IB WJEC Biology: Comprehensive Formula Handbook | IB WJEC 生物：公式汇总手册

📚 IB WJEC Biology: Comprehensive Formula Handbook | IB WJEC 生物：公式汇总手册

Mastering key formulae is essential for both IB Higher Level and WJEC A-level biology assessments. This handbook brings together the most important equations from cell biology, genetics, ecology, and physiology, providing clear explanations, worked examples, and bilingual support to strengthen your quantitative skills and exam confidence.

掌握关键公式对于 IB 高级水平和 WJEC A-level 生物考试都至关重要。本手册汇集了细胞生物学、遗传学、生态学和生理学中最核心的方程，提供清晰的解释、例题和中英双语支持，以增强你的量化技能与应试信心。

1. Magnification and Scale | 放大率与实际大小

Magnification is the ratio between the size of an image and the real size of the object. It is used to interpret drawings, micrographs, and electron micrographs. The formula rearranges easily to find any missing variable.

放大率是图像尺寸与物体实际尺寸的比值，用于解释手绘图、光学显微照片和电子显微照片。该公式可轻松变形以求解任一未知量。

Magnification = Image size ÷ Actual size

放大率 = 图像尺寸 ÷ 实际尺寸

metres. For instance, if an image of a chloroplast measures 20 mm and its true length is 5 µm, first convert 20 mm to 20 000 µm, then magnification = 20 000 ÷ 5 = 4000×.

务必将所有测量值转换为相同单位，通常为毫米或微米。例如，若一个叶绿体图像长度为 20 mm，其实际长度为 5 µm，应先将 20 mm 转换为 20 000 µm，然后放大率 = 20 000 ÷ 5 = 4000 倍。

ual size = Image size ÷ Magnification. This skill is frequently tested in paper 1 multiple‑choice and paper 2 data‑response questions.

计算实际尺寸时，可变形为 实际尺寸 = 图像尺寸 ÷ 放大率。这一技能在试卷一选择题和试卷二数据分析题中屡见不鲜。

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

The Hardy‑Weinberg model describes allele and genotype frequencies in a population that is not evolving. It provides a null hypothesis for detecting evolutionary change.

哈迪‑温伯格模型描述了不演化种群中的等位基因频率和基因型频率，为检测演化变化提供了零假设。

p + q = 1   and   p² + 2pq + q² = 1

p + q = 1   以及   p² + 2pq + q² = 1

Here p represents the frequency of the dominant allele, q the frequency of the recessive allele. p² is the frequency of homozygous dominant individuals, 2pq of heterozygotes, and q² of homozygous recessives. These equations hold only under five conditions: large population, random mating, no mutations, no gene flow, and no natural selection.

其中 p 代表显性等位基因的频率，q 为隐性等位基因频率。p² 为纯合显性个体的频率，2pq 为杂合子频率，q² 为纯合隐性个体频率。这些等式仅在满足五大条件时成立：大种群、随机交配、无突变、无基因流动、无自然选择。

ulation of 1000 individuals, 160 show a recessive trait. Thus q² = 160/1000 = 0.16, so q = √0.16 = 0.4. Then p = 1 – individuals are carriers. Such calculations appear in both IB and WJEC genetics questions.

例题：在一个 1000 人的种群中，160 人表现出隐性性状。因此 q² = 160/1000 = 0.16，故 q = √0.16 = 0.4。则 p = 1 – 0.4 = 0.6。携带者频率 2pq = 2 × 0.6 × 0.4 = 0.48，即 480 人为携带者。此类计算同时出现在 IB 与 WJEC 的遗传学考题中。

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3. Respiratory Quotient (RQ) | 呼吸商

The respiratory quotient indicates the type of substrate being respired and can be used to infer metabolic activity. It is calculated as the ratio of carbon dioxide produced to oxygen consumed.

呼吸商可指示被呼吸的底物类型，并用于推断代谢活动。其计算式为产生的二氧化碳与消耗的氧气的比值。

RQ = CO₂ produced ÷ O₂ consumed

RQ = 产生的 CO₂ ÷ 消耗的 O₂

carbohydrate respiration, RQ = 1.0; for lipid respiration, RQ ≈ 0.7; for protein, RQ ≈ 0.8– of carbohydrate to lipid. Measurements are made using a respirometer with a CO₂ absorber.

纯碳水化合物呼吸时，RQ = 1.0；脂质呼吸时 RQ 约 0.7；蛋白质约 0.8–0.9。RQ 值高于 1.0 暗示无氧呼吸或碳水化合物向脂质的转换。测量时需使用装有 CO₂ 吸收剂的呼吸计。

In experiments, the volume change of air in a respirometer over time yields O₂ consumption. If a separate tube without KOH is used, the net gas change gives (CO₂ produced – O₂ consumed), and combining the two allows RQ to be determined. Both IB and WJEC practical assessments may ask for this calculation.

实验中，呼吸计内空气体积随时间的变化可得出 O₂ 消耗量。若另设一管不加 KOH，净气体变化给出（产生的 CO₂ – 消耗的 O₂），两者结合即可求算 RQ。IB 和 WJEC 的实操评估都可能要求此项计算。

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4. Net Primary Production (NPP) | 净初级生产量

Productivity is a core concept in ecology, describing energy flow through ecosystems. Net primary production is the energy remaining after autotrophs have met their own respiratory needs.

生产力是生态学核心概念，描述生态系统中的能量流动。净初级生产量是自养生物满足自身呼吸需求后剩余的能量。

NPP = GPP – R

NPP = GPP – R

GPP (gross primary production) is the total light energy converted to chemical energy by photosynthesis. R is the respiratory energy loss. NPP is the biomass available to consumers. Units are typically kJ m⁻² yr⁻¹ or g m⁻² yr⁻¹.

GPP（总初级生产量）是通过光合作用转化为化学能的总光能。R 为呼吸能量损失。NPP 为可供消费者利用的生物量，单位常用 kJ m⁻² yr⁻¹ 或 g m⁻² yr⁻¹。

el ÷ energy ingested) × 100. Students must be comfortable with unit conversions and chain calculations in IB data‑based questions and WJEC extended response.

随后常进行效率计算：净生产效率 = (NPP ÷ GPP) × 100，或生态效率 = (传递至下一营养级的能量 ÷ 摄入能量) × 100。学生需在 IB 数据题和 WJEC 拓展应答中熟练进行单位换算与连锁计算。

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5. Lincoln Index (Capture‑Mark‑Recapture) | 林可指数（标记重捕法）

The Lincoln index estimates population size for motile organisms. It relies on the assumption that marked and unmarked individuals mix randomly.

林可指数用于估算移动生物种群的大小，其前提是标记与未标记个体随机混合。

Population size N = (M × C) ÷ R

种群大小 N = (M × C) ÷ R

M is the number initially captured, marked, and released; C is the number captured in the second sample; R is the number of marked individuals recaptured in the second sample.

M 为初次捕获、标记并释放的个体数；C 为第二次取样捕获的个体数；R 为第二次取样中重捕到的标记个体数。

Assumptions include: no births, deaths, immigration or emigration between samples; marks are not lost and do not affect survival; and sampling is random. If these are violated, estimates lose accuracy. WJEC often asks for evaluation of the method, while IB may embed the calculation in data analysis.

假设条件包括：两次取样之间无出生、死亡、迁入或迁出；标记不丢失且不影响存活；取样随机。若违背条件，估算值将失准。WJEC 常要求学生评估方法，而 IB 可能将计算融入数据分析。

marked individuals, then N = (40 × 50) ÷ 10 = 200. This simple ratio is worth memorising.

例如，若标记释放 40 只潮虫，随后取样 50 只中有 10 只带标记，则 N = (40 × 50) ÷ 10 = 200。此简单比值值得牢记。

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6. Simpson's Diversity Index | 西蒙斯多样性指数

Simpson's index quantifies biodiversity by accounting for both species richness and evenness. A higher value indicates greater diversity.

西蒙斯指数通过考虑物种丰富度和均匀度来量化生物多样性，指数越高代表多样性越大。

D = 1 – Σ (n/N)²

D = 1 – Σ (n/N)²

ty) to 1 (high diversity). An alternative form D = N(N−1) / Σ n(n−1) is sometimes used; exam questions will specify which version to apply.

其中 n 为某一特定物种的个体数，N 为所有物种的总个体数。D 值范围从 0（低多样性）到 1（高多样性）。有时也会用到 D = N(N−1) / Σ n(n−1) 的形式；考题会指明使用哪个版本。

To calculate: list each species, square each n, sum these squares, divide by N², and subtract from 1. Many IB and WJEC practical assessments involve sampling with quadrats or pitfall traps and then calculating D to compare habitats.

计算步骤：列出每个物种，平方各自的 n，求和，除以 N²，再从 1 中减去。许多 IB 和 WJEC 实践评估都涉及样方或陷阱取样，随后计算 D 来比较栖息地。

Interpreting the index requires linking high diversity to ecosystem stability and resilience, a common discussion point in longer-answer sections.

解读指数时需将高多样性与生态系统稳定性和恢复力联系起来，这是长答题部分常见的讨论要点。

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7. Exponential Growth Model | 指数增长模型

When resources are unlimited, populations grow exponentially. This pattern is described mathematically and appears in microbial growth curves and introductory ecology.

当资源无限时，种群呈指数增长。这一模式可用数学描述，常见于微生物生长曲线和生态学入门内容。

Nₜ = N₀ × 2ⁿ   or   Nₜ = N₀ e^rt

Nₜ = N₀ × 2ⁿ   或   Nₜ = N₀ e^rt

ate, and e is the base of natural logarithms. For bacteria, the doubling time formula is commonly used: Nₜ = N₀ × 2^t/t_double.

N₀ 为起始种群大小，Nₜ 为 t 时刻后的种群大小，n 为世代数（用于二分裂），r 为内禀增长率，e 为自然对数底数。对于细菌，常用倍增时间公式：Nₜ = N₀ × 2^t/t_double。

Students should be able to linearise such data using logarithms to estimate growth rates.

若细菌培养起始为 100 个细胞，每 30 分钟倍增一次，3 小时后（6 次倍增）N = 100 × 2⁶ = 6400 个细胞。学生应能通过取对数将数据线性化，以估算增长率。

While exponential growth is unrealistic long-term, it underpins the early phase of sigmoid population curves and is essential for understanding r‑selected strategies.

虽然指数增长在长期是不现实的，但它支撑着 S 形种群曲线的早期阶段，对理解 r‑选择策略至关重要。

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8. Chi‑Squared (χ²) Test | 卡方检验

The chi‑squared test is a statistical tool for comparing observed and expected frequencies. In genetics, it checks whether offspring ratios match Mendelian predictions.

卡方检验是用于比较观测频率和期望频率的统计工具。在遗传学中，它用以检验子代表现型比率是否符合孟德尔预期。

χ² = Σ (O – E)² ÷ E

χ² = Σ (O – E)² ÷ E

The sum is calculated over all categories. Degrees of freedom (df) = number of categories – ² value is compared against a critical value at a chosen probability level (usually 0.05).

O 为观测数，E 为期望数。对所有类别求和，自由度（df）= 类别数 – 1。将算得的 χ² 值与选定概率水平（通常为 0.05）下的临界值进行比较。

If χ² < critical value, accept the null hypothesis; if χ² > critical value, reject the null hypothesis, indicating a significant difference. Both IB and WJEC ask students to perform this test, state conclusions, and suggest reasons for deviations.

若 χ² < 临界值，接受零假设；若 χ² > 临界值，拒绝零假设，表明存在显著差异。IB 和 WJEC 均要求学生完成此项检验，陈述结论，并提出偏差的可能原因。

Example: In a dihybrid cross, observed phenotypes are 315, 108, 101, 32; expected 9:3:3:1 ratio gives 312.75, 104.25, 104.25, 34.75. χ² calculation yields a low value, confirming the data fit Mendel's second law. Always show your working.

示例：一项双因子杂交，观测表型为 315、108、101、32；预期 9:3:3:1 比例给出 312.75、104.25、104.25、34.75。卡方计算得到较小值，证实数据符合孟德尔第二定律。务必展示计算过程。

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9. Cardiac Output | 心输出量

Cardiac output is a fundamental physiological parameter that describes the volume of blood pumped by the heart per minute. It links heart rate and stroke volume.

心输出量是一项基本生理参数，描述心脏每分钟泵出的血量，它将心率与每搏输出量联系起来。

Cardiac Output (CO) = Heart Rate (HR) × Stroke Volume (SV)

心输出量 (CO) = 心率 (HR) × 每搏输出量 (SV)

g cardiac output ≈ 70 bpm × 70 mL = 4900 mL/min ≈ 5 L/min. These values can rise dramatically during exercise.

心率以每分钟搏动次数计；每搏输出量是每个心室每次搏动的射血量，成人静息时通常约 70 mL。因此静息心输出量 ≈ 70 bpm × 70 mL = 4900 mL/min ≈ 5 L/min。运动时这些数值会大幅增高。

nitoring traces or from data tables, such as determining stroke volume after measuring end‑diastolic and end‑systolic volumes (SV = EDV – ncludes this in the circulatory system topic, and IB may present it in option D or in data‑based questions.

考题可能要求根据心电监测轨迹或数据表进行计算，如通过测量舒张末期容积与收缩末期容积来求每搏输出量（SV = EDV – ESV）。WJEC 通常在循环系统课题中包含此内容，IB 可能在选修 D 或数据分析题中呈现。

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10. Water Potential | 水势

Water potential (Ψ) is the measure of free energy of water and determines the direction of water movement in plant and animal cells. It is central to understanding osmosis.

水势（Ψ）是水自由能的度量，决定了水分在植物和动物细胞中的移动方向，是理解渗透作用的核心。

Ψ = Ψₛ + Ψₚ

水势 = 溶质势 + 压力势

hysical pressure on the cell contents, often positive when a cell wall exerts turgor pressure. Pure water at atmospheric pressure has Ψ = 0 MPa.

Ψₛ（溶质势或渗透势）在活细胞中恒为负值，因溶质降低了自由能；Ψₚ（压力势）是施加在细胞内容物上的物理压力，当细胞壁施加膨压时常为正值。纯水在大气压下的水势为 0 MPa。

In a turgid plant cell, Ψₛ is negative but balanced by a positive Ψₚ, giving a total Ψ of zero or slightly negative. When tissue is placed in a solution, water moves from higher Ψ to lower Ψ. This principle underlies the classic potato cylinder osmosis experiments.

在膨胀的植物细胞中，Ψₛ 为负值，但它被正值的 Ψₚ 平衡，总水势 Ψ 为零或略负。当组织浸入溶液时，水分从水势较高处移向水势较低处。这一原理是经典土豆条渗透实验的基础。

he ionisation constant, C is molar concentration, R is the pressure constant and T is temperature in kelvin. Both IB and WJEC may ask for simple numerical applications or qualitative predictions.

计算时常需将溶质浓度换算为溶质势，使用公式 Ψₛ = −iCRT，其中 i 为电离常数，C 为摩尔浓度，R 为压力常数，T 为开尔文温度。IB 和 WJEC 都可能要求简单的数值应用或定性预测。

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

The surface area to volume ratio (SA:V) governs the rate of exchange and heat loss in organisms. As an organism or cell increases in size, the SA:V decreases, imposing constraints on diffusion.

表面积与体积比（SA:V）支配着生物体内的物质交换速率和热量散失。随着生物体或细胞增大，SA:V 下降，从而限制扩散效率。

SA:V = Surface Area ÷ Volume

SA:V = 表面积 ÷ 体积

ssed per unit length, e.g. 6:1 for a 1 cm cube becomes 3:1 for a 2 cm cube. This simple geometry explains why large organisms require specialised exchange systems.

对于一个边长为 L 的立方体，SA = 6L²，V = L³，因而 SA:V = 6/L。该比值以单位长度表示，例如边长 1 cm 的立方体为 6:1，边长 2 cm 即变为 3:1。这一简单几何关系解释了为何大型生物需要特化的交换系统。

apes approximated by blocks. Students should be able to deduce the effect of size on heat retention and metabolic rate, a common IB and WJEC topic linking cell biology to physiology.

计算常涉及球体（SA = 4πr²，V = ⁴⁄₃πr³）或用块状结构近似的不规则形状。学生应能推导体型大小对保温能力和代谢率的影响，这一主题将细胞生物学与生理学相连，常见于 IB 和 WJEC 试题。

Adaptations like villi, alveoli, and lamellae all increase SA:V to maximise exchange—concepts examined through data interpretation and diagram analysis.

绒毛、肺泡、鳃片等适应性结构均通过增大 SA:V 来最大化交换——这些概念常通过数据解读和图解分析进行考查。

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