📚 Year 13 CIE Biology: Formula & Theorem Quick Reference Handbook | CIE A2 生物公式定理速查手册
Quantitative skills are an essential part of CIE Year 13 Biology. From magnification and Hardy–Weinberg to respiratory quotients and Simpson’s Index, being confident with formulae and theorems can make the difference between a good grade and a top grade. This quick-reference handbook brings together every key formula and principle you need to memorise for the A2 exam, complete with worked examples and practical tips.
定量技能是 CIE 生物 A2 考试的核心组成部分。从显微镜放大倍数、哈代－温伯格定律到呼吸商和辛普森多样性指数，熟练掌握公式与定理是冲击高分的制胜法宝。本速查手册汇集了 Year 13 必须记住的所有关键公式和原理，并配以典型示例与解题要点。
1. Magnification and Actual Size | 放大倍数与实际尺寸
triangle. You must be able to rearrange it and use consistent units (usually micrometres, µm, or millimetres, mm). 1 mm = 1000 µm.
放大倍数、图像尺寸和标本实际尺寸之间存在一个简单的三角形关系。你必须能够灵活变形公式，并统一单位（通常用微米 µm 或毫米 mm）。1 mm = 1000 µm。
Magnification = Image size / Actual size
o the same unit before calculating. If a scale bar is given, measure its length on the paper, convert to the real length, and compute M.
该公式也可写作 M = I ÷ A。计算前一定要将所有长度换算为同一单位。如果题目给出了比例尺，先测量图上的比例尺长度，换算成实际长度，再计算放大倍数。
- Image size = Magnification × Actual size
- Actual size = Image size ÷ Magnification
When drawing a scale bar, choose a sensible real length (e.g., 10 µ calculate the drawn length using the formula: drawn length = real length × magnification.
绘制比例尺时，选择一个合理的实际长度（如 10 µm），然后用公式“图上长度 = 实际长度 × 放大倍数”计算绘制长度。
2. Hardy–Weinberg Principle | 哈代－温伯格定律
The Hardy–Weinberg principle states that allele and genotype frequencies in a large, randomly mating population will remain constant from generation to generation in the absence of mutation, migration, genetic drift, and natural selection. It provides a null model for detecting evolutionary change.
哈代－温伯格定律指出，在一个无限大、随机交配、没有突变、迁移、遗传漂变和自然选择的种群中，等位基因频率和基因型频率世代保持不变。它是检验进化的零假设模型。
p + q = 1
p² + 2pq + q² = 1
2pq = frequency of heterozygotes, and q² = frequency of homozygous recessive individuals.
其中 p = 显性等位基因频率，q = 隐性等位基因频率，p² = 显性纯合子频率，2pq = 杂合子频率，q² = 隐性纯合子频率。
To test if a population is in Hardy– xams, you are often given the number or frequency of recessive homozygotes (q²). Take the square root to find q, then calculate p = 1 – q.
检验是否符合哈代－温伯格平衡时，先从观测到的等位基因频率算出期望基因型数目，再用卡方检验（见第3节）比较。考试中常给出隐性纯合子的数目或频率 (q²)，开平方得到 q，再求 p = 1 – q。
3. Chi-Squared (χ²) Test | 卡方（χ²）检验
The chi-squared test determines whether there is a significant difference between observed and expected frequencies. In Biology, it is commonly used for genetic crosses, ecological data, and Hardy–Weinberg equilibrium testing.
卡方检验用于判断观测值与期望值之间是否存在显著差异。在生物学中常用于遗传杂交、生态学数据和哈代－温伯格平衡检验。
χ² = Σ (O – E)² / E
ution table at a given probability level (usually p < 0.05) and for the correct degrees of freedom (d.f. = number of categories – 1).
O = 观测值，E = 期望值，对所有分类求和。计算得到的 χ² 值与临界值（通常在 p < 0.05 水平）进行比较，自由度 d.f. = 类别数 – 1。
- If χ² < critical value: the difference is not significant; accept the null hypothesis.
- 如果 χ² < 临界值：差异不显著，接受零假设。
- If χ² ≥ critical value: the difference is significant; reject the null hypothesis.
- 如果 χ² ≥ 临界值：差异显著，拒绝零假设。
Always state the null hypothesis clearly. When doing a genetics cross, the expected ratio (e.g., 9:3:3:1) must be converted into expected numbers based on the total count.
务必先明确零假设。在遗传学实验中，需将预期比例（如 9:3:3:1）根据总数换算成预期数目。
4. Respiratory Quotient (RQ) | 呼吸商
The respiratory quotient (RQ) indicates the type of respiratory substrate being used and whether respiration is aerobic or anaerobic. It is defined as the ratio of carbon dioxide produced to oxygen consumed over a given period.
呼吸商 (RQ) 反映机体正在使用何种呼吸底物以及呼吸方式。它是一定时间内二氧化碳产生量与氧气消耗量的比值。
RQ = Volume of CO₂ produced / Volume of O₂ consumed
| Substrate / 底物 | RQ |
| Carbohydrate / 碳水化合物 | 1.0 |
| Lipids / 脂质 | ~0.7 |
| Protein / 蛋白质 | ~0.9 |
RQ values greater than 1.0 suggest anaerobic respiration because CO₂ is produced without corresponding O₂ consumption. Measurements are often made using a respirometer, with carbon dioxide absorbed by soda lime or potassium hydroxide, allowing O₂ consumption to be measured directly.
RQ 大于 1.0 通常暗示发生了无氧呼吸，因为产生了 CO₂ 却没有消耗等量的 O₂。实验常用呼吸计测量，通过钠石灰或 KOH 溶液吸收 CO₂，直接测得 O₂ 消耗量。
5. Productivity: NPP = GPP – R | 生产力：净初级生产力 = 总初级生产力 – 呼吸消耗
In ecosystems, the rate at which producers convert light energy into chemical energy is gross primary productivity (GPP). Plants use some of this energy in respiration (R); the remaining energy available to the next trophic level is net primary productivity (NPP).
在生态系统中，生产者将光能转化为化学能的速率称为总初级生产力 (GPP)。植物呼吸作用消耗一部分能量 (R)，剩下的可供下一营养级利用的能量即为净初级生产力 (NPP)。
NPP = GPP – R
erms if two are known. Net secondary productivity (NSP) follows a similar logic: NSP = GSP – R for consumers.
单位通常为 kJ m⁻² 年⁻¹ 或 g m⁻² 年⁻¹。已知其中两项即可计算第三项。消费者净次级生产力 (NSP) 的逻辑相似：NSP = GSP – R。
- GPP is the total fixation of energy by photosynthesis.
- GPP 是光合作用固定的总能量。
- NPP is the energy available for growth and reproduction, or for herbivores.
- NPP 是可供植物生长、繁殖或被植食动物取食的能量。
When given a Sankey diagram or energy flow chart, NPP is usually the arrow leaving the producer box after respiration has been subtracted.
在桑基图或能量流图中，NPP 通常是生产者方框中减去呼吸后的流出箭头。
6. Simpson’s Index of Diversity | 辛普森多样性指数
Simpson’s Index of Diversity (D) measures the biodiversity of a habitat, taking into account both species richness and evenness. A higher value indicates greater diversity.
辛普森多样性指数 (D) 用于衡量一个栖息地的生物多样性，同时考虑物种丰富度和均匀度。数值越高，多样性越大。
D = 1 – Σ (n/N)²
of organisms of all species. The sum is calculated for each species present. The formula 1 – Σ(n/N)² gives a value between 0 and 1, but an alternative formula D = N(N–1) / Σ n(n–1) may also be used in some syllabuses — check your specification.
式中 n = 某一物种的个体总数，N = 所有物种的个体总数，对每个物种计算 (n/N)² 并求和。1 – Σ(n/N)² 的结果在 0 到 1 之间，个别教材也使用 D = N(N–1) / Σ n(n–1)，请确认考纲要求。
Steps for calculation: list each species, record the number of individuals (n), calculate n/N for each, square it, sum all (n/N)², then subtract from 1. Always show your working.
计算步骤：列出每个物种及其个体数量 n，计算各物种的 n/N，求平方后求和，得到 Σ(n/N)²，最后用 1 减去该和。必须展示完整的演算过程。
7. Serial Dilution | 序列稀释
Serial dilutions are used to reduce a dense culture of microorganisms to a countable number. Each step dilutes the previous concentration by a fixed factor, making it possible to estimate the original population density.
序列稀释用于将浓密的微生物培养液逐步稀释至可计数的浓度。每一步都按固定倍数稀释前一管溶液，从而能回推出原始菌液密度。
Dilution factor = Volume of stock / Total volume after dilution
For example, if 1 cm³ of bacterial suspension is added to 9 cm³ of sterile water, the dilution factor is 1/10, written as 10⁻¹. A further 1 cm³ taken from this tube and added to another 9 cm³ gives 10⁻², and so on.
例如，将 1 cm³ 菌液加入 9 cm³ 无菌水中，稀释倍数为 1/10，记作 10⁻¹。再从该管取 1 cm³ 加入 9 cm³ 水中，得到 10⁻²，依此类推。
- Total dilution factor = product of all individual dilutions.
- 总稀释倍数 = 每一步稀释倍数的乘积。
- Original cell count = colony count on plate × (1 ÷ volume plated) × total dilution factor.
- 原始菌数 = 平板菌落数 × (1 ÷ 涂布体积) × 总稀释倍数。
Plates with 30–300 colonies are considered statistically reliable. Numbers outside this range should not be used for calculations.
菌落数在 30–300 之间的平板具有统计学可靠性，超出此范围的不应用于计算。
8. Surface Area to Volume Ratio (SA:V) | 表面积与体积比
As an organism or cell increases in size, its volume increases faster than its surface area. The surface area to volume ratio has profound implications for diffusion rate, heat exchange, and nutrient uptake.
随着生物体或细胞体积增大，体积的增长速度快于表面积。表面积与体积之比直接影响扩散速率、热量交换和营养吸收。
SA:V ratio = Surface Area / Volume
tio decreases with increasing size, explaining why large organisms need specialised transport systems (e.g., lungs, gills, circulatory systems) and why cells are microscopic.
对于边长为 L 的立方体：表面积 = 6L²，体积 = L³，SA:V = 6/L。对于半径为 r 的球体：SA:V = 3/r。该比率随尺寸增大而减小，这解释了为何大型生物需要专门的运输系统（如肺、鳃、循环系统），以及细胞为何是微观的。
- High SA:V ratio → rapid diffusion, efficient exchange.
- 高 SA:V 比率 → 快速扩散，高效交换。
- Low SA:V ratio → diffusion becomes limiting; transport systems evolve.
- 低 SA:V 比率 → 扩散受限；进化出运输系统。
In the exam, you may be asked to calculate surface area and volume, and then comment on adaptations such as flattened shape, villi, or extensive branching.
考试中可能要求计算表面积和体积，然后解释扁平形态、绒毛或广泛分支等适应性特征的意义。
9. Temperature Coefficient (Q₁₀) | 温度系数 (Q₁₀)
Q₁₀ is a measure of the effect of a 10 °C temperature increase on the rate of a reaction or metabolic process. It indicates how much faster the process runs when the temperature is raised by 10 °C.
Q₁₀ 用于衡量温度升高 10 °C 对反应或代谢过程速率的影响。它表示温度每升高 10 °C 时，过程的速率加快了多少倍。
Q₁₀ = (Rate at T + 10 °C) / (Rate at T °C)
For most enzyme-controlled reactions, the Q₁₀ is around 2 to 3 between 0 °C and 40 °C. Above the optimum temperature, enzymes denature and the rate falls sharply, making Q₁₀ less than 1.
在 0 °C 到 40 °C 之间，大多数酶促反应的 Q₁₀ 在 2 到 3 左右。超过最适温度后，酶变性失活，速率急剧下降，此时 Q₁₀ 将小于 1。
C and 45 U at 30 °C, Q₁₀ = 45/15 = 3. This suggests the process is largely enzyme-limited.
示例：若 20 °C 时呼吸速率为 15 单位，30 °C 时为 45 单位，则 Q₁₀ = 45/15 = 3，表明该过程主要由酶控制。
10. Water Potential (ψ) | 水势 (ψ)
Water potential determines the direction of water movement. Water moves from a region of higher water potential (less negative) to a region of lower water potential (more negative) across a partially permeable membrane.
水势决定水分的运动方向。水分子总是通过部分透性膜从水势较高（负值较小）的区域流向水势较低（负值更大）的区域。
ψ = ψₛ + ψₚ
ψₛ (solute potential) is always negative or zero, and becomes more negative as solute concentration increases. ψₚ (pressure potential) is usually zero or positive (turgor pressure in plant cells). Pure water at atmospheric pressure has a water potential of zero.
ψₛ（溶质势）总是负值或零，且随溶质浓度增大变得更负。ψₚ（压力势）通常为零或正值（植物细胞的膨压）。纯水在标准大气压下水势为 0。
- In a turgid plant cell, ψₚ is positive and offsets the negative ψₛ, raising total ψ.
- 在质壁分离的植物细胞中，ψₚ 接近零，ψ ≈ ψₛ（极负），导致失水。
- In an animal cell, ψₚ = 0, so ψ = ψₛ; water potential is essentially solute potential.
- 在动物细胞中，ψₚ = 0，所以 ψ = ψₛ；水势就等于溶质势。
Water potential is measured in pressure units, usually kPa. You may be asked to calculate ψ from given component values or to predict the direction of net water movement.
水势的单位是压强单位，通常为 kPa。考题可能要求根据给定的溶质势和压力势计算水势，或预测水分子净移动方向。
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