📚 Essential Formulae and Theorems Quick Reference for CIE A2 Chemistry | CIE A2 化学公式定理速查手册
This quick reference guide compiles the key quantitative relationships and theoretical equations required for the CIE A Level Chemistry (Year 13) syllabus. Each entry includes the formula, symbol definitions, relevant conditions, and typical applications. Mastering these relationships is essential for solving numerical problems and interpreting chemical data in the examination.
本速查手册涵盖了 CIE A Level 化学（Year 13）课程中必须掌握的定量关系与理论方程。每个条目均给出了公式、符号定义、适用条件和典型应用场景。熟练掌握这些关系式是解题和解读实验数据的关键。
1. Ideal Gas Equation and Partial Pressure | 理想气体状态方程与分压
The ideal gas law relates pressure, volume, temperature and amount of a gas: pV = nRT. For gas mixtures, the total pressure is the sum of the partial pressures, and the partial pressure of a component is given by PA = χA × Ptotal, where χA is the mole fraction.
理想气体状态方程为 pV = nRT。在混合气体中，总压等于各组分分压之和；某一组分的分压 PA = χA × P总，其中 χA 为该气体的摩尔分数。
| Symbol | Meaning | Units / Notes |
| p | Pressure | Pa or atm |
| V | Volume | m³ or dm³ |
| n | Amount of substance | mol |
| R | Molar gas constant | 8.31 J mol⁻¹ K⁻¹ |
| T | Absolute temperature | K |
The mole fraction χA = nA / ntotal, and the partial pressure PA = χA Ptotal. This is widely used in equilibrium calculations involving gaseous systems (Kp).
摩尔分数 χA = nA / n总，分压 PA = χA P总。此关系常用于涉及气体的平衡常数 Kp 计算。
2. Enthalpy Changes and Hess’s Law | 焓变与赫斯定律
Hess’s Law states that the total enthalpy change for a reaction is independent of the pathway, provided the initial and final states are the same. This allows the calculation of ΔH for reactions that are difficult to measure directly.
赫斯定律指出，反应的总焓变只取决于始态和终态，与途径无关。利用该定律可以间接计算难以直接测得的反应焓变。
Standard enthalpy changes can be combined via cycles:
ΔH°reaction = Σ ΔH°f(products) − Σ ΔH°f(reactants)
ΔH°reaction = Σ bond energies broken − Σ bond energies formed (for gaseous species)
标准焓变可通过以下关系组合：
ΔH°反应 = Σ ΔH°f(生成物) − Σ ΔH°f(反应物)
ΔH°反应 = Σ 断裂键能 − Σ 形成键能（适用于气态物质）
Common standard enthalpy changes include ΔH°c (combustion), ΔH°f (formation), ΔH°neut (neutralisation), ΔH°sol (solution), and ΔH°hyd (hydration).
常见的标准焓变有：标准燃烧焓 ΔH°c、标准生成焓 ΔH°f、标准中和焓 ΔH°neut、标准溶解焓 ΔH°sol 和标准水合焓 ΔH°hyd。
3. Entropy and Gibbs Free Energy | 熵与吉布斯自由能
The spontaneity of a process at constant temperature and pressure is determined by the Gibbs free energy change: ΔG° = ΔH° − TΔS°. The standard entropy change for a reaction is ΔS° = Σ S°(products) − Σ S°(reactants).
恒温恒压下，过程的自发性由吉布斯自由能变判定：ΔG° = ΔH° − TΔS°。反应的标准熵变 ΔS° = Σ S°(生成物) − Σ S°(反应物)。
ΔG° is related to the equilibrium constant by ΔG° = −RT ln K, where R = 8.31 J mol⁻¹ K⁻¹ and T in kelvin. A negative ΔG° corresponds to K > 1 (equilibrium favours products).
ΔG° 与平衡常数 K 的关系为 ΔG° = −RT ln K，其中 R = 8.31 J mol⁻¹ K⁻¹，T 为热力学温度。ΔG° < 0 时，K > 1，平衡偏向生成物方向。
For non-standard conditions, ΔG = ΔG° + RT ln Q, where Q is the reaction quotient.
对于非标准状态，有 ΔG = ΔG° + RT ln Q，其中 Q 为反应商。
4. Equilibrium Constants Kc and Kp | 平衡常数 Kc 与 Kp
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
For gases, in terms of partial pressures (in atm or Pa):
Kp = PCc PDd / PAa PBb
对于均相反应 aA + bB ⇌ cC + dD，浓度平衡常数表达为 Kc = [C]c[D]d / [A]a[B]b。对于气体反应，分压平衡常数 Kp = PCc PDd / PAa PBb。
Kc and Kp are related by Kp = Kc (RT)Δn, where Δn = (c + d) − (a + b) for gaseous species only.
Kc 与 Kp 的关系为 Kp = Kc (RT)Δn，其中 Δn = (c + d) − (a + b)，仅考虑气体物种。
The magnitude of K indicates the extent of the reaction: K >> 1 – products favoured, K << 1 – reactants favoured.
K 值的大小反映反应程度：K >> 1 表示平衡偏向生成物；K << 1 表示平衡偏向反应物。
5. Acid–Base Equilibria: pH, pKa, pKb and Kw | 酸碱平衡：pH、pKa、pKb 与 Kw
pH is defined as pH = −log₁₀ [H⁺], and pOH = −log₁₀ [OH⁻]. In pure water at 298 K, Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ mol² dm⁻⁶, so pH + pOH = 14.
pH 定义为 pH = −log₁₀ [H⁺]，pOH = −log₁₀ [OH⁻]。298 K 纯水中，Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ mol² dm⁻⁶，因此 pH + pOH = 14。
For a weak acid HA: Ka = [H⁺][A⁻] / [HA] and pKa = −log₁₀ Ka. For a weak base B: Kb = [BH⁺][OH⁻] / [B] and pKb = −log₁₀ Kb. Furthermore, pKa + pKb = 14 for a conjugate acid–base pair at 298 K.
对于弱酸 HA：Ka = [H⁺][A⁻] / [HA]，pKa = −log₁₀ Ka。对于弱碱 B：Kb = [BH⁺][OH⁻] / [B]，pKb = −log₁₀ Kb。共轭酸碱对在 298 K 时满足 pKa + pKb = 14。
6. Buffer Solutions and Henderson–Hasselbalch Equation | 缓冲溶液与亨德森–哈塞尔巴赫方程
Buffer solutions resist changes in pH upon addition of small amounts of acid or alkali. Their pH can be calculated using the Henderson–Hasselbalch equation for an acid buffer (HA/A⁻):
pH = pKa + log₁₀ ( [A⁻] / [HA] )
缓冲溶液能够抵抗少量酸或碱加入引起的 pH 变化。对于酸性缓冲体系 (HA/A⁻)，其 pH 由亨德森–哈塞尔巴赫方程给出：pH = pKa + log₁₀ ( [A⁻] / [HA] )。
This equation assumes that the amount of dissociation of HA is negligible and that all A⁻ comes from the salt. Maximum buffer capacity occurs when [A⁻] = [HA], i.e. pH = pKa. For basic buffers, a similar expression applies using pKb.
该方程假设 HA 自身电离可忽略，且所有 A⁻ 来自共轭碱的盐。当 [A⁻] = [HA] 时缓冲能力最大，此时 pH = pKa。碱性缓冲体系可使用类似公式，其中用 pKb 替换。
7. Solubility Product Ksp | 溶度积 Ksp
For a sparingly soluble salt with formula MmXn(s) ⇌ mMn+(aq) + nXm−(aq), the solubility product is Ksp = [Mn+]m[Xm−]n. It applies only to saturated solutions at a given temperature.
对于难溶盐 MmXn(s) ⇌ mMn+(aq) + nXm−(aq)，溶度积 Ksp = [Mn+]m[Xm−]n。该表达式仅适用于特定温度下的饱和溶液。
Precipitation occurs when the ionic product exceeds Ksp. The common ion effect reduces solubility by shifting the equilibrium left, which can be quantitatively predicted using Ksp.
当离子积超过 Ksp 时生成沉淀。同离子效应通过使平衡向左移动而降低溶解度，该效应可通过 Ksp 进行定量预测。
Solubility (s) can be calculated from Ksp: for a 1:1 salt, s = √Ksp; for a 1:2 salt, s = (Ksp/4)1/3.
已知 Ksp 可求溶解度 s：对于 1:1 型盐，s = √Ksp；对于 1:2 型盐，s = (Ksp/4)1/3。
8. Electrode Potentials and the Nernst Equation | 电极电势与能斯特方程
The standard electrode potential, E°, is measured under standard conditions (298 K, 1 mol dm⁻³, 100 kPa). The cell potential is E°cell = E°cathode − E°anode, where both potentials are written as reduction potentials.
标准电极电势 E° 在标准条件下（298 K, 1 mol dm⁻³, 100 kPa）测定。电池电动势 E°cell = E°阴极 − E°阳极，其中两个电势均取还原电势。
Under non-standard conditions, the Nernst equation gives the electrode potential:
E = E° − (RT / nF) ln Q or at 298 K, E = E° − (0.059 / n) log₁₀ Q, where n is moles of electrons transferred and F = 96 500 C mol⁻¹.
非标准状况下，能斯特方程给出电极电势：E = E° − (RT / nF) ln Q，在 298 K 时可写为 E = E° − (0.059 / n) log₁₀ Q，其中 n 为转移电子数，F = 96 500 C mol⁻¹。
The relationship between cell potential and Gibbs free energy is ΔG° = −n F E°cell. A positive E°cell corresponds to a spontaneous reaction.
电池电动势与吉布斯自由能变的关系为 ΔG° = −n F E°cell。E°cell > 0 意味着反应可自发进行。
9. Rate Equations and the Arrhenius Equation | 速率方程与阿伦尼乌斯方程
The rate equation for a reaction aA + bB → products is rate = k [A]m[B]n, where m and n are the orders of reaction with respect to A and B. The overall order is m + n, and k is the rate constant.
反应 aA + bB → 生成物的速率方程为 速率 = k [A]m[B]n，其中 m 和 n 分别为对 A 和 B 的反应级数，总级数为 m + n，k 为速率常数。
The temperature dependence of k is described by the Arrhenius equation:
k = A exp(−Ea / (RT)) or ln k = ln A − Ea / (RT).
A is the pre-exponential factor, Ea is the activation energy (J mol⁻¹), R = 8.31 J mol⁻¹ K⁻¹, and T is in kelvin.
速率常数 k 与温度的关系由阿伦尼乌斯方程描述：k = A exp(−Ea / (RT)) 及其对数形式 ln k = ln A − Ea / (RT)。A 为指前因子，Ea 为活化能（J mol⁻¹），R = 8.31 J mol⁻¹ K⁻¹。
A plot of ln k against 1/T gives a straight line with slope = −Ea/R.
以 ln k 对 1/T 作图可得斜率为 −Ea/R 的直线。
10. Born–Haber Cycle and Lattice Energy | Born–Haber 循环与晶格能
The Born–Haber cycle is an application of Hess’s Law to the formation of an ionic compound. It relates the standard enthalpy of formation (ΔH°f) to atomisation enthalpies, ionisation energies, electron affinities, and the lattice enthalpy (ΔH°L).
Born–Haber 循环是赫斯定律在离子化合物形成过程中的应用，它将标准生成焓 ΔH°f 与原子化焓、电离能、电子亲和能和晶格焓 ΔH°L 联系起来。
For NaCl(s):
ΔH°f(NaCl) = ΔH°at(Na) + I.E.(Na) + ½ΔH°at(Cl₂) + E.A.(Cl) + ΔH°L(NaCl)
Lattice enthalpy (always exothermic for stable ionic solids) is often calculated as the unknown by rearranging the Born–Haber cycle.
对于 NaCl(s)：ΔH°f(NaCl) = ΔH°at(Na) + I.E.(Na) + ½ΔH°at(Cl₂) + E.A.(Cl) + ΔH°L(NaCl)。晶格焓（稳定离子固体的形成过程总是放热）常作为 Born–Haber 循环中的未知量被求解出来。
11. Partition Coefficient Kpc | 分配系数 Kpc
The partition coefficient (or distribution coefficient) describes how a solute distributes itself between two immiscible solvents at equilibrium: Kpc = [solute]organic layer / [solute]aqueous layer, usually measured at a specified temperature.
分配系数描述溶质在两种互不相溶的溶剂间达到平衡时的浓度比：Kpc = [溶质]有机层 / [溶质]水层，通常在指定温度下测定。
Kpc is used in solvent extraction to predict the number of extractions required to isolate a compound. It assumes the solute exists in the same molecular form in both solvents.
Kpc 常用于溶剂萃取中，预测分离化合物所需的萃取次数。该系数假定溶质在两种溶剂中以相同分子形式存在。
12. Stability Constants of Complex Ions Kstab | 配离子的稳定常数 Kstab
For a complex ion formation reaction such as Cu²⁺(aq) + 4NH₃(aq) ⇌ [Cu(NH₃)₄]²⁺(aq), the stability constant is Kstab = [Cu(NH₃)₄²⁺] / ([Cu²⁺][NH₃]⁴). A large Kstab indicates a very stable complex.
对于配离子形成反应，如 Cu²⁺(aq) + 4NH₃(aq) ⇌ [Cu(NH₃)₄]²⁺(aq)，稳定常数 Kstab = [Cu(NH₃)₄²⁺] / ([Cu²⁺][NH₃]⁴)。Kstab 值越大，表明配合物越稳定。
Successive stepwise stability constants K₁, K₂, K₃ describe the formation of intermediate complexes, and the overall Kstab is the product of all stepwise constants. These values are useful for understanding redox and substitution behaviour of transition metal complexes.
逐级稳定常数 K₁、K₂、K₃ 描述了中间配合物的生成过程，总稳定常数 Kstab 是各逐级常数的乘积。这些数值有助于理解过渡金属配合物的氧化还原和取代行为。
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