📚 Year 12 Cambridge Biology: Quick Reference Handbook of Formulas and Principles | 剑桥Year 12生物学：公式定理速查手册
Success in Cambridge Year 12 Biology requires not only understanding core concepts but also applying key formulas and principles with confidence. From calculating actual cell sizes to predicting heart function, these quantitative tools underpin practical work and exam questions. This quick-reference handbook brings together the essential equations, ratios and laws you need to memorise, explained clearly with worked applications.
在剑桥Year 12生物学中取得好成绩，不仅需要理解核心概念，还需要自信地应用关键公式和定理。从计算细胞的实际大小到预测心脏功能，这些定量工具是实验工作和考试题目中的基础。本速查手册汇集了您必须牢记的所有重要方程、比值和定律，并通过实用的示例加以清晰解释。
1. Magnification Formula | 放大倍率公式
The magnification equation links image size, actual size and magnification. It is fundamental when working with light or electron micrographs. The formula is:
放大倍率公式将镜像尺寸、实物尺寸和放大倍率联系起来。这是使用光学或电子显微照片时的基础公式：
Image size = Actual size × Magnification
This can be rearranged:
可以变形为：
Magnification = Image size ÷ Actual size or Actual size = Image size ÷ Magnification
l Always ensure that image and actual size are expressed in the same units. Typically, actual sizes are given in micrometres (µ or millimetres (mm), while image sizes may be in mm. Convert carefully using: 1 mm = 1000 µm, and 1 µm = 1000 nm.
务必确保镜像尺寸和实物尺寸使用相同的单位。通常，实物尺寸以微米（µm）或毫米（mm）表示，而镜像尺寸可能是毫米。请仔细换算：1 mm = 1000 µm，1 µm = 1000 nm。
ures 25 mm and the magnification is ×5000, the actual length = 25 mm / 5000 = 0.005 mm = 5 µm.
例如，若图中的线粒体长度为25 mm，放大倍率为5000倍，则实际长度 = 25 mm / 5000 = 0.005 mm = 5 µm。
2. Percentage Change | 百分比变化
Percentage change is frequently used to compare data before and after a treatment, or to express growth over time. The formula is:
百分比变化常用于比较处理前后的数据，或表示随时间的增长。公式为：
Percentage change = (Final value – Initial value) ÷ Initial value × 100%
A positive result indicates an increase, while a negative result indicates a decrease. When interpreting results involving osmosis experiments, egg readiness in different solutions or population changes, this simple formula gives a standardised comparison.
结果为正表示增加，结果为负表示减少。在解释渗透实验、不同溶液中卵的变化或种群变化的结果时，这个简单的公式提供了一种标准化的比较方式。
ip changes from 5.0 g to 5.8 g in a sucrose solution, the percentage change in mass = (5.8 – 5.0) / 5.0 × 100% = +16%.
例如，若马铃薯条的质量在蔗糖溶液中从5.0 g变成5.8 g，则质量变化百分比 = (5.8 – 5.0) / 5.0 × 100% = +16%。
3. Mitotic Index | 有丝分裂指数
The mitotic index is a measure of the proportion of cells actively dividing in a tissue. It is especially useful when comparing root tips or tumour cells. The formula is:
有丝分裂指数是衡量组织中处于活跃分裂状态的细胞比例的一个指标。在比较根尖或肿瘤细胞时特别有用。公式为：
Mitotic index = (Number of cells in mitosis ÷ Total number of cells observed) × 100
Cells in mitosis include those in prophase, metaphase, anaphase and telophase. Interphase nuclei are not counted. A high mitotic index indicates rapid cell division, which is typical of meristematic or cancerous tissues.
处于有丝分裂的细胞包括前期、中期、后期和末期的细胞。不将间期细胞核计算在内。有丝分裂指数高表明细胞分裂迅速，这在分生组织或癌组织中很常见。
count 12 mitotic figures in a field of 200 cells, the mitotic index = (12 / 200) × 100 = 6%.
例如，若在200个细胞的一个视野中数出12个分裂相，则有丝分裂指数 = (12 / 200) × 100 = 6%。
4. Fick's Law of Diffusion | 菲克扩散定律
Fick's Law describes the rate of diffusion across a membrane or any exchange surface. It is most commonly expressed as:
菲克定律描述了物质通过膜或任何交换表面的扩散速率。它通常表达为：
Rate of diffusion ∝ (Surface area × Concentration difference) ÷ Diffusion distance
In symbols: R ∝ (A × ΔC) / d. The faster the diffusion, the greater the surface area and concentration gradient, and the thinner the diffusion path. This explains why gas exchange surfaces – such as alveoli, gills and mesophyll cell surfaces – are large, thin and moist.
用符号表示为：R ∝ (A × ΔC) / d。扩散越快，要求表面积和浓度梯度越大，扩散路径越短。这解释了为什么气体交换表面（如肺泡、鳃和叶肉细胞表面）都大而薄且湿润。
Adaptations that maximise Fick's Law include the flattened shape of the red blood cell, the extensive branching of the tracheal system in insects, and the microvilli of the small intestine.
最大化菲克定律的适应性结构包括红细胞的扁平形状、昆虫气管系统的广泛分支以及小肠的微绒毛。
5. Water Potential and Solute Potential | 水势与溶质势
Water potential (Ψ) describes the tendency of water to move from one area to another. It is the sum of the solute potential (Ψₛ) and the pressure potential (Ψₚ). The overall equation is:
水势（Ψ）描述水分从一个区域移动到另一个区域的趋势。它是溶质势（Ψₛ）与压力势（Ψₚ）之和。总体方程为：
Ψ = Ψₛ + Ψₚ
Pure water at standard pressure has a water potential of zero. Adding solutes makes the solute potential negative, thus lowering the total water potential. For a dilute solution, the solute potential can be estimated using:
纯水在标准压力下的水势为零。加入溶质会使溶质势变为负值，从而降低总水势。对于稀溶液，溶质势可用以下公式估算：
Ψₛ = –iCRT
Here i is the ionisation constant (1 for sucrose, 2 for NaCl), C is the molar concentration (mol L⁻¹), R is the pressure constant (0.00831 kPa L mol⁻¹ K⁻¹), and T is the absolute temperature in Kelvin (typically 293 K at 20°C). The resulting Ψₛ is in kPa.
其中 i 是电离常数（蔗糖为1，NaCl为2），C 是摩尔浓度（mol L⁻¹），R 是压力常数（0.00831 kPa L mol⁻¹ K⁻¹），T 是绝对温度（开尔文，通常在20°C时为293 K）。计算得到的 Ψₛ 单位为 kPa。
In plant cells, a large positive pressure potential generated by the cell wall can offset the negative solute potential, allowing water to enter down a water potential gradient.
在植物细胞中，细胞壁产生的较高正压力势可以抵消负的溶质势，使水能够顺着水势梯度进入细胞。
6. Temperature Coefficient (Q₁₀) | 温度系数 (Q₁₀)
Q₁₀ measures how much the rate of a biological process increases when the temperature is raised by 10°C. It is particularly useful for enzyme-controlled reactions and respiration rates. The formula is:
Q₁₀ 衡量温度每升高10°C时生物过程的速率增加多少。它对酶促反应和呼吸速率的分析特别有用。公式为：
Q₁₀ = Rate at (T + 10)°C ÷ Rate at T°C
For most chemical reactions in living systems, Q₁₀ typically lies between 2 and 3 – meaning the rate roughly doubles or triples for every 10°C rise within the optimal range. Above the optimum temperature, enzymes denature, and Q₁₀ falls below 1, indicating a severe drop in rate.
对于生命系统中的大多数化学反应，Q₁₀ 通常在2到3之间，即在最适温度范围内，温度每升高10°C，速率大约增加一倍或两倍。超过最适温度后，酶会变性，Q₁₀ 会降至1以下，表明速率急剧下降。
For example, if a reaction proceeds at 15 µmol min⁻¹ at 25°C and at 30 µmol min⁻¹ at 35°C, then Q₁₀ = 30 / 15 = 2.
例如，若某反应在25°C时以15 µmol min⁻¹ 的速率进行，在35°C时以30 µmol min⁻¹ 进行，则 Q₁₀ = 30 / 15 = 2。
7. Cardiac Output | 心输出量
Cardiac output is the volume of blood pumped by one ventricle per minute. It is determined by two factors:
心输出量是每分钟一个心室泵出的血液体积。它由两个因素决定：
Cardiac output (CO) = Heart rate (HR) × Stroke volume (SV)
Heart rate is the number of beats per minute (bpm), and stroke volume is the volume pumped per beat (typically around 70 mL in a resting adult). The resulting CO is measured in mL min⁻¹ or L min⁻¹.
心率是每分钟心跳次数（bpm），每搏输出量是每次搏动泵出的体积（静息成人通常约为70 mL）。计算得到的心输出量单位为 mL min⁻¹ 或 L min⁻¹。
During exercise, both heart rate and stroke volume increase, causing cardiac output to rise dramatically, sometimes to five times the resting level. This adaptation delivers more oxygen and glucose to active muscles.
运动时，心率和每搏输出量都会增加，导致心输出量显著上升，有时可达静息水平的五倍。这一适应为活跃的肌肉提供了更多的氧气和葡萄糖。
8. Minute Ventilation | 分钟通气量
Minute ventilation measures the volume of air moved into the lungs per minute. It is analogous to cardiac output but for the respiratory system:
分钟通气量衡量每分钟进入肺部的空气体积。它与心输出量类似，但用于呼吸系统：
Minute ventilation (VE) = Breathing rate × Tidal volume
Breathing rate is the number of breaths per minute, and tidal volume is the volume of air inhaled or exhaled with each normal breath (typically about 0.5 L at rest). The product is given in L min⁻¹.
呼吸频率是每分钟呼吸次数，而潮气量是每次正常呼吸吸入或呼出的空气体积（静息时通常约为0.5 L）。两者的乘积单位为 L min⁻¹。
During vigorous activity, minute ventilation can increase from around 6 L min⁻¹ to over 100 L min⁻¹ in a trained athlete, achieved by increasing both breathing rate and tidal volume.
在剧烈运动中，分钟通气量可从约6 L min⁻¹ 增加到训练有素运动员的100 L min⁻¹ 以上，这是通过同时增加呼吸频率和潮气量来实现的。
9. Chi-squared Test | 卡方检验
The chi-squared (χ²) test is used to determine whether observed data differ significantly from expected data in genetic crosses or other categorical experiments. The formula is:
卡方（χ²）检验用于判断在遗传杂交或其他分类实验中，观察数据是否与期望数据存在显著差异。公式为：
χ² = Σ (O – E)² ÷ E
ency. The sum is taken over all categories. Degrees of freedom (df) = number of categories – lue, the null hypothesis is rejected, meaning the difference is statistically significant.
其中 O 是观察频数，E 是期望频数，并对所有类别求和。自由度（df）= 类别数 – 1。将计算得到的 χ² 值与 p = 0.05 时统计表中的临界值进行比较。如果 χ² 大于临界值，则拒绝原假设，即差异具有统计学显著性。
d with df = 1, the critical value is 3.84. Used correctly, this test lets biologists separate chance variation from real biological effects.
对于典型的3:1单因子杂交，若总共80个后代，期望值应为60个显性和20个隐性。计算出 χ² 值，当自由度为1时，临界值为3.84。正确使用这一检验可帮助生物学家区分偶然变异与真实的生物学效应。
10. Surface Area to Volume Ratio | 表面积与体积比
Although not a numeric formula, the concept of surface area to volume ratio (SA:V) is a fundamental principle governing cell size, body shapes and specialised exchange surfaces. As an organism or cell gets larger, volume increases much faster than surface area. This means that the SA:V ratio decreases, limiting the rate at which materials can enter or leave the organism.
虽然不是一个数值公式，但表面积与体积比（SA:V）的概念是控制细胞大小、体型和特化交换表面的基本原理。随着生物体或细胞增大，体积的增长速度远快于表面积。这意味着 SA:V 比值减小，限制了物质进出生物体的速率。
Small organisms, such as bacteria or single-celled protists, have a high SA:V ratio and can rely solely on simple diffusion for gas exchange and waste removal. Larger animals, however, have evolved complex transport systems (blood circulation) and specialised exchange organs (lungs, gills, villi) to bypass the constraints of a low SA:V ratio. This principle also explains why actively growing tissues have flattened or folded shapes to maximise their surface area.
小型生物，如细菌或单细胞原生生物，具有较高的 SA:V 比值，可以仅依靠简单的扩散进行气体交换和废物排泄。而较大的动物则进化出了复杂的运输系统（血液循环）和专门的交换器官（肺、鳃、绒毛）来克服低 SA:V 比值的限制。这一原理也解释了为什么活跃生长的组织具有扁平或折叠的形状，以最大限度地增加表面积。
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