Pre-U Cambridge Science: Quick Reference Handbook of Formulas and Theorems | Pre-U Cambridge 科学：公式定理速查手册

📚 Pre-U Cambridge Science: Quick Reference Handbook of Formulas and Theorems | Pre-U Cambridge 科学：公式定理速查手册

This concise handbook compiles the essential formulas, laws, and theorems required for the Cambridge Pre-U Science syllabus. It covers key principles from Physics, Chemistry, and Biology, providing a rapid reference for revision and problem-solving. Each entry is presented with clear English and Chinese definitions, ensuring bilingual learners can grasp the concepts firmly. Use this guide to solidify your understanding and tackle examination questions with confidence.

这本简明手册汇编了剑桥大学预科科学课程所需的核心公式、定律和定理。内容涵盖物理、化学和生物的关键原理，为复习与解题提供快速参考。每一条目均配以清晰的英文和中文释义，确保双语学习者牢固掌握概念。请使用本指南巩固理解，自信应对考试题目。

1. Mechanics and Motion | 力学与运动

Mechanics forms the bedrock of physics. The following equations define linear and accelerated motion, forces, energy, and momentum. They are essential for solving problems involving constant acceleration and collisions.

力学是物理学的基础。以下方程定义了直线运动、加速运动、力、能量和动量，对于求解涉及匀加速和碰撞的问题至关重要。

v = u + at

Final velocity equals initial velocity plus acceleration multiplied by time.

末速度等于初速度加上加速度乘以时间。

s = ut + ½at²

Displacement is given by initial velocity times time plus half the acceleration times the square of time.

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。

F = ma

Newton’s second law: net force equals mass times acceleration.

牛顿第二定律：合力等于质量乘以加速度。

p = mv

Momentum is the product of mass and velocity. Conservation of momentum applies in closed systems.

动量是质量与速度的乘积。孤立系统中动量守恒。

W = Fs cos θ

Work done is the product of the force component along the displacement and the displacement magnitude.

做功等于位移方向上的力分量与位移大小的乘积。

Eₖ = ½mv²

Kinetic energy depends on mass and the square of speed.

动能取决于质量和速度的平方。

Eₚ = mgh

Gravitational potential energy near Earth’s surface.

地表附近的重力势能。

Power = ΔW/Δt = Fv

Power is the rate of doing work, also expressed as force times velocity when force and velocity are parallel.

功率是做功的速率，当力与速度平行时也可表示为力乘以速度。

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2. Thermal Physics | 热物理

Thermal physics connects temperature, heat, internal energy, and the behaviour of gases. The kinetic model and the laws of thermodynamics are fundamental to understanding energy transfer and gas properties.

热物理将温度、热量、内能与气体行为联系起来。分子动理论模型和热力学定律是理解能量传递与气体性质的基础。

ΔU = Q + W

The first law of thermodynamics: change in internal energy equals heat added to the system plus work done on the system. Sign conventions must be applied consistently.

热力学第一定律：内能的变化等于系统吸收的热量加上对系统所做的功。必须一致使用符号规定。

pV = nRT

The ideal gas equation links pressure, volume, amount in moles, the molar gas constant R, and absolute temperature.

理想气体方程将压强、体积、物质的量（摩尔）、摩尔气体常数 R 和绝对温度联系起来。

pV = ⅓ N m⟨c²⟩

Kinetic theory equation: pressure times volume equals one-third of the number of molecules times molecular mass times mean square speed.

分子动理论方程：压强乘以体积等于分子数乘以分子质量乘以均方速度的三分之一。

Eₖ = ³⁄₂ kT

Average kinetic energy per molecule for an ideal monatomic gas, where k is Boltzmann’s constant.

理想单原子气体每个分子的平均动能，k 是玻尔兹曼常数。

Efficiency η = W/Qₕₒₜ

The efficiency of a heat engine is the ratio of work output to heat input at the hot reservoir.

热机效率是输出功与从高温热源吸收的热量之比。

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3. Waves and Optics | 波与光学

Wave phenomena include reflection, refraction, diffraction, and interference. The wave equation and Snell’s law are crucial for describing how waves propagate and interact with boundaries.

波动现象包括反射、折射、衍射和干涉。波动方程和斯涅尔定律对于描述波如何传播以及与边界相互作用至关重要。

v = fλ

The 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 of refraction, relating the angles of incidence and refraction to the refractive indices of the two media.

斯涅尔折射定律，将入射角和折射角与两种介质的折射率联系起来。

n = c / v

Refractive index is the ratio of the speed of light in vacuum to its speed in the medium.

折射率是光在真空中的速度与在介质中的速度之比。

d sin θ = nλ

Constructive interference for a diffraction grating, where d is the slit separation and n is the order integer.

衍射光栅的相长干涉条件，d 是狭缝间距，n 是级次整数。

Δx = λL / d

Fringe spacing in Young’s double-slit experiment.

杨氏双缝实验中的条纹间距。

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4. Electricity and Magnetism | 电与磁

This section covers the fundamental laws of circuits, fields, and electromagnetic induction. Ohm’s law, Kirchhoff’s rules, and Faraday’s law are indispensable tools for analysing electrical systems.

本节介绍电路、场和电磁感应的基本定律。欧姆定律、基尔霍夫定则和法拉第定律是分析电气系统不可或缺的工具。

V = IR

Ohm’s law: potential difference equals current times resistance, valid for ohmic conductors at constant temperature.

欧姆定律：电势差等于电流乘以电阻，适用于恒温下的欧姆导体。

P = IV = I²R = V²/R

Electrical power dissipated in a resistor can be expressed in these three equivalent forms.

电阻器耗散的电功率可用这三种等效形式表示。

F = BIl sin θ

Force on a current-carrying conductor in a magnetic field; the angle θ is between the current direction and the magnetic field.

载流导体在磁场中所受的力；角度 θ 是电流方向与磁场方向之间的夹角。

F = qvB sin θ

Force on a charged particle moving in a magnetic field.

运动带电粒子在磁场中所受的力。

ε = – N (ΔΦ/Δt)

Faraday’s law of electromagnetic induction: induced e.m.f. is proportional to the rate of change of magnetic flux linkage. The negative sign indicates Lenz’s law.

法拉第电磁感应定律：感应电动势与磁通链变化率成正比。负号表示楞次定律。

Vₚ/Vₛ = Nₚ/Nₛ = Iₛ/Iₚ

Ideal transformer equations linking primary and secondary voltage, turns ratio, and current.

理想变压器公式，将初级与次级电压、匝数比和电流联系起来。

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5. Atomic and Nuclear Physics | 原子与核物理

The quantum nature of matter and radiation is described by photoelectric equations and nuclear decay laws. The concept of mass-energy equivalence and radioactive half-life are central to this topic.

物质和辐射的量子性质由光电方程和核衰变定律描述。质能等价和放射性半衰期的概念是本章的核心。

E = hf = hc/λ

Photon energy: Planck’s constant times frequency, also expressed in terms of wavelength.

光子能量：普朗克常数乘以频率，也用波长表示。

hf = φ + Eₖₘₐₓ

Einstein’s photoelectric equation: incident photon energy equals work function plus maximum kinetic energy of emitted electron.

爱因斯坦光电方程：入射光子能量等于逸出功加上发射电子的最大动能。

N = N₀ e⁻λᵗ

Radioactive decay law, where λ is the decay constant and N₀ is the initial number of nuclei.

放射性衰变定律，λ 是衰变常数，N₀ 是初始原子核数目。

T½ = ln2 / λ

Half-life is related to the decay constant by the natural logarithm of 2.

半衰期与衰变常数的关系是自然对数 ln2。

E = mc²

Mass-energy equivalence: energy released is equal to mass defect times the square of the speed of light.

质能等价：释放的能量等于质量亏损乘以光速的平方。

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6. Chemical Stoichiometry and Gas Laws | 化学计量与气体定律

Stoichiometry connects the amounts of reactants and products through balanced equations and the mole concept. Combining gas laws and molar volume allows quantitative predictions of reactions involving gases.

化学计量通过配平的方程式和摩尔概念将反应物与产物的量联系起来。结合气体定律和摩尔体积，可对涉及气体的反应进行定量预测。

n = m/M

Amount in moles equals mass divided by molar mass.

物质的量（摩尔数）等于质量除以摩尔质量。

c = n/V

Concentration of a solution: moles of solute per unit volume of solution.

溶液浓度：单位体积溶液中溶质的摩尔数。

p₁V₁ / T₁ = p₂V₂ / T₂

Combined gas law for a fixed amount of gas, linking pressure, volume, and absolute temperature.

固定量气体的综合气体定律，联系压强、体积和绝对温度。

gas volume at RTP = n × 24.0 dm³ mol⁻¹

Under room temperature and pressure conditions, the molar volume of an ideal gas is approximately 24.0 dm³ per mole.

在常温常压条件下，理想气体的摩尔体积约为每摩尔 24.0 dm³。

% yield = (actual yield / theoretical yield) × 100%

Percentage yield measures the efficiency of a chemical reaction.

产率百分比衡量化学反应的效率。

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7. Acids, Bases, and Equilibrium | 酸碱与平衡

Acid-base concepts rely on the Brønsted–Lowry theory and the equilibrium constant expressions. pH, Kₐ, Kₗ, and Kₑq provide a quantitative framework for understanding proton transfer and dynamic equilibrium.

酸碱概念基于布朗斯特-劳里理论和平衡常数表达式。pH、Kₐ、Kₗ 和 Kₑq 为理解质子转移和动态平衡提供了定量框架。

pH = –log₁₀[H⁺]

pH is the negative base-10 logarithm of the hydrogen ion concentration.

pH 是氢离子浓度的以10为底的负对数。

[H⁺] = 10⁻ᵖᴴ

The hydrogen ion concentration can be found from the pH value using this inverse relationship.

氢离子浓度可利用此逆关系由 pH 值求得。

Kₐ = [H⁺][A⁻] / [HA]

Acid dissociation constant for a weak acid, at equilibrium.

弱酸在平衡时的酸解离常数。

Kₙ = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ mol² dm⁻⁶

The ionic product of water at 298 K.

298 K 时水的离子积。

Kₑq = [C]ᶜ[D]ᵈ / [A]ᵃ[B]ᵇ

General equilibrium constant expression for the reaction aA + bB ⇌ cC + dD.

对于反应 aA + bB ⇌ cC + dD 的通用平衡常数表达式。

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8. Organic Chemistry Fundamentals | 有机化学基础

Organic chemistry is built upon functional groups, systematic nomenclature, and key reaction mechanisms. Recognising homologous series and isomerism is vital for synthesis and analysis.

有机化学建立在官能团、系统命名和关键反应机理之上。识别同系物和异构现象对于合成与分析至关重要。

Homologous Series / 同系物	Functional Group / 官能团	General Formula / 通式
Alkanes / 烷烃	C–C, C–H only	CₙH₂ₙ₊₂
Alkenes / 烯烃	C=C double bond	CₙH₂ₙ
Alcohols / 醇	–OH	CₙH₂ₙ₊₁OH
Carboxylic acids / 羧酸	–COOH	CₙH₂ₙ₊₁COOH
Esters / 酯	–COO–	RCOOR’

Isomers are compounds with the same molecular formula but different structural arrangements. Structural isomers differ in the connectivity of atoms, while stereoisomers have the same connectivity but a different spatial arrangement.

异构体是分子式相同但结构排列不同的化合物。结构异构体的原子连接方式不同，而立体异构体连接方式相同但空间排列不同。

% Atom Economy = (mass of desired product / total mass of reactants) × 100%

Atom economy measures how efficiently reactant atoms are incorporated into the desired product, important for green chemistry.

原子经济性衡量反应物原子结合到目标产物的效率，对绿色化学很重要。

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9. Cell Biology and Genetics | 细胞生物学与遗传学

The cell is the basic unit of life. Genetic information flows from DNA to protein, and Mendelian inheritance patterns can be predicted using probability and monohybrid crosses.

细胞是生命的基本单位。遗传信息从DNA流向蛋白质，孟德尔遗传模式可利用概率和单因子杂交来预测。

Magnification = image size / actual size

This formula is essential for calibrating microscope drawings and calculating dimensions of organelles.

该公式对于校准显微镜绘图和计算细胞器尺寸至关重要。

Surface area : volume ratio = SA / V

A higher surface area to volume ratio facilitates efficient exchange of materials; a key constraint on cell size.

较高的表面积与体积比有利于物质的高效交换；这是限制细胞大小的关键因素。

Phenotypic ratio for monohybrid cross of heterozygotes (Aa × Aa) = 3:1

When dominant and recessive alleles are involved, the expected phenotype ratio in the F₂ generation is 3 dominant to 1 recessive.

当涉及显性和隐性等位基因时，F₂代预期的表现型比例为3显性:1隐性。

Chi-squared test: χ² = Σ (O – E)² / E

The chi-squared statistic compares observed and expected frequencies to determine if deviations are due to chance.

卡方统计量比较观察频数与期望频数，以判断偏差是否由偶然因素造成。

Hardy-Weinberg equilibrium: p² + 2pq + q² = 1

For a gene locus with two alleles, p and q represent allele frequencies, and this equation predicts genotype frequencies in a non-evolving population.

对于具有两个等位基因的基因座，p 和 q 代表等位基因频率，该方程用于预测非进化种群中的基因型频率。

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10. Ecology and Evolution | 生态与进化

Ecological communities are described by population dynamics, energy flow, and nutrient cycles. Evolution by natural selection is driven by genetic variation and differential reproductive success.

生态群落由种群动态、能量流动和营养循环来描述。自然选择驱动的进化由遗传变异和差异繁殖成功率推动。

population growth rate = (births – deaths) + (immigration – emigration)

The rate of change in population size depends on both natural increase and net migration.

种群数量的变化率取决于自然增长量和净迁移量。

Net primary production (NPP) = GPP – respiration

NPP is the energy available to primary consumers after plants have used some for their own respiration.

NPP是植物用于自身呼吸后可供初级消费者利用的能量。

Efficiency of energy transfer = (energy in higher trophic level / energy in lower trophic level) × 100%

Typically only about 10% of energy is passed from one trophic level to the next; the rest is lost as heat or in waste.

通常只有大约10%的能量从一个营养级传递到下一个营养级；其余以热或废物形式散失。

Lincoln index: N = (n₁ × n₂) / m

A capture-mark-recapture method for estimating population size, where n₁ is the first capture, n₂ the second, and m the number of marked recaptures.

一种通过标记重捕法估算种群大小的方法，n₁ 是第一次捕获数，n₂ 是第二次捕获数，m 是重捕中带有标记的个体数。

Simpson’s diversity index: D = 1 – Σ (n/N)²

This index measures biodiversity, where n is the number of individuals of a particular species and N is the total number of organisms.

该指数衡量生物多样性，n 是某一特定物种的个体数，N 是生物体总数。

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