📚 IB Science: Essential Formula Handbook | IB 科学：核心公式汇总手册
Mastering the essential formulas is the key to success in IB sciences. This handbook compiles the most important equations from IB Physics and IB Chemistry - , covering mechanics, thermal physics, waves, electricity, atomic physics, stoichiometry, energetics, equilibrium and data analysis - with a short note on when each one applies. Knowing not only the formula but also its conditions and units is what turns a memorised equation into a reliable exam tool for your internal assessment and final papers.
掌握核心公式是 IB 科学成功的关键。本手册汇编了 IB 物理与 IB 化学中最重要的方程式——涵盖力学、热物理、波动、电学、原子物理、化学计量、能量学、平衡与数据分析——并说明每个公式的适用条件。不仅记住公式本身，还理解它的适用条件与单位，才能把死记硬背的算式变成内部评估与考试中可靠的得分工具。
1. Mechanics - Kinematics | 力学——运动学
Kinematics describes motion without asking what causes it. The four suvat equations hold only when acceleration is constant, and they link displacement s, initial velocity u, final velocity v, acceleration a and time t.
运动学描述物体的运动，而不追问运动的原因。四个匀变速直线运动公式仅在加速度恒定时成立，它们把位移 s、初速度 u、末速度 v、加速度 a 与时间 t 联系起来。
v = u + at
The final velocity v equals the initial velocity u plus acceleration a multiplied by time t; valid for constant acceleration.
末速度 v 等于初速度 u 加上加速度 a 乘以时间 t，适用于匀加速运动。
s = ut + ½at²
Displacement s under uniform acceleration is the sum of ut and half the product of acceleration and the square of time.
匀加速下位移 s 等于 ut 加上加速度与时间平方乘积的一半。
v² = u² + 2as
This relates final velocity squared to initial velocity squared, acceleration and displacement, without involving time directly.
该式将末速度平方与初速度平方、加速度和位移关联起来，不直接涉及时间。
s = ½(u + v)t
Displacement equals average velocity multiplied by time; useful when acceleration is constant but not explicitly given.
位移等于平均速度乘以时间；当加速度恒定但未直接给出时非常实用。
ives displacement; on a displacement-time graph the gradient gives velocity (v = Δs ÷ Δt).
在速度-时间图像中，斜率给出加速度（a = Δv ÷ Δt），图线下的面积给出位移；在位移-时间图像中，斜率给出速度（v = Δs ÷ Δt）。
2. Forces, Energy and Momentum | 力、能量与动量
Newton's laws connect force to the rate of change of momentum, while work and energy give an alternative, often quicker route to the same answer.
牛顿定律把力与动量的变化率联系起来，而功与能量给出另一条常常更快的解题路径。
Fnet = ma    W = mg
Resultant force equals mass times acceleration; near the Earth's surface gravitational field strength g is taken as 9.81 N kg⁻¹.
合力等于质量乘以加速度；在地表附近，重力场强度 g 取 9.81 N kg⁻¹。
W = Fs cosθ   Ek = ½mv²   Ep = mgh
Work done is the component of force along the displacement; kinetic energy depends on speed squared and gravitational potential energy on height.
功等于力沿位移方向的分量与位移的乘积；动能取决于速度的平方，重力势能取决于高度。
p = mv    FΔt = Δp
Momentum is mass times velocity, and impulse equals the change in momentum - the basis of crash-safety and collision questions.
动量等于质量乘以速度，冲量等于动量的变化量——这是碰撞与安全设计题目的基础。
In an isolated system total momentum is conserved, and for elastic collisions kinetic energy is conserved as well; in inelastic collisions it is not.
在孤立系统中总动量守恒；弹性碰撞中动能同样守恒，非弹性碰撞中动能不守恒。
F = Gm₁m₂ ÷ r²    g = GM ÷ r²
Newton's law of gravitation gives the attractive force between two point masses, and the same relation yields the field strength of a planet.
牛顿万有引力定律给出两个质点间的引力，同一关系也可用来求某行星的场强。
3. Thermal Physics | 热物理
Thermal questions usually involve either a temperature change, a phase change, or an ideal gas - and sometimes all three in one structured problem.
热学题目通常涉及温度变化、物态变化或理想气体，有时一道结构化大题三者全包。
Q = mcΔT    Q = mL
The first equation gives the energy needed to change temperature, the second the energy needed to change state at constant temperature.
第一式给出改变温度所需的能量，第二式给出在恒定温度下改变物态所需的能量。
pV = nRT
ume, amount and absolute temperature; T must always be in kelvin and R = 8.31 J mol⁻¹ K⁻¹.
理想气体方程把压强、体积、物质的量与绝对温度联系起来；T 必须用开尔文，R = 8.31 J mol⁻¹ K⁻¹。
ΔU = Q + W
In the IB convention, W is the work done on the gas, so compression is positive and expansion negative; internal energy of an ideal gas depends only on temperature.
按 IB 约定，W 为外界对气体做的功，压缩为正、膨胀为负；理想气体的内能只与温度有关。
Efficiency = useful energy output ÷ total energy input × 100% body equals the energy gained by the cold body, ignoring heat losses to the surroundings.
效率 = 有用能量输出 ÷ 总能量输入 × 100%；在量热实验中，忽略向环境散热时，高温物体放出的能量等于低温物体吸收的能量。
4. Waves and Oscillations | 波动与振动
Waves transfer energy without transferring matter, and almost every wave question begins with the wave equation.
波传递能量而不传递物质，几乎所有波动题都从波速公式开始。
v = fλ    T = 1 ÷ f
Wave speed equals frequency times wavelength, and the period is the reciprocal of frequency.
波速等于频率乘以波长，周期是频率的倒数。
n₁ sinθ₁ = n₂ sinθ₂
Snell' eflection occurs when the angle of incidence exceeds the critical angle, sinθ_c = n₂ ÷ n₁.
斯涅耳定律描述界面处的折射；当入射角超过临界角时发生全内反射，sinθ_c = n₂ ÷ n₁。
v = √T ÷ μ   I = P ÷ A   I ∝ 1 ÷ r²
The speed of a wave on a stretched string depends on tension and mass per unit length; intensity is power per unit area and falls off with the square of distance for a point source.
张紧弦上波的传播速度取决于张力与线密度；强度是单位面积上的功率，点源情况下随距离平方衰减。
For the Doppler effect, f' = f(v ± vo) ÷ (v ∓ vs source, and the upper sign in the denominator when the source moves towards the observer.
多普勒效应中，f' = f(v ± vo) ÷ (v ∓ vs)：观察者靠近声源时分子取上面的符号，声源靠近观察者时分母取上面的符号。
5. Electricity and Circuits | 电学与电路
Circuit questions reward a systematic approach: simplify series and parallel sections, then apply Ohm's law and Kirchhoff's rules.
电路题需要系统的方法：先化简串联与并联部分，再应用欧姆定律与基尔霍夫定律。
I = q ÷ t    V = IR
Current is the rate of flow of charge, and Ohm's law applies to a component whose resistance is constant at fixed temperature.
电流是电荷的流动速率；在温度不变时，欧姆定律适用于电阻恒定的元件。
P = VI = I²R = V² ÷ R
Electrical power can be written in three equivalent forms; choose the one that matches the quantities you already know.
电功率有三种等价写法，选择与已知量相匹配的那一种最省时间。
Rseries = R₁ + R₂    1 ÷ R_parallel = 1 ÷ R₁ + 1 ÷ R₂
Resistances add in series; in parallel the reciprocals add, so the combined resistance is always smaller than the smallest single resistor.
串联电阻相加；并联时倒数相加，因此并联总电阻总小于其中最小的单个电阻。
ε = I(R + r)    I = nAvq
EMF equals the terminal p.d. plus the lost volts across internal resistance r; the drift-velocity equation links current to charge carrier density and cross-sectional area.
电动势等于路端电压加上内阻 r 上的损耗电压；漂移速度公式把电流与载流子数密度和横截面积联系起来。
6. Atomic, Nuclear and Quantum Physics | 原子、核与量子物理
This topic mixes quantum ideas with nuclear decay, and the constants from the data booklet must be used carefully.
本主题把量子概念与核衰变结合在一起，使用数据手册中的常量时必须格外小心。
E = hf = hc ÷ λ
proportional to frequency and inversely proportional to wavelength; h = 6.63 × 10⁻³⁴ J s.
光子能量与频率成正比、与波长成反比；h = 6.63 × 10⁻³⁴ J s。
hf = Φ + Ek(max)
The photoelectric equation states that photon energy equals the work function plus the maximum kinetic energy of the emitted electron.
光电效应方程表明：光子能量等于逸出功加上逸出电子的最大动能。
λ = h ÷ p    E = mc²
De Broglie's relation gives the wavelength of any particle with momentum p; mass-energy equivalence explains the energy released in nuclear reactions.
德布罗意关系给出动量为 p 的任何粒子的波长；质能等价解释了核反应中释放的能量。
N = N₀(½)t ÷ t₁/₂   A = λN
is proportional to the number of undecayed nuclei, and the decay constant λ = ln2 ÷ t₁/₂.
放射性衰变是指数式的；活度与未衰变核数成正比，衰变常数 λ = ln2 ÷ t₁/₂。
nverting between electronvolts and joules is one of the most common sources of lost marks.
记住 1 eV = 1.60 × 10⁻¹⁹ J；电子伏与焦耳之间的换算是失分最常见的环节之一。
7. Stoichiometry and the Mole | 化学计量与物质的量
Almost every quantitative chemistry question passes through the mole, so these three relationships should be automatic.
几乎所有定量化学题都要经过"物质的量"这一环节，因此下面三个关系式应当做到条件反射式熟练。
n = m ÷ M
The amount in moles equals mass in grams divided by molar mass in g mol⁻¹.
物质的量（摩尔）等于质量（克）除以摩尔质量（g mol⁻¹）。
n = cV
For solutions, amount equals concentration in mol dm⁻³ multiplied by volume in dm³ - convert cm³ to dm³ by dividing by 1000 first.
对于溶液，物质的量等于浓度（mol dm⁻³）乘以体积（dm³）——先把 cm³ 除以 1000 换算为 dm³。
n = V ÷ Vm   N = nN_A
For gases at standard conditions Vm = 22.7 dm³ mol⁻¹; one mole contains 6.02 × 10²³ particles.
标准状况下气体的摩尔体积 Vm = 22.7 dm³ mol⁻¹；1 摩尔含 6.02 × 10²³ 个粒子。
Percentage yield = actual yield ÷ theoretical yield × 100%, and atom economy = mass of desired product ÷ total mass of all products × 100%; the two are often confused in Paper 2.
产率 = 实际产量 ÷ 理论产量 × 100%；原子经济性 = 目标产物的质量 ÷ 所有产物总质量 × 100%。Paper 2 中这两者常被混淆。
8. Energetics and Thermochemistry | 能量学与热化学
Enthalpy questions combine a calorimetry calculation with Hess's law, so keep the sign conventions clear throughout.
焓变题目通常把量热计算与盖斯定律结合，全程务必保持符号约定清晰。
q = mcΔT    ΔH = -q ÷ n
Heat energy gained or lost equals mcΔT, and the molar enthalpy change is that heat divided by the amount of substance, with a negative sign for exothermic solution reactions.
吸收或放出的热量等于 mcΔT，摩尔焓变是该热量除以物质的量；放热的溶液反应取负号。
ΔH°_r = ΣΔH°_f(products) - ΣΔH°_f(reactants)
Standard enthalpy of reaction is found from standard enthalpies of formation, multiplying each by its stoichiometric coefficient.
标准反应焓由标准生成焓求得，每一项都要乘以相应的化学计量系数。
ΔH = Σ bonds broken - Σ bonds formed
Using average bond enthalpies, breaking bonds is endothermic and forming bonds is exothermic; the result is an average value and ignores the specific environment of each bond.
使用平均键能时，断键吸热、成键放热；所得结果是平均值，忽略了每根键所处的具体环境。
Hess's law allows an unknown enthalpy change to be found by adding known steps, provided the equations are reversed or scaled consistently.
盖斯定律允许通过叠加已知步骤求出未知焓变，前提是方程式的反向与倍乘要保持一致。
9. Kinetics and Equilibrium | 动力学与平衡
Kinetics asks how fast a reaction goes; equilibrium asks how far it goes. Both are expressed with concentration terms.
动力学研究反应有多快；平衡研究反应进行得多彻底。两者都用浓度项来表达。
rate = Δ[concentration] ÷ Δt
