📚 Year 12 CIE Biology: Formula & Theorem Quick Reference Handbook | Year 12 CIE 生物：公式定理速查手册
This quick reference handbook brings together all the essential equations, relationships and quantitative principles required for the Year 12 Cambridge International AS Biology course. Whether you are preparing for Paper 2 structured questions or Paper 3 practical skills, mastering these tools will help you handle calculations confidently and explain biological phenomena with precision.
本速查手册汇集了 Year 12 剑桥国际 AS 生物学课程中所有关键的公式、关系式和定量原理。无论你是在备战 Paper 2 结构化试题还是 Paper 3 实验技能，掌握这些工具都将帮助你自信地应对计算题，并精准地解释生物现象。
1. Microscope Magnification and Actual Size | 显微镜放大率与实际尺寸
The relationship between magnification, image size and actual size is fundamental in microscopy. Always convert all measurements to the same unit before substituting into the formula.
放大率、图像尺寸和实际尺寸之间的关系是显微学的基础。在代入公式前，务必将所有测量值转换为同一单位。
Magnification = Image size ÷ Actual size
放大率 = 图像尺寸 ÷ 实际尺寸
0 mm at a magnification of ×400, the actual size is 20 mm ÷ 400 = 0.05 mm, which equals 50 µm.
该式可变形为：实际尺寸 = 图像尺寸 ÷ 放大率。若显微照片中一个细胞在×400 下测得长度为 20 mm，则实际尺寸为 20 mm ÷ 400 = 0.05 mm，即 50 µm。
2. Unit Conversions in Biology | 生物学中的单位换算
Many calculations require conversions between metres, centimetres, millimetres, micrometres and nanometres. Knowing these relationships eliminates common mistakes.
许多计算需要在米、厘米、毫米、微米和纳米之间进行换算。熟记这些关系可以避免常见错误。
1 cm = 10⁻² m; 1 mm = 10⁻³ m; 1 µm = 10⁻⁶ m; 1 nm = 10⁻⁹ m
1 cm = 10⁻² m; 1 mm = 10⁻³ m; 1 µm = 10⁻⁶ m; 1 nm = 10⁻⁹ m
For example, a mitochondrion that is 0.002 mm wide can be expressed as 2 µm. When using the magnification formula, always select the same unit for image size and actual size – perhaps both in mm or both in µm.
例如，一个宽 0.002 mm 的线粒体可表示为 2 µm。使用放大率公式时，图像尺寸与实际尺寸必须采用同一单位——比如都用 mm 或都用 µm。
3. Fick's Law of Diffusion | 菲克扩散定律
Fick's Law describes the rate of diffusion across a membrane. It is especially important when explaining adaptations of gas exchange surfaces, such as alveoli and gill lamellae.
菲克定律描述了跨膜扩散的速率。在解释气体交换表面（如肺泡和鳃瓣）的适应性时，这一定律尤为重要。
Rate of diffusion ∝ (Surface area × Concentration difference) ÷ Thickness of membrane
扩散速率 ∝ (表面积 × 浓度差) ÷ 膜厚度
A large surface area, a steep concentration gradient and a thin diffusion path all increase the rate of diffusion. In CIE exam answers, linking structural features to each factor earns full marks.
较大的表面积、陡峭的浓度梯度和较短的扩散路径都能提高扩散速率。在 CIE 考试答案中，将结构特征与各个因素联系起来方能获得满分。
4. Surface Area to Volume Ratio | 表面积与体积之比
The surface area to volume ratio (SA : V) dictates how efficiently an organism can exchange materials with its environment. As size increases, SA : V decreases, placing limits on diffusion alone.
表面积与体积之比（SA : V）决定了生物体与周围环境进行物质交换的效率。随着体积增大，SA : V 下降，从而限制了单纯依赖扩散的能力。
SA : V = Surface area ÷ Volume
SA : V = 表面积 ÷ 体积
al decrease explains why large multicellular organisms need specialised transport systems.
一个边长为 1 cm 的立方体，表面积 = 6 cm²，体积 = 1 cm³，SA : V = 6 : 1。边长为 3 cm 的立方体，表面积 = 54 cm²，体积 = 27 cm³，SA : V = 2 : 1。这一数学上的下降解释了为何大型多细胞生物需要专门的运输系统。
5. Water Potential | 水势
Water potential (ψ) is the measure of the tendency of water to move from one region to another. It is a key concept in osmosis and plant cell turgor.
水势（ψ）衡量水从一个区域向另一区域移动的趋势。这是渗透作用和植物细胞膨压中的关键概念。
ψ = ψₛ + ψₚ
水势 = 溶质势 + 压力势
The solute potential (ψₛ) is always negative or zero, while the pressure potential (ψₚ) can be positive (turgid cell) or zero (flaccid cell, plasmolysed cell). Water moves from an area of higher (less negative) water potential to an area of lower (more negative) water potential.
溶质势（ψₛ）总是负值或零，压力势（ψₚ）可为正值（膨压细胞）或零（萎蔫细胞、质壁分离细胞）。水从较高（负值较小）的水势区域向较低（负值更大）的水势区域移动。
6. Mitotic Index | 有丝分裂指数
The mitotic index is used to estimate the proportion of cells undergoing mitosis in a tissue. It is often calculated to assess the growth rate of root tips or to detect cancerous tissues.
有丝分裂指数用于估算组织中处于有丝分裂阶段的细胞比例。通常被用来评估根尖的生长速率或检测癌变组织。
Mitotic index = (Number of cells in mitosis ÷ Total number of cells observed) × 100
有丝分裂指数 = (正在分裂的细胞数 ÷ 观察到的细胞总数) × 100
at prophase, metaphase, anaphase or telophase, the mitotic index is (18 ÷ 120) × 100 = 15 %.
结果以百分数表示。例如，计数 120 个细胞，其中 18 个在前期、中期、后期或末期出现可见的染色体，则有丝分裂指数 = (18 ÷ 120) × 100 = 15 %。
7. Cardiac Output | 心输出量
Cardiac output is the volume of blood pumped by one ventricle per minute. It is a direct measure of heart performance and is commonly examined in relation to exercise.
心输出量是指一个心室每分钟泵出的血液体积。它是衡量心脏机能的直接指标，常结合运动进行考查。
Cardiac output = Heart rate × Stroke volume
心输出量 = 心率 × 每搏输出量
Heart rate is the number of beats per minute; stroke volume is the volume of blood ejected per beat. During strenuous exercise, both heart rate and stroke volume increase, raising cardiac output to deliver more oxygen to muscles. Typical resting values are 70 bpm and 70 cm³, giving a cardiac output of about 4.9 dm³ min⁻¹.
心率为每分钟跳动次数，每搏输出量为每次搏动射出的血容量。剧烈运动时，心率和每搏输出量均上升，从而增大心输出量，向肌肉输送更多氧气。静息时一般心率为 70 bpm，每搏输出量为 70 cm³，心输出量约为 4.9 dm³ min⁻¹。
8. Respiratory Quotient (RQ) | 呼吸商
Respiratory quotient indicates the type of respiratory substrate being metabolised. It is calculated from the volumes of carbon dioxide produced and oxygen consumed.
呼吸商指示正在被代谢的呼吸底物类型。它由产生的二氧化碳体积与消耗的氧气体积之比求得。
RQ = Volume of CO₂ produced ÷ Volume of O₂ consumed
RQ = 产生的 CO₂ 体积 ÷ 消耗的 O₂ 体积
For carbohydrate respiration RQ is close to 1.0; for lipid it is about 0.7; for protein it is around 0.9. When a respirometer is used with soda lime to absorb CO₂, the distance moved by a manometer fluid can be used to determine O₂ consumption and, in a parallel set‑up without soda lime, to find CO₂ production.
碳水化合物的呼吸商接近 1.0；脂质约 0.7；蛋白质约 0.9。当使用含有钠石灰（吸收 CO₂）的呼吸计，可通过压力计液柱的移动测定 O₂ 消耗量，并在另一套无钠石灰的装置中测定 CO₂ 生成量。
9. Chi‑squared Test | 卡方检验
The chi‑squared (χ²) test is used in genetics to determine whether observed phenotypic ratios deviate significantly from expected Mendelian ratios. It helps decide whether any difference is due to chance or to a real genetic effect.
卡方（χ²）检验用于遗传学中，以判断观察到的表型比例是否与预期的孟德尔比例存在显著差异。它帮助确定差异是由偶然性还是真正的遗传效应所致。
χ² = Σ (O - E)² ÷ E
χ² = Σ (O - E)² ÷ E
btain χ², then compare it with a critical value from the χ² table at a given probability (typically p = 0.05) and degrees of freedom (number of phenotypic classes minus 1). If χ² is smaller than the critical value, you accept the null hypothesis; any deviation is not significant.
O 代表观察值，E 代表每个类别的期望值。对每种表型计算 (O - E)² ÷ E，将这些数值求和得到 χ²，然后将其与特定概率（通常 p = 0.05）和自由度（表型类别数减 1）下的 χ² 表临界值比较。若 χ² 小于临界值，则接受原假设，此时的偏差不显著。
10. Hardy–Weinberg Equilibrium | 哈代–温伯格平衡
The Hardy–Weinberg principle provides a mathematical baseline against which evolutionary change can be detected. It applies when a large population mates randomly with no mutation, migration or natural selection.
哈代–温伯格原理提供了一个数学基线，用于检测进化变化。适用于随机交配、无限大、无突变、无迁移且无自然选择的种群。
p + q = 1
p + q = 1
p² + 2pq + q² = 1
p² + 2pq + q² = 1
Here, p is the frequency of the dominant allele, q is the frequency of the recessive allele. p² represents the frequency of homozygous dominant individuals, 2pq the frequency of heterozygotes, and q² the frequency of homozygous recessive individuals. If the frequency of a recessive condition is known, q can be calculated as √frequency of affected individuals, and from q, p can be derived. These are used to predict carrier frequencies in populations.
其中，p 为显性等位基因的频率，q 为隐性等位基因的频率。p² 代表显性纯合子的频率，2pq 代表杂合子的频率，q² 代表隐性纯合子的频率。如果已知隐性遗传病的发病率，可由 q = √患者频率 求出 q，进而求得 p。由此可预测人群中的携带者频率。
11. Probability in Genetics | 遗传概率
In monohybrid and dihybrid crosses, the probability of a particular genotype or phenotype can be calculated using the product rule and the sum rule.
在单因子和双因子杂交中，可利用乘法定律与加法定律计算特定基因型或表型的概率。
Product rule: Probability of two independent events occurring together = P(A) × P(B)
乘法定律：两独立事件同时发生的概率 = P(A) × P(B)
Sum rule: Probability of either of two mutually exclusive events occurring = P(A) + P(B)
加法定律：任意一个互斥事件发生的概率 = P(A) + P(B)
obability of being either homozygous dominant or heterozygous is ¼ + ½ = ¾. These rules also help construct Punnett squares efficiently.
例如，在双杂合子亲本（Aa × Aa）的杂交中，子代为隐性纯合子（aa）的概率为 ½ × ½ = ¼。子代为显性纯合子或杂合子的概率为 ¼ + ½ = ¾。这些定律也有助于高效构建庞纳特方格。
12. Serial Dilution and Standard Curves | 系列稀释与标准曲线
Serial dilution is a stepwise dilution of a stock solution, widely used in enzyme assays, reducing‑sugar quantification with the Benedict’s test, and protein determination with the biuret test. A standard curve then allows the concentration of an unknown sample to be interpolated.
系列稀释是对储备液进行逐步稀释，广泛应用于酶测定、用本尼迪克特试剂定量还原糖以及用双缩脲法测定蛋白质含量。标准曲线则可通过内插法求得未知样品的浓度。
Dilution factor = Total final volume ÷ Volume of stock solution used
稀释因子 = 最终总体积 ÷ 所用储备液体积
equent step can produce concentrations of 1/10, 1/100, 1/1000 and so on. When the absorbance or colour intensity of these known concentrations is plotted, the best‑ curve. Reading the absorbance of an unknown sample off the curve gives its concentration.
要配制 1 : 10 的稀释液，可将 1 cm³ 储备液与 9 cm³ 水混合，稀释因子 = (1 + 9) ÷ 1 = 10。后续每一步可依次得到 1/10、1/100、1/1000 等的浓度。将这些已知浓度的吸光度或颜色强度作图，最佳拟合线即构成标准曲线。从曲线上读取未知样品的吸光度，即可得到其浓度。
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