📚 Year 13 CIE Science: Formula & Theorem Quick Reference Handbook | Year 13 CIE 科学：公式定理速查手册
This handbook compiles the essential formulas and theorems that every Year 13 CIE science student needs at their fingertips. It covers physics and chemistry topics likely to appear in A2 examinations, presented in paired English–Chinese explanations for quick, dual‑language revision.
本手册汇编了每位 Year 13 CIE 科学考生必须烂熟于心的核心公式与定理，涵盖 A2 阶段物理和化学中的高频考点。每个要点都使用英中双语对照解释，帮助你快速、准确地复习。
1. Kinematics & Mechanics Formulas | 运动学与力学公式
Uniformly accelerated motion is described by the SUVAT equations. The first relates final velocity v, initial velocity u, acceleration a, and time t.
匀加速运动由 SUVAT 方程描述。第一个方程联系末速度 v、初速度 u、加速度 a 和时间 t。
v = u + at
Displacement s is given by the product of average velocity and time.
位移 s 等于平均速度与时间的乘积。
s = ut + ½at²
An object in circular motion of radius r experiences a centripetal acceleration a = v²/r = ω²r.
做半径为 r 的圆周运动的物体具有向心加速度 a = v²/r = ω²r。
a = v²/r = ω²r
The centripetal force is F = mv²/r = mω²r.
向心力为 F = mv²/r = mω²r。
F = mv²/r = mω²r
Newton’s law of gravitation states that the force between two masses m₁ and m₂ separated by a distance r is F = Gm₁m₂/r².
牛顿引力定律指出，两个质量 m₁ 和 m₂ 相距 r 时的引力为 F = Gm₁m₂/r²。
F = Gm₁m₂/r²
Gravitational field strength at a distance r from a point mass M is g = GM/r².
距离点质量 M 为 r 处的重力场强度为 g = GM/r²。
g = GM/r²
2. Waves and Oscillations | 波与振动
In simple harmonic motion (SHM), the acceleration is proportional to the displacement from equilibrium and opposite in direction: a = -ω²x.
在简谐运动（SHM）中，加速度与离开平衡位置的位移成正比且方向相反：a = -ω²x。
a = -ω²x
The velocity at a displacement x is v = ±ω√(A² – x²), where A is the amplitude.
位移为 x 时的速度为 v = ±ω√(A² – x²)，其中 A 为振幅。
v = ±ω√(A² – x²)
The period of a simple pendulum is T = 2π√(l/g), and for a mass‑spring system T = 2π√(m/k).
单摆的周期为 T = 2π√(l/g)，弹簧振子的周期为 T = 2π√(m/k)。
T = 2π√(l/g) T = 2π√(m/k)
The wave speed is given by v = fλ, and the intensity of a wave is proportional to the square of its amplitude, I ∝ A².
波速为 v = fλ，波的强度与其振幅的平方成正比，I ∝ A²。
v = fλ I ∝ A²
The Doppler effect for sound when the source moves at speed vₛ relative to a stationary observer is f’ = f c/(c – vₛ), where c is the speed of sound.
声源相对于静止观察者以速度 vₛ 运动时的多普勒效应为 f’ = f c/(c – vₛ)，c 为声速。
f’ = f c/(c ∓ vₛ)
3. Electric Circuits and Fields | 电路与电场
For a uniform electric field between two parallel plates separated by distance d, the field strength is E = V/d, and the force on a charge q is F = qE.
对于间距为 d 的两平行板间的匀强电场，场强为 E = V/d，电荷 q 受力 F = qE。
E = V/d F = qE
The capacitance C of a parallel‑plate capacitor is C = Q/V, and the energy stored is U = ½QV = ½CV² = ½Q²/C.
平行板电容器的电容 C = Q/V，储存的能量为 U = ½QV = ½CV² = ½Q²/C。
C = Q/V U = ½CV²
The time constant for an RC circuit is τ = RC. The charge on a capacitor during charging is Q = Q₀(1 – e⁻ᵗ/ᴿᶜ).
RC 电路的时间常数为 τ = RC。充电时电容上的电荷为 Q = Q₀(1 – e⁻ᵗ/ᴿᶜ)。
τ = RC Q = Q₀(1 – e⁻ᵗ/ᴿᶜ)
The force on a current‑carrying conductor of length L in a magnetic field B is F = BIL sinθ; for a moving charge it becomes F = Bqv sinθ.
在磁场 B 中，长度为 L 的载流导体受力为 F = BIL sinθ；对运动电荷则为 F = Bqv sinθ。
F = BIL sinθ F = Bqv sinθ
Faraday’s law of electromagnetic induction states that the induced emf is ε = -dΦ/dt, where Φ is magnetic flux.
法拉第电磁感应定律指出，感应电动势为 ε = -dΦ/dt，其中 Φ 是磁通量。
ε = -dΦ/dt
4. Thermal Physics and Gases | 热物理与气体
The ideal gas equation relates pressure p, volume V, amount n, and temperature T: pV = nRT, where R = 8.31 J mol⁻¹ K⁻¹.
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，R = 8.31 J mol⁻¹ K⁻¹。
pV = nRT
The kinetic theory gives the average translational kinetic energy of a gas molecule as ½m
气体动理论给出气体分子的平均平动动能为 ½m
½m
The first law of thermodynamics is ΔU = Q + W, where ΔU is the change in internal energy, Q the heat added to the system, and W the work done on the system.
热力学第一定律为 ΔU = Q + W，ΔU 是内能的变化，Q 是系统吸收的热量，W 是对系统做的功。
ΔU = Q + W
For an ideal gas undergoing an adiabatic change, pV^γ = constant, where γ = Cₚ/Cᵥ.
理想气体经历绝热变化时，pV^γ = 常数，其中 γ = Cₚ/Cᵥ。
pV^γ = constant
5. Nuclear and Particle Physics | 核与粒子物理
The radioactive decay law is N = N₀e⁻λᵗ, where N₀ is the initial number of nuclei, λ the decay constant, and t the time.
放射性衰变定律为 N = N₀e⁻λᵗ，其中 N₀ 是初始原子核数，λ 是衰变常数，t 是时间。
N = N₀e⁻λᵗ
Activity A is defined as A = λN, and the half‑life is T½ = ln2/λ.
活度 A 定义为 A = λN，半衰期为 T½ = ln2/λ。
A = λN T½ = ln2/λ
Einstein’s mass–energy equivalence is E = mc². The energy released in a nuclear reaction is ΔE = Δm c², where Δm is the mass defect.
爱因斯坦质能等价式为 E = mc²。核反应中释放的能量为 ΔE = Δm c²，Δm 为质量亏损。
E = mc² ΔE = Δm c²
The binding energy of a nucleus is the energy required to separate it into its constituent nucleons.
原子核的结合能是将其分离成组成核子所需的能量。
6. Quantum and Atomic Physics | 量子与原子物理
The energy of a photon is E = hf, where h is Planck’s constant.
光子的能量为 E = hf，h 为普朗克常数。
E = hf
Einstein’s photoelectric equation relates photon energy to the work function Φ and maximum kinetic energy of photoelectrons: hf = Φ + Eₖ_max.
爱因斯坦光电方程联系光子能量与功函数 Φ 以及光电子的最大动能：hf = Φ + Eₖ_max。
hf = Φ + ½mv²_max
The de Broglie wavelength of a particle with momentum p is λ = h/p.
动量为 p 的粒子的德布罗意波长为 λ = h/p。
λ = h/p
When an electron transitions between two atomic energy levels, ΔE = hf = hc/λ.
电子在两个原子能级之间跃迁时，ΔE = hf = hc/λ。
ΔE = hf = hc/λ
The minimum wavelength of X‑rays produced by electrons accelerated through a voltage V is λ_min = hc / (eV).
电子被电压 V 加速后产生的 X 射线的最短波长为 λ_min = hc / (eV)。
λ_min = hc / (eV)
7. Thermochemistry & Thermodynamics | 热化学与热力学
The standard enthalpy change of a reaction, ΔH°, can be calculated from standard enthalpies of formation: ΔH° = Σ ΔH_f°(products) – Σ ΔH_f°(reactants).
反应的标准焓变 ΔH° 可由标准生成焓计算：ΔH° = Σ ΔH_f°(产物) – Σ ΔH_f°(反应物)。
ΔH° = Σ ΔH_f°(products) – Σ ΔH_f°(reactants)
The Gibbs free energy change is ΔG = ΔH – TΔS. A reaction is spontaneous when ΔG < 0.
吉布斯自由能变为 ΔG = ΔH – TΔS。当 ΔG < 0 时，反应自发进行。
ΔG = ΔH – TΔS
The standard Gibbs free energy change is related to the equilibrium constant K by ΔG° = -RT ln K.
标准吉布斯自由能变与平衡常数 K 的关系为 ΔG° = -RT ln K。
ΔG° = -RT ln K
The total entropy change of the universe is ΔS_total = ΔS_system + ΔS_surroundings, where ΔS_surroundings = -ΔH/T.
宇宙的总熵变为 ΔS_total = ΔS_system + ΔS_surroundings，其中 ΔS_surroundings = -ΔH/T。
ΔS_total = ΔS_system – ΔH/T
8. Chemical Kinetics | 化学动力学
For a reaction aA + bB → products, the rate law can be expressed as r = k[A]ᵐ[B]ⁿ, where m and n are the orders with respect to A and B.
对于反应 aA + bB → 产物，速率方程可表示为 r = k[A]ᵐ[B]ⁿ，m 和 n 分别为对 A 和 B 的级数。
r = k[A]ᵐ[B]ⁿ
The Arrhenius equation relates the rate constant k to the absolute temperature T: k = A e^(-Eₐ/RT), or in logarithmic form ln k = -Eₐ/(RT) + ln A.
阿伦尼乌斯方程将速率常数 k 与绝对温度 T 联系起来：k = A e^(-Eₐ/RT)，对数形式为 ln k = -Eₐ/(RT) + ln A。
k = A e^(-Eₐ/RT) ln k = -Eₐ/(RT) + ln A
For a first‑order reaction, the half‑life is independent of concentration: t½ = ln2/k.
一级反应的半衰期与浓度无关：t½ = ln2/k。
t½ = ln2 / k
The integrated rate law for a first‑order reaction is ln[A] = ln[A]₀ – kt.
一级反应的积分速率方程为 ln[A] = ln[A]₀ – kt。
ln[A] = ln[A]₀ – kt
9. Chemical Equilibrium | 化学平衡
The equilibrium constant Kc for a homogeneous reaction aA + bB ⇌ cC + dD is given by Kc = [C]ᶜ[D]ᵈ / ([A]ᵃ[B]ᵇ).
均相反应 aA + bB ⇌ cC + dD 的平衡常数 Kc 为 Kc = [C]ᶜ[D]ᵈ / ([A]ᵃ[B]ᵇ)。
Kc = [C]ᶜ[D]ᵈ / ([A]ᵃ[B]ᵇ)
For gaseous reactions, Kp = (p
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