📚 Pre-U CAIE Chemistry: Quick Reference Handbook of Formulas & Theorems | Pre-U CAIE 化学：公式定理速查手册
This handbook compiles the essential formulas, equations and relationships required for the Cambridge Pre-U Chemistry syllabus. It serves as a rapid revision tool for key quantitative and conceptual topics, aiding students in mastering calculations and reinforcing theoretical understanding across physical, inorganic and organic chemistry.
本手册汇编了剑桥 Pre-U 化学大纲所需的必备公式、方程式和关系式，覆盖物理化学、无机化学与有机化学三大板块，旨在为关键定量和概念性主题提供快速复习工具，帮助学生掌握计算并加深理论理解。
1. Stoichiometry and the Mole Concept | 化学计量与摩尔概念
n = m / M
The amount of substance n (mol) is calculated as mass m (g) divided by molar mass M (g mol⁻¹). This is the fundamental link between a measured mass and the number of particles it represents, and it underpins every subsequent titration, gas and thermochemical calculation.
物质的量 n (mol) 等于质量 m (g) 除以摩尔质量 M (g mol⁻¹)，这是连接可测质量与粒子数量的基本关系，也是后续滴定、气体与热化学计算的基础。
N = n × NA
The total number of particles N is obtained by multiplying the amount of substance n by the Avogadro constant NA e same relationship converts between moles and ions, atoms or electrons in redox problems.
粒子总数 N 等于物质的量 n 乘以阿伏伽德罗常数 NA = 6.022 × 10²³ mol⁻¹。同一关系也可用于在氧化还原问题中换算摩尔数与离子、原子或电子数目。
c = n / V
hat volume in these expressions always refers to the total solution, never the solvent alone.
物质的量浓度 c (mol dm⁻³) 等于溶质的物质的量 n (mol) 除以溶液体积 V (dm³)。注意 1 dm³ = 1000 cm³，且公式中的体积始终指溶液总体积，而非溶剂体积。
c1V1 = c2V2
For dilution problems, the number of moles of solute is unchanged, so the product of concentration and volume before dilution equals the product after dilution. This relationship also applies to acid-base titrations when the reaction stoichiometry is 1:1.
稀释计算中溶质物质的量不变，因此稀释前浓度与体积的乘积等于稀释后的乘积。当酸碱中和的化学计量比为 1:1 时，该关系式同样适用于滴定计算。
pV = nRT
ged to find the molar mass of a gas: M = mRT / pV, a standard Pre-U practical calculation.
理想气体状态方程关联压强 p (Pa)、体积 V (m³)、物质的量 n、气体常数 R (8.31 J K⁻¹ mol⁻¹) 与温度 T (K)，可变形用于求气体摩尔质量：M = mRT / pV，这是 Pre-U 实验题的常规计算。
% Yield = (actual yield / theoretical yield) × 100%
Percentage yield measures the efficiency of a preparation and reflects losses from incomplete reaction, side reactions and transfer. It is always based on the limiting reagent, which must be identified before the calculation begins.
百分比产率衡量制备反应的效率，反映反应不完全、副反应与转移损失。计算必须以限量试剂为基准，因此在计算前须先确定限量试剂。
tom economy = (molar mass of desired product / sum of molar masses of all reactants) × 100%
Atom economy evaluates green credentials by showing how much of the reactant mass ends up in the desired product. Addition reactions have 100% atom economy, whereas substitution and elimination reactions are inherently less efficient.
原子经济性 = (目标产物摩尔质量 / 所有反应物摩尔质量总和) × 100%，用于评估绿色化学特征，反映反应物质量进入目标产物的比例。加成反应原子经济性为 100%，而取代与消去反应则天然较低。
| Constant | 常数 | Symbol | 符号 | Value | 数值 |
| Avogadro constant | 阿伏伽德罗常数 | NA | 6.022 × 10²³ mol⁻¹ |
| Gas constant | 气体常数 | R | 8.31 J K⁻¹ mol⁻¹ |
| Faraday constant | 法拉第常数 | F | 96 500 C mol⁻¹ |
| Planck constant | 普朗克常数 | h | 6.63 × 10⁻³⁴ J s |
| Speed of light | 光速 | c | 3.00 × 10⁸ m s⁻¹ |
| Molar volume at RTP | 室温常压摩尔体积 | Vm | 24.0 dm³ mol⁻¹ |
2. Gas Laws and Kinetic Theory | 气体定律与分子运动论
p1V1 / T1 = p2V2 / T2
The combined gas law holds for a fixed mass of gas behaving ideally. At constant temperature, p1V1 = p2V2 (Boyle's law); at constant pressure, V1 / T1 = V2 / T2 (Charles's law); at constant volume, p1 / T1 = p2 / T2 (the pressure law).
固定质量且近似理想的气体遵循联合气体定律。恒温时 p1V1 = p2V2（玻意耳定律）；恒压时 V1 / T1 = V2 / T2（查理定律）；恒容时 p1 / T1 = p2 / T2（压力定律）。
pV = ⅓ N m c̄²
rom molecular motion, where N is the number of molecules, m the mass of one molecule and c̄ lecule is given by KE = 3/2 kT, so temperature is a direct measure of mean kinetic energy.
分子运动论由分子运动导出理想气体方程，其中 N 为分子数，m 为单分子质量，c̄² 为均方速率。分子的平均动能为 KE = 3/2 kT，因此温度是平均动能的直接量度。
Rate of diffusion ∝ 1 / √M
Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. This explains why helium escapes through a porous pot far faster than carbon dioxide, and why ²³⁵UF₆ and ²³⁸UF₆ can be separated by gaseous diffusion.
格雷姆定律指出气体的扩散速率与其摩尔质量的平方根成反比。这解释了氦气通过多孔罐逸出的速率远大于二氧化碳，也解释了 ²³⁵UF₆ 与 ²³⁸UF₆ 可用气体扩散法分离。
(p + an² / V²)(V − nb) = nRT
Real gases deviate from ideality at high pressure and low temperature because of intermolecular forces and finite molecular volume. The van der Waals equation corrects pressure for attractive forces (a) and volume for molecular size (b), and the largest deviations occur near the conditions of liquefaction.
真实气体在高压与低温下因分子间作用力和分子体积不可忽略而偏离理想行为。范德华方程对压强作吸引作用修正 (a)、对体积作分子体积修正 (b)，偏离最显著的条件接近气体液化点。
3. Atomic Structure, Ionisation Energy and Periodicity | 原子结构、电离能与周期性
ΔE = hν = hc / λ
ewritten as ΔE = hc / λ, allowing an emission wavelength to be converted into an electron energy gap.
吸收或放出光子的能量由 ΔE = hν 给出，其中 ν 为频率 (Hz)，h 为普朗克常数。由 c = λν 可得 ΔE = hc / λ，据此可由发射波长换算电子能级差。
The first ionisation energy is the energy required to remove one mole of electrons from one mole of gaseous atoms to form one mole of gaseous 1+ ions. Successive ionisation energies always increase because the remaining electrons are held by a progressively larger effective nuclear charge with less shielding.
第一电离能是指从一摩尔气态原子中移去一摩尔电子，形成一摩尔气态 1+ 离子所需的能量。逐级电离能总是升高，因为剩余电子受到的有效核电荷更大且屏蔽更少。
Ar = Σ (isotope mass × abundance) / Σ abundance
Relative atomic mass from a mass spectrum is a weighted mean of isotopic masses. In a mass spectrometer, the x-axis is m/z and the y-axis is relative abundance; the molecular ion peak (M⁺) gives the relative molecular mass, while fragmentation peaks reveal structural features.
由质谱求相对原子质量即同位素质量的加权平均值。质谱图中横轴为质荷比 m/z，纵轴为相对丰度；分子离子峰 (M⁺) 给出相对分子质量，碎片峰则揭示结构信息。
Periodic trends follow directly from structure: atomic radius decreases across a period as nuclear charge rises with constant shielding, and increases down a group as shells are added. First ionisation energy rises across a period with dips at Group 3 (new p sub-shell, higher energy) and Group 6 (electron pairing repulsion in a p orbital).
周期性趋势可直接由结构解释：同周期从左到右核电荷增加而屏蔽基本不变，原子半径减小；同主族自上而下增加电子层，半径增大。第一电离能同周期总体升高，但在第 3 族（新填 p 亚层、能量更高）与第 6 族（p 轨道电子成对排斥）出现下降。
4. Bonding, Lattice Enthalpy and Born-Haber Cycles | 化学键、晶格能与玻恩-哈伯循环
ΔHlatt = −(ΔHf − ΔHatom − ΔHIE − ½ΔHdiss − ΔHEA)
A Born-Haber cycle applies Hess's law to the formation of an ionic solid from its elements in their standard states. The lattice enthalpy of formation is the enthalpy change when one mole of a solid ionic lattice is formed from its gaseous ions, and it is always strongly exothermic.
玻恩-哈伯循环把赫斯定律应用于由标准态单质生成离子晶体的过程。晶格生成焓是指由气态离子生成一摩尔固态离子晶格时的焓变，其值总是显著放热。
Comparison of theoretical lattice enthalpy from the perfect ionic model with the experimental value reveals covalent character. The greater the polarising power of the cation (high charge density) and the greater the polarisability of the anion (large, electron-rich), the more the experimental value exceeds the theoretical one, because electron density is distorted toward the cation.
把理想离子模型
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