📚 AS CAIE Science: Formula & Theorem Quick Reference Guide | AS CAIE 科学：公式定理速查手册
This concise handbook gathers the key formulas, equations and theorems you need for the CAIE AS Level Science syllabus. Covering physics, chemistry and biology, it is designed as a rapid revision tool. Use it to check relationships, practice calculations and build confidence before your examinations.
这本速查手册汇集了 CAIE AS Level 科学课程必备的核心公式、方程与定理，涵盖物理、化学和生物三部分。它是一款简明复习工具，可用于核对关系式、练习计算，并在考前建立信心。
1. SI Units and Prefixes | 国际单位制与词头
The SI base units are metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd). Prefixes like kilo (k = 10³), centi (c = 10⁻²), milli (m = 10⁻³) and micro (μ = 10⁻⁶) scale quantities. Always convert data to base units before substituting into formulas. Derived units include the newton (N = kg m s⁻²), joule (J = N m) and watt (W = J s⁻¹). The relationship Power = Work done / Time gives the watt.
SI 基本单位包括米 (m)、千克 (kg)、秒 (s)、安培 (A)、开尔文 (K)、摩尔 (mol) 和坎德拉 (cd)。词头如千 (k = 10³)、厘 (c = 10⁻²)、毫 (m = 10⁻³) 和微 (μ = 10⁻⁶) 用来缩放数值。在代入公式前，务必先将数据换算为基本单位。导出单位有牛顿 (N = kg m s⁻²)、焦耳 (J = N m) 和瓦特 (W = J s⁻¹)。关系式 功率 = 做功 / 时间 定义了瓦特。
P = W / t
2. Kinematics Equations | 运动学公式
For uniformly accelerated motion along a straight line, the four SUVAT equations link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). Choose the equation that excludes the unknown quantity. Acceleration is constant in these expressions.
对于直线上的匀加速运动，四个 SUVAT 方程将位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t) 联系起来。选择不含待求量的那个方程。在这些表达式中加速度是恒定的。
v = u + at
s = ut + ½ at²
v² = u² + 2as
s = (u + v) t / 2
3. Dynamics and Newton’s Laws | 动力学与牛顿定律
Newton’s second law states that the net force acting on a body equals the rate of change of its momentum. For constant mass, it simplifies to F = ma, where F is resultant force, m is mass and a is acceleration. Weight is the gravitational force W = mg, with g = 9.81 m s⁻² on Earth. Momentum p = mv is conserved in isolated systems. Impulse is force × time, equal to the change in momentum: FΔt = Δp.
牛顿第二定律指出，作用在物体上的合力等于其动量的变化率。当质量恒定时，简化为 F = ma，其中 F 为合力，m 为质量，a 为加速度。重力即重量 W = mg，地球上 g = 9.81 m s⁻²。动量 p = mv 在孤立系统中守恒。冲量为力 × 时间，等于动量的变化量：FΔt = Δp。
F = ma
W = mg
p = mv
4. Work, Energy and Power | 功、能和功率
Work done by a constant force is W = Fd cos θ, where θ is the angle between force and displacement. Kinetic energy Eₖ = ½ mv² and gravitational potential energy Eₚ = mgh (near Earth’s surface). Power is the rate of doing work: P = W / t. For an object moving at constant speed against a force, power can also be written as P = Fv. Efficiency is the ratio of useful output to total input; it is always less than 1.
恒力做功为 W = Fd cos θ，其中 θ 为力与位移之间的夹角。动能 Eₖ = ½ mv²，重力势能 Eₚ = mgh（近地表）。功率是做功的速率：P = W / t。对于恒速抵抗外力运动的物体，功率也可写为 P = Fv。效率为有用输出与总输入之比，总小于 1。
W = Fd cos θ
Eₖ = ½ mv²
Eₚ = mgh
efficiency = useful output / total input
5. Waves and Optics | 波动与光学
The wave equation v = fλ links speed (v), frequency (f) and wavelength (λ). For light travelling from one medium to another, the refractive index n = sin i / sin r = c₀ / c, where i is the angle of incidence, r is the angle of refraction and c₀ is the speed of light in a vacuum. The critical angle θc satisfies sin θc = 1 / n. The thin lens formula is 1/f = 1/u + 1/v, with magnification M = v/u (real-is-positive sign convention).
波动方程 v = fλ 将波速 (v)、频率 (f) 和波长 (λ) 联系起来。光从一种介质进入另一种介质时，折射率 n = sin i / sin r = c₀ / c，其中 i 是入射角，r 是折射角，c₀ 是真空中的光速。临界角 θc 满足 sin θc = 1 / n。薄透镜公式为 1/f = 1/u + 1/v，放大率 M = v/u（采用实正符号法则）。
v = fλ
n = sin i / sin r
sin θc = 1 / n
1/f = 1/u + 1/v
6. Electricity Fundamentals | 电学基础
Ohm’s law states V = IR for a metallic conductor at constant temperature. Electrical power dissipated in a resistor is P = IV = I²R = V²/R. For series resistors, total resistance Rₜ = R₁ + R₂ + …; for parallel resistors, 1/Rₜ = 1/R₁ + 1/R₂ + … The electromotive force (emf) ε of a source equals the terminal voltage plus the internal voltage drop: ε = V + Ir. Kirchhoff’s first law (junction rule) states that the sum of currents entering a junction equals the sum leaving.
欧姆定律指出，在恒温下金属导体满足 V = IR。电阻器消耗的电功率为 P = IV = I²R = V²/R。串联电阻的总电阻 Rₜ = R₁ + R₂ + …；并联电阻满足 1/Rₜ = 1/R₁ + 1/R₂ + …。电源电动势 ε 等于端电压加上内电压降：ε = V + Ir。基尔霍夫第一定律（节点定律）指出，流入节点的电流之和等于流出节点的电流之和。
V = IR
P = IV
ε = V + Ir
7. Chemical Formulae and Moles | 化学式与摩尔
The mole is the amount of substance containing Avogadro’s number (6.02 × 10²³) of particles. The number of moles n can be found from mass and molar mass: n = m / M. For gases at room temperature and pressure (rtp), the molar volume Vₘ ≈ 24 dm³ mol⁻¹, so n = V(gas) / Vₘ. Concentration c (mol dm⁻³) is given by c = n / V(solution). Percentage yield = (actual yield / theoretical yield) × 100%.
摩尔是包含阿伏伽德罗常数 (6.02 × 10²³) 个粒子的物质的量。摩尔数 n 可由质量和摩尔质量求得：n = m / M。对于室温和常压下的气体，摩尔体积 Vₘ ≈ 24 dm³ mol⁻¹，因此 n = V(气体) / Vₘ。浓度 c（mol dm⁻³）为 c = n / V(溶液)。产率百分数 = (实际产量 / 理论产量) × 100%。
n = m / M
n = V(gas) / Vₘ
c = n / V
8. Chemical Energetics | 化学能量学
Enthalpy change ΔH in a reaction can be calculated from bond energies: ΔH = Σ(bond energies broken) − Σ(bond energies formed). In calorimetry, the heat transferred q is measured by temperature change: q = mcΔT, where m is mass, c is specific heat capacity and ΔT is temperature change. To find ΔH in kJ mol⁻¹, divide the heat by the number of moles reacting: ΔH = −q / n (the negative sign indicates exothermic if heat is released to the surroundings).
反应焓变 ΔH 可通过键能计算：ΔH = Σ(断裂键的键能) － Σ(形成键的键能)。在量热法中，传递的热量 q 通过温度变化测量：q = mcΔT，其中 m 是质量，c 是比热容，ΔT 是温度变化。为求得 ΔH (kJ mol⁻¹)，用热量除以反应摩尔数：ΔH = −q / n（负号表示放热，热量释放到环境中）。
ΔH = ΣE(broken) − ΣE(formed)
q = mcΔT
ΔH = −q / n
9. Rates and Equilibrium | 反应速率与平衡
For a reaction aA + bB ⇌ cC + dD, the equilibrium constant Kc is expressed as Kc = ([C]ᶜ [D]ᵈ) / ([A]ᵃ [B]ᵇ), where square brackets denote equilibrium concentrations. Le Chatelier’s principle states that a system at equilibrium, when subjected to a change, will shift to oppose the change. For a first-order reaction, rate = k[A]; the units of k depend on the overall order.
对于反应 aA + bB ⇌ cC + dD，平衡常数 Kc 表示为 Kc = ([C]ᶜ [D]ᵈ) / ([A]ᵃ [B]ᵇ)，方括号表示平衡浓度。勒夏特列原理指出，处于平衡的系统在受到扰动时会向削弱该扰动的方向移动。对于一级反应，速率 = k[A]；k 的单位取决于总反应级数。
Kc = [C]ᶜ [D]ᵈ / [A]ᵃ [B]ᵇ
rate = k[A]
10. Cell Biology and Magnification | 细胞生物学与放大倍数
Magnification is the ratio of image size to actual size: M = I / A. Always convert both measurements to the same unit. The surface area to volume ratio (SA:V) limits cell size; a higher ratio allows faster diffusion. For approximate calculations, surface area of a sphere is 4πr² and volume is (4/3)πr³, giving SA:V = 3/r. This relationship explains why cells are microscopic.
放大倍数是图像尺寸与实际尺寸之比：M = I / A。务必先将两者换算为同一单位。表面积与体积比 (SA:V) 限制了细胞大小；比值越大，扩散越快。近似计算中，球的表面积为 4πr²，体积为 (4/3)πr³，因此 SA:V = 3/r。这一关系解释了细胞为什么很小。
Magnification = image size / actual size
SA:V = 3/r (for a sphere)
11. Genetics and Probability | 遗传学与概率
Monohybrid crosses between heterozygous parents (Tt × Tt) produce a phenotypic ratio of 3 : 1 (dominant : recessive). The probability of each genotype can be calculated with a Punnett square: ¼ TT, ½ Tt, ¼ tt. For a dihybrid cross (TtYy × TtYy) with independent assortment, the expected phenotypic ratio is 9 : 3 : 3 : 1. Test crosses with a homozygous recessive individual reveal unknown genotypes.
杂合亲本 (Tt × Tt) 的单因子杂交产生 3 : 1 的表现型比例（显性：隐性）。每种基因型的概率可用庞纳特方格计算：¼ TT、½ Tt、¼ tt。对于独立分配的双因子杂交 (TtYy × TtYy)，预期表现型比例为 9 : 3 : 3 : 1。与纯合隐性个体的测交可揭示未知基因型。
Monohybrid F₂ ratio: 3 : 1
Dihybrid F₂ ratio: 9 : 3 : 3 : 1
12. Transport and Exchange Formulae | 运输与交换公式
Fick’s law describes the rate of diffusion: rate ∝ (surface area × concentration difference) / thickness of membrane. In the human circulatory system, cardiac output = stroke volume × heart rate. For efficient gas exchange, biological surfaces are thin, moist and have a large surface area. Diffusion gradients are maintained by ventilation and blood flow.
菲克定律描述了扩散速率：速率 ∝ (表面积 × 浓度差) / 膜厚度。在人体循环系统中，心输出量 = 搏出量 × 心率。为了实现高效的气体交换，生物表面通常很薄、湿润且具有很大的表面积。通风和血流维持了扩散浓度梯度。
rate of diffusion ∝ (SA × ΔC) / d
cardiac output = stroke volume × heart rate
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