📚 Pre-U Cambridge Chemistry: Formula & Theorem Quick Reference Handbook | Pre-U Cambridge 化学：公式定理速查手册
This handbook collects the essential formulas, equations, and theorems required for Pre-U Cambridge Chemistry. It is designed as a rapid revision tool, linking quantitative reasoning to core concepts across physical, inorganic, and organic topics. Use it to reinforce memory, check derivations, and build confidence in calculations for the examinations.
本手册汇集了 Pre-U Cambridge 化学所需的核心公式、方程式与定理，旨在作为快速复习工具，将定量推理与物理化学、无机化学和有机化学的核心概念联系起来。通过它巩固记忆、核验推导，并增强考试中的计算信心。
1. Mole Concept & Stoichiometry | 摩尔概念与化学计量学
The mole is the SI unit for amount of substance, linking mass to particle number via the Avogadro constant L ≈ 6.02 × 10²³ mol⁻¹. The key equation is n = m / M, where n is amount (mol), m is mass (g), and M is molar mass (g mol⁻¹). For gases at RTP, molar volume Vₘ ≈ 24.0 dm³ mol⁻¹, while at STP it is 22.4 dm³ mol⁻¹.
摩尔是物质的量的 SI 单位，通过阿伏伽德罗常数 L ≈ 6.02 × 10²³ mol⁻¹ 将质量与粒子数联系起来。关键方程为 n = m / M，其中 n 为物质的量 (mol)，m 为质量 (g)，M 为摩尔质量 (g mol⁻¹)。在常温常压下，气体摩尔体积 Vₘ ≈ 24.0 dm³ mol⁻¹，而在标准状况下为 22.4 dm³ mol⁻¹。
Concentration of a solution: c = n / V (mol dm⁻³). Dilution follows c₁V₁ = c₂V₂. Percentage yield = (actual yield / theoretical yield) × 100%. Atom economy = (molar mass of desired product / total molar mass of reactants) × 100%.
溶液的浓度：c = n / V (mol dm⁻³)。稀释遵循 c₁V₁ = c₂V₂。产率百分数 = (实际产量 / 理论产量) × 100%。原子经济性 = (目标产物摩尔质量 / 反应物总摩尔质量) × 100%。
Empirical and molecular formulae: empirical formula is the simplest whole-number ratio of atoms; the molecular formula is a multiple of it, n = Mr / empirical mass.
实验式与分子式：实验式是原子最简单整数比；分子式是其倍数，n = Mᵣ / 实验式质量。
2. Gas Laws & Ideal Gas Equation | 气体定律与理想气体状态方程
The ideal gas equation combines Boyle’s, Charles’s, and Avogadro’s laws: pV = nRT. p is pressure (Pa), V is volume (m³), n is amount (mol), R = 8.31 J K⁻¹ mol⁻¹, and T is temperature (K). At constant n and T, p ∝ 1/V (Boyle). At constant n and p, V ∝ T (Charles). At constant p and T, V ∝ n (Avogadro).
理想气体状态方程综合了玻义耳定律、查理定律和阿伏伽德罗定律：pV = nRT。p 为压力 (Pa)，V 为体积 (m³)，n 为物质的量 (mol)，R = 8.31 J K⁻¹ mol⁻¹，T 为温度 (K)。在 n 与 T 恒定时，p ∝ 1/V（玻义耳）。在 n 与 p 恒定时，V ∝ T（查理）。在 p 与 T 恒定时，V ∝ n（阿伏伽德罗）。
pV = nRT
For a mixture of ideal gases, Dalton’s law of partial pressures: ptotal = Σ pi, with pi = xi ptotal, where xi = ni / ntotal is the mole fraction.
对于理想气体混合物，道尔顿分压定律：ptotal = Σ pi，且 pi = xi ptotal，其中 xi = ni / ntotal 为摩尔分数。
Real gases deviate from ideality at high pressure and low temperature; the van der Waals equation introduces corrections: (p + a n²/V²)(V – n b) = nRT, where a and b are gas-specific constants.
实际气体在高压低温下偏离理想状态；范德华方程引入修正：(p + a n²/V²)(V – n b) = nRT，其中 a、b 为气体特征常数。
3. Thermochemistry & Enthalpy | 热化学与焓变
Enthalpy change ΔH is the heat transferred at constant pressure. Standard enthalpy of formation ΔHf° refers to the formation of 1 mol of a compound from its elements under standard conditions. Standard enthalpy of combustion ΔHc° is the enthalpy change when 1 mol of a substance is completely burned in oxygen.
焓变 ΔH 是恒压下传递的热量。标准生成焓 ΔHf° 指在标准条件下由元素生成 1 mol 化合物时的焓变。标准燃烧焓 ΔHc° 是 1 mol 物质在氧气中完全燃烧时的焓变。
Hess’s law: The total enthalpy change for a reaction is independent of the route taken. ΔHreaction = Σ ΔHf°(products) – Σ ΔHf°(reactants). It can also be found from ΔHc° values: ΔHreaction = Σ ΔHc°(reactants) – Σ ΔHc°(products).
盖斯定律：反应的总焓变与途径无关。ΔH反应 = Σ ΔHf°(生成物) – Σ ΔHf°(反应物)。也可由燃烧焓求得：ΔH反应 = Σ ΔHc°(反应物) – Σ ΔHc°(生成物)。
Bond enthalpy (mean): ΔH ≈ Σ (bond energies broken) – Σ (bond energies formed). This is an approximation because mean bond energies are averaged over different environments.
键焓（平均）：ΔH ≈ Σ (断裂键的键能) – Σ (生成键的键能)。由于平均键焓是对不同环境的平均值，此为近似值。
q = mcΔT
Calorimetry: Heat transferred q = mcΔT, where m is mass, c is specific heat capacity, ΔT is temperature change. ΔH = –q / n (exothermic gives negative ΔH).
量热法：传递的热量 q = mcΔT，其中 m 为质量，c 为比热容，ΔT 为温度变化。ΔH = –q / n（放热时 ΔH 为负）。
4. Entropy & Gibbs Free Energy | 熵与吉布斯自由能
Entropy S (J K⁻¹ mol⁻¹) measures the dispersal of energy. Standard entropy change ΔS° = Σ S°(products) – Σ S°(reactants). A positive ΔS indicates increased disorder.
熵 S (J K⁻¹ mol⁻¹) 衡量能量的分散程度。标准熵变 ΔS° = Σ S°(生成物) – Σ S°(反应物)。ΔS 为正表示无序度增加。
Gibbs free energy change determines spontaneity at constant T and p: ΔG = ΔH – TΔS. A reaction is feasible when ΔG < 0. Standard free energy change ΔG° relates to the equilibrium constant: ΔG° = –RT ln K.
吉布斯自由能变决定恒温恒压下的自发性：ΔG = ΔH – TΔS。当 ΔG < 0 时反应可行。标准自由能变与平衡常数的关系：ΔG° = –RT ln K。
ΔG° = –RT ln K
The temperature at which a reaction becomes just feasible (ΔG = 0) is T = ΔH / ΔS (with ΔH and ΔS assumed constant). The van ‘t Hoff equation describes the temperature dependence of K: ln(K₂/K₁) = –(ΔH°/R)(1/T₂ – 1/T₁).
反应恰好可行 (ΔG = 0) 时的温度 T = ΔH / ΔS（假设 ΔH 与 ΔS 为常数）。范特霍夫方程描述 K 对温度的依赖：ln(K₂/K₁) = –(ΔH°/R)(1/T₂ – 1/T₁)。
5. Chemical Equilibrium | 化学平衡
For a homogeneous reaction aA + bB ⇌ cC + dD, the equilibrium constant in terms of concentration is Kc = [C]c[D]d / ([A]a[B]b). Units of Kc depend on stoichiometry. In terms of partial pressures, Kp = (pC)c(pD)d / (pA)a(pB)b with each pressure expressed relative to standard pressure (1 bar or 10⁵ Pa).
对于均相反应 aA + bB ⇌ cC + dD，浓度平衡常数 Kc = [C]c[D]d / ([A]a[B]b)。Kc 的单位取决于计量数。对于分压平衡常数，Kp = (pC)c(pD)d / (pA)a(pB)b，各压力均相对于标准压力 (1 bar 或 10⁵ Pa) 表达。
Relationship between Kp and Kc: Kp = Kc(RT)Δn, where Δn = (c+d) – (a+b) for gaseous species.
Kp 与 Kc 的关系：Kp = Kc(RT)Δn，其中 Δn = (c+d) – (a+b)（仅针对气态物种）。
Le Chatelier’s principle: If a system at equilibrium is subjected to a change, the equilibrium shifts to partially oppose the change. Temperature changes alter K; pressure changes and concentration changes do not alter K but shift the position of equilibrium.
勒夏特列原理：若对平衡体系施加改变，平衡将向减弱该改变的方向移动。温度改变会改变 K 值；压力或浓度的改变不会改变 K 值，但会使平衡位置移动。
For heterogeneous equilibria, pure solids and liquids are omitted from K expressions because their activities are taken as 1.
对于多相平衡，纯固体和纯液体不出现在 K 的表达式中，因其活度视为 1。
6. Acid–Base Equilibria & pH | 酸碱平衡与 pH
Bransted–Lowry acids are proton donors; bases are proton acceptors. The ionic product of water Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ mol² dm⁻⁶ at 298 K. pH = –log₁₀[H⁺]; pOH = –log₁₀[OH⁻]; pH + pOH = 14 (at 298 K).
布朗斯特–劳里酸是质子给体，碱是质子受体。水的离子积 Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ mol² dm⁻⁶ (298 K 时)。pH = –log₁₀[H⁺]；pOH = –log₁₀[OH⁻]；pH + pOH = 14（298 K 时）。
pH = –log₁₀[H⁺]
Acid dissociation constant Ka = [H⁺][A⁻] / [HA]; pKa = –log₁₀Ka. For a weak acid, [H⁺] ≈ √(Ka ca). Base dissociation constant Kb = [BH⁺][OH⁻] / [B]; for a weak base, [OH⁻] ≈ √(Kb cb). Relation: Ka × Kb = Kw for a conjugate acid–base pair.
酸解离常数 Ka = [H⁺][A⁻] / [HA]；pKa = –log₁₀Ka。对于弱酸，[H⁺] ≈ √(Ka ca)。碱解离常数 Kb = [BH⁺][OH⁻] / [B]；对于弱碱，[OH⁻] ≈ √(Kb cb)。对于共轭酸碱对，Ka × Kb = Kw。
Buffer solutions resist pH change. Henderson–Hasselbalch equation: pH = pKa + log₁₀([A⁻]/[HA]). Maximum buffering capacity when [A⁻] ≈ [HA].
缓冲溶液能抵抗 pH 改变。亨德森–哈塞尔巴尔赫方程：pH = pKa + log₁₀([A⁻]/[HA])。当 [A⁻] ≈ [HA] 时缓冲能力最强。
Indicators are weak acids with distinct colours; their transition range is around pKa ± 1. In titrations, the endpoint pH should match the indicator’s range.
指示剂为具有不同颜色的弱酸，其变色范围约在 pKa ± 1。滴定终点 pH 应与指示剂变色范围吻合。
7. Reaction Kinetics | 反应速率
Rate of reaction is the change in concentration per unit time: rate = –(1/a) d[A]/dt = (1/b) d[B]/dt. The rate equation for a reaction aA + bB → products is often of the form: rate = k [A]m[B]n, where m and n are orders with respect to A and B, and k is the rate constant. Overall order = m + n.
反应速率是浓度随时间的改变：rate = –(1/a) d[A]/dt = (1/b) d[B]/dt。反应 aA + bB → 产物的速率方程常为形式：rate = k [A]m[B]n，其中 m 和 n 分别为对 A、B 的反应级数，k 为速率常数。总级数为 m + n。
First-order kinetics: integrated rate law ln[A] = ln[A]₀ – kt; half-life t₁/₂ = ln2 / k (constant). Second-order (single reactant): 1/[A] = 1/[A]₀ + kt; half-life t₁/₂ = 1 / (k[A]₀). Zero-order: [A] = [A]₀ – kt.
一级反应：积分速率定律 ln[A] = ln[A]₀ – kt；半衰期 t₁/₂ = ln2 / k （恒定）。二级反应（单一反应物）：1/[A] = 1/[A]₀ + kt；半衰期 t₁/₂ = 1/(k[A]₀)。零级反应：[A] = [A]₀ – kt。
t₁/₂ = ln 2 / k (first order)
Arrhenius equation links rate constant to temperature: k = A e–Eₐ/RT. Linear form: ln k = ln A – Eₐ/(RT). Eₐ is activation energy (J mol⁻¹), A is pre-exponential factor, R = 8.31 J K⁻¹ mol⁻¹.
阿伦尼乌斯方程将速率常数与温度关联：k = A e–Eₐ/RT。线性形式：ln k = ln A – Eₐ/(RT)。Eₐ 为活化能 (J mol⁻¹)，A 为指前因子，R = 8.31 J K⁻¹ mol⁻¹。
Catalysts lower Eₐ by providing an alternative pathway, increasing rate without being consumed. Homogeneous and heterogeneous catalysts operate differently but both are explained by Arrhenius behaviour.
催化剂通过提供替代路径以降低 Eₐ，从而提高速率且不被消耗。均相与多相催化剂作用方式不同，但均可用阿伦尼乌斯行为解释。
8. Electrochemistry & Nernst Equation | 电化学与能斯特方程
Standard electrode potential E° measures the tendency of a half-cell to gain electrons relative to the standard hydrogen electrode (SHE). Cell EMF under standard conditions: E°cell = E°cathode – E°anode. A positive E°cell indicates a spontaneous reaction.
标准电极电势 E° 衡量半电池相对于标准氢电极 (SHE) 得电子的倾向。标准条件下的电池电动势：E°cell = E°阴极 – E°阳极。E°cell 为正表示反应自发。
The Nernst equation for a half-reaction aOx + ne⁻ ⇌ bRed at 298 K simplifies to: E = E° – (0.0592/n) log₁₀Q, where Q = [Red]b/[Ox]a. General form: E = E° – (RT/nF) ln Q, F = 9.65 × 10⁴ C mol⁻¹.
能斯特方程，对于半反应 aOx + ne⁻ ⇌ bRed 在 298 K 下简化为：E = E° – (0.0592/n) log₁₀Q，其中 Q = [Red]b/[Ox]a。一般形式：E = E° – (RT/nF) ln Q，F = 9.65 × 10⁴ C mol⁻¹。
E = E° – (RT/nF) ln Q
Relationship between E°cell and ΔG°: ΔG° = –nFE°cell. This links thermodynamic feasibility to cell potential.
E°cell 与 ΔG° 的关系：ΔG° = –nFE°cell，将热力学可行性与电池电势联系起来。
Faraday’s laws of electrolysis: mass deposited or reacted m = (M I t) / (n F), where I is current (A), t is time (s), M is molar mass, n is number of electrons in the half-equation.
法拉第电解定律：沉积或反应的质量 m = (M I t) / (n F)，其中 I 为电流 (A)，t 为时间 (s)，M 为摩尔质量，n 为半反应中的电子数。
9. Spectroscopy & Quantum Theory | 光谱学与量子理论
Electromagnetic radiation energy: E = hν = hc/λ, where h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹, λ is wavelength (m), and ν is frequency (Hz). Wave number ṽ = 1/λ, often in cm⁻¹.
电磁辐射能量：E = hν = hc/λ，其中 h = 6.63 × 10⁻³⁴ J s，c = 3.00 × 10⁸ m s⁻¹，λ 为波长 (m)，ν 为频率 (Hz)。波数 ṽ = 1/λ，常用 cm⁻¹。
E = hν = hc/λ
Beer–Lambert law: A = εcl, where A is absorbance (no units), ε is molar absorption coefficient (dm³ mol⁻¹ cm⁻¹), c is concentration (mol dm⁻³), and l is path length (cm).
比尔–朗伯定律：A = εcl，A 为吸光度（无量纲），ε 为摩尔吸光系数 (dm³ mol⁻¹ cm⁻¹)，c 为浓度 (mol dm⁻³)，l 为光程长 (cm)。
In hydrogen-like atoms, energy levels are given by Bohr’s formula (approximate): Eₙ = –(2.18 × 10⁻¹⁸ J) Z²/n², with Z nuclear charge, n principal quantum number. Transitions ΔE = E₂ – E₁ = hν.
在类氢原子中，能级由玻尔公式（近似）
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