Essential Formulas for the UK Chemistry Olympiad | UKCHO化学竞赛必备公式梳理

📚 Essential Formulas for the UK Chemistry Olympiad | UKCHO化学竞赛必备公式梳理

The UK Chemistry Olympiad (UKChO) challenges students to apply chemical principles to unfamiliar, often olympiad-style problems. While the competition is not purely a test of memorisation, a confident grasp of key formulas — from thermodynamics to kinetics, equilibria, electrochemistry and structure — is essential for fast and accurate problem solving.

UKCHO（英国化学奥林匹克竞赛）不仅考察知识记忆，更要求学生将化学原理灵活运用于陌生情境。然而，熟练掌握核心公式——从热力学到动力学、平衡、电化学与结构化学——仍然是快速准确解题的基础。本文为你系统梳理UKChO中最常出现的必备公式，助你冲刺高分。

1. Stoichiometry and the Mole | 化学计量与物质的量

The mole concept forms the backbone of almost every UKChO calculation. The number of moles \(n\) is related to mass \(m\), molar mass \(M\), volume of gas \(V\), and molar gas volume \(V_m\) at room temperature and pressure (24 dm³ mol⁻¹ in many olympiad contexts).

物质的量是UKChO几乎所有计算的基础。物质的量 \(n\) 与质量 \(m\)、摩尔质量 \(M\)、气体体积 \(V\) 以及摩尔气体体积 \(V_m\)（室温常压下常取 24 dm³ mol⁻¹）密切相关。

n = m / M = V / V_m = c × V_solution (dm³)

For solutions, concentration \(c\) is expressed in mol dm⁻³, and volume must be in dm³ when using \(n = cV\).

溶液浓度 \(c\) 的单位为 mol dm⁻³，使用 \(n = cV\) 时体积须以 dm³ 为单位。
For gases under non-standard conditions, use the ideal gas equation rather than fixed molar volumes.

非标准状况下的气体应使用理想气体状态方程，而不是固定摩尔体积。

Always balance chemical equations before using mole ratios. In UKChO, redox equations and organic combustion equations often require extra care with oxygen atoms.

使用物质的量比例前务必配平方程式。UKChO中的氧化还原反应与有机物燃烧方程常需格外留意氧原子守恒。

2. Ideal Gas Equation | 理想气体状态方程

The ideal gas equation links pressure, volume, temperature and moles. It is indispensable for calculating molar masses of gases or volatile liquids, and for analysing gas-phase reactions.

理想气体状态方程将压力、体积、温度与物质的量联系起来。它常用于计算气体或挥发性液体的摩尔质量，以及分析气相反应。

pV = nRT

Units must be consistent: \(p\) in Pa, \(V\) in m³, \(R = 8.314\) J K⁻¹ mol⁻¹, \(T\) in K.

单位必须一致：\(p\) 以 Pa 为单位，\(V\) 以 m³ 为单位，\(R = 8.314\) J K⁻¹ mol⁻¹，\(T\) 以 K 为单位。
If pressure is given in kPa and volume in dm³, then \(R = 8.314\) kPa dm³ K⁻¹ mol⁻¹ works directly.

若压力以 kPa、体积以 dm³ 给出，可直接使用 \(R = 8.314\) kPa dm³ K⁻¹ mol⁻¹。
Convert temperatures from Celsius to Kelvin: \(T(K) = T(°C) + 273.15\).

温度须从摄氏度换算为开尔文：\(T(K) = T(°C) + 273.15\)。

A common exam trap is using the molar volume 24 dm³ at RTP without checking whether the gas is at RTP. The ideal gas equation is safer for all conditions.

常见陷阱是默认气体处于室温常压而直接使用 24 dm³ 的摩尔体积。遇到非标准条件时，用理想气体状态方程更加稳妥。

3. Thermodynamics: Enthalpy and Hess’s Law | 热力学：焓变与盖斯定律

UKChO frequently asks students to calculate reaction enthalpies from bond enthalpies, formation enthalpies, or combustion enthalpies. Hess’s law states that the overall enthalpy change depends only on initial and final states.

UKChO经常要求利用键焓、生成焓或燃烧焓计算反应焓变。盖斯定律指出，总焓变只取决于反应的始态与终态，而与路径无关。

Δ_rH = Σ Δ_fH(products) − Σ Δ_fH(reactants)

Δ_rH = Σ (bonds broken) − Σ (bonds formed)

The bond enthalpy method treats bond breaking as endothermic and bond making as exothermic, so the formula is: ΔH = energy required to break bonds − energy released when forming bonds.

键焓法视断键为吸热、成键为放热，故公式为：ΔH = 断键所需能量 − 成键释放能量。
Hess cycles can be solved using energy level diagrams or algebraic addition of equations.

盖斯循环可通过能级图或方程式的代数相加来求解。
Remember to multiply enthalpy changes by stoichiometric coefficients when applying Hess’s law.

运用盖斯定律时，不要忘记将焓变乘上化学计量系数。

Watch out for bond enthalpy values being average values over different compounds; they produce approximate ΔH values. Formation enthalpies give more precise results.

注意键焓是不同化合物中的平均值，因此计算结果为近似值；利用生成焓计算则更为精确。

4. Entropy and Gibbs Free Energy | 熵与吉布斯自由能

Predicting whether a reaction is spontaneous requires combining enthalpy and entropy at a given temperature. Entropy \(S\) measures disorder; the total entropy change of the universe must be positive for a spontaneous process.

判断反应是否自发需要综合特定温度下的焓变与熵变。熵 \(S\) 衡量体系的混乱程度；自发过程的宇宙总熵变必须为正值。

ΔS_system = Σ S(products) − Σ S(reactants)

ΔG = ΔH − TΔS

If ΔG < 0, the process is spontaneous; if ΔG = 0, the system is at equilibrium; if ΔG > 0, the process is non-spontaneous under standard conditions.

若ΔG < 0，过程自发；若ΔG = 0，体系处于平衡；若ΔG > 0，则标准条件下过程非自发。
Temperature can reverse the sign of ΔG when both ΔH and ΔS are positive (or both negative).

当ΔH与ΔS同号时，温度可使ΔG的符号反转。例如ΔH和ΔS均为正时，高温使反应自发。
Entropy changes can be estimated qualitatively: gas production increases entropy, while gas consumption or formation of more ordered solids decreases it.

熵变可定性判断：产生气体使熵增大，消耗气体或生成更加有序的固体则使熵减小。

In UKChO, you are often given entropy values in J K⁻¹ mol⁻¹ but enthalpies in kJ mol⁻¹ — always convert to the same energy unit before using ΔG = ΔH − TΔS.

UKChO常给出以 J K⁻¹ mol⁻¹ 为单位的熵值和以 kJ mol⁻¹ 为单位的焓值——使用 ΔG = ΔH − TΔS 前务必统一能量单位。

5. Equilibrium Constants | 平衡常数

For a general reaction \(aA + bB ⇌ cC + dD\), the equilibrium constant in terms of concentration is expressed using activities or concentrations raised to their stoichiometric coefficients.

对于一般反应 \(aA + bB ⇌ cC + dD\)，浓度平衡常数通过各物质浓度以其化学计量系数为指数来表述。

K_c = [C]^c[D]^d / ([A]^a[B]^b)

Pure solids and pure liquids do not appear in K_c or K_p expressions because their activities are unity.

纯固体与纯液体不出现在K_c或K_p的表达式中，因其活度为1。
For gas-phase equilibria, \(K_p\) uses partial pressures: \(K_p = (p_C^c p_D^d) / (p_A^a p_B^b)\).

气相平衡中，\(K_p\) 使用分压：\(K_p = (p_C^c p_D^d) / (p_A^a p_B^b)\)。
Partial pressure of a gas = mole fraction × total pressure: \(p_A = x_A P_{total}\).

气体分压 = 摩尔分数 × 总压：\(p_A = x_A P_{total}\)。

The equilibrium constant depends only on temperature, not on concentration, pressure or the presence of a catalyst. Changing concentration or pressure shifts the position of equilibrium but leaves K unchanged.

平衡常数只取决于温度，与浓度、压力或催化剂无关。改变浓度或压力只能移动平衡位置，而不能改变K值。

6. Acids, Bases and pH | 酸碱与pH

Aqueous acid–base equilibria are a rich source of UKChO problems, often involving polyprotic acids or buffer solutions. The key definitions are:

水溶液中的酸碱平衡是UKChO的重要考点，常涉及多元酸或缓冲溶液。核心定义如下：

pH = −log₁₀[H⁺] ; pOH = −log₁₀[OH⁻] ; pH + pOH = 14 (at 25 °C)

K_a = [H⁺][A⁻] / [HA] ; pK_a = −log₁₀ K_a

For a weak acid, the approximation \([H⁺] ≈ √(K_a × C)\) is valid when the acid is weak and not too dilute.

对于弱酸，当酸很弱且浓度不太低时，近似式 \([H⁺] ≈ √(K_a × C)\) 成立。
For a buffer solution, use the Henderson–Hasselbalch equation: pH = pK_a + log₁₀([A⁻]/[HA]).

对于缓冲溶液，使用亨德森-哈塞尔巴尔赫方程：pH = pK_a + log₁₀([A⁻]/[HA])。
Ionic product of water: \(K_w = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴\) at 25 °C.

水的离子积：\(K_w = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴\)（25 °C）。

For polyprotic acids like H₃PO₄, successive K_a values usually decrease by orders of magnitude. Only the first deprotonation generally contributes significantly to pH.

像H₃PO₄这样的多元酸，逐级K_a通常按数量级递减。一般只有第一步去质子化对pH有显著贡献。

7. Solubility Product | 溶度积

Solubility product \(K_{sp}\) is a special equilibrium constant for sparingly soluble salts. For a salt \(A_mB_n\), the dissociation is \(A_mB_n(s) ⇌ mA^{n+}(aq) + nB^{m-}(aq)\).

溶度积 \(K_{sp}\) 是难溶盐的一种特殊平衡常数。对于盐 \(A_mB_n\)，溶解平衡为 \(A_mB_n(s) ⇌ mA^{n+}(aq) + nB^{m-}(aq)\)。

K_sp = [Aⁿ⁺]^m × [B^m−]ⁿ

To find molar solubility \(s\) of a pure salt, substitute relationships between ion concentrations and \(s\). For example, for AgCl: \(s = √K_{sp}\). For PbCl₂: \(4s³ = K_{sp}\).

求纯盐的摩尔溶解度 \(s\) 时，将离子浓度与 \(s\) 的关系式代入。例如AgCl：\(s = √K_{sp}\)；PbCl₂：\(4s³ = K_{sp}\)。
When a common ion is present, solubility decreases. Calculate the new ion concentrations and solve for the unknown ion.

存在同离子效应时溶解度会降低。可先求已知离子浓度，再解出未知离子浓度。
Precipitation occurs when the ion product \(Q\) exceeds \(K_{sp}\); if \(Q < K_{sp}\), no precipitate forms.

当离子积 \(Q\) 超过 \(K_{sp}\) 时产生沉淀；若 \(Q < K_{sp}\)，则不形成沉淀。

Solubility products are strongly temperature-dependent, so comparison of \(K_{sp}\) values must always be made at the same temperature.

溶度积受温度影响很大，因此比较 \(K_{sp}\) 值时必须在相同温度下进行。

8. Electrochemistry and Nernst Equation | 电化学与能斯特方程

Electrode potentials determine the direction of redox reactions. The standard cell potential is the difference between the reduction potentials of the cathode and anode.

电极电势决定氧化还原反应的方向。标准电池电动势等于阴极还原电势减去阳极还原电势。

E^θ_cell = E^θ_cathode − E^θ_anode

ΔG^θ = −nFE^θ_cell ; ΔG^θ = −RT ln K

F is the Faraday constant, \(F = 96485\) C mol⁻¹ (often rounded to 96500 C mol⁻¹); \(n\) is the number of electrons transferred per mole of reaction.

F为法拉第常数，\(F = 96485\) C mol⁻¹（常取96500 C mol⁻¹）；\(n\) 为每摩尔反应转移的电子数。
Combining the two expressions gives \(\ln K = nFE^{θ}_{cell}/RT\), allowing prediction of equilibrium constants from cell potentials.

联立两式可得 \(\ln K = nFE^{θ}_{cell}/RT\)，从而用电池电动势预测平衡常数。
Under non-standard conditions, use the Nernst equation: \(E = E^θ − (RT/nF) \ln Q\). At 25 °C, this can be written as \(E = E^θ − (0.0592/n) \log₁₀ Q\).

非标准状况下使用能斯特方程：\(E = E^θ − (RT/nF) \ln Q\)。在25 °C时可写作 \(E = E^θ − (0.0592/n) \log₁₀ Q\)。

Remember that the electrode with the more negative \(E^θ\) is oxidised at the anode. In a galvanic cell, electrons flow from anode to cathode.

记住：\(E^θ\) 更负的电极在阳极被氧化。在原电池中，电子从阳极流向阴极。

9. Kinetics: Rate Laws and Integrated Equations | 化学动力学：速率方程与积分式

UKChO often asks students to deduce rate orders from data or from proposed mechanisms. The rate law for a reaction \(aA + bB → products\) is determined experimentally, not from stoichiometry.

UKChO常要求从实验数据或机理推导反应级数。速率方程 \(rate = k[A]^x[B]^y\) 必须由实验确定，而不能根据化学计量式直接写出。

rate = k[A]^x[B]^y ; overall order = x + y

For a first-order reaction: \(\ln[A]_t = \ln[A]_0 − kt\), and half-life \(t_{1/2} = \ln 2 / k\), independent of initial concentration.

一级反应：\(\ln[A]_t = \ln[A]_0 − kt\)，半衰期 \(t_{1/2} = \ln 2 / k\)，与初始浓度无关。
For a second-order reaction: \(1/[A]_t = 1/[A]_0 + kt\), with half-life \(t_{1/2} = 1/(k[A]_0)\).

二级反应：\(1/[A]_t = 1/[A]_0 + kt\)，半衰期 \(t_{1/2} = 1/(k[A]_0)\)。
For a zero-order reaction: \([A]_t = [A]_0 − kt\), with half-life \(t_{1/2} = [A]_0/(2k)\).

零级反应：\([A]_t = [A]_0 − kt\)，半衰期 \(t_{1/2} = [A]_0/(2k)\)。

The rate-determining step in a mechanism controls the overall rate. Intermediates should not appear in the rate law; use pre-equilibrium approximations when necessary.

决速步控制总反应速率。速率方程中不应出现中间体；必要时需使用平衡近似。

10. Arrhenius Equation | 阿伦尼乌斯方程

The temperature dependence of the rate constant \(k\) is described by the Arrhenius equation. UKChO may ask you to determine activation energy graphically or to compare rates at two temperatures.

速率常数 \(k\) 随温度的变化由阿伦尼乌斯方程描述。UKChO可能要求图解求活化能，或比较两个温度下的反应速率。

k = A e^−E_a/RT

ln k = ln A − E_a / (RT)

A plot of \(\ln k\) against \(1/T\) gives a straight line with slope \(-E_a/R\) and intercept \(\ln A\).

以 \(\ln k\) 对 \(1/T\) 作图得直线，斜率为 \(-E_a/R\)，截距为 \(\ln A\)。
Comparing two temperatures:

比较两个温度时：

ln(k₂/k₁) = (E_a/R)(1/T₁ − 1/T₂)

The pre-exponential factor \(A\) is related to the frequency and orientation of molecular collisions. For simple reactions, a small change in temperature can dramatically change \(k\) when \(E_a\) is large.

指前因子 \(A\) 与分子碰撞频率和取向有关。对于简单反应，当活化能较大时，温度微小的变化就可使 \(k\) 发生显著改变。

11. Quantum Numbers and Electron Configurations | 量子数与电子排布

Understanding electronic structure requires the four quantum numbers: principal \(n\), angular momentum \(l\), magnetic \(m_l\), and spin \(m_s\). The order of orbital filling can be predicted by the Aufbau principle.

理解电子结构需要掌握四个量子数：主量子数 \(n\)、角量子数 \(l\)、磁量子数 \(m_l\) 和自旋量子数 \(m_s\)。轨道填充顺序可由构造原理预测。

For a given \(n\), \(l\) ranges from 0 to \(n−1\). The number of orbitals in a subshell is \(2l+1\). The maximum number of electrons in a shell is \(2n²\).

对给定 \(n\)，\(l\) 取0到 \(n−1\)。一个亚壳层的轨道数为 \(2l+1\)。一个主壳层最多容纳 \(2n²\) 个电子。
Hund’s rule states that electrons occupy degenerate orbitals singly before pairing; Pauli’s exclusion principle forbids two electrons in the same atom from having identical sets of four quantum numbers.

洪特规则指出电子先单独占据简并轨道再配对；泡利不相容原理禁止同一原子中两个电子拥有完全相同的四个量子数。
Isoelectronic species have the same number of electrons; ionic radii and ionisation energies can be compared using effective nuclear charge \(Z_{eff}\).

等电子体具有相同的电子数；离子半径与电离能可通过有效核电荷 \(Z_{eff}\) 比较。

Periodic trends such as ionisation energy, electron affinity and electronegativity are governed by shielding and \(Z_{eff}\). UKChO often uses these trends to explain anomalous cases like oxygen vs nitrogen.

电离能、电子亲和能、电负性等周期性趋势由屏蔽效应与 \(Z_{eff}\) 决定。UKChO常用这些趋势解释如氧与氮的异常现象。

12. Ideal Bond Angles and VSEPR | 理想键角与VSEPR

VSEPR theory predicts molecular shapes by minimising electron-pair repulsion around the central atom. The total number of bonding pairs and lone pairs determines the electron-pair geometry.

VSEPR理论通过最小化中心原子周围电子对的排斥力来预测分子形状。成键电子对与孤电子对总数决定电子对几何构型。

Electron pairs / 电子对数	Geometry shape / 几何构型	Ideal bond angle / 理想键角
2	Linear 直线形	180°
3	Trigonal planar 平面三角形	120°
4	Tetrahedral 正四面体形	109.5°
5	Trigonal bipyramidal 三角双锥形	90°, 120°
6	Octahedral 正八面体形	90°

Lone pairs repel more strongly than bonding pairs, compressing bond angles. For example, NH₃ has a bond angle of about 107° instead of 109.5°.

孤电子对的排斥力大于成键电子对，会压缩键角。例如NH₃的键角约为107°，而非109.5°。
Multiple bonds count as one electron domain but exert slightly greater repulsion than single bonds.

重键视为一个电子域，但其排斥力略大于单键。
Electronegativity differences may distort bond angles further in mixed halogen compounds.

在混合卤素化合物中，电负性差异可能进一步扭曲键角。

Beyond VSEPR, UKChO may ask you to compare bond angles using hybridisation: \(sp\) → 180°, \(sp²\) → 120°, \(sp³\) → 109.5°, with deviations caused by lone pairs.

除VSEPR外，UKChO还可能利用杂化方式比较键角：\(sp\) 为180°，\(sp²\) 为120°，\(sp³\) 为109.5°，孤电子对会导致偏差。

Mastering these formulas is only the first step. In UKChO, you must also know when to apply each formula, how to handle non-ideal conditions, and how to combine multiple concepts in a single problem. Consistent practice with past papers and olympiad-style problems will help you internalise these tools.

掌握以上公式只是第一步。在UKChO中，你还需要知道何时使用每个公式、如何处理非理想条件，以及如何在同一个题目中综合多个概念。通过真题与奥林匹克风格题目的持续练习，你才能真正内化这些工具。

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Essential Formulas for the UK Chemistry Olympiad | UKCHO化学竞赛必备公式梳理