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

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

This handbook provides a concise yet comprehensive collection of the essential formulas, equations, and theorems required for the Cambridge Pre-U Science syllabus. Covering physics, chemistry, and biology, it is designed as a rapid revision tool, enabling students to recall and apply core quantitative and logical principles under examination conditions. Each entry is paired with a dual English–Chinese explanation to reinforce understanding across both languages, ensuring that no fundamental concept is lost in translation.

本手册简明扼要地汇集了剑桥 Pre-U 科学课程所必需的核心公式、方程和定理，涵盖物理、化学和生物三大领域。它旨在成为快速复习的工具，帮助学生在考试环境下迅速回想并运用关键的定量与逻辑原理。每一条目均配有英中双语解释，以加深对概念的理解，确保在语言转换过程中不丢失任何基础知识。

1. Kinematics and Motion | 运动学与运动

The equations of uniformly accelerated motion form the bedrock of mechanics. For an object moving with constant acceleration a, initial velocity u, and final velocity v after time t, the displacement s is given by a set of four interrelated equations. The first, v = u + a t, directly relates velocity to time. The second, s = (u + v) t / 2, expresses displacement as a function of the average velocity. The third, s = u t + ½ a t², useful when final velocity is not known, and the fourth, v² = u² + 2 a s, eliminates time altogether.

匀加速直线运动的方程组是力学的基础。对于以恒定加速度 a 运动的物体，初速度为 u，经时间 t 后的末速度为 v，位移 s 可由以下四个相互关联的方程求得。第一式 v = u + a t 直接建立了速度与时间的关系。第二式 s = (u + v) t / 2 将位移表示为平均速度的函数。第三式 s = u t + ½ a t² 在未知末速度时尤为有用，而第四式 v² = u² + 2 a s 则完全消去了时间变量。

2. Newton’s Laws and Momentum | 牛顿定律与动量

Newton’s three laws of motion govern classical dynamics. The first law states that an object remains at rest or in uniform motion in a straight line unless acted upon by a resultant force. The second law quantifies force as the rate of change of momentum: F = Δp / Δt, which for constant mass simplifies to F = m a. The third law asserts that every action has an equal and opposite reaction. The principle of conservation of momentum holds that in a closed system, total momentum before an event equals total momentum after: m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂.

牛顿运动三定律支配着经典动力学。第一定律指出，除非受到合外力作用，否则物体将保持静止或匀速直线运动状态。第二定律将力定量为动量的变化率：F = Δp / Δt，在质量恒定时简化为 F = m a。第三定律断言，每一个作用力都有一个大小相等、方向相反的反作用力。动量守恒定律指出，在一个封闭系统中，事件发生前的总动量等于事件发生后的总动量：m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂。

3. Work, Energy, and Power | 功、能与功率

Work is done when a force moves its point of application: W = F d cos θ, where θ is the angle between force and displacement. The principle of conservation of energy states that energy can neither be created nor destroyed, only transformed. Kinetic energy Eₖ = ½ m v², and gravitational potential energy Eₚ = m g h. Power is the rate of doing work: P = W / t or P = F v for constant force and velocity. Efficiency is the ratio of useful output power to input power, expressed as a percentage.

力在作用点发生位移时便做了功：W = F d cos θ，其中 θ 是力与位移的夹角。能量守恒定律指出，能量既不能被创造也不能被消灭，只能从一种形式转化为另一种形式。动能 Eₖ = ½ m v²，重力势能 Eₚ = m g h。功率是做功的速率：P = W / t，或在恒力和恒速条件下 P = F v。效率是有用输出功率与输入功率之比，通常以百分比表示。

4. Waves and Optics | 波与光学

The wave equation v = f λ connects wave speed v, frequency f, and wavelength λ. The refractive index n = c / v, where c is the speed of light in vacuum. Snell’s law describes refraction: n₁ sin θ₁ = n₂ sin θ₂. The critical angle θ꜀ is given by sin θ꜀ = n₂ / n₁ when n₁ > n₂. For a thin lens, the lens formula is 1/f = 1/u + 1/v, with magnification m = v/u. In double-slit interference, constructive interference occurs when d sin θ = nλ, where d is the slit separation and n is an integer.

波动方程 v = f λ 将波速 v、频率 f 和波长 λ 联系起来。折射率 n = c / v，其中 c 是真空中的光速。斯涅尔定律描述了折射现象：n₁ sin θ₁ = n₂ sin θ₂。全内反射的临界角 θ꜀ 当 n₁ > n₂ 时满足 sin θ꜀ = n₂ / n₁。对于薄透镜，透镜公式为 1/f = 1/u + 1/v，放大率 m = v/u。在双缝干涉中，加强干涉发生在 d sin θ = nλ 时，其中 d 为缝间距，n 为整数。

5. Electricity and Circuits | 电学与电路

Ohm’s law states V = I R for a conductor at constant temperature. Resistance of a wire is R = ρ L / A, where ρ is resistivity, L length, and A cross-sectional area. Power dissipated is P = I V = I² R = V² / R. In series circuits, R_total = R₁ + R₂, and in parallel, 1/R_total = 1/R₁ + 1/R₂. Kirchhoff’s first law (junction rule) states that the sum of currents entering a junction equals the sum of currents leaving. Kirchhoff’s second law (loop rule) states that the sum of the e.m.f.s around any closed loop equals the sum of the p.d. drops.

欧姆定律指出，对于恒温下的导体有 V = I R。导线的电阻为 R = ρ L / A，其中 ρ 为电阻率，L 为长度，A 为横截面积。耗散功率为 P = I V = I² R = V² / R。在串联电路中，总电阻 R_total = R₁ + R₂；在并联电路中，1/R_total = 1/R₁ + 1/R₂。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。基尔霍夫第二定律（回路定律）指出，任一闭合回路中电动势的代数和等于电压降的代数和。

6. Radioactivity and Nuclear Physics | 放射性及核物理

The activity A of a radioactive source is given by A = –dN/dt = λ N, where λ is the decay constant and N is the number of undecayed nuclei. The exponential decay law is N = N₀ e^(–λ t). The half-life t½ is related to the decay constant by t½ = ln 2 / λ ≈ 0.693 / λ. The unified atomic mass unit u is defined relative to the mass of carbon-12. In nuclear reactions, mass–energy equivalence is expressed by E = m c², which governs the energy released in fission and fusion.

放射性源的活度 A 由 A = –dN/dt = λ N 给出，其中 λ 为衰变常数，N 为未衰变原子核的数目。指数衰变定律为 N = N₀ e^(–λ t)。半衰期 t½ 与衰变常数的关系为 t½ = ln 2 / λ ≈ 0.693 / λ。统一原子质量单位 u 是相对于碳-12 的质量定义的。在核反应中，质能等价关系由 E = m c² 表示，它支配着裂变与聚变反应中释放的能量。

7. Moles and Stoichiometry | 摩尔与化学计量学

The mole is the amount of substance that contains 6.022 × 10²³ particles (Avogadro’s constant, Nₐ). The number of moles n can be calculated from mass m and molar mass M: n = m / M. For solutions, concentration c = n / V, and the molar volume of an ideal gas at room temperature and pressure is approximately 24 dm³ mol⁻¹. Stoichiometric coefficients in a balanced chemical equation give the exact mole ratios of reactants and products. The ideal gas equation is p V = n R T, where R = 8.31 J mol⁻¹ K⁻¹.

摩尔是含有 6.022 × 10²³ 个粒子（阿伏伽德罗常数 Nₐ）的物质的数量。物质的量 n 可以通过质量 m 与摩尔质量 M 计算：n = m / M。对于溶液，浓度 c = n / V；理想气体在室温常压下的摩尔体积约为 24 dm³ mol⁻¹。配平的化学方程式中的计量系数给出了反应物与生成物之间的精确摩尔比。理想气体状态方程为 p V = n R T，其中 R = 8.31 J mol⁻¹ K⁻¹。

8. Energetics and Thermodynamics | 能量学与热力学

In chemistry, the enthalpy change ΔH for a reaction is calculated using ΔH = H_products – H_reactants. Standard enthalpy of combustion and formation are tabulated. Hess’s law states that the total enthalpy change for a reaction is independent of the route taken. The heat change in a solution is given by q = m c ΔT, where m is mass, c is specific heat capacity, and ΔT is the temperature change. The Gibbs free energy change ΔG⁰ determines reaction feasibility: ΔG⁰ = ΔH⁰ – T ΔS⁰. A reaction is spontaneous if ΔG < 0.

在化学中，反应的焓变 ΔH 通过 ΔH = H_生成物 – H_反应物 计算。标准燃烧焓和标准生成焓可从数据表中查得。赫斯定律指出，一个反应的总焓变与途径无关，只取决于始态和终态。溶液中的热量变化由 q = m c ΔT 给出，其中 m 为质量，c 为比热容，ΔT 为温度变化。吉布斯自由能变 ΔG⁰ 决定反应的可行性：ΔG⁰ = ΔH⁰ – T ΔS⁰。当 ΔG < 0 时反应可自发进行。

9. Equilibrium and Rates | 平衡与速率

For a reversible reaction aA + bB ⇌ cC + dD, the equilibrium constant K꜀ is defined as K꜀ = ([C]^c [D]^d) / ([A]^a [B]^b), where square brackets denote equilibrium concentrations. The magnitude of K꜀ indicates the position of equilibrium. The rate of a chemical reaction can be expressed as rate = k [A]^m [B]^n, where k is the rate constant and m, n are the orders with respect to each reactant. The Arrhenius equation k = A e^(–Eₐ/RT) relates the rate constant to temperature and activation energy Eₐ.

对于可逆反应 aA + bB ⇌ cC + dD，平衡常数 K꜀ 定义为 K꜀ = ([C]^c [D]^d) / ([A]^a [B]^b)，其中方括号表示平衡时的浓度。K꜀ 的大小指示了平衡的位置。化学反应的速率可表示为 rate = k [A]^m [B]^n，其中 k 为速率常数，m、n 分别为对各反应物的反应级数。阿伦尼乌斯方程 k = A e^(–Eₐ/RT) 将速率常数与温度及活化能 Eₐ 联系起来。

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

Magnification of a microscope is calculated as magnified size / actual size. The surface area to volume ratio influences the rate of diffusion in cells. The DNA molecule is a double helix held together by complementary base pairing: adenine (A) with thymine (T), and cytosine (C) with guanine (G). In genetics, monohybrid crosses follow Mendel’s law of segregation, where the phenotypic ratio in the F₂ generation for a heterozygous cross is 3:1. The Hardy–Weinberg principle states that allele frequencies in a large, randomly-mating population remain constant in the absence of evolutionary forces: p² + 2pq + q² = 1, where p and q are the frequencies of two alleles.

显微镜的放大倍数由放大后的尺寸除以实际尺寸计算。表面积与体积比影响着细胞内的扩散速率。DNA 分子是一个双螺旋结构，由互补碱基对维系：腺嘌呤 (A) 与胸腺嘧啶 (T) 配对，胞嘧啶 (C) 与鸟嘌呤 (G) 配对。在遗传学中，单因子杂交遵循孟德尔的分离定律，杂合子自交后的 F₂ 代表型比为 3:1。哈代–温伯格原理指出，在一个足够大的随机交配种群中，若没有进化力量的影响，等位基因频率将世代保持不变：p² + 2pq + q² = 1，其中 p 和 q 是两种等位基因的频率。

11. Biochemistry and Enzymes | 生物化学与酶

The lock-and-key model and the induced-fit model explain enzyme specificity. The rate of an enzyme-catalysed reaction follows Michaelis–Menten kinetics, described by v = (V_max [S]) / (Kₘ + [S]), where V_max is the maximum rate, [S] is the substrate concentration, and Kₘ is the Michaelis constant, which reflects the affinity of the enzyme for its substrate. Competitive inhibition increases the apparent Kₘ without affecting V_max, while non-competitive inhibition reduces V_max without altering Kₘ. Proteins are polymers of amino acids linked by peptide bonds, and their primary structure determines higher order folding.

锁钥模型和诱导契合模型解释了酶的专一性。酶催化反应的速率遵循米氏动力学，用 v = (V_max [S]) / (Kₘ + [S]) 描述，其中 V_max 是最大反应速率，[S] 是底物浓度，Kₘ 是米氏常数，反映酶对底物的亲和力。竞争性抑制使表观 Kₘ 增大但不影响 V_max，而非竞争性抑制使 V_max 降低而不改变 Kₘ。蛋白质是由氨基酸通过肽键连接而成的多聚体，其一级结构决定了高级折叠方式。

12. Ecology and Populations | 生态学与种群

The capture-mark-recapture method estimates population size N using the Lincoln index: N = (M × C) / R, where M is the number initially captured and marked, C is the total captured in the second sample, and R is the number of marked individuals recaptured. Population growth can be described by the exponential model dN/dt = r N, where r is the intrinsic rate of increase. Carrying capacity K is introduced in the logistic growth model: dN/dt = r N (1 – N/K). Ecological efficiency between trophic levels is typically around 10%. Simpson’s Index of Diversity D = 1 – (Σ n(n – 1) / N(N – 1)) reflects biodiversity, where n is the number of individuals of a species and N is the total number of organisms.

标记重捕法使用林肯指数估计种群大小 N：N = (M × C) / R，其中 M 为最初捕获并标记的个体数，C 为第二次捕获的总个体数，R 为重捕的标记个体数。种群增长可用指数模型 dN/dt = r N 描述，其中 r 为内禀增长率。在逻辑斯谛增长模型中引入了环境容纳量 K：dN/dt = r N (1 – N/K)。营养级间的生态效率通常约为 10%。辛普森多样性指数 D = 1 – (Σ n(n – 1) / N(N – 1)) 反映生物多样性，其中 n 为某一物种的个体数，N 为所有物种的总个体数。

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