📚 IGCSE Biology: Essential Formulas & Equations Handbook | IGCSE 生物：公式汇总手册
Mastering the key formulas in IGCSE Biology is essential for tackling calculations on magnification, rates, energy efficiency, and more. This handbook brings together all the essential equations, explains each variable clearly, and provides worked examples to build your confidence.
掌握 IGCSE 生物中的关键公式对于应对放大倍率、速率、能量效率等计算题至关重要。本手册汇集了所有必备公式，清晰解释每个变量，并提供典型例题，帮助你树立信心。
1. Magnification and Size | 放大倍率与尺寸
The relationship between image size, actual size, and magnification is fundamental in microscopy.
图像大小、实际大小与放大倍率之间的关系是显微镜技术的基础。
Magnification = Image size ÷ Actual size
放大倍率 = 图像尺寸 ÷ 实际尺寸
Image size is the measured length of the specimen in a drawing or micrograph, actual size is the true length of the object, both in the same units.
图像尺寸是指绘图或显微照片中测得的标本长度，实际尺寸是物体的真实长度，两者单位需一致。
To convert between millimetres (mm), micrometres (µm), and nanometres (nm): 1 mm = 1000 µm, 1 µm = 1000 nm.
进行单位换算时：1 毫米 = 1000 微米，1 微米 = 1000 纳米。
Example: A cell image measures 30 mm under a microscope, the actual cell is 0.03 mm. Magnification = 30 ÷ 0.03 = 1000×.
示例：显微镜下细胞图像为 30 毫米，实际细胞为 0.03 毫米。放大倍率 = 30 ÷ 0.03 = 1000 倍。
You may also need to rearrange: Actual size = Image size ÷ Magnification.
也可能需要转换公式：实际尺寸 = 图像尺寸 ÷ 放大倍率。
2. Rate of Reaction | 反应速率
The rate of a biological reaction (e.g. enzyme activity, photosynthesis) is often measured as the amount of product formed or substrate used per unit time.
生物反应速率（如酶活性、光合作用）通常以单位时间内生成产物的量或消耗底物的量来表示。
Rate = Change in quantity ÷ Time taken
速率 = 变化量 ÷ 所用时间
Quantity can be volume of gas produced (cm³), mass change (g), or colour change in arbitrary units.
变化量可以是产气体积（cm³）、质量变化（g）或任意单位的颜色变化。
For example, if 20 cm³ of oxygen is produced over 5 minutes, the rate = 20 ÷ 5 = 4 cm³/min.
例如，5 分钟内产生 20 cm³ 氧气，则速率 = 20 ÷ 5 = 4 cm³/min。
When plotting a graph of product vs time, the rate at any point is the gradient (slope) of the tangent to the curve.
绘制产物-时间曲线时，任意点的速率即为该点切线的斜率（梯度）。
3. Photosynthesis Rate Calculations | 光合作用速率计算
In photosynthesis experiments using aquatic plants, the rate is often measured by counting oxygen bubbles per minute or by measuring gas volume.
在使用水生植物的光合作用实验中，常用每分钟产生的氧气气泡数或气体体积来衡量光合速率。
Rate = Number of bubbles ÷ Time (min)
速率 = 气泡数 ÷ 时间（分钟）
When using a gas syringe or inverted measuring cylinder, volume of gas (cm³) collected per minute gives a more precise rate.
使用气体注射器或倒置量筒时，以每分钟收集的气体体积（cm³）表示速率更为精确。
To calculate the rate of change in carbon dioxide uptake or oxygen production: Rate = (Final volume − Initial volume) ÷ Time.
计算二氧化碳吸收或氧气释放的速率：速率 = （最终体积 − 初始体积）÷ 时间。
Controlling variables such as light intensity, temperature, and CO₂ concentration is crucial for valid comparisons.
控制光照强度、温度和 CO₂ 浓度等变量对于有效比较至关重要。
4. Respiratory Quotient (RQ) | 呼吸商（RQ）
The respiratory quotient is used to deduce which substrate is being respired aerobically.
呼吸商用于推断有氧呼吸中利用的底物类型。
RQ = Volume of CO₂ produced ÷ Volume of O₂ consumed
RQ = 产生的 CO₂ 体积 ÷ 消耗的 O₂ 体积
| Substrate 底物 | RQ value RQ 值 |
|---|---|
| Carbohydrate 碳水化合物 | 1.0 |
| Lipid 脂肪 | ~0.7 |
| Protein 蛋白质 | ~0.9 |
An RQ close to 1.0 suggests carbohydrate respiration; lower values indicate lipid or protein use.
RQ 接近 1.0 表明以碳水化合物为呼吸底物；较低的值提示脂肪或蛋白质的被利用。
It is measured using a respirometer, where CO₂ is absorbed by soda lime and the change in volume of oxygen is recorded.
RQ 使用呼吸计测量，其中的 CO₂ 被碱石灰吸收，记录氧气体积的变化。
5. Efficiency of Biomass and Energy Transfer | 生物量与能量传递效率
Efficiency of transfer between trophic levels is a key concept in ecology and food chains.
营养级之间的传递效率是生态学和食物链中的关键概念。
Efficiency (%) = (Energy or biomass in higher level ÷ Energy or biomass in lower level) × 100
效率 (%) = （较高营养级的能量或生物量 ÷ 较低营养级的能量或生物量）× 100
Biomass is usually measured as dry mass (g/m² or kg/m²); energy is in kJ/m²/year.
生物量通常以干重（g/m² 或 kg/m²）衡量；能量以 kJ/m²/年 表示。
For example, if grass contains 20 000 kJ and a cow assimilates 2 000 kJ, efficiency = (2000 ÷ 20000) × 100 = 10%.
例如，若草地含能 20 000 kJ，奶牛同化了 2 000 kJ，效率 = (2000 ÷ 20000) × 100 = 10%。
Typical transfers range from about 10% to 20%; losses occur through respiration, egestion, and uneaten parts.
典型的传递效率约为 10% 到 20%；呼吸、排泄和未摄食部分造成能量损失。
6. Population Density and Distribution | 种群密度与分布
Estimating population size is often done using quadrats and transects. The basic formula for population density is simple.
估算种群大小通常使用样方和样线法。种群密度的基本公式很简单。
Population density = Number of individuals ÷ Area sampled
种群密度 = 个体数 ÷ 取样面积
To estimate total population from a quadrat count: Estimated total = (Mean count per quadrat) × (Total area ÷ Quadrat area).
根据样方计数估算总种群数：估计总数 = 每样方平均个体数 ×（总面积 ÷ 样方面积）。
The Lincoln Index (capture-mark-recapture) is used for motile animals and is expressed as:
对于活动性动物，采用林肯指数（标记重捕法），公式为：
N = (M × C) ÷ R
N =（M × C）÷ R
Where N = population estimate, M = number initially marked, C = total caught on second occasion, R = number of marked individuals recaptured.
其中 N = 种群估算值，M = 初次标记数，C = 第二次捕获总数，R = 第二次捕获中带标记的个体数。
7. Genetic Probability (Punnett Squares) | 遗传概率（庞尼特方格）
Genetic crosses use probability rules to predict offspring ratios. The probability of a particular genotype is expressed as a fraction or percentage.
遗传杂交利用概率规则预测后代比例。特定基因型的概率以分数或百分数表示。
Probability = Number of offspring with trait ÷ Total number of offspring
概率 = 具有某性状的后代数 ÷ 后代总数
For a monohybrid cross (Aa × Aa), the Punnett square shows AA (1/4), Aa (2/4), aa (1/4).
对于单因子杂交（Aa × Aa），庞尼特方格显示：AA 占 1/4，Aa 占 2/4，aa 占 1/4。
If alleles show codominance or multiple alleles (e.g. ABO blood groups), the same probability principles apply.
如果等位基因呈共显性或多等位基因（如 ABO 血型），同样适用概率原理。
8. Surface Area to Volume Ratio | 表面积与体积比
This ratio is critical for understanding diffusion, heat loss, and why cells are microscopic. It is a simple mathematical relationship.
该比值对于理解扩散、散热以及细胞为何微小至关重要，是一种简单的数学关系。
Surface area : Volume ratio = Surface area ÷ Volume
表面积与体积比 = 表面积 ÷ 体积
For a cube-shaped organism with side length L, surface area = 6L², volume = L³, ratio = 6L²/L³ = 6/L.
对于边长为 L 的立方体生物，表面积 = 6L²，体积 = L³，比值 = 6L²/L³ = 6/L。
As an organism gets larger, the ratio decreases, meaning less surface area per unit volume for exchange – hence large organisms need specialised exchange surfaces.
随着生物体变大，比值减小，意味着单位体积用于物质交换的表面积减少——因此大型生物需要特化的交换表面。
9. Percentage Change in Mass (Osmosis Experiments) | 质量变化百分比（渗透实验）
Osmosis experiments with plant tissue often require calculating percentage change in mass to allow comparison between samples of differing initial masses.
使用植物组织的渗透实验通常需要计算质量变化百分比，以便比较初始质量不同的样本。
Percentage change in mass = ((Final mass − Initial mass) ÷ Initial mass) × 100
质量变化百分比 = (（最终质量 − 初始质量）÷ 初始质量) × 100
A positive percentage indicates water gain (lower solute concentration outside), negative indicates water loss.
正百分比表示吸水（外部溶质浓度较低），负百分比表示失水。
Plotting % change in mass against solute concentration helps determine the point of incipient plasmolysis (where no net change occurs).
以溶质浓度为横轴、质量变化百分比为纵轴作图，可确定初始质壁分离点（无净变化处）。
10. Cardiac Output and Heart Rate | 心输出量与心率
Cardiovascular calculations link heart rate, stroke volume, and cardiac output in the circulatory system.
心血管系统的计算将心率、每搏输出量和心输出量联系在一起。
Cardiac output (cm³/min) = Heart rate (beats/min) × Stroke volume (cm³/beat)
心输出量（cm³/min）= 心率（次/分）× 每搏输出量（cm³/次）
Heart rate can be determined by pulse measurement: Heart rate = Number of beats counted ÷ Time (min).
心率可通过脉搏测量得出：心率 = 计数的搏动次数 ÷ 时间（分钟）。
For example, a resting heart rate of 70 bpm and stroke volume of 70 cm³ gives a cardiac output of 4900 cm³/min.
例如，静息心率 70 次/分，每搏输出量 70 cm³，则心输出量为 4900 cm³/min。
11. Diluting Solutions for Investigations | 实验中的溶液稀释
When making concentration series for enzyme or osmosis practicals, the dilution formula is handy.
在配制酶或渗透实验的浓度梯度时，稀释公式非常实用。
C₁V₁ = C₂V₂
C₁V₁ = C₂V₂
Where C₁ = initial concentration, V₁ = volume needed of the stock, C₂ = desired final concentration, V₂ = total final volume.
其中 C₁ = 初始浓度，V₁ = 所需母液体积，C₂ = 目标最终浓度，V₂ = 最终总体积。
Example: To prepare 100 cm³ of 0.2 mol/dm³ from a 1.0 mol/dm³ stock, V₁ = (0.2 × 100) ÷ 1.0 = 20 cm³ stock plus water to 100 cm³.
示例：用 1.0 mol/dm³ 母液配制 100 cm³ 的 0.2 mol/dm³ 溶液，V₁ = (0.2 × 100) ÷ 1.0 = 20 cm³ 母液加水至 100 cm³。
12. Mean, Median, and Percentage for Data Analysis | 数据分析中的平均值、中位数与百分比
Processing data in biological investigations involves basic statistics: mean, median, and percentages are commonly required.
生物实验的数据处理涉及基本统计量：平均值、中位数和百分比是常见要求。
Mean = Sum of all values ÷ Number of values
平均值 = 所有数值之和 ÷ 数值个数
The median is the middle value when data are ordered; useful when outliers are present.
中位数是将数据排序后的中间值；存在异常值时更为适用。
Percentage increase: ((New − Original) ÷ Original) × 100.
增长百分比：((新值 – 原值) ÷ 原值) × 100。
These are essential for evaluating differences between control and experimental groups.
这些是评估对照组与实验组之间差异的基础。
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