Year 13 Edexcel Biology: Formula and Theorem Quick Reference Handbook

📚 Year 13 Edexcel Biology: Formula and Theorem Quick Reference Handbook | 公式定理速查手册

This quick reference handbook presents all the essential formulas, equations, and key principles you need to master for the Edexcel A Level Biology Year 13 course. From magnification and surface area to volume ratios to statistical tests and Hardy‑Weinberg equilibrium, each section provides the formula, explains the meaning of symbols, and clarifies how to apply them in examination questions. Use this guide to consolidate your revision and boost your confidence in calculation‑based topics.

本速查手册汇集了 Edexcel A Level 生物 Year 13 课程必须掌握的所有核心公式、方程和重要定理。从放大倍数与表面积体积比，到统计学检验和哈迪‑温伯格平衡，每个小节均给出公式、符号含义，并阐明如何在考试题目中应用。用这份指南巩固复习，提升你在计算类题目中的信心。

1. Magnification and Scale Bar Calculations | 放大倍数与比例尺计算

Magnification is the number of times larger an image is than the actual object. The formula is: Magnification = Image size / Actual size (M = I / A). Image size and actual size must be in the same units; convert all measurements to micrometres (µm) or millimetres (mm) before calculating. For scale bars, measure the length of the bar in the image and divide by the value it represents in real units to find the magnification.

放大倍数是图像比实际物体大的倍数。公式为：放大倍数 = 图像大小 / 实际大小 (M = I / A)。图像大小和实际大小必须使用相同的单位；计算前通常将所有测量值转换为微米 (µm) 或毫米 (mm)。对于比例尺，测量图像中比例尺的长度，除以它所代表的实际长度，即可得到放大倍数。

If an eyepiece graticule is calibrated with a stage micrometer, each eyepiece unit (epu) corresponds to a known length. Actual size = (number of epu) × (calibration value per epu).

若用目镜测微尺配合镜台测微尺校准，每个目镜刻度单位 (epu) 对应已知长度。实际大小 = epu 数 × 每 epu 校准值。

M = I / A

2. Surface Area to Volume Ratio | 表面积与体积比

The surface area to volume ratio (SA:V) decreases as an object increases in size. For a cube: SA = 6 × side², V = side³, so SA:V = 6 / side. For a sphere: SA = 4πr², V = (4/3)πr³, so SA:V = 3 / r. A large SA:V is favourable for efficient diffusion, osmosis, and heat exchange – this explains why organisms often have flattened, folded or highly branched structures (e.g. alveoli, villi, root hairs).

表面积与体积比 (SA:V) 随着物体体积增大而减小。对于立方体：表面积 = 6 × 边长²，体积 = 边长³，因此 SA:V = 6 / 边长。对于球体：表面积 = 4πr²，体积 = (4/3)πr³，所以 SA:V = 3 / r。较大的 SA:V 有利于高效扩散、渗透和热量交换——这解释了为什么生物体常具有扁平、折叠或高度分枝的结构（如肺泡、绒毛、根毛）。

3. Fick’s Law of Diffusion | 菲克扩散定律

Fick’s Law describes the rate of diffusion across a membrane or tissue. The rate of diffusion is proportional to (surface area × concentration difference) / diffusion distance.

菲克定律描述了跨膜或跨组织的扩散速率。扩散速率与 (表面积 × 浓度差) / 扩散距离成正比。

Rate of diffusion ∝ (Surface area × ΔC) / Thickness of exchange surface

To maximise diffusion, organisms increase surface area (e.g. microvilli), maintain a steep concentration gradient (ventilation, blood flow), and minimise the distance (thin epithelia). In the lungs, the alveolar epithelium and capillary endothelium are both only one cell thick.

为使扩散最大化，生物体增大表面积（如微绒毛），维持陡峭的浓度梯度（通风、血流），并缩短扩散距离（薄的上皮组织）。在肺中，肺泡上皮和毛细血管内皮都只有一层细胞厚。

4. Water Potential (Ψ) | 水势 (Ψ)

Water potential (Ψ) measures the tendency of water to move from one region to another. Pure water has a water potential of 0 kPa under standard conditions. The addition of solutes lowers water potential (makes it more negative). Water moves from an area of higher (less negative) water potential to an area of lower (more negative) water potential. The water potential of a plant cell is given by:

水势 (Ψ) 衡量水分从一个区域移动到另一个区域的趋势。纯水在标准条件下的水势为 0 kPa。溶质的加入会降低水势（使其变得更负）。水分从水势较高（较不负）的区域向水势较低（较负）的区域移动。植物细胞的水势可表示为：

Ψ = Ψₛ + Ψₚ

Where Ψₛ = solute (osmotic) potential, always negative or zero; Ψₚ = pressure potential, usually positive inside a turgid cell, zero in a flaccid cell. In plasmolysis, Ψₚ becomes negative as the plasma membrane pulls away from the cell wall.

其中 Ψₛ = 溶质势（渗透势），总是负值或零；Ψₚ = 压力势，在硬挺细胞中通常为正值，在萎蔫细胞中为零。质壁分离时，由于质膜与细胞壁分离，Ψₚ 变为负值。

5. Cardiac Output and Minute Ventilation | 心输出量与每分钟通气量

Cardiac output (CO) is the volume of blood pumped by one ventricle of the heart in one minute. It is calculated as:

心输出量 (CO) 是指心脏一个心室在一分钟内泵出的血液量。计算公式为：

Cardiac output (cm³ min⁻¹) = Stroke volume × Heart rate

Stroke volume is the volume of blood pumped per beat (cm³). Heart rate is beats per minute (bpm). Remember to multiply units correctly – often expressed in dm³ min⁻¹ by dividing cm³ min⁻¹ by 1000.

每搏输出量是每次心跳泵出的血量 (cm³)。心率为每分钟心跳次数 (bpm)。注意单位换算——结果常以 dm³ min⁻¹ 表示，需将 cm³ min⁻¹ 除以 1000。

Minute ventilation (pulmonary ventilation) is the total volume of air breathed in and out in one minute:

每分钟通气量（肺通气量）是一分钟内吸入和呼出的空气总量：

Minute ventilation (dm³ min⁻¹) = Tidal volume × Breathing rate

Tidal volume is the volume of air moved in or out per breath at rest. Breathing rate is the number of breaths per minute.

潮气量是静息时每次呼吸吸入或呼出的空气量。呼吸速率是每分钟呼吸次数。

6. Respiratory Quotient (RQ) | 呼吸商 (RQ)

The respiratory quotient is the ratio of carbon dioxide produced to oxygen consumed during respiration over a given time period:

呼吸商是在一定时间内，呼吸作用产生的二氧化碳量与消耗的氧气量之比：

RQ = CO₂ produced / O₂ consumed

RQ values reflect the respiratory substrate being used. For carbohydrates, RQ ≈ 1; for lipids, RQ ≈ 0.7; for proteins, RQ ≈ 0.9. Values outside the range 0.7–1.0 may indicate mixed substrate use or anaerobic respiration. RQ is determined using a respirometer – CO₂ production is measured by a change in gas volume when CO₂ is absorbed by KOH.

RQ 值可以反映正在被使用的呼吸底物。碳水化合物的 RQ ≈ 1，脂类的 RQ ≈ 0.7，蛋白质的 RQ ≈ 0.9。超出 0.7–1.0 范围的值可能表示混合底物使用或有无氧呼吸参与。RQ 可用呼吸计测定——用 KOH 吸收 CO₂，通过气体体积变化可求得 CO₂ 的产生量。

7. Productivity: GPP and NPP | 生产力：总初级生产力与净初级生产力

In an ecosystem, gross primary productivity (GPP) is the total chemical energy converted from light energy by plants per unit area per year. Net primary productivity (NPP) is the energy remaining after respiratory losses (R):

在生态系统中，总初级生产力 (GPP) 是植物每年每单位面积将光能转化为化学能的总量。净初级生产力 (NPP) 是扣除呼吸消耗 (R) 后剩余的能量：

NPP = GPP – R

Units are usually kJ m⁻² yr⁻¹. NPP represents the energy available to the next trophic level. For consumers, net production is calculated as: N = I – (F + R), where I = ingested chemical energy, F = energy lost in faeces, and R = respiratory losses.

单位通常为 kJ m⁻² yr⁻¹。NPP 代表可供下一个营养级利用的能量。对消费者而言，净生产量计算公式为：N = I – (F + R)，其中 I = 摄入的化学能，F = 粪便中损失的能量，R = 呼吸消耗。

8. Lincoln Index (Capture‑Mark‑Recapture) | 林肯指数 (标记重捕法)

The Lincoln Index estimates the total population size of a motile species. A sample of individuals (n₁) is captured, marked, and released. After allowing time for reintegration, a second sample (n₂) is captured, and the number of marked individuals in that sample (m₂) is recorded. The population size (N) is estimated as:

林肯指数用于估算移动性物种的种群总数。先捕获、标记并释放 n₁ 个个体。待其重新混合后，再捕获 n₂ 个个体，记录其中带有标记的个体数 m₂。种群数量 (N) 的估算公式为：

N = n₁ × n₂ / m₂

Assumptions include: no migration, no births or deaths during the study, marks are not lost or overlooked, and marking does not affect survival or recapture probability. The method becomes less reliable if the population is very small or if marked individuals become trap‑shy.

假设包括：研究期间无迁入迁出、无出生死亡；标记不脱落也不被忽略；标记不影响存活率或重捕概率。若种群非常小或已标记个体产生陷阱回避行为，该方法的可靠性会下降。

9. Hardy‑Weinberg Equilibrium | 哈迪‑温伯格平衡

Hardy‑Weinberg equilibrium describes the allele and genotype frequencies in a population that is not evolving. For a single gene with two alleles (dominant A and recessive a), the allele frequencies are p (frequency of A) and q (frequency of a), such that:

哈迪‑温伯格平衡描述了一个不处于进化中的种群的等位基因和基因型频率。对于具有两个等位基因（显性 A 和隐性 a）的单基因，等位基因频率分别为 p (A 的频率) 和 q (a 的频率)，满足：

p + q = 1

The genotype frequencies are then:

基因型频率则为：

p² + 2pq + q² = 1

Where p² = frequency of homozygous dominant (AA), 2pq = frequency of heterozygous (Aa), q² = frequency of homozygous recessive (aa). If any genotype frequency is known, the others can be calculated. Use q = √q² to find allele frequency when the recessive phenotype is expressed. The equilibrium holds only if there is no mutation, no migration, random mating, a large population size, and no natural selection.

其中 p² = 显性纯合子 (AA) 的频率，2pq = 杂合子 (Aa) 的频率，q² = 隐性纯合子 (aa) 的频率。若已知任一种基因型频率，即可推算出其余频率。当存在隐性表型时，可用 q = √q² 求得等位基因频率。平衡仅适用于无突变、无迁移、随机交配、大种群且无自然选择的情况。

10. Chi‑squared (χ²) Test | 卡方 (χ²) 检验

The chi‑squared test is used to determine whether there is a significant difference between observed (O) and expected (E) frequencies in categorical data (e.g. phenotypic ratios in a genetic cross):

卡方检验用于判断分类数据中观测频数 (O) 与期望频数 (E) 之间是否存在显著差异（如遗传杂交中的表型比率）：

χ² = Σ (O – E)² / E

Calculate (O – E)² / E for each category, sum these values to obtain χ². Degrees of freedom (df) = number of categories – 1 (or more for contingency tables). Compare the calculated χ² value with a critical value from the chi‑squared distribution table at p=0.05. If χ² > critical value, reject the null hypothesis – the difference is significant and not due to chance alone.

对每个类别计算 (O – E)² / E，求和即得 χ²。自由度 (df) = 类别数 – 1（列联表更复杂）。将算得的 χ² 值与 p=0.05 下卡方分布表中的临界值比较。若 χ² > 临界值，则拒绝零假设——差异显著，不能仅归因于随机。

11. Student’s t‑test | 学生 t 检验

The unpaired (independent) t‑test is used to compare the means of two sets of normally distributed, continuous data with similar variances. The formula for the t‑statistic is:

非配对（独立）t 检验用于比较两组服从正态分布、方差相近的连续数据的均值。t 统计量的公式为：

t = (x̄₁ – x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Where x̄₁ and x̄₂ are the sample means, s₁² and s₂² are the sample variances, n₁ and n₂ are the sample sizes. Degrees of freedom = n₁ + n₂ – 2. Compare the calculated |t| with the critical value at p=0.05. If |t| > critical value, the means are significantly different. For paired data, a paired t‑test (difference test) is used, requiring a slightly different calculation.

其中 x̄₁ 和 x̄₂ 为样本均值，s₁² 和 s₂² 为样本方差，n₁ 和 n₂ 为样本大小。自由度 = n₁ + n₂ – 2。将计算所得的 |t| 值与 p=0.05 下的临界值比较。若 |t| > 临界值，则均值存在显著差异。对于配对数据，需使用配对 t 检验，计算稍有不同。

12. Spearman’s Rank Correlation Coefficient | 斯皮尔曼秩相关系数

Spearman’s rank test measures the strength and direction of association between two ranked variables. The coefficient rₛ is calculated as:

斯皮尔曼秩检验用于衡量两个排序变量之间关联的强度和方向。相关系数 rₛ 的计算公式为：

rₛ = 1 – (6 Σ D²) / (n(n² – 1))

D is the difference between the ranks of each pair of observations, and n is the number of paired observations. rₛ ranges from +1 (perfect positive correlation) to –1 (perfect negative correlation). Compare the absolute value of rₛ with the critical value in the Spearman’s rank table for the given n and p=0.05. If |rₛ| > critical value, the correlation is statistically significant.

D 为每对观测值的秩次差，n 为配对数据的对数。rₛ 的值域为 +1（完全正相关）至 –1（完全负相关）。将 |rₛ| 与给定 n 和 p=0.05 下的斯皮尔曼秩相关表中的临界值比较。若 |rₛ| > 临界值，则该相关性具有统计学显著性。

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Year 13 Edexcel Biology: Formula and Theorem Quick Reference Handbook | 公式定理速查手册