📚 CAIE IGCSE Science Formula & Theorem Quick Reference Handbook | CAIE IGCSE 科学公式定理速查手册
This quick reference handbook brings together the essential equations, formulas and key theorems across Physics, Chemistry and Biology for the CAIE IGCSE Co-ordinated Sciences (Double Award) or Combined Science syllabus. Use it to rapidly check definitions, manipulate relationships, and strengthen your problem-solving confidence before the examination.
本速查手册汇集了 CAIE IGCSE 协同科学（双奖）或综合科学考纲中物理、化学和生物的核心方程、公式及关键定理。可用于快速查阅定义、变换关系，并在考前提升解题信心。
1. Motion & Forces | 运动与力
Average speed is total distance travelled divided by time taken. When an object moves with constant acceleration, three equations of motion link initial velocity u, final velocity v, acceleration a, time t and displacement s.
平均速率等于总路程除以时间。当物体以恒定加速度运动时，三条运动学方程将初速度 u、末速度 v、加速度 a、时间 t 和位移 s 联系起来。
v = s / t
平均速率 v 等于距离 s 除以时间 t：v = s / t。
a = (v − u) / t
加速度 a 等于速度变化量除以时间：a = (v − u) / t。
v = u + a t | s = u t + ½ a t² | v² = u² + 2 a s
匀加速运动方程：v = u + a t，s = u t + ½ a t²，v² = u² + 2 a s。
Newton’s second law states that resultant force F equals mass m multiplied by acceleration a. Weight is the gravitational force on a mass, and momentum is the product of mass and velocity.
牛顿第二定律指出合力 F 等于质量 m 乘以加速度 a。重量是作用于物体质量的引力，动量是质量与速度的乘积。
F = m a | W = m g
合力 F = m a，重量 W = m g（g = 9.8 m/s² 地球表面重力场强）。
p = m v | F = Δp / Δt
动量 p = m v；力等于动量变化率，F = Δp / Δt。在碰撞中，若外力为零，总动量守恒。
2. Energy, Work & Power | 能量、功与功率
Energy cannot be created or destroyed, only transferred. Kinetic energy depends on mass and the square of speed; gravitational potential energy depends on mass, height and gravitational field strength. Work is done when a force moves an object in the direction of the force.
能量既不能创造也不能消灭，只能转移。动能取决于质量与速率的平方；重力势能取决于质量、高度与重力场强度。力沿其方向移动物体时做功。
KE = ½ m v² | GPE = m g h
动能 KE = ½ m v²，重力势能 GPE = m g h。
W = F d | P = W / t = F v
功 W = F d（力与位移平行）；功率 P 为做功速率：P = W / t，也可表示为 P = F v。
Efficiency is the ratio of useful output energy to total input energy, multiplied by 100 %.
效率是有用输出能量与总输入能量之比，乘以 100 %。
Efficiency = (Useful output energy / Total input energy) × 100 %
效率 = (有用输出能量 ÷ 总输入能量) × 100 %。
3. Electricity | 电学
Electric charge Q is the product of current I and time t. Potential difference V across a conductor is related to current and resistance by Ohm’s law. Power dissipated in a resistor can be expressed in three equivalent forms.
电荷量 Q 等于电流 I 与时间 t 的乘积。导体两端电势差 V 通过欧姆定律与电流、电阻关联。电阻器耗散的功率可用三种等价形式表达。
Q = I t | V = I R
电荷 Q = I t；欧姆定律 V = I R（在恒温条件下适用）。
P = I V = I² R = V² / R
电功率 P = I V = I² R = V² / R。电能 E = P t = I V t。
For resistors in series, total resistance is the sum; for resistors in parallel, the reciprocal of total resistance equals the sum of the reciprocals of individual resistances.
串联电阻总阻值为各电阻之和；并联时总电阻的倒数等于各电阻倒数之和。
Rtotal = R₁ + R₂ + … | 1/Rtotal = 1/R₁ + 1/R₂ + …
R总 = R₁ + R₂ + … ；1/R总 = 1/R₁ + 1/R₂ + …。
An ideal transformer relates primary and secondary voltages to the turns ratio. For a 100 % efficient transformer, input power equals output power.
理想变压器将原、副线圈电压与匝数比相联系。对于效率100%的变压器，输入功率等于输出功率。
Vp / Vs = Np / Ns | Pinput = Poutput ⇒ Vp Ip = Vs Is
变压比：Vp / Vs = Np / Ns；理想时 P输入 = P输出，即 Vp Ip = Vs Is。
4. Waves | 波
The wave equation links wave speed v, frequency f and wavelength λ. Refractive index n of a medium can be found from the ratio of the sine of the angle of incidence i to the sine of the angle of refraction r, or from the speed of light in vacuum c and in the medium.
波速方程将波速 v、频率 f 和波长 λ 联系起来。介质的折射率 n 可由入射角 i 的正弦与折射角 r 的正弦之比求得，也可由光在真空中的速度 c 与介质中的速度之比求得。
v = f λ
波速 v = f λ。
n = sin i / sin r | n = c / v
折射率 n = sin i / sin r，也等于 c / v。
The critical angle c for total internal reflection is the angle of incidence for which the angle of refraction is 90°.
全内反射的临界角 c 是折射角为 90° 时的入射角。
sin c = 1 / n
sin c = 1 / n（从光密介质到光疏介质）。
5. Thermal Physics | 热物理
The thermal energy required to change the temperature of a substance depends on its mass m, specific heat capacity c and temperature change Δθ. During a change of state, the energy required depends on mass and specific latent heat L.
改变物体温度所需的热能取决于其质量 m、比热容 c 和温度变化 Δθ。在物态变化过程中，所需能量取决于质量与比潜热 L。
Q = m c Δθ
显热 Q = m c Δθ。
Q = m L (Lf for fusion, Lv for vaporisation)
潜热 Q = m L（Lf 为熔解比潜热，Lv 为汽化比潜热）。
The pressure of a fixed mass of gas at constant volume is proportional to its absolute temperature (Kelvin).
定容下，一定质量气体的压强与其绝对温度（开尔文）成正比。
p₁ / T₁ = p₂ / T₂ (V constant)
p₁ / T₁ = p₂ / T₂（体积不变）。
6. Matter & Density | 物质与密度
Density is mass per unit volume. Pressure exerted by a force on a surface is force per unit area. For a liquid column, pressure depends on density, gravitational field strength and depth.
密度是单位体积的质量。力在表面上产生的压强是单位面积上的力。液柱产生的压强取决于密度、重力场强度和深度。
ρ = m / V
密度 ρ = m / V。
p = F / A | p = ρ g h
压强 p = F / A；液体压强 p = ρ g h。
For a gas, pressure and volume at constant temperature follow Boyle’s law: pV = constant.
对于气体，恒温下压强与体积遵循玻意耳定律：pV = 常数。
p₁ V₁ = p₂ V₂
p₁ V₁ = p₂ V₂（温度恒定）。
7. Chemistry: Moles & Stoichiometry | 化学：摩尔与化学计量
The amount of substance n in moles is mass m divided by molar mass M. The concentration c of a solution is the amount of solute divided by the volume of solution V in dm³.
物质的量 n（单位摩尔）等于质量 m 除以摩尔质量 M。溶液的浓度 c 为溶质的物质的量除以溶液体积 V（dm³）。
n = m / M | c = n / V
n = m / M；浓度 c = n / V (mol/dm³)。
At room temperature and pressure (r.t.p.), one mole of any gas occupies 24 dm³. The volume of gas Vgas is therefore proportional to amount in moles.
在室温和常压下（r.t.p.），1 mol 任何气体占据 24 dm³。因此气体体积 V气体 与物质的量成正比。
Vgas (dm³) = n × 24
气体体积 V气体 (dm³) = n × 24（r.t.p.）。
Percentage yield compares the actual mass of product obtained to the theoretical mass predicted from the balanced equation.
产率百分数比较实际获得的产品质量与根据配平方程预测的理论质量。
% yield = (Actual mass / Theoretical mass) × 100
产率 = (实际质量 ÷ 理论质量) × 100 %。
8. Chemistry: Rates & Equilibrium | 化学：速率与平衡
The rate of a reaction can be expressed as the change in concentration of a reactant or product per unit time. For a gaseous system, rate may be followed via volume of gas produced per second.
反应速率可表示为反应物或生成物浓度在单位时间内的变化。对于气体体系，速率可通过每秒生成的气体体积来跟踪。
Rate = Δ(quantity) / Δt
速率 = 变化量 / 时间间隔。
For a reversible reaction at equilibrium, the equilibrium constant Kc is given by the ratio of the product concentrations to reactant concentrations, each raised to the power of its stoichiometric coefficient.
对于可逆反应达到平衡，平衡常数 Kc 等于生成物浓度幂的乘积除以反应物浓度幂的乘积，幂次等于化学计量系数。
aA + bB ⇌ cC + dD ⇒ Kc = [C]ᶜ [D]ᵈ / ([A]ᵃ [B]ᵇ)
对于 aA + bB ⇌ cC + dD，Kc = [C]ᶜ [D]ᵈ / ([A]ᵃ [B]ᵇ)。（注意：Unicode中上标c d a b可能不是所有环境都支持，这里使用ᶜ等。保险起见，可写 Kc = [C]^c [D]^d / ([A]^a [B]^b)，但需要用Unicode。我使用 [C]ᶜ，或者直接用 [C]^c？指令要求用Unicode，^不是上标，最好用上标数字，但c,d,a,b不是数字。可以写成文字描述。重新调整：可以用文字说明，或者用 [C]c？不行，只能用Unicode。Unicode没有上标字母c,d等。因此使用Kc = [C] raised to power c等等。为了符合Unicode，改用公式用文字描述，或者使用[产物]系数。我们改为稳妥：用下标数字，上标数字示例，这里避免上标字母。可重新写为 Kc = [C]ᶜ [D]ᵈ / ([A]ᵃ [B]ᵇ) 使用上标字母可能无法显示。最好用 Kc = [products]coefficients… 但必须用Unicode。我们可以这样：Kc = [C]ᶜ [D]ᵈ 但ᶜ是Unicode MODIFIER LETTER SMALL C，理论上可用。我使用ᶜ ᵈ ᵃ ᵇ。可以。但可能不支持，但尝试。或者改用：Kc = [C]ʳ [D]ˢ / ([A]ᵐ [B]ⁿ) 但实际系数用字母。我可以不使用字母上标，而是给出通用形式：Kc = [C]ᵖ [D]ᵠ / ([A]ʳ [B]ˢ)，但不确定。还是用描述性公式：Kc = [products]^coefficients / [reactants]^coefficients。但指令要求公式用Unicode居中，可以写成文字结合。我决定采用安全方式：写成Kc = ([C] × [D]) / ([A] × [B]) with powers, 但不够准确。我们使用上标数字示例，通常教材中Kc表达式可能涉及上标字母。我可以接受Kc = [C]ᶜ [D]ᵈ / ([A]ᵃ [B]ᵇ) 这些上标字符在Unicode中存在。就这样。)
9. Biology: Magnification & Size | 生物：放大率与尺寸
Whenever you use a microscope or examine a diagram, the magnification M is the ratio of the image size to the actual size of the specimen. Always convert all lengths to the same unit before calculating.
使用显微镜或检查示意图时，放大率 M 是图像尺寸与标本实际尺寸之比。计算前务必将所有长度换算为相同单位。
M = Image size / Actual size or M = I / A
放大率 M = 图像尺寸 ÷ 实际尺寸，或简写为 M = I / A。
When given a scale bar, measure its length in the image, convert to the same units, and apply the formula.
若给出比例尺，测量其在图像中的长度，换算至相同单位后再代入公式。
10. Biology: Photosynthesis & Respiration | 生物：光合作用与呼吸
The overall word equations and balanced chemical equations summarise the two key biochemical processes. Photosynthesis uses light energy to convert carbon dioxide and water into glucose and oxygen. Aerobic respiration releases energy from glucose in the presence of oxygen.
总文字方程式和配平的化学方程式概括了两个关键的生物化学过程。光合作用利用光能将二氧化碳和水转化为葡萄糖和氧气。有氧呼吸在有氧条件下从葡萄糖中释放能量。
6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂ (photosynthesis)
光合作用：6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂。
C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O (aerobic respiration)
有氧呼吸：C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O。
While these are not calculation formulas in the strict sense, balancing them correctly is essential for stoichiometry questions in the Biology and Chemistry overlap.
严格来说这些并非计算公式，但正确配平对于生物与化学交叉部分的化学计量题至关重要。
11. Key Principles & Theorems | 关键原理与定理
Beyond mathematical formulas, the CAIE Science syllabus expects you to recall and apply fundamental laws that govern physical, chemical and biological systems.
除数学公式外，CAIE 科学考纲要求你回忆并应用支配物理、化学和生物系统的基本定律。
Newton’s First Law (Inertia): An object remains at rest or in uniform motion in a straight line unless acted upon by a resultant external force.
牛顿第一定律（惯性定律）：除非受到净外力作用，物体将保持静止或匀速直线运动状态。
Newton’s Third Law: When body A exerts a force on body B, body B exerts an equal and opposite force on body A; these forces act on different bodies.
牛顿第三定律：当物体 A 对物体 B 施加一个力时，物体 B 同时对物体 A 施加一个大小相等、方向相反的力；这两个力作用在不同物体上。
Ohm’s Law: At constant temperature, the current through a metallic conductor is directly proportional to the potential difference across it.
欧姆定律：在恒温下，通过金属导体的电流与其两端电势差成正比。
Conservation of Energy: Energy cannot be created or destroyed; it can only be transferred, stored or dissipated.
能量守恒：能量既不能创造也不能消灭，只能转移、储存或耗散。
Conservation of Momentum: In a closed system with no external resultant force, the total momentum before a collision equals the total momentum after.
动量守恒：在没有外净力的封闭系统中，碰撞前的总动量等于碰撞后的总动量。
Le Chatelier’s Principle: If a system at dynamic equilibrium is subjected to a change in concentration, temperature or pressure, the equilibrium position shifts to oppose the change.
勒夏特列原理：处于动态平衡的体系若受到浓度、温度或压强的变化，平衡位置将朝削弱该变化的方向移动。
Law of Reflection: Angle of incidence i equals angle of reflection r, both measured from the normal.
反射定律：入射角 i 等于反射角 r，两者均从法线量度。
Published by TutorHao |
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导