📚 Year 12 CIE Science: Formula and Theorem Quick Reference Handbook | CIE 12年级科学：公式定理速查手册
This quick reference handbook brings together the essential formulae and theorems from the CIE AS Level sciences – Physics, Chemistry and Biology. Whether you are tackling mechanics, calculating equilibrium constants or working out magnifications, you will find the core relationships clearly stated and explained. Use this guide to consolidate your knowledge and build confidence for your examinations.
这本速查手册汇集了 CIE AS 阶段科学（物理、化学和生物）中的基本公式与定理。无论你是在解决力学问题、计算平衡常数还是练习放大倍率，都能在这里找到清晰的核心关系式与说明。利用这份指南巩固知识，为考试建立信心。
1. Kinematics Equations | 运动学方程
For objects moving with constant acceleration along a straight line, four equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). They are often called the suvat equations.
对于匀加速直线运动的物体，有四个方程将位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 与时间 (t) 联系起来，常被称为 suvat 方程。
v = u + a t
This gives the velocity after time t.
该式给出时间 t 后的速度。
s = u t + ½ a t²
This is the displacement when initial velocity and constant acceleration are known.
当已知初速度与恒定加速度时，用此式计算位移。
v² = u² + 2 a s
Useful when time is not given; it links velocity, acceleration and displacement.
当不涉及时间时非常有用，该式连接了速度、加速度与位移。
2. Forces and Newton’s Laws | 力与牛顿定律
Newton’s three laws of motion form the foundation of classical mechanics. The second law gives the quantitative relationship between force, mass and acceleration.
牛顿运动三定律是经典力学的基础。第二定律给出了力、质量与加速度之间的定量关系。
F = m a
The net force acting on an object equals its mass times its acceleration. For equilibrium the net force must be zero.
作用在物体上的合力等于其质量乘以加速度。处于平衡状态时，合力必须为零。
The weight of an object is the force due to gravity: W = m g e. In problems on inclined planes, resolve forces parallel and perpendicular to the slope.
物体的重量是重力引起的：W = m g，在地球表面 g 取 9.81 m s⁻²。在斜面上的问题中，需将力沿斜面平行和垂直方向分解。
The principle of conservation of momentum states that for a system with no external forces, total momentum before an event equals total momentum after: Σ pbefore = Σ pafter, where p = m v.
动量守恒定律指出，对于不受外力的系统，事件前的总动量等于事件后的总动量：Σ p前 = Σ p后，其中 p = m v。
3. Work, Energy and Power | 功、能与功率
Work is done when a force moves its point of application in the direction of the force. It is a scalar quantity measured in joules (J).
当一个力使其作用点沿力方向移动时，该力做了功。功是标量，单位是焦耳 (J)。
W = F d cos θ
θ is the angle between the force and the displacement vector.
θ 是力与位移矢量之间的夹角。
Kinetic energy is the energy an object possesses due to its motion: Ek = ½ m v². Gravitational potential energy near Earth’s surface is: Ep = m g h.
动能是物体由于运动而具有的能量：Ek = ½ m v²。地表附近的重力势能为：Ep = m g h。
Power is the rate of doing work or transferring energy: P = W / t or P = F v for a force moving at constant velocity v.
功率是做功或能量转移的速率：P = W / t；若力以恒定速度 v 移动，则 P = F v。
4. Waves and Optics | 波与光学
The wave equation links the speed (v), frequency (f) and wavelength (λ) of a periodic wave.
波的方程将周期波的速度 (v)、频率 (f) 和波长 (λ) 联系起来。
v = f λ
For electromagnetic waves in a vacuum, v = c = 3.00×10⁸ m s⁻¹.
对于真空中的电磁波，v = c = 3.00×10⁸ m s⁻¹。
Refractive index n of a medium is defined as the ratio of the speed of light in vacuum to the speed in the medium: n = c / v. Snell’s law relates angles of incidence and refraction: n₁ sin θ₁ = n₂ sin θ₂.
介质的折射率 n 定义为真空中光速与介质中光速之比：n = c / v。斯涅尔定律将入射角和折射角联系起来：n₁ sin θ₁ = n₂ sin θ₂。
When a wave passes through a diffraction grating, maxima occur at angles satisfying: d sin θ = n λ, where d is the grating spacing and n is the order.
当波通过衍射光栅时，极大值出现的角度满足：d sin θ = n λ，其中 d 是光栅间距，n 是级数。
5. Electricity | 电学
Basic electric quantities are linked by Ohm’s law and the definitions of power and resistance.
基本电学量通过欧姆定律以及功率和电阻的定义相互关联。
V = I R
Potential difference (V) across a conductor equals current (I) times resistance (R).
导体两端的电势差 (V) 等于电流 (I) 乘以电阻 (R)。
Power dissipated in a component can be calculated using: P = I V = I² R = V² / R.
用电器消耗的功率可用以下公式计算：P = I V = I² R = V² / R。
For resistors combined in series: Rtotal = R₁ + R₂ + …. In parallel: 1/Rtotal = 1/R₁ + 1/R₂ + ….
电阻串联时：R总 = R₁ + R₂ + …。并联时：1/R总 = 1/R₁ + 1/R₂ + …。
Kirchhoff’s first law: the sum of currents entering a junction equals the sum leaving. Second law: the sum of e.m.f.s around any closed loop equals the sum of p.d.s around the loop.
基尔霍夫第一定律：流入节点的电流之和等于流出电流之和。第二定律：沿任一闭合回路的电动势之和等于该回路中电势差之和。
6. Materials: Stress, Strain and Young Modulus | 材料力学：应力、应变与杨氏模量
When a solid is deformed, stress and strain describe the response. For a wire under tension:
固体发生形变时，可以用应力和应变来描述其响应。对于拉伸的金属丝：
stress = F / A
F is the applied force and A is the cross-sectional area.
F 是施加的力，A 是横截面积。
strain = ΔL / L₀
ΔL is the extension and L₀ the original length.
ΔL 是伸长量，L₀ 是原始长度。
Young modulus E is a measure of stiffness: E = stress / strain = (F L₀) / (A ΔL). This applies within the elastic limit where Hooke’s law holds (F = k ΔL).
杨氏模量 E 是刚度的量度：E = 应力 / 应变 = (F L₀) / (A ΔL)。该关系适用于弹性极限内，此时胡克定律成立 (F = k ΔL)。
7. Atomic, Nuclear and Particle Physics | 原子、核与粒子物理
Einstein’s mass–energy equivalence is central to nuclear processes.
爱因斯坦质能方程是核过程的核心。
E = m c²
When a nucleus undergoes alpha, beta or gamma decay, the total nucleon number A and proton number Z are conserved. The activity A of a radioactive sample is the number of decays per unit time: A = λ N, where λ is the decay constant and N is the number of undecayed nuclei.
当原子核发生 α、β 或 γ 衰变时，总核子数 A 和质子数 Z 守恒。放射性样品的活度 A 是单位时间的衰变次数：A = λ N，其中 λ 是衰变常数，N 是未衰变核的数目。
The exponential decay law gives: N = N₀ e–λt. Half-life t½ is related to λ by: t½ = ln 2 / λ.
指数衰变规律为：N = N₀ e–λt。半衰期 t½ 与 λ 的关系为：t½ = ln 2 / λ。
8. Moles and Stoichiometry | 摩尔与化学计量
The mole is the SI unit for amount of substance. The number of moles n can be found from mass (m) and molar mass (M), or for solutions from concentration (c) and volume (V).
摩尔是国际单位制中物质的量的单位。物质的量 n 可以通过质量 (m) 与摩尔质量 (M) 计算，或对于溶液，通过浓度 (c) 与体积 (V) 计算。
n = m / M
n = c V
tio of reactants and products. Percentage yield = (actual yield / theoretical yield) × 100%.
在配平的化学方程式中，系数表示反应物与产物之间的摩尔比。产率 = (实际产量 / 理论产量) × 100%。
The ideal gas equation connects pressure, volume, temperature and amount:
理想气体状态方程将压力、体积、温度与物质的量联系起来：
p V = n R T
R = 8.31 J K⁻¹ mol⁻¹; T must be in kelvin.
R = 8.31 J K⁻¹ mol⁻¹；T 必须以开尔文为单位。
9. Thermochemistry and Energetics | 热化学与能量学
Enthalpy change ΔH for a reaction can be determined using Hess’s law or from experimental data: q = m c ΔT, where q is heat transferred, m is mass, c is specific heat capacity and ΔT is the temperature change. Then ΔH = –q / n.
反应的焓变 ΔH 可通过赫斯定律或实验数据确定：q = m c ΔT，其中 q 是传递的热量，m 是质量，c 是比热容，ΔT 是温度变化。然后 ΔH = –q / n。
Standard enthalpy of combustion and formation are used in Hess cycles:
标准燃烧焓和标准生成焓可用于赫斯循环：
ΔHreaction = Σ ΔH°f (products) – Σ ΔH°f (reactants)
For spontaneity, the Gibbs free energy change is given by:
对于反应的自发性，吉布斯自由能变由下式给出：
ΔG = ΔH – T ΔS
A reaction is feasible when ΔG < 0.
当 ΔG < 0 时，反应是可行的。
10. Chemical Equilibrium | 化学平衡
For a reversible reaction aA + bB ⇌ cC + dD, the equilibrium constant Kc in terms of concentration is:
对于可逆反应 aA + bB ⇌ cC + dD，以浓度表示的平衡常数 Kc 定义为：
Kc = [C]c[D]d / [A]a[B]b
Only gases and aqueous species appear in the expression; solids and pure liquids are omitted.
只有气体和水溶液中的物种出现在表达式中；固体与纯液体被省略。
When all reactants and products are gases, the equilibrium constant Kp can be expressed in terms of partial pressures:
当反应物和产物均为气体时，平衡常数 Kp 可以用分压表示：
Kp = (pC)c (pD)d / (pA)a (pB)b
Le Chatelier’s principle predicts how equilibrium shifts when conditions change. Kc and Kp are temperature dependent only.
勒夏特列原理可预测条件改变时平衡的移动方向。Kc 和 Kp 仅受温度影响。
| Change | Shift |
|---|---|
| Increase concentration of reactant | Shifts right to form more products |
| Increase temperature (exothermic) | Shifts left; Kc decreases |
| Increase pressure (gas reactions) | Shifts towards side with fewer gas moles |
变化：增加反应物浓度，平衡右移生成更多产物；升高温度（放热反应），平衡左移，Kc 减小；增大压力（气体反应），平衡向气体摩尔数较少的方向移动。
11. Biological Calculations: Magnification and Size | 生物计算：放大倍率与尺寸
In biology, magnification is the factor by which an image appears larger than the actual object. Use the formula:
在生物学中，放大倍率是图像相对于实物尺寸的放大倍数。使用公式：
Magnification = Image size / Actual size
or M = I / A, where both measurements must be in the same units.
即 M = I / A，两个测量值必须使用相同单位。
A graticule is calibrated with a stage micrometer to measure actual sizes. The conversion requires determining the length represented by each eyepiece graticule unit.
目镜测微尺需要用镜台测微尺校准以测量实际尺寸。校准时需确定每个测微尺单位所代表的长度。
For microscopes, total magnification is the product of the objective lens magnification and the eyepiece lens magnification.
对于显微镜，总放大倍率是物镜放大倍率与目镜放大倍率的乘积。
12. Statistical Tests in Biology | 生物统计检验
The chi-squared (χ²) test compares observed and expected frequencies to determine if differences are significant.
卡方 (χ²) 检验通过比较观察频数与期望频数，判断差异是否显著。
χ² = Σ (O – E)² / E
t a chosen probability level (usually p = 0.05) and appropriate degrees of freedom (df = n – 1 for categories).
O = 观察值，E = 期望值。将算得的 χ² 值与所选概率水平（通常 p = 0.05）及相应自由度下的临界值比较（分类数据自由度 df = n – 1）。
The Student’s t-test compares the means of two samples:
学生 t 检验用于比较两组样本的平均值：
t = (x̄₁ – x̄₂) / √s₁²/n₁ + s₂²/n₂
where x̄ are sample means, s² are sample variances and n are sample sizes. It is used when sample sizes are small and population variance is unknown.
其中 x̄ 是样本平均值，s² 是样本方差，n 是样本量。当样本量较小且总体方差未知时使用。
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