📚 A-Level CCEA Biology Formula Handbook | A-Level CCEA 生物公式汇总手册
Welcome to your quick-reference guide for all the essential quantitative relationships in the CCEA A-Level Biology specification. This handbook brings together the key formulae for microscopy, physiology, ecology, genetics and population biology, with clear definitions and worked examples of how each equation is applied. Mastering these formulae will not only boost your confidence in data-response and practical questions but also deepen your understanding of the underlying biological principles.
欢迎使用这份CCEA A-Level生物学定量关系速查手册。本手册汇集了显微镜、生理学、生态学、遗传学和种群生物学中的关键公式，对每个方程都给出了清晰的定义和计算示例。掌握这些公式不仅能提升你解答数据分析和实验题的信心，还能加深你对背后生物学原理的理解。
1. Microscopy and Cell Size Calculations | 显微镜与细胞大小计算
The core magnification formula links the size of an image to the real size of the specimen. All measurements must be expressed in the same units before calculation, and careful calibration of the eyepiece graticule against a stage micrometer is essential for accuracy.
核心放大倍数公式将图像的尺寸与标本的真实尺寸联系起来。计算前必须将所有测量值换算成相同单位，并且必须用台尺仔细校准目镜测微尺，才能获得准确结果。
Magnification = Image size / Actual size
This equation can be rearranged: Actual size = Image size / Magnification and Image size = Actual size × Magnification.
该方程可以变形为：实际大小 = 图像尺寸 / 放大倍数以及图像尺寸 = 实际大小 × 放大倍数。
Unit conversions:
单位换算：
- 1 cm = 10 mm
- 1 mm = 1000 µm
- 1 µm = 1000 nm
When using an eyepiece graticule, calibrate it for each objective lens by counting how many graticule divisions match a known length on the stage micrometer. One eyepiece unit = (number of stage divisions × length of one stage division) / number of eyepiece divisions.
使用目镜测微尺时，需要对每个物镜进行校准：数出多少个目镜分度正好等于台尺上的已知长度。一个目镜单位 = (台尺分度数 × 一个台尺分度的长度) / 目镜分度数。
2. Cardiac Output | 心输出量
Cardiac output is the volume of blood pumped by one ventricle per minute. It is determined by how fast the heart beats and how much blood is ejected with each beat.
心输出量是指一个心室每分钟泵出的血液体积。它由心跳的快慢和每次搏动射出的血量共同决定。
Cardiac output = Heart rate × Stroke volume
CO = HR × SV
| CO | Cardiac output (dm³ min⁻¹ or L min⁻¹) | 心输出量（dm³ min⁻¹ 或 L min⁻¹） |
| HR | Heart rate (beats min⁻¹) | 心率（次 min⁻¹） |
| SV | Stroke volume (dm³ or L) | 每搏输出量（dm³ 或 L） |
For example, if a person has a resting heart rate of 70 beats min⁻¹ and a stroke volume of 0.07 dm³, their cardiac output is 70 × 0.07 = 4.9 dm³ min⁻¹. During exercise both heart rate and stroke volume can increase, dramatically raising cardiac output.
例如，某人安静时心率为70次 min⁻¹，每搏输出量为0.07 dm³，则心输出量为 70 × 0.07 = 4.9 dm³ min⁻¹。运动时心率和每搏输出量均可增加，从而使心输出量显著升高。
3. Lung Volumes and Ventilation | 肺容量与通气量
Pulmonary ventilation is the total volume of air moved into and out of the lungs per minute. It depends on how deeply and how frequently we breathe.
肺通气量是指每分钟进出肺部的空气总体积。它取决于呼吸的深度和频率。
Minute ventilation = Tidal volume × Breathing rate
分钟通气量 = 潮气量 × 呼吸频率
Tidal volume (TV) is the volume of air inhaled or exhaled in one normal breath. Breathing rate (f) is the number of breaths per minute. Vital capacity is the maximum volume that can be exhaled after a maximal inhalation and can be expressed as:
潮气量（TV）是一次正常呼吸吸入或呼出的气体体积。呼吸频率（f）是每分钟呼吸的次数。肺活量是最大吸气后能够呼出的最大气体量，可表示为：
Vital capacity = Tidal volume + Inspiratory reserve volume + Expiratory reserve volume
肺活量 = 潮气量 + 补吸气量 + 补呼气量
4. Respiratory Quotient (RQ) | 呼吸商
The respiratory quotient indicates which type of respiratory substrate is being metabolised. It is the ratio of carbon dioxide produced to oxygen consumed over a given time.
呼吸商揭示了机正在代谢的是哪种呼吸底物。它是特定时间内产生的二氧化碳与消耗的氧气的体积比。
RQ = Volume of CO₂ produced / Volume of O₂ consumed
RQ = 产生的CO₂体积 / 消耗的O₂体积
Typical RQ values: carbohydrate = 1.0; lipid = 0.7; protein ≈ 0.9. An RQ above 1.0 suggests anaerobic respiration, as additional CO₂ is released without consuming O₂.
典型的RQ值：糖类 = 1.0；脂质 = 0.7；蛋白质 ≈ 0.9。若RQ高于1.0则提示存在无氧呼吸，因为有额外的CO₂释放而不消耗O₂。
5. Productivity and Energy Transfer | 生产力与能量传递
In ecosystems, the net primary production (NPP) represents the energy available to consumers after plants have used some energy for their own respiration.
在生态系统中，净初级生产力（NPP）是指植物将一部分能量用于自身呼吸后、可供消费者利用的能量。
NPP = GPP − R
净初级生产力 = 总初级生产力 − 呼吸消耗
where GPP is gross primary production (total energy fixed by photosynthesis) and R is respiratory loss. For secondary productivity, the efficiency of energy transfer between trophic levels can be calculated as:
其中GPP是总初级生产力（光合作用固定的总能量），R是呼吸消耗。对于次级生产力，营养级之间的能量传递效率可按下式计算：
Efficiency (%) = (Energy in one trophic level / Energy in the previous trophic level) × 100
效率（%）=（某一营养级的能量 / 上一营养级的能量）× 100
Alternatively, use the ecological efficiency form: Efficiency = (Energy available after transfer / Energy available before transfer) × 100. These values are typically low because energy is lost as heat, in respiration and in uneaten parts.
也可以使用生态效率公式：效率 =（传递后可利用的能量 / 传递前可利用的能量）× 100。这些数值通常很低，因为能量会以热量、呼吸消耗和未食用部分等形式散失。
6. Population Estimation – Mark-Release-Recapture | 种群估算 – 标记重捕法
The Lincoln index provides an estimate of population size for mobile organisms. It assumes random mixing, no migration, no births or deaths, and that marks are not lost or harmful.
林肯指数用于估算移动生物种群的大小。其前提假设包括：随机混合、无迁徙、无出生或死亡，且标记不会丢失或对生物造成伤害。
N = (M × C) / R
| N | Estimated total population | 估计种群总数 |
| M | Number captured, marked and released in first sample | 第一次捕获、标记并释放的数量 |
| C | Total number captured in second sample | 第二次捕获的总数 |
| R | Number of marked individuals recaptured in second sample | 第二次捕获中带有标记的个体数 |
7. Population Growth – Exponential Model | 种群增长 – 指数增长模型
When resources are unlimited, populations of bacteria and other organisms can grow exponentially. The number of individuals after a given time depends on the initial population and the number of generations.
当资源不受限制时，细菌和其他生物的种群可以呈指数增长。给定时间后的个体数取决于初始种群和繁殖的代数。
Nₜ = N₀ × 2n
or, using doubling time td and elapsed time t:
或者，使用倍增时间td和经历时间t：
Nₜ = N₀ × 2(t / td)
where Nₜ = population after time t, N₀ = initial population, n = number of generations, td = doubling (generation) time. The mean generation time can also be calculated as g = t / n.
其中 Nₜ = 时间t后的种群数量，N₀ = 初始种群数量，n = 世代数，td = 倍增时间。平均世代时间也可通过 g = t / n 求得。
8. Hardy–Weinberg Principle | 哈代–温伯格定律
The Hardy–Weinberg equations predict allele and genotype frequencies in a large, randomly mating population that is not subject to mutation, migration or natural selection. They provide a null model for detecting evolutionary change.
哈代–温伯格方程预测一个大且随机交配、没有突变、迁移或自然选择的种群中，等位基因频率和基因型频率。它提供了一种检测进化改变的无效模型。
p + q = 1
p² + 2pq + q² = 1
p = frequency of the dominant allele; q = frequency of the recessive allele. p² = frequency of homozygous dominant genotype; 2pq = frequency of heterozygous genotype; q² = frequency of homozygous recessive genotype. When only the recessive phenotype frequency (q²) is known, take its square root to find q, then calculate p = 1 − q.
p = 显性等位基因频率；q = 隐性等位基因频率。p² = 纯合显性基因型频率；2pq = 杂合子基因型频率；q² = 纯合隐性基因型频率。若仅知隐性表型频率(q²)，可对其开方求q，再由 p = 1 − q 计算p。
9. Chi-Squared (χ²) Test | 卡方检验
The chi-squared test is used to determine whether there is a significant difference between observed and expected categorical data. In biology, it is frequently applied to genetic crosses and ecological sampling.
卡方检验用于判断观测数据与期望分类数据之间是否存在显著差异。在生物学中，它常用于遗传杂交实验和生态取样分析。
χ² = Σ (O − E)2 / E
O = observed frequency; E = expected frequency. The sum is taken over all categories. After calculating χ², the value is compared with a critical value at the appropriate degrees of freedom (df = number of categories − 1, or (rows−1)×(columns−1) for contingency tables) and a probability level (usually p = 0.05).
O = 观测值；E = 期望值。对所有类别求和。计算出χ²值后，将其与对应自由度（df = 类别数−1，或列联表中(行−1)×(列−1)）和概率水平（通常 p = 0.05）下的临界值进行比较。
10. Genetic Linkage and Recombination Frequency | 遗传连锁与重组频率
When two genes are located on the same chromosome, they tend to be inherited together. The recombination frequency from a test cross allows the distance between genes to be estimated and linkage maps to be constructed.
当两个基因位于同一染色体上时，它们倾向于一起遗传。测交中获得的重组率可用于估计基因间的距离并构建连锁图谱。
Recombination frequency (%) = (Number of recombinant offspring / Total number of offspring) × 100
重组率（%）=（重组子代数 / 子代总数）× 100
A recombination frequency of 0 % means complete linkage; a frequency of 50 % indicates independent assortment (genes far apart on the same chromosome or on different chromosomes). One map unit (centimorgan) is equivalent to 1 % recombination.
重组率为0%表明完全连锁；50%表明独立分配（基因位于同一染色体上距离很远或位于不同染色体）。1个图距单位（厘摩）相当于1%的重组率。
11. Water Potential (ψ) | 水势
Water potential describes the tendency of water to move from one area to another. It is affected by the concentration of solutes and by physical pressure. Water always moves from a region of higher (less negative) water potential to a region of lower (more negative) water potential.
水势描述水分从一个区域向另一区域移动的趋势。它受溶质浓度和物理压力的影响。水总是由水势较高（负值较小）的区域向水势较低（负值较大）的区域移动。
ψ = ψs + ψp
水势 = 溶质势 + 压力势
ψs (solute potential) is always negative or zero; dissolving solutes lowers water potential. ψp (pressure potential) is usually positive inside plant cells (turgor pressure) and can be negative in the xylem under tension. In animal cells, the term osmotic potential (often equivalent to solute potential) is used, and the net movement of water is governed by differences in osmolarity.
ψs (溶质势) 总是负值或零；溶质溶解会降低水势。ψp (压力势) 在植物细胞内部通常为正值（膨压），而在木质部受到张力时可呈负值。在动物细胞中，使用渗透势（通常等同于溶质势）术语，水分的净流动取决于渗透浓度的差异。
12. Simpson’s Diversity Index | 辛普森多样性指数
Simpson’s index quantifies the biodiversity of a habitat, taking into account both species richness and evenness. A higher value indicates greater diversity.
辛普森指数量化生境的生物多样性，同时考虑物种丰富度和均匀度。指数值越高代表多样性越高。
D = 1 − Σ n(n−1) / N(N−1)
Where n = total number of organisms of a particular species, N = total number of organisms of all species. The index ranges from 0 (no diversity) to a maximum value approaching 1 (high diversity). Alternatively, in some specifications the simpler form D = 1 − Σ (n/N)² is used; always confirm with CCEA mark schemes, but the n(n−1) form is the more statistically robust version.
其中 n = 某一物种的个体总数，N = 所有物种的个体总数。指数范围为0（无多样性）到接近1的最高值（高多样性）。有些大纲也会使用简化形式 D = 1 − Σ (n/N)²；请以CCEA评分方案为准，但 n(n−1) 的形式在统计学上更为稳健。
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