📚 Year 12 Cambridge Science: Formula & Theorem Quick Reference Handbook | 12年级剑桥科学：公式定理速查手册
This quick reference handbook brings together the essential formulas, theorems, constants and key relationships you will encounter across physics, chemistry and quantitative work in Year 12 Cambridge science. Whether you are revising for assessments, tackling problem sets or building a strong foundation for the full A Level, having these core tools at your fingertips saves time and deepens understanding. Each entry is presented with clear definitions, typical units and concise explanatory notes that link the mathematical expression to the scientific concept behind it.
这本速查手册汇集了12年级剑桥科学（涵盖物理、化学以及定量分析）中必须掌握的公式、定理、常数和关键关系。无论你是在备考、完成习题集，还是为完整的A Level学习打下坚实基础，将这些核心工具放在手边都能节省时间并加深理解。每一条目都配有清晰的定义、常用单位以及精炼的说明，把数学表达式和背后的科学概念紧密联系起来。
1. Kinematics Equations | 运动学公式
For uniformly accelerated motion along a straight line, the following four equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). Air resistance is assumed negligible.
对于匀加速直线运动，以下四个方程给出了位移(s)、初速度(u)、末速度(v)、加速度(a)和时间(t)之间的关系。假设空气阻力可忽略不计。
v = u + at
v = u + at
s = ut + ½at²
s = ut + ½at²
v² = u² + 2as
v² = u² + 2as
s = ½(u + v)t
s = ½(u + v)t
Tip: Always choose the equation that contains the unknown you need and all the quantities you already know. Use + and – consistently for direction.
提示：始终选择包含你要求的未知量，并且你已经知道所有其他量的方程。使用正负号时要保持方向的一致性。
2. Dynamics and Newton's Laws | 动力学与牛顿定律
Newton's second law gives the relationship between net force (F), mass (m) and acceleration (a): F = ma. The newton (N) is the unit of force, equivalent to kg·m·s⁻².
牛顿第二定律给出了合力(F)、质量(m)和加速度(a)之间的关系：F = ma。力的单位是牛顿(N)，相当于kg·m·s⁻²。
Momentum (p) is defined as p = mv. The impulse-momentum relationship states that FΔt = Δp = m(v – u), where Δt is the time during which the force acts.
动量(p)定义为 p = mv。冲量-动量关系表明 FΔt = Δp = m(v – u)，其中Δt是力作用的时间。
Weight is the gravitational force: W = mg ·s⁻². Frictional forces often oppose motion and can be modelled by coefficients such as μ.
重量是引力作用：W = mg。在地球上g ≈ 9.81 m·s⁻²。摩擦力通常阻碍运动，可以用摩擦系数μ等模型来描述。
3. Work, Energy and Power | 功、能量与功率
Work done (W) when a constant force (F) acts over a displacement (s) in the direction of the force: W = Fs cos θ. When θ = 0°, this simplifies to W = Fs.
恒力(F)沿其方向作用一段位移(s)所做的功(W)：W = Fs cos θ。当θ = 0°时简化为W = Fs。
Kinetic energy: Ek = ½mv². Gravitational potential energy near Earth's surface: Ep = mgh, where h is the vertical height.
动能：Ek = ½mv²。近地表的重力势能：Ep = mgh，其中h为竖直高度。
The principle of conservation of energy states that energy cannot be created or destroyed, only transferred. Power (P) is the rate of doing work: P = W/t = Fv (for constant force and velocity in the same direction).
能量守恒定律表明能量既不能被创造也不能被消灭，只能转移。功率(P)是做功的速率：P = W/t = Fv（适用于恒力且速度与力同向的情况）。
4. Waves and Optics | 波与光学
Wave speed (v) is related to frequency (f) and wavelength (λ): v = fλ. Period (T) is the reciprocal of frequency: T = 1/f.
波速(v)与频率(f)、波长(λ)的关系：v = fλ。周期(T)是频率的倒数：T = 1/f。
Snell's law of refraction: n₁ sin θ₁ = n₂ sin θ₂, where n is the refractive index. Critical angle (θc) for total internal reflection: sin θc = n₂/n₁ (n₁ > n₂).
斯涅耳折射定律：n₁ sin θ₁ = n₂ sin θ₂，其中n为折射率。全内反射的临界角(θc)：sin θc = n₂/n₁（n₁ > n₂）。
For a diffraction grating, maxima occur at d sin θ = mλ, where d is the slit spacing and m is the order (integer).
对于衍射光栅，极大值出现在 d sin θ = mλ，其中d为狭缝间距，m为级数（整数）。
5. Electricity and Circuits | 电学与电路
Ohm's law for a conductor at constant temperature: V = IR, where V is potential difference (voltage), I is current and R is resistance.
恒温下导体的欧姆定律：V = IR，其中V为电势差（电压），I为电流，R为电阻。
Power dissipated in a resistor: P = IV = I²R = V²/R. Electromotive force (emf, ε) is the energy supplied per unit charge: ε = E/Q.
电阻器耗散的功率：P = IV = I²R = V²/R。电动势(ε)是每单位电荷提供的能量：ε = E/Q。
Resistors in series: Rtotal = R₁ + R₂ + .... Resistors in parallel: 1/Rtotal = 1/R₁ + 1/R₂ + .... For two resistors in parallel, a useful shortcut is Rtotal = (R₁R₂)/(R₁+R₂).
电阻串联：R总 = R₁ + R₂ + ...。电阻并联：1/R总 = 1/R₁ + 1/R₂ + ...。两个电阻并联时，便捷公式为 R总 = (R₁R₂)/(R₁+R₂)。
6. Thermal Physics | 热物理
Temperature change and heat: Q = mcΔθ, where Q is heat energy, m is mass, c is specific heat capacity and Δθ is temperature change.
温度变化与热量：Q = mcΔθ，其中Q为热量，m为质量，c为比热容，Δθ为温度变化。
Latent heat for a change of state: Q = mL, where L is specific latent heat. No temperature change occurs during the phase transition.
状态变化时的潜热：Q = mL，其中L为比潜热。相变过程中温度保持不变。
Ideal gas equation: pV = nRT, where p is pressure, V is volume, n is amount (mol), R is the molar gas constant (8.31 J·mol⁻¹·K⁻¹) and T is thermodynamic temperature in kelvin.
理想气体方程：pV = nRT，其中p为压强，V为体积，n为物质的量(mol)，R为摩尔气体常数(8.31 J·mol⁻¹·K⁻¹)，T为热力学温度（开尔文）。
The kinetic theory relates the average kinetic energy of gas molecules to absolute temperature: Ek (average) = (3/2)kT, where k is the Boltzmann constant.
分子动理论将气体分子的平均动能与绝对温度联系起来：Ek(平均) = (3/2)kT，其中k为玻尔兹曼常数。
7. Atomic and Nuclear Physics | 原子与核物理
Photon energy: E = hf = hc/λ, where h is Planck's constant (6.63 × 10⁻³⁴ J·s), f is frequency and c is the speed of light.
光子能量：E = hf = hc/λ，其中h为普朗克常数(6.63 × 10⁻³⁴ J·s)，f为频率，c为光速。
Radioactive decay follows an exponential law: N = N₀ e⁻λt, where N is the number of undecayed nuclei at time t, N₀ is the initial number and λ is the decay constant. Half-life: t½ = ln2 / λ.
放射性衰变遵循指数规律：N = N₀ e⁻λt，其中N为t时刻未衰变的原子核数，N₀为初始数目，λ为衰变常数。半衰期：t½ = ln2 / λ。
The mass-energy equivalence principle: E = mc². This is central to understanding nuclear binding energy and energy released in fission and fusion.
质能方程：E = mc²。这是理解核结合能以及裂变与聚变中释放能量的核心。
8. Chemical Quantities and the Mole | 化学计量与摩尔
Amount of substance: n = m/M vogadro constant, 6.02 × 10²³ mol⁻¹, links the number of particles to the amount in moles.
物质的量：n = m/M，其中m为质量(g)，M为摩尔质量(g·mol⁻¹)。阿伏伽德罗常数6.02 × 10²³ mol⁻¹将粒子数与摩尔数联系起来。
Concentration of a solution: c = n/V, where V is the volume of solution (dm³). The unit is mol·dm⁻³. In titrations, at the equivalence point: c₁V₁ = c₂V₂ (for a 1:1 reaction).
溶液浓度：c = n/V，其中V为溶液体积(dm³)，单位为mol·dm⁻³。在滴定中，当量点时：c₁V₁ = c₂V₂（针对1:1反应）。
ses at room temperature and pressure (RTP), molar volume ≈ 24 dm³·mol⁻¹. The ideal gas law pV = nRT applies as in physics, with the same R.
在常温常压(RTP)下，气体摩尔体积约为24 dm³·mol⁻¹。理想气体定律 pV = nRT 同样适用，R值相同。
9. Energetics and Thermochemistry | 能量学与热化学
Enthalpy change (ΔH) from a calorimetry experiment: Q = –mcΔT, where Q is heat released or absorbed, m is total mass of the solution, c is specific heat capacity and ΔT is temperature change. Then ΔH = Q / n (with sign convention).
由量热实验计算焓变(ΔH)：Q = –mcΔT，其中Q为释放或吸收的热量，m为溶液的总质量，c为比热容，ΔT为温度变化。然后ΔH = Q / n（注意符号规定）。
Hess's law states that the total enthalpy change for a reaction is independent of the route taken. It allows the calculation of ΔH using enthalpy changes of formation (ΔHf°) or combustion (ΔHc°).
赫斯定律指出反应的总焓变与途径无关。可以利用生成焓(ΔHf°)或燃烧焓(ΔHc°)来计算ΔH。
Bond enthalpy calculations: ΔH ≈ Σ (bond energies broken) – Σ (bond energies formed). This provides an estimate of reaction enthalpies.
键能计算：ΔH ≈ Σ（断裂键的键能）– Σ（形成键的键能）。这提供了反应焓的估计值。
10. Equilibrium and Rates | 平衡与速率
For a reversible reaction aA + bB ⇌ cC + dD, the equilibrium constant in terms of concentration is Kc = [C]ᶜ[D]ᵈ / [A]ᵃ[B]ᵇ. Kc is temperature dependent.
对于可逆反应 aA + bB ⇌ cC + dD，以浓度表示的平衡常数为 Kc = [C]ᶜ[D]ᵈ / [A]ᵃ[B]ᵇ。Kc与温度有关。
Le Chatelier's principle: if a system at equilibrium is disturbed, the position of equilibrium shifts to counteract the change. This guides predictions about pressure, temperature and concentration effects.
勒夏特列原理：如果平衡系统受到干扰，平衡位置会向削弱该干扰的方向移动。这有助于预测压强、温度和浓度变化的影响。
Rate equations are determined experimentally: rate = k[A]ᵐ[B]ⁿ, where m and n are orders of reaction and k is the rate constant. The overall order is m+n. The half-life of a first-order reaction is constant: t½ = ln2 / k.
速率方程由实验确定：rate = k[A]ᵐ[B]ⁿ，其中m和n为反应级数，k为速率常数。总反应级数为m+n。一级反应的半衰期为常数：t½ = ln2 / k。
11. Important Constants, Units and Conversions | 重要常数、单位与换算
Frequently used constants: Planck' s constant R = 8.31 J·mol⁻¹·K⁻¹; gravitational field strength g = 9.81 N·kg⁻¹ (Earth); specific heat capacity of water c = 4.18 J·g⁻¹·K⁻¹.
常用常数：普朗克常数h = 6.63 × 10⁻³⁴ J·s；光速c = 3.00 × 10⁸ m·s⁻¹；元电荷e = 1.60 × 10⁻¹⁹ C；阿伏伽德罗常数NA = 6.02 × 10²³ mol⁻¹；摩尔气体常数R = 8.31 J·mol⁻¹·K⁻¹；引力场强度g = 9.81 N·kg⁻¹（地球）；水的比热容c = 4.18 J·g⁻¹·K⁻¹。
Unit conversions: 1 dm³ = 1 L = 1000 cm³; 1 atm = 1.01 × 10⁵ Pa; absolute zero = –273 °C = 0 K. When using the gas law, temperature must be in kelvin.
单位换算：1 dm³ = 1 L = 1000 cm³；1 atm = 1.01 × 10⁵ Pa；绝对零度 = –273 °C = 0 K。使用气体定律时温度必须用开尔文。
12. Key Theorems and Conservation Laws | 关键定理与守恒定律
Conservation of energy: energy is neither created nor destroyed; it transforms from one form to another. This underlies all calculations in mechanics, thermodynamics and electrical circuits.
能量守恒：能量既不能被创造也不能被消灭，只能从一种形式转化为另一种。这是力学、热力学和电路所有计算的基础。
Conservation of momentum: in a closed system with no external forces, total momentum before an event equals total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. This is essential for collision problems.
动量守恒：在无外力作用的封闭系统中，事件前后的总动量相等：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。这对碰撞问题至关重要。
The law of conservation of mass: in a chemical reaction, total mass of reactants equals total mass of products. Combined with the mole concept, it enables stoichiometric predictions.
质量守恒定律：在化学反应中，反应物的总质量等于生成物的总质量。结合摩尔概念，它可以进行化学计量预测。
The first law of thermodynamics: ΔU = Q – W, where ΔU is the change in internal energy of a system, Q is the heat added and W is the work done by the system. For many processes, this simplifies to analysis of heat transfer.
热力学第一定律：ΔU = Q – W，其中ΔU是系统内能的变化，Q是加入的热量，W是系统做的功。对许多过程，这简化为对热传递的分析。
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