📚 Year 12 CCEA Biology: Formula and Theorem Quick Reference Handbook | Year 12 CCEA 生物：公式定理速查手册
This concise handbook presents all the key formulae, equations and fundamental principles required for the CCEA Year 12 (AS) Biology course. It is designed as a rapid revision tool to help you master quantitative skills and conceptual relationships across topics including microscopy, cell physiology, genetics, ecology and experimental data analysis. Each entry delivers the core formula in a clear, centered format, followed by a brief explanation of its meaning and application. For every English paragraph, a parallel Chinese version is provided to support bilingual learning and deepen understanding.
本手册浓缩了 CCEA 考试局 Year 12（AS）生物学课程中所有核心公式、方程和基本原理。它作为快速复习工具，帮助你掌握显微镜、细胞生理、遗传、生态和实验数据分析等主题中的定量技能和概念关系。每一条目均以醒目的居中格式给出核心公式，并附简洁解释说明其意义和应用。每个英文段落后紧跟对应的中文段落，支持双语学习并加深理解。
1. Microscopy and Magnification | 显微镜与放大率
The most frequently used equation in microscopy links image size, actual size and magnification. It allows you to calculate the real dimensions of a specimen from a calibrated microscope image or measure a scale bar.
显微镜学中最常用的方程将图像大小、实际大小和放大率联系起来。你可以用它从已校准的显微镜图像计算标本的真实尺寸，或测量比例尺。
Magnification = Image size ÷ Actual size (M = I / A)
ion and image size, actual size = I ÷ M. For scale bars, measure the bar in mm, convert to µ de by the stated size to find the magnification. Typical AS tasks involve converting mm to µm (×1000) or to nm (×1 000 000).
三个量必须使用相同单位。记住三角形关系：I 在上端，A 和 M 在下端。如果已知放大率和图像大小，实际大小 = I ÷ M。对于比例尺，先以 mm 为单位测量尺长，必要时转换为 µm，再除以标称尺寸即可得到放大率。AS 考试中常见的单位换算包括将 mm 转换为 µm（乘以 1000）或转换为 nm（乘以 1 000 000）。
2. Cell Size and Surface Area to Volume Ratio | 细胞大小与表面积／体积比
As a cell grows, its volume increases faster than its surface area. This relationship is critical for understanding transport limits and why cells remain microscopic or develop specialised structures.
细胞生长时，其体积增加速度快于表面积。这一关系对于理解运输限制以及细胞为何保持微小尺寸或发育出特化结构至关重要。
Surface Area : Volume ratio = Total surface area ÷ Volume
he larger the organism or cell, the smaller the SA:V ratio, which reduces the efficiency of diffusion of oxygen, nutrients and wastes. This principle explains why multicellular organisms require transport systems and why cells such as those in the proximal convoluted tubule have microvilli to increase surface area.
对于边长为 L 的立方体：表面积 = 6L²，体积 = L³，因此比值 = 6 / L。对于球体：表面积 = 4πr²，体积 = ⁴⁄₃πr³，比值为 3 / r。生物体或细胞越大，SA:V 比越小，这会降低氧气、营养物质和废物的扩散效率。该原理解释了为什么多细胞生物需要运输系统，以及为什么近曲小管等细胞具有微绒毛以增加表面积。
3. Water Potential and Osmosis | 水势与渗透
Water potential (Ψ, the Greek letter psi) is the measure of the tendency of water to move from one region to another. It integrates the effects of solutes and physical pressure, and is expressed in pressure units (kPa or MPa).
水势（Ψ，希腊字母 psi）是衡量水从一个区域向另一个区域移动趋势的量度。它综合了溶质和物理压力的影响，用压力单位（kPa 或 MPa）表示。
Ψ = Ψₛ + Ψₚ
sure has Ψ = 0. Ψₚ (pressure potential) is the physical pressure exerted by the cell wall; it is usually positive inside turgid plant cells and zero in an open system. Water moves from regions of higher (less negative) water potential to regions of lower (more negative) water potential. In animal cells, the absence of a cell wall means Ψₚ is typically zero, so water potential is determined solely by solute concentration, often discussed in terms of osmotic pressure.
Ψₛ（溶质势）在植物细胞中始终为负值或零，因为添加溶质会降低水势。纯水在标准压力下 Ψ = 0。Ψₚ（压力势）是细胞壁施加的物理压力；在膨胀的植物细胞内部通常为正值，在开放系统中为零。水从水势较高（负值较小）的区域向水势较低（负值较大）的区域移动。在动物细胞中，由于缺乏细胞壁，Ψₚ 通常为零，因此水势仅由溶质浓度决定，通常用渗透压来讨论。
4. Cardiac Output and Heart Rate | 心输出量与心率
Cardiac output (CO) is the volume of blood pumped by one ventricle per minute. It is a fundamental measure of cardiovascular performance and is directly influenced by heart rate and stroke volume.
心输出量（CO）是一个心室每分钟泵出的血液体积。它是衡量心血管功能的基本指标，直接受心率和每搏输出量的影响。
Cardiac Output = Heart Rate × Stroke Volume (CO = HR × SV)
per minute (bpm). Stroke volume (SV) is the volume of blood ejected per beat, typically 70– s often examined in the context of the effects of adrenaline (increases HR and SV) or changes in venous return (Starling’s law). You may be asked to calculate any missing variable, so ensure you can rearrange the equation: HR = CO / SV; SV = CO / HR.
心率（HR）以每分钟心跳次数（bpm）表示。每搏输出量（SV）是每次心跳射出的血量，静息成年人通常为 70–80 cm³。因此静息心输出量约为 5 dm³ min⁻¹，但在运动时可增加数倍。考试常围绕肾上腺素（增加 HR 和 SV）或静脉回流量变化（斯塔林定律）考查此关系。你可能需要计算任意未知量，因此务必掌握公式变形：HR = CO / SV，SV = CO / HR。
5. Breathing and Lung Volumes | 呼吸与肺容积
Pulmonary volumes and capacities describe the amount of air moved into and out of the lungs during different phases of the breathing cycle. These are recorded on a spirometer trace.
肺容积和肺容量描述在呼吸周期不同阶段进出肺的空气量，这些数据记录在肺活量计描记图上。
+ Inspiratory Reserve Volume (IRV) + Expiratory Reserve Volume (ERV) = Vital Capacity (VC)
Total Lung Capacity (TLC) = VC + Residual Volume (RV)
m³). Inspiratory reserve volume is the extra air that can be forcibly inhaled. Expiratory reserve volume is the extra air that can be forcibly exhaled. Vital capacity is the maximum volume of air that can be moved into and out of the lungs in one breath. Residual volume, which cannot be exhaled, remains in the lungs after a maximal expiration to prevent alveolar collapse. These terms are frequently tested in interpretation of spirograms and in the context of exercise or lung disease.
潮气量是静息时每次正常呼吸吸入或呼出的气体量（约 0.5 dm³）。补吸气量是用力补吸气的额外气体量。补呼气量是用力补呼气的额外气体量。肺活量是一次呼吸中能够进出肺的最大气体体积。残气量是无法呼出、在最大呼气后仍留存于肺内以防止肺泡塌陷的气体量。这些术语常在肺活量图解读以及运动或肺部疾病的情境中考查。
6. Mitotic Index | 有丝分裂指数
The mitotic index is a simple ratio used to quantify the proportion of cells in a tissue that are undergoing mitosis. It is a powerful diagnostic indicator for growth rate and, in medicine, for the aggressiveness of tumours.
有丝分裂指数是一个简单的比值，用于量化组织中正在进行有丝分裂的细胞比例。它是衡量生长速率的有效指标，并在医学上用于判断肿瘤的侵袭性。
Mitotic index = Number of cells in mitosis ÷ Total number of cells observed
In a root tip squash or micrograph, cells in prophase, metaphase, anaphase and telophase are counted as dividing. The mitotic index is determined by scanning several fields of view and pooling the counts to improve reliability. A high mitotic index indicates rapid cell division, typical of meristems and cancerous tissues. Values range from nearly zero (fully differentiated tissue) to more than 0.1 in very active regions. It can be expressed as a decimal or percentage.
在根尖压片或显微照片中，处于前期、中期、后期和末期的细胞计为分裂细胞。通过扫描多个视野并汇总计数以计算有丝分裂指数，可提高可靠性。高有丝分裂指数表明细胞分裂迅速，常见于分生组织和癌组织。数值范围从接近零（完全分化的组织）到活跃区域中超过 0.1 不等，可用小数或百分数表示。
7. Monohybrid Inheritance and Probability Rules | 单基因杂交与概率法则
Monohybrid crosses involve a single gene with two alleles. Mendel’s law of segregation and the laws of probability allow prediction of genotype and phenotype ratios.
单基因杂交涉及一个基因的两个等位基因。孟德尔分离定律和概率法则可用来预测基因型和表型比例。
F₂ phenotypic ratio for a heterozygous cross: 3 : 1
F₂ genotypic ratio: 1 homozygous dominant : 2 heterozygous : 1 homozygous recessive
For example, the chance of obtaining a homozygous recessive offspring from two heterozygous parents (Aa × Aa) is ½ (from mother) × ½ (from father) = ¼. This multiplicative rule also applies when considering dihybrid crosses or sex‑ or observed, and be prepared to compare observed counts with expected ratios using the chi‑squared test.
在构建庞纳特方格时，独立事件同时发生的概率是各自概率的乘积。例如，两个杂合亲本（Aa × Aa）产生纯合隐性后代的概率为 ½（母方）× ½（父方）= ¼。此乘法规则同样适用于双基因杂交或性连锁遗传。记得说明比例是基于期望值还是观测值，并做好利用卡方检验比较观测计数与期望比例的准备。
8. Hardy–Weinberg Principle | 哈代–温伯格定律
The Hardy–Weinberg principle is a mathematical model that predicts allele and genotype frequencies in a large, randomly mating population that is not subject to evolutionary forces. It provides a null hypothesis for studying microevolution.
哈代–温伯格定律是一个数学模型，用于预测一个大且随机交配、不受进化力影响的种群中的等位基因频率和基因型频率，为研究微进化提供了零假设。
p + q = 1
p² + 2pq + q² = 1
Here p represents the frequency of the dominant allele and q the frequency of the recessive allele. The terms p², 2pq and q² correspond to the frequencies of homozygous dominant, heterozygous and homozygous recessive genotypes respectively. In CCEA AS examinations, you will typically be given the frequency of the recessive phenotype (q²), from which you can calculate q as √q², then p = 1 – no mutation, no migration, random mating, large population size and no natural selection.
这里 p 代表显性等位基因的频率，q 代表隐性等位基因的频率。p²、2pq 和 q² 分别对应纯合显性、杂合和纯合隐性基因型的频率。在 CCEA AS 考试中，通常会给出隐性表型的频率 (q²)，由此可按 √q² 计算 q，再得出 p = 1 – q，进而求算各基因型频率。所有基因型频率之和必须为 1。该定律假设没有突变、没有迁移、随机交配、种群规模大且没有自然选择。
9. Chi‑squared (χ²) Test | 卡方 (χ²) 检验
The chi‑squared test is a statistical tool used to evaluate whether there is a significant difference between observed and expected categorical data. In biology, it is routinely applied to genetic cross results and ecological distribution studies.
卡方检验是一种统计工具，用于评估分类数据的观测值与期望值之间是否存在显著差异。在生物学中，它常用于遗传杂交结果和生态分布研究。
χ² = Σ (O - E)² / E
e from tables at a chosen probability level (usually p = 0.05) and appropriate degrees of freedom. Degrees of freedom (df) = number of categories – 1 for simple tests. If χ²calc > critical value, the null hypothesis is rejected, indicating a significant difference. You must be able to state a null hypothesis, perform the arithmetic, interpret the result and conclude whether any deviation is due to chance or a real effect.
O 为观测频率，E 为期望频率，对所有类别求和。计算所得的 χ² 值与给定概率水平（通常 p = 0.05）及适当自由度下的临界值比较。简单检验中自由度 (df) = 类别数 – 1。如果 χ² 计算值 > 临界值，则拒绝零假设，说明存在显著差异。你必须能够陈述零假设、进行计算、解读结果并判断偏差是由偶然还是真实效应引起。
10. Species Diversity – Simpson’s Index | 物种多样性 – 辛普森指数
Simpson’s Diversity Index quantifies the biodiversity of a habitat, taking into account both the number of species (richness) and the relative abundance of each species (evenness). CCEA uses a form where higher values indicate greater diversity.
辛普森多样性指数量化了栖息地的生物多样性，同时考虑物种数目（丰富度）和各物种的相对丰度（均匀度）。CCEA 所采用的形式中，数值越高表示多样性越大。
D = 1 - Σ (n / N)²
Here n is the number of individuals of a particular species, and N is the total number of individuals of all species. The sum is taken over all species present. The value of D ranges from 0 (no diversity) to just below 1 (very high diversity). A low D indicates that a few species dominate the community. When comparing two habitats, the one with the higher D is considered more diverse. You may be asked to calculate D from raw count data and to discuss the limitations of the index, such as its reliance on accurate identification and sampling effort.
其中 n 为某一特定物种的个体数，N 为所有物种的个体总数，对所有存在的物种求和。D 值范围从 0（无多样性）到接近 1（极高的多样性）。D 值低表示少数物种在群落中占优势。比较两个栖息地时，D 值较高的被认为多样性更高。你可能会被要求从原始计数数据计算 D，并讨论该指数的局限性，例如它依赖于准确鉴定和采样力度。
11. Capture-Mark-Recapture (Lincoln Index) | 标记重捕法（林肯指数）
The Lincoln Index estimates the size of a motile population by capturing, marking, releasing and recapturing individuals. It assumes a closed population and random mixing of marked animals.
林肯指数通过对个体进行捕捉、标记、释放和重捕来估算活动种群的大小，其假设种群封闭且标记动物均匀混合。
N = (M × C) / R
f marked individuals recaptured in the second sample. This method works best when the time between captures is short and mortality, birth and migration are negligible. The accuracy depends on marks not affecting survival, marks not being lost, and the population being randomly mixed. You should be able to evaluate the reliability of the estimate by considering these assumptions.
N = 估算的种群总大小。M = 第一次捕获并标记的个体数。C = 第二次样本中捕获的总个体数。R = 第二次样本中重捕到的标记个体数。当两次捕捉之间时间短、死亡率、出生率和迁移可忽略时，该方法效果最佳。其准确性取决于标记不影响存活、标记不脱落以及种群均匀混合。你应能通过考虑这些假设来评价估算值的可靠性。
12. Population Growth Rate and Ecological Efficiency | 种群增长率与生态效率
Population growth rate reflects the net change in population size over a given period. At the ecosystem level, energy transfer between trophic levels is governed by ecological efficiency, often expressed as a percentage.
种群增长率反映了给定时间段内种群数量的净变化。在生态系统层面，营养级之间的能量传递受生态效率控制，常以百分数表示。
Population growth rate = (Births + Immigration) - (Deaths + Emigration)
Transfer efficiency (%) = (Energy available to next level ÷ Energy available to current level) × 100
In the CCEA specification, energy flow through ecosystems is often examined through food chains and pyramids of energy. Typical transfer efficiencies are around 10%, with most energy lost as heat through respiration, uneaten matter and excretion. These losses limit the length of food chains and explain why pyramids of energy are always upright. You may need to calculate efficiency values from given data tables and discuss reasons for the observed values.
在 CCEA 大纲中，生态系统中的能量流动常通过食物链和能量锥体进行考察。典型的传递效率约为 10%，大部分能量通过呼吸作用、未食用物质和排泄以热的形式散失。这些损失限制了食物链的长度，并解释了为什么能量锥体总是正立的。你可能需要从给定的数据表计算效率值并讨论观察值的原因。
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