📚 OCR A-Level Science: Essential Formulas and Theorems Quick Reference | OCR A-Level 科学：公式定理速查手册
This handbook provides a concise summary of the key formulas, equations, and theorems that Year 13 students need to master for OCR A-Level Science examinations, covering Physics, Chemistry, and Biology. It is designed as a rapid revision tool to help you recall and apply essential quantitative relationships.
这本手册提供了 Year 13 学生参加 OCR A-Level 科学考试需要掌握的关键公式、方程和定理的简明总结，涵盖物理、化学和生物。它是一份快速复习工具，帮助你记忆和应用基本的数量关系。
1. Physical Quantities and Units | 物理量与单位
All physical quantities in the OCR specification are expressed in SI base units or appropriate derived units. It is vital to check unit consistency before using any formula. Common base quantities include mass (kg), length (m), time (s), electric current (A), temperature (K), and amount of substance (mol).
OCR 规范中的所有物理量均采用国际基本单位或适当的导出单位表示。在使用任何公式之前检查单位的一致性至关重要。常见的基本量包括质量 (kg)、长度 (m)、时间 (s)、电流 (A)、温度 (K) 和物质的量 (mol)。
Prefixes such as pico (p, 10⁻¹²), nano (n, 10⁻⁹), micro (µ, 10⁻⁶), milli (m, 10⁻³), kilo (k, 10³), mega (M, 10⁶) and giga (G, 10⁹) are frequently used. Homogeneity of equations is tested: every term in a valid physical equation must have the same base units.
前缀如皮可 (p, 10⁻¹²)、纳诺 (n, 10⁻⁹)、微 (µ, 10⁻⁶)、毫 (m, 10⁻³)、千 (k, 10³)、兆 (M, 10⁶) 和吉 (G, 10⁹) 经常使用。还要检验方程的单位一致性：一个有物理意义的方程中，每一项必须具有相同的基本单位。
2. Mechanics: Motion and Forces | 力学：运动与力
SUVAT equations apply to motion with constant acceleration in a straight line:
v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t.
SUVAT 方程 适用于匀加速直线运动：
v = u + at，s = ut + ½at²，v² = u² + 2as，s = ½(u + v)t。
Newton’s second law states that the net force acting on an object is equal to the rate of change of its momentum:
F = Δp / Δt
and for constant mass, F = ma.
牛顿第二定律指出，作用在物体上的净力等于其动量变化率：
F = Δp / Δt
质量不变时，F = ma。
Momentum p = mv is always conserved in the absence of external forces. Impulse is the change in momentum and equals the area under a force–time graph.
动量 p = mv 在没有外力时总是守恒的。冲量是动量的变化量，等于力–时间曲线下的面积。
The coefficient of restitution e = (speed of separation) / (speed of approach) is used for collisions in one dimension. For a perfectly elastic collision, e = 1; for a perfectly inelastic collision, e = 0.
恢复系数 e =（分离速度）/（接近速度），用于一维碰撞。对于完全弹性碰撞，e = 1；对于完全非弹性碰撞，e = 0。
3. Work, Energy and Power | 功、能量和功率
Work done by a constant force is W = F s cosθ, where θ is the angle between the force and displacement vectors. When the force varies, work done is the area under a force–displacement graph.
恒力所做的功为 W = F s cosθ，其中 θ 是力矢量与位移矢量之间的夹角。当力变化时，功等于力–位移曲线下的面积。
Kinetic energy Eₖ = ½mv² and gravitational potential energy Eₚ = mgh (near Earth’s surface). The principle of conservation of energy states that energy cannot be created or destroyed, only transferred into different forms.
动能为 Eₖ = ½mv²，重力势能为 Eₚ = mgh（近地面）。能量守恒定律指出，能量不能凭空产生或消灭，只能转化为不同形式。
Power is the rate of doing work: P = ΔW / Δt = Fv for a constant force acting on an object moving at speed v. Efficiency = (useful power output) / (total power input) × 100%.
功率是做功的快慢：P = ΔW / Δt，如果恒力作用在速度为 v 的物体上，则 P = Fv。效率 =（有用输出功率）/（总输入功率）× 100%。
4. Electric Circuits | 电路
Ohm’s law: V = IR, where resistance R is constant for an ohmic conductor at constant temperature. Resistivity ρ relates resistance to geometry: R = ρL / A.
欧姆定律：V = IR，其中对于欧姆导体，在恒温下电阻 R 恒定。电阻率 ρ 将电阻与几何形状关联：R = ρL / A。
For series circuits, total resistance R_total = R₁ + R₂ + … . For parallel circuits, 1/R_total = 1/R₁ + 1/R₂ + … . Potential dividers produce an output voltage V_out = R₂/(R₁ + R₂) × V_in.
串联电路：总电阻 Rₜₒₜₐₗ = R₁ + R₂ + … 。并联电路：1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + … 。分压器产生输出电压 Vₒᵤₜ = R₂/(R₁ + R₂) × Vᵢₙ。
Kirchhoff’s first law: total current entering a junction = total current leaving. Second law: the sum of e.m.f.s around any closed loop equals the sum of p.d.s (energy conservation).
基尔霍夫第一定律：流入节点的总电流等于流出节点的总电流。第二定律：任一闭合回路中电动势的代数和等于电压降的代数和（能量守恒）。
Internal resistance r of a source reduces terminal p.d.: V = ε − Ir. The electromotive force (e.m.f.) ε is the energy supplied per unit charge.
电源内阻 r 降低路端电压：V = ε − Ir。电动势 ε 是每单位电荷提供的能量。
5. Electric and Magnetic Fields | 电场与磁场
For a uniform electric field between parallel plates, field strength E = V / d. Force on a charge q is F = qE. Coulomb’s law gives the force between two point charges: F = kQ₁Q₂ / r², where k = 1/(4πε₀).
平行板间的匀强电场场强 E = V / d。作用在电荷 q 上的力为 F = qE。库仑定律给出两点电荷之间的力：F = kQ₁Q₂ / r²，其中 k = 1/(4πε₀)。
A magnetic field exerts a force on a moving charge: F = BQv sinθ, and on a current‑carrying conductor: F = BIL sinθ. Fleming’s left‑hand rule determines the direction.
磁场对运动电荷施加力：F = BQv sinθ，对载流导体：F = BIL sinθ。弗莱明左手定则确定方向。
Magnetic flux density B is measured in teslas (T). Magnetic flux Φ = BA cosθ, where θ is the angle between the field lines and the normal to the area. Faraday’s law: induced e.m.f. = −N ΔΦ/Δt. Lenz’s law gives the direction of the induced current.
磁通密度 B 以特斯拉 (T) 为单位。磁通量 Φ = BA cosθ，其中 θ 是磁力线与面积法线之间的夹角。法拉第定律：感应电动势 = −N ΔΦ/Δt。楞次定律给出感应电流的方向。
6. Thermal Physics | 热物理
The ideal gas equation links pressure p, volume V, amount n and temperature T:
pV = nRT
where R = 8.31 J mol⁻¹ K⁻¹. An alternative form is pV = NkT, where N is the number of molecules and k is the Boltzmann constant.
理想气体状态方程联系压强 p、体积 V、物质的量 n 和温度 T：
pV = nRT
其中 R = 8.31 J mol⁻¹ K⁻¹。另一种形式为 pV = NkT，其中 N 是分子数，k 是玻尔兹曼常数。
The mean kinetic energy of a gas molecule is proportional to absolute temperature: ½ m (c_rms)² = (3/2) kT. The root mean square speed c_rms = √(3RT/M).
气体分子的平均动能与绝对温度成正比：½ m (c_rms)² = (3/2) kT。均方根速率 c_rms = √(3RT/M)。
Specific heat capacity c relates energy to temperature change: Q = mcΔθ. Specific latent heat L relates energy to change of state: Q = mL. First law of thermodynamics: ΔU = Q − W (change in internal energy equals heat added minus work done by the system).
比热容 c 联系能量与温度变化：Q = mcΔθ。比潜热 L 联系能量与物态变化：Q = mL。热力学第一定律：ΔU = Q − W（内能增量等于吸收的热量减去系统对外做的功）。
7. Waves and Quantum Physics | 波与量子物理
Wave speed v = fλ. For electromagnetic waves in vacuum, v = c = 3.00 × 10⁸ m s⁻¹. Refractive index n = c / v, and Snell’s law: n₁ sinθ₁ = n₂ sinθ₂. Critical angle θ_c satisfies sinθ_c = n₂ / n₁ (n₂ < n₁).
波速 v = fλ。对于真空中的电磁波，v = c = 3.00 × 10⁸ m s⁻¹。折射率 n = c / v，斯涅耳定律：n₁ sinθ₁ = n₂ sinθ₂。临界角 θ_c 满足 sinθ_c = n₂ / n₁ (n₂ < n₁)。
Photon energy E = hf = hc/λ, where h is the Planck constant. The photoelectric effect is described by hf = Φ + KE_max, where Φ is the work function. Threshold frequency f₀ = Φ/h.
光子能量 E = hf = hc/λ，其中 h 是普朗克常数。光电效应由 hf = Φ + KE_max 描述，其中 Φ 是逸出功。截止频率 f₀ = Φ/h。
De Broglie wavelength λ = h/p, where p is momentum. This wave–particle duality applies to all particles. Electron diffraction confirms the wave nature of electrons.
德布罗意波长 λ = h/p，其中 p 是动量。这种波粒二象性适用于所有粒子。电子衍射证实了电子的波动性。
8. Chemical Equilibria and Kinetics | 化学平衡与动力学
For a reversible reaction aA + bB ⇌ cC + dD, the equilibrium constant Kc = [C]^c[D]^d / [A]^a[B]^b, where concentrations are in mol dm⁻³ at equilibrium. Kc is temperature dependent only.
对于可逆反应 aA + bB ⇌ cC + dD，平衡常数 Kc = [C]^c[D]^d / [A]^a[B]^b，其中浓度为平衡时的 mol dm⁻³。Kc 仅取决于温度。
The rate of a reaction can be expressed by a rate equation: rate = k[A]^m[B]^n. The orders m and n are determined experimentally, not from the stoichiometric coefficients. The overall order is m+n.
反应速率可用速率方程表示：rate = k[A]^m[B]^n。级数 m 和 n 由实验确定，并非来自化学计量系数。总级数为 m+n。
Integrated rate laws: for a first‑order reaction, ln[A] = ln[A]₀ − kt, and the half‑life t₁/₂ = ln 2/k. For second‑order, 1/[A] = 1/[A]₀ + kt.
积分速率定律：对于一级反应，ln[A] = ln[A]₀ − kt，半衰期 t₁/₂ = ln 2/k。对于二级反应，1/[A] = 1/[A]₀ + kt。
The Arrhenius equation links rate constant k to temperature: k = A e^(−Eₐ/RT) or ln k = ln A − Eₐ/(RT). A plot of ln k against 1/T gives a straight line of slope −Eₐ/R.
阿伦尼乌斯方程将速率常数 k 与温度关联：k = A e^(−Eₐ/RT) 或 ln k = ln A − Eₐ/(RT)。ln k 对 1/T 作图得到一条斜率为 −Eₐ/R 的直线。
9. Acids, Bases and Thermodynamics | 酸、碱与热力学
For a weak acid HA, the acid dissociation constant Kₐ = [H⁺][A⁻] / [HA]. pKₐ = −log₁₀Kₐ. The pH of a weak acid is calculated using [H⁺] = √(Kₐ[HA]). The ionic product of water Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ mol² dm⁻⁶ at 298 K, giving pH + pOH = 14.
对于弱酸 HA，酸解离常数 Kₐ = [H⁺][A⁻] / [HA]。pKₐ = −log₁₀Kₐ。弱酸的 pH 使用 [H⁺] = √(Kₐ[HA]) 计算。水的离子积 Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ mol² dm⁻⁶ (298 K)，因此 pH + pOH = 14。
Buffer solutions resist changes in pH. For an acidic buffer consisting of a weak acid and its salt, the Henderson–Hasselbalch equation is pH = pKₐ + log([A⁻]/[HA]).
缓冲溶液能抵抗 pH 的变化。对于由弱酸及其盐组成的酸性缓冲溶液，亨德森–哈塞尔巴尔赫方程为 pH = pKₐ + log([A⁻]/[HA])。
Gibbs free energy change ΔG determines reaction spontaneity: ΔG = ΔH − TΔS. For a reaction to be feasible, ΔG < 0. The relationship with equilibrium constant is ΔG° = −RT ln K.
吉布斯自由能变 ΔG 决定反应的自发性：ΔG = ΔH − TΔS。反应可行的条件是 ΔG < 0。与平衡常数的关系为 ΔG° = −RT ln K。
Enthalpy change ΔH can be determined experimentally via q = mcΔT and then scaled to moles. Hess’s law allows calculation of ΔH for reactions that are difficult to measure directly.
焓变 ΔH 可通过实验 q = mcΔT 测定并换算为每摩尔量。赫斯定律允许计算难以直接测量的反应的 ΔH。
10. Biology: Genetics and Population Growth | 生物：遗传与种群增长
The Hardy–Weinberg principle predicts allele frequencies in a non‑evolving population: p² + 2pq + q² = 1, where p and q are the frequencies of the dominant and recessive alleles respectively. It assumes no mutation, random mating, no gene flow, infinite population size, and no selection.
哈迪–温伯格定律预测非进化种群中的等位基因频率：p² + 2pq + q² = 1，其中 p 和 q 分别是显性和隐性等位基因的频率。它假设没有突变、随机交配、没有基因流动、无限大的种群规模且没有选择。
Population growth can be modelled by the exponential growth equation dN/dt = rN, where N is the population size and r is the per capita growth rate. When resources are limited, the logistic growth model applies: dN/dt = rN(1 − N/K), where K is the carrying capacity.
种群增长可用指数增长方程 dN/dt = rN 建模，其中 N 为种群大小，r 为人均增长率。当资源有限时，适用逻辑斯蒂增长模型：dN/dt = rN(1 − N/K)，其中 K 为环境容纳量。
In ecological sampling, Simpson’s Index of Diversity 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. This measures biodiversity within a habitat.
在生态取样中，辛普森多样性指数 D = 1 − Σ(n/N)²，其中 n 是某一物种的个体数，N 是所有物种的总个体数。该指数衡量栖息地内的生物多样性。
In microbial growth, the number of cells after n generations is N = N₀ × 2ⁿ. The generation time can be determined from a logarithmic growth curve and linked to the rate of binary fission.
在微生物生长中，经过 n 代后的细胞数为 N = N₀ × 2ⁿ。代时可从对数生长曲线确定，并与二分裂速率相关联。
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