📚 Formula and Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference handbook compiles the essential formulas and theorems required for Year 13 CCEA Science, covering Physics, Chemistry and Biology. It is designed to help you revise efficiently for AS examinations by presenting the key quantitative relationships and principles in one place.
本速查手册汇编了 Year 13 CCEA 科学课程所需的物理、化学和生物核心公式与定理，旨在帮助您高效备考 AS 阶段考试，将关键的定量关系与原理集中呈现。
1. Kinematics Equations | 运动学方程
Kinematics equations describe uniformly accelerated motion along a straight line. These four equations are known as the SUVAT equations, where s is displacement, u is initial velocity, v is final velocity, a is constant acceleration and t is time.
运动学方程描述匀加速直线运动。以下四个方程被称为 SUVAT 方程，其中 s 为位移，u 为初速度，v 为末速度，a 为恒定加速度，t 为时间。
Equation 1: v = u + at (final velocity equals initial velocity plus acceleration multiplied by time).
方程1：v = u + at（末速度等于初速度加加速度与时间的乘积）。
Equation 2: s = ut + ½ at² (displacement equals initial velocity × time plus half the acceleration × time squared).
方程2：s = ut + ½ at²（位移等于初速度乘时间加上加速度乘时间平方的一半）。
Equation 3: s = ½ (u + v) t (displacement equals average velocity multiplied by time).
方程3：s = ½ (u + v) t（位移等于平均速度乘时间）。
Equation 4: v² = u² + 2as (relates final velocity, initial velocity, acceleration and displacement without time).
方程4：v² = u² + 2as（联系末速度、初速度、加速度和位移，不含时间）。
2. Newton’s Laws and Momentum | 牛顿定律与动量
Newton’s three laws of motion and the principle of conservation of momentum are fundamental to dynamics.
牛顿运动三定律与动量守恒原理是动力学的基础。
Newton’s second law: the net force on an object equals its mass times its acceleration, F = ma. Resultant force is measured in newtons (N).
牛顿第二定律：物体受到的合力等于其质量与加速度的乘积，F = ma。合力单位为牛顿 (N)。
Momentum p is defined as p = mv, where m is mass and v is velocity. Momentum is a vector quantity.
动量 p 定义为 p = mv，其中 m 为质量，v 为速度。动量是矢量。
In a closed system, total momentum before a collision equals total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
在封闭系统中，碰撞前总动量等于碰撞后总动量：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
Impulse is the change in momentum: FΔt = Δp = mv – mu.
冲量等于动量的变化：FΔt = Δp = mv – mu。
3. Work, Energy and Power | 功、能与功率
Work done by a constant force is the product of the force and the displacement in the direction of the force: W = Fd cos θ.
恒力做功等于力与沿着该力方向位移的乘积：W = Fd cos θ。
Kinetic energy: Eₖ = ½ mv². Gravitational potential energy near the Earth’s surface: Eₚ = mgh.
动能：Eₖ = ½ mv²。地表附近的重力势能：Eₚ = mgh。
The work–energy principle states that the net work done on an object equals its change in kinetic energy: W_net = ΔEₖ.
功能原理表明，物体所受合力做的功等于其动能的变化：W_net = ΔEₖ。
Power is the rate of doing work: P = W / t. For an object moving at constant velocity under a force, power can also be expressed as P = Fv.
功率是做功的快慢：P = W / t。物体在力作用下匀速运动时，功率也可表示为 P = Fv。
Efficiency is the ratio of useful output power to total input power, often given as a percentage.
效率是有用输出功率与总输入功率的比值，通常用百分数表示。
4. Electricity Basics | 电学基础
Ohm’s law states that the current through a conductor is directly proportional to the potential difference across it, provided temperature remains constant: V = IR.
欧姆定律指出，在温度不变的条件下，通过导体的电流与其两端的电势差成正比：V = IR。
Resistance of a wire: R = ρL / A, where ρ is resistivity, L is length and A is cross‑sectional area.
导线电阻：R = ρL / A，其中 ρ 为电阻率，L 为长度，A 为横截面积。
Power dissipated in a circuit component: P = IV = I²R = V² / R.
电路元件消耗的功率：P = IV = I²R = V² / R。
Resistors in series: R_total = R₁ + R₂ + …. Resistors in parallel: 1 / R_total = 1 / R₁ + 1 / R₂ + ….
电阻串联：R_total = R₁ + R₂ + …。电阻并联：1 / R_total = 1 / R₁ + 1 / R₂ + …。
5. Wave Equations | 波的公式
The wave equation relates wave speed v, frequency f and wavelength λ: v = fλ.
波动方程将波速 v、频率 f 和波长 λ 联系起来：v = fλ。
Refractive index n of a medium: n = c / v, where c is the speed of light in a vacuum and v is the speed of light in the medium.
介质的折射率 n：n = c / v，其中 c 为真空中光速，v 为介质中光速。
Snell’s law: n₁ sin θ₁ = n₂ sin θ₂. When light enters an optically denser medium, it bends towards the normal.
斯涅尔定律：n₁ sin θ₁ = n₂ sin θ₂。光进入光密介质时，会向法线偏折。
The critical angle for total internal reflection: sin C = 1 / n, valid when light travels from a medium of refractive index n into air.
全反射的临界角：sin C = 1 / n，仅当光从折射率为 n 的介质射向空气时成立。
6. Mole Concept and Stoichiometry | 摩尔概念与化学计量
The amount of substance n is measured in moles. One mole contains Avogadro’s number, 6.02 × 10²³, of specified entities.
物质的量 n 以摩尔为单位。1 mol 物质含有阿伏伽德罗常数 6.02 × 10²³ 个指定粒子。
Mole–mass relationship: n = m / M, where m is mass and M is molar mass (g mol⁻¹).
摩尔与质量的关系：n = m / M，其中 m 为质量，M 为摩尔质量 (g mol⁻¹)。
For gases at room temperature and pressure (RTP), one mole occupies 24 dm³. Mole–volume relationship: n = V / 24 (or n = V / Vₘ) when V is measured in dm³.
在常温常压 (RTP) 下，1 mol 气体占据 24 dm³。摩尔与体积的关系：当体积 V 单位为 dm³ 时，n = V / 24（或 n = V / Vₘ）。
Concentration of a solution: c = n / V, with V in dm³, giving concentration in mol dm⁻³.
溶液的浓度：c = n / V，V 单位为 dm³，浓度单位为 mol dm⁻³。
7. Thermochemistry and Hess’s Law | 热化学与赫斯定律
Enthalpy change ΔH is the heat energy transferred in a reaction at constant pressure. In calorimetry experiments, ΔH = – (mcΔT) / n, where m is the mass of the solution, c is specific heat capacity (usually 4.18 J g⁻¹ K⁻¹ for water), ΔT is the temperature change, and n is the amount of reactant that actually reacted.
焓变 ΔH 是恒压下反应中传递的热量。在量热实验中，ΔH = – (mcΔT) / n，其中 m 是溶液质量，c 是比热容（水一般为 4.18 J g⁻¹ K⁻¹），ΔT 是温度变化，n 是参加反应的反应物的物质的量。
Hess’s law states that the total enthalpy change for a reaction is independent of the route taken, provided the initial and final conditions are the same. It allows the calculation of ΔH using enthalpy cycles.
赫斯定律指出，只要始终态相同，反应的总焓变与其途径无关。可利用焓循环计算 ΔH。
Standard enthalpy of combustion (ΔH_c°) and standard enthalpy of formation (ΔH_f°) are useful data. For a reaction: ΔH_reaction = ∑ ΔH_f°(products) – ∑ ΔH_f°(reactants).
标准燃烧焓 (ΔH_c°) 和标准生成焓 (ΔH_f°) 很有用。对某反应：ΔH_reaction = ∑ ΔH_f°(产物) – ∑ ΔH_f°(反应物)。
8. Equilibrium Constant and Le Chatelier’s Principle | 平衡常数与勒夏特列原理
For a reversible reaction at dynamic equilibrium, the equilibrium constant Kc is expressed using equilibrium concentrations. For a general reaction aA + bB ⇌ cC + dD, Kc = [C]ᶜ [D]ᵈ / [A]ᵃ [B]ᵇ, where square brackets denote concentration in mol dm⁻³.
对于达到动态平衡的可逆反应，平衡常数 Kc 用平衡浓度表示。对于一般反应 aA + bB ⇌ cC + dD，Kc = [C]ᶜ [D]ᵈ / [A]ᵃ [B]ᵇ，方括号表示浓度，单位为 mol dm⁻³。
Kc is constant at a given temperature. If Kc >> 1, the equilibrium lies to the right (products favoured); if Kc << 1, it lies to the left (reactants favoured).
在温度一定时 Kc 是常数。若 Kc >> 1，平衡偏向右侧（生成物有利）；若 Kc << 1，平衡偏向左侧（反应物有利）。
Le Chatelier’s principle: when a system at equilibrium is subjected to a change in concentration, pressure or temperature, the position of equilibrium shifts to oppose that change.
勒夏特列原理：当平衡体系受到浓度、压强或温度变化时，平衡将向削弱该变化的方向移动。
Increasing temperature favours the endothermic direction; increasing pressure favours the side with fewer gas molecules.
升高温度有利于吸热方向；增大压强有利于气体分子总数较少的一侧。
9. Magnification and Actual Size in Microscopy | 显微镜放大率与实际尺寸
In biological microscopy, the relationship between image size, actual size and magnification is given by: Magnification = Image size / Actual size, often written as M = I / A.
在生物显微镜学中，图像大小、实际大小和放大率的关系为：放大率 = 图像大小 / 实际大小，常写作 M = I / A。
It is essential to ensure that image size and actual size are in the same units before calculation. The actual size of a specimen can be estimated using an eyepiece graticule calibrated with a stage micrometer.
计算前必须确保图像大小与实际大小单位一致。可使用目镜测微尺经台微尺校准后估算标本的实际大小。
For example, if a cell image measures 50 mm under a microscope and its magnification is 400×, the actual size is 50 mm / 400 = 0.125 mm (125 µm).
例如，某细胞图像在显微镜下测量为 50 mm，放大率为 400×，则实际大小为 50 mm / 400 = 0.125 mm（125 µm）。
10. Chi-Squared Test | 卡方检验
The chi-squared (χ²) test is used in biology to determine if there is a significant difference between observed and expected frequencies. The formula is: χ² = Σ (O – E)² / E, where O is the observed value, E is the expected value, and Σ means the sum over all categories.
卡方 (χ²) 检验在生物学中用于确定观测频率与预期频率之间是否存在显著差异。公式为：χ² = Σ (O – E)² / E，其中 O 为观测值，E 为预期值，Σ 表示对所有类别求和。
The calculated χ² value is compared against a critical value from a table at a chosen probability level (e.g. p = 0.05) and for a given degrees of freedom (df = number of categories – 1). If χ²_calculated > χ²_critical, the null hypothesis is rejected.
将计算出的 χ² 值与选定概率水平（如 p = 0.05）及相应自由度（df = 类别数 – 1）下的临界值比较。若 χ²_计算值 > χ²_临界值，则拒绝零假设。
This test is commonly applied in genetics (e.g. Mendelian ratios) and ecological studies.
该检验常应用于遗传学（如孟德尔比例）和生态研究。
11. Simpson’s Diversity Index | 辛普森多样性指数
Simpson’s Diversity Index (D) measures the biodiversity of a habitat by considering both species richness and evenness. One common form used in CCEA is: D = 1 – Σ (n / N)², where n is the number of individuals of a particular species, and N is the total number of individuals of all species.
辛普森多样性指数 (D) 通过考虑物种丰富度和均匀度来衡量栖息地的生物多样性。CCEA 常用的一种形式为：D = 1 – Σ (n / N)²，其中 n 为某一特定物种的个体数，N 为所有物种的总个体数。
The value of D ranges from 0 (low diversity) to almost 1 (high diversity). A higher value indicates a more diverse and stable ecosystem.
D 值范围在 0（低多样性）到接近 1（高多样性）之间。较高的值表明生态系统更具多样性和稳定性。
When comparing two communities, the one with the larger D index is considered more diverse.
比较两个群落时，D 指数较大的群落被视为更具多样性。
12. Capture-Mark-Recapture (Lincoln Index) | 标记重捕法（林肯指数）
The Lincoln index estimates the population size of motile organisms. The formula is: N = (M × C) / R, where M is the number of individuals captured, marked and released in the first sample, C is the total number captured in the second sample, and R is the number of marked individuals recaptured in the second sample.
林肯指数用于估算能自由移动的生物的种群大小。公式为：N = (M × C) / R，其中 M 是第一次取样时捕获、标记并释放的个体数，C 是第二次取样捕获的总数，R 是第二次取样中再捕到的标记个体数。
This method assumes that the marked individuals have mixed randomly with the population, that no migration, births or deaths occur between samples, and that marks are not lost or overlooked.
该方法假设标记个体已与种群随机混合，两次取样期间无迁入迁出、出生或死亡，且标记不脱落也不被忽略。
The estimated population size N is a useful index of abundance in ecological fieldwork.
估算出的种群大小 N 是生态野外调查中衡量丰度的一个有用指标。
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