IB & AQA Science Formula Compendium | IB与AQA科学公式汇编

📚 IB & AQA Science Formula Compendium | IB与AQA科学公式汇编

This comprehensive handbook collates essential formulae from Physics, Chemistry, and Biology covered in both the IB Diploma Programme and AQA A-level specifications. Each section presents key equations with brief explanations, designed for rapid revision. Keep this guide at your fingertips to reinforce conceptual understanding and solve problems efficiently.

本手册汇聚了IB文凭课程与AQA A-level科学（物理、化学、生物）的核心公式。每一节都列出关键等式并辅以简明解释，便于快速复习。将这份指南放在手边，可强化概念理解并高效解题。

1. Kinematics | 运动学

For uniform acceleration in one dimension, the following SUVAT equations apply. The symbols are s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time).

在一维匀加速运动中可使用下列SUVAT方程。符号含义：s（位移），u（初速度），v（末速度），a（加速度），t（时间）。

v = u + a t

Velocity after time t equals initial velocity plus acceleration multiplied by time.

经过时间t后的速度等于初速度加上加速度乘以时间。

s = u t + ½ a t²

Displacement is the product of initial velocity and time plus half the product of acceleration and the square of time.

位移等于初速度与时间之积加上加速度与时间平方之积的一半。

v² = u² + 2 a s

Relates final velocity, initial velocity, acceleration and displacement without using time.

将末速度、初速度、加速度和位移联系起来，不显含时间。

s = ½ (u + v) t

Displacement equals the average velocity multiplied by time, useful when acceleration is constant.

位移等于平均速度乘以时间，适用于加速度恒定的情况。

2. Forces & Newton’s Laws | 力与牛顿定律

Newton’s second law defines the relationship between net force, mass and acceleration. The direction of acceleration is the same as the net force.

牛顿第二定律定义了净力、质量与加速度之间的关系。加速度方向与净力方向一致。

F = m a

Net force equals mass times acceleration. For equilibrium, net force is zero.

净力等于质量乘以加速度。平衡时净力为零。

The force of friction is proportional to the normal reaction force, with the coefficient of friction μ.

摩擦力与正压力成正比，比例系数为摩擦因数μ。

F_friction ≤ μ R

Static friction maximum equals μ_s R; kinetic friction is μ_k R.

最大静摩擦力为μ_s R；动摩擦力为μ_k R。

Weight (gravitational force) near Earth’s surface:

地球表面附近的重力：

W = m g

g is the acceleration of free fall, approximately 9.81 m s⁻².

g为自由落体加速度，约为9.81 m s⁻²。

3. Work, Energy & Power | 功、能量与功率

Work done by a constant force is the product of the force component along the displacement and the displacement magnitude.

恒力做功等于力在位移方向上的分量与位移大小的乘积。

W = F s cos θ

Here θ is the angle between the force and displacement vectors.

其中θ为力与位移矢量之间的夹角。

Kinetic energy and gravitational potential energy:

动能与重力势能：

Eₖ = ½ m v²

Eₚ = m g h

Conservation of mechanical energy holds when only conservative forces (e.g., gravity) do work.

只有保守力（例如重力）做功时，机械能守恒。

Power is the rate of doing work or energy transfer:

功率是做功或能量转换的速率：

P = W / t = F v

Efficiency is the ratio of useful output energy to total input energy, often expressed as a percentage.

效率是有用输出能量与总输入能量之比，常用百分比表示。

4. Waves & Optics | 波与光学

The wave equation connects wave speed (v), frequency (f), and wavelength (λ).

波速方程将波速（v）、频率（f）和波长（λ）联系起来。

v = f λ

For refraction, Snell’s law relates the angles of incidence and refraction to the refractive indices of the two media.

对于折射，斯涅耳定律将入射角和折射角与两种介质的折射率联系起来。

n₁ sin θ₁ = n₂ sin θ₂

Critical angle for total internal reflection occurs when n₁ > n₂ and sin θ_c = n₂ / n₁.

全内反射的临界角发生在n₁ > n₂时，且sin θ_c = n₂ / n₁。

For a converging lens or mirror, the thin lens equation:

对于会聚透镜或反射镜，薄透镜方程：

1/f = 1/u + 1/v

Here f is focal length, u is object distance, v is image distance. The sign convention follows the system used (real is positive).

其中f为焦距，u为物距，v为像距。符号约定遵循使用体系（实为正）。

5. Electricity & Magnetism | 电磁学

Ohm’s law in its simplest form links potential difference, current and resistance.

欧姆定律的最简形式将电势差、电流和电阻联系起来。

V = I R

Power dissipated in a resistor can be expressed in alternative forms:

电阻器消耗的功率可用不同形式表达：

P = V I = I² R = V² / R

For resistors in series (R_total = R₁ + R₂ + …) and in parallel (1/R_total = 1/R₁ + 1/R₂ + …).

电阻串联时R_total = R₁ + R₂ + …；并联时1/R_total = 1/R₁ + 1/R₂ + …。

Magnetic force on a moving charge in a magnetic field:

运动电荷在磁场中所受的磁力：

F = q v B sin θ

For a current-carrying straight conductor of length L, the force is F = B I L sin θ.

对于长度为L的载流直导线，力为F = B I L sin θ。

6. Thermal Physics | 热物理

Internal energy is the sum of the random kinetic and potential energies of constituent particles. Key relations include specific heat capacity and latent heat.

内能是组成粒子的无规则动能与势能之和。关键关系包括比热容和潜热。

Q = m c ΔT

Q is thermal energy, m is mass, c is specific heat capacity, ΔT is temperature change.

Q为热量，m为质量，c为比热容，ΔT为温度变化。

Q = m L

L is specific latent heat (fusion or vaporisation) for phase changes at constant temperature.

L为比潜热（熔化或汽化），用于恒温相变。

For an ideal gas, the pressure p, volume V and absolute temperature T are related by the ideal gas equation:

理想气体的压强p、体积V和热力学温度T由理想气体方程关联：

p V = n R T

n is amount (mol), R = 8.31 J mol⁻¹ K⁻¹.

n为物质的量，R=8.31 J mol⁻¹ K⁻¹。

7. Atomic & Nuclear Physics | 原子与核物理

Photon energy E is proportional to frequency f, with Planck’s constant h.

光子能量E与频率f成正比，普朗克常数h。

E = h f

The photoelectric effect: maximum kinetic energy of emitted electrons is given by Einstein’s equation.

光电效应：逸出电子的最大动能由爱因斯坦方程给出。

Eₖ_max = h f – Φ

Φ is the work function of the metal.

Φ为金属的逸出功。

In nuclear reactions, mass-energy equivalence and activity are fundamental:

在核反应中，质能等价和活度是基本概念：

E = m c²

A = λ N

A is activity, λ is decay constant, N is number of radioactive nuclei. The half-life T₁/₂ = ln 2 / λ.

A为活度，λ为衰变常数，N为放射性原子核数。半衰期T₁/₂ = ln 2 / λ。

8. Mole Concept & Stoichiometry | 摩尔概念与化学计量

The mole links mass to number of particles. Avogadro’s constant N_A = 6.022 × 10²³ mol⁻¹.

摩尔将质量与粒子数联系起来。阿伏伽德罗常数 N_A = 6.022 × 10²³ mol⁻¹。

n = m / M

Amount n equals mass m divided by molar mass M.

物质的量n等于质量m除以摩尔质量M。

For solutions, concentration c in mol dm⁻³ is:

对于溶液，浓度c（mol dm⁻³）为：

c = n / V

V is volume in dm³. In gas calculations at RTP/STP, molar volume is used.

V为体积（dm³）。在常温常压/标准状况气体计算中，使用摩尔体积。

V_gas = n × V_m

At RTP (25 °C, 1 atm) V_m ≈ 24 dm³ mol⁻¹; at STP V_m = 22.7 dm³ mol⁻¹.

常温常压下V_m ≈ 24 dm³ mol⁻¹；标准状况下V_m = 22.7 dm³ mol⁻¹。

9. Thermodynamics & Hess’s Law | 热力学与赫斯定律

Hess’s law states that the enthalpy change for a reaction is independent of the route taken.

赫斯定律表明，反应焓变与反应途径无关。

ΔH_reaction = ΣΔH_f(products) – ΣΔH_f(reactants)

ΔH_f denotes standard enthalpy of formation. For combustion, ΔH_c can be used similarly.

ΔH_f表示标准生成焓。对于燃烧，可类似使用ΔH_c。

Bond enthalpy calculation:

键焓计算：

ΔH = Σ(bond energies broken) – Σ(bond energies formed)

Remember that bond breaking is endothermic (+) and bond making exothermic (-).

记住断键吸热（+），成键放热（-）。

Gibbs free energy change determines spontaneity:

吉布斯自由能变决定反应自发性：

ΔG = ΔH – T ΔS

A reaction is spontaneous when ΔG is negative. ΔS is entropy change.

当ΔG为负时反应自发。ΔS为熵变。

10. Equilibrium & Acid-Base | 化学平衡与酸碱

The equilibrium constant Kc for a homogeneous reaction aA + bB ⇌ cC + dD is:

均相反应aA + bB ⇌ cC + dD的平衡常数Kc为：

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

Concentrations are in mol dm⁻³. Kc is temperature dependent.

浓度单位为mol dm⁻³。Kc随温度变化。

Acid dissociation constant Ka for a weak acid HA ⇌ H⁺ + A⁻:

弱酸HA ⇌ H⁺ + A⁻的酸解离常数Ka：

Ka = [H⁺][A⁻] / [HA]

pKa = -log Ka. The Henderson–Hasselbalch equation relates pH and pKa for buffer solutions:

pKa = -log Ka。亨德森-哈塞尔巴尔赫方程关联缓冲溶液的pH与pKa：

pH = pKa + log ([A⁻] / [HA])

The ionic product of water Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ at 25 °C; pH + pOH = 14.

水的离子积Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴（25°C）；pH + pOH = 14。

11. Biological Processes & Genetics | 生物过程与遗传学

In biology, quantitative relationships are found in microscopy, genetics and population growth. Magnification:

生物学中的定量关系见于显微镜、遗传学和种群增长。放大率：

Magnification = Image size / Actual size

Ensure both measurements are in the same units. For genetic crosses, the Hardy–Weinberg principle describes allele frequencies in non-evolving populations:

确保测量值单位一致。对于遗传杂交，哈迪-温伯格定律描述非进化种群中的等位基因频率：

p² + 2pq + q² = 1

p and q are frequencies of dominant and recessive alleles. p + q = 1. This predicts genotype frequencies: homozygous dominant (p²), heterozygous (2pq), homozygous recessive (q²).

p和q分别为显性和隐性等位基因的频率。p + q = 1。这预测基因型频率：显性纯合(p²)、杂合(2pq)、隐性纯合(q²)。

Chi-squared test for goodness of fit:

卡方适合度检验：

χ² = Σ (O – E)² / E

O = observed frequency, E = expected frequency. Compare χ² with critical value.

O = 观测值，E = 期望值。将χ²与临界值比较。

12. Ecology & Populations | 生态与种群

Population growth can be modelled exponentially or logistically. Exponential growth rate:

种群增长可呈指数或逻辑斯谛模型。指数增长率：

dN/dt = r N

r is the intrinsic rate of increase. Logistic growth incorporates carrying capacity K:

r为内禀增长率。逻辑斯谛增长引入环境容纳量K：

dN/dt = r N (K – N) / K

Simpson’s diversity index D measures biodiversity:

辛普森多样性指数D衡量生物多样性：

D = 1 – Σ (n / N)²

n = number of individuals of a particular species, N = total number of organisms.

n = 某特定物种个体数，N = 总生物体数。

Lincoln index for estimating population size via mark-release-recapture:

林肯指数用于通过标记重捕法估算种群数量：

N = (M × C) / R

M = number initially marked, C = number caught in second sample, R = number of marked individuals in recapture.

M = 初次标记数，C = 第二次捕获数，R = 重捕中标记个体数。

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