²/R，适用于纯电阻电路。
Resistors in series: total resistance Rₛ = R₁ + R₂ + … . In parallel: 1/Rₚ = 1/R₁ + 1/R₂ + … .
Rₛ = R₁ + R₂ + … | 1/Rₚ = 1/R₁ + 1/R₂ + …
电阻串联：总电阻 Rₛ = R₁ + R₂ + … ；电阻并联：1/Rₚ = 1/R₁ + 1/R₂ + … 。
Electromotive force (emf) ε and internal resistance r: terminal pd V = ε − Ir.
V = ε − Ir
电动势 ε 与内阻 r：端电压 V = ε − Ir，其中 I 为回路电流。
7. Stoichiometry & Gas Laws | 化学计量与气体定律
The mole is the central unit to relate mass, molar mass and number of particles. Amount of substance n = m/M.
n = m/M
摩尔是联系质量与粒子数的核心单位。物质的量 n = m/M，m 为质量，M 为摩尔质量。
Concentration of a solution: c = n/V, where V is the volume of solution in dm³.
c = n/V
溶液浓度：c = n/V，V 为溶液的体积（单位 dm³），n 为溶质的物质的量。
The ideal gas equation links pressure, volume, amount and temperature: pV = nRT. R = 8.31 J mol⁻¹ K⁻¹.
pV = nRT
理想气体状态方程：pV = nRT，p 为压强，V 为体积，n 为物质的量，T 为热力学温度，R = 8.31 J mol⁻¹ K⁻¹。
8. Chemical Energetics | 化学能量学
Enthalpy change ΔH is the heat absorbed or released at constant pressure. It can be calculated from standard enthalpies of formation.
ΔH = ΣΔH°f(products) − ΣΔH°f(reactants)
焓变 ΔH 表示恒压下的热交换。可由标准生成焓计算：ΔH = ΣΔH°f(生成物) − ΣΔH°f(反应物)。
Using bond energies: ΔH ≈ Σ(bond energies broken) − Σ(bond energies formed).
ΔH ≈ ΣE(bonds broken) − ΣE(bonds formed)
利用键能估算：ΔH ≈ 断裂键的总键能 − 形成键的总键能。
Calorimetry: q = mcΔT relates heat transferred to temperature change. For reaction enthalpy, ΔH = −q/n or −q/m (per mole or mass).
q = mcΔT
量热法：q = mcΔT，m 为质量，c 为比热容，ΔT 为温度变化。反应焓变 ΔH = −q/n（每摩尔）。
9. Chemical Equilibrium & Kinetics | 化学平衡与动力学
For a reversible reaction aA + bB ⇌ cC + dD, the equilibrium constant in terms of concentration is Kc.
Kc = ([C]ᶜ[D]ᵈ) / ([A]ᵃ[B]ᵇ)
对于可逆反应 aA + bB ⇌ cC + dD，浓度平衡常数 Kc = ([C]ᶜ[D]ᵈ) / ([A]ᵃ[B]ᵇ)。
Rate equations take the form rate = k[A]ᵐ[B]ⁿ, where m and n are orders with respect to each reactant.
rate = k[A]ᵐ[B]ⁿ
速率方程：rate = k[A]ᵐ[B]ⁿ，m 和 n 分别为对 A 和 B 的反应级数，k 为速率常数。
The Arrhenius equation links rate constant to temperature: k = Ae⁻ᴱᵃ/ᴿᵀ. The logarithmic form ln k = ln A − Ea/(RT) can be used to find activation energy.
k = Ae⁻ᴱᵃ/ᴿᵀ ; ln k = ln A − Ea/(RT)
阿伦尼乌斯方程：k = Ae⁻ᴱᵃ/ᴿᵀ，对数形式 ln k = ln A − Ea/(RT)，用于求算活化能 Ea。
10. Organic Chemistry Core Reactions | 有机化学核心反应
Alkanes undergo free-radical substitution with halogens in UV light: CH₄ + Cl₂ → CH₃Cl + HCl (chain reaction).
CH₄ + Cl₂ → CH₃Cl + HCl
烷烃在紫外光下与卤素发生自由基取代反应：CH₄ + Cl₂ → CH₃Cl + HCl（链反应）。
Alkenes undergo electrophilic addition. Ethene with bromine: C₂H₄ + Br₂ → C₂H₄Br₂.
C₂H₄ + Br₂ → C₂H₄Br₂
烯烃发生亲电加成反应。乙烯与溴反应：C₂H₄ + Br₂ → C₂H₄Br₂。
Oxidation of alcohols: primary alcohol → aldehyde → carboxylic acid using K₂Cr₂O₇/H⁺. The functional group changes are important to recognise.
RCH₂OH → RCHO → RCOOH
醇的氧化：伯醇经酸化重铬酸钾依次氧化为醛（RCHO）和羧酸（RCOOH）。
11. Biology: Magnification & Chi-Squared Test | 生物：放大率与卡方检验
Magnification is the ratio of image size to actual object size. Ensure units are the same.
Magnification = Image size / Actual size
放大率 = 图像尺寸 / 实际尺寸，计算时需保持单位一致。
In genetics and ecology, the chi-squared test (χ²) compares observed and expected frequencies. The formula sums the squared relative differences.
χ² = Σ (O − E)² / E
卡方检验（χ²）用于比较观测频数与期望频数：χ² = Σ (O − E)² / E。O 为观测值，E 为期望值。
Degrees of freedom for a monohybrid cross goodness-of-fit test is typically n − 1, where n is the number of phenotypic classes.
df = n − 1
单因子杂交适合度检验的自由度通常为 n−1，n 为表现型类别数。将计算所得的 χ² 值与临界值比较以判断显著性。
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