📚 Year 12 CAIE Biology: Quick-Reference Handbook of Formulas & Theorems | Year 12 CAIE 生物：公式定理速查手册
This handbook compiles the essential quantitative relationships, equations and statistical tests that every Year 12 student following the Cambridge International AS Level Biology (9700) syllabus must have at their fingertips. Use it while tackling numerical problems, required practicals and data-analysis examination questions to ensure you apply the correct logic and units.
本手册汇总了剑桥国际 AS Level 生物学 (9700) 每位 Year 12 学生必须烂熟于心的定量关系、方程和统计检验。在你解答数字计算题、完成规定实验和应对数据分析考题时，可以用它来确保逻辑正确、单位恰当。
1. Microscope Measurements & Magnification | 显微镜测量与放大倍数
Magnification describes how many times larger an image is compared with the real object. The key formula is Magnification = Image size / Actual size. Rearranged, Actual size = Image size / Magnification. Always convert all lengths to the same unit (commonly millimetres, micrometres or nanometres: 1 mm = 1000 µm, 1 µm = 1000 nm) before substituting.
Magnification = Image size ÷ Actual size
放大倍数表示图像比实际物体放大了多少倍。核心公式为 放大倍数 = 图像尺寸 / 实际尺寸。变形后，实际尺寸 = 图像尺寸 / 放大倍数。代入前务必将所有长度换算为相同单位（常用毫米、微米或纳米：1 mm = 1000 µm，1 µm = 1000 nm）。
When using a light microscope with an eyepiece graticule, you must calibrate the graticule against a stage micrometer. Use the formula:
One eyepiece unit (µm) = (Number of stage micrometer divisions × known length per division) ÷ Number of eyepiece graticule divisions aligned
使用光学显微镜带目镜测微尺时，必须用镜台测微尺对其进行校准。公式为：
一个目镜单位 = (镜台测微尺格数 × 每格已知长度) ÷ 对齐的目镜格数
Record the calibration for each objective lens. All subsequent measurements are then made in eyepiece units and converted to real length using this calibrated value.
为每个物镜记录校准值。此后的所有测量均以目镜单位计数，再用校准值换算成真实长度。
2. Cell Cycle & Mitotic Index | 细胞周期与有丝分裂指数
The mitotic index gives the proportion of cells undergoing mitosis in a tissue section. It is calculated as:
Mitotic Index (%) = (Number of cells in mitosis ÷ Total number of cells counted) × 100
有丝分裂指数表示组织切片中正在进行有丝分裂的细胞的比例。计算方法为：
有丝分裂指数 (%) = (处于有丝分裂的细胞数 ÷ 计数的细胞总数) × 100
You can use the mitotic index to estimate the duration of a specific stage of the cell cycle if the total cycle time is known. For example, Time spent in mitosis = Mitotic index × Total cycle time. This assumes the cell population is asynchronous and uniformly distributed around the cycle.
Duration of a phase = (Number of cells in that phase ÷ Total cells) × Total cycle time
若已知整个细胞周期的时长，便可利用有丝分裂指数估算某一阶段所占的时间。例如，处于有丝分裂的时间 = 有丝分裂指数 × 周期总时间。前提是细胞群体不同步且各时期分布均匀。
某一时期时长 = (处于该时期的细胞数 ÷ 细胞总数) × 周期总时间
Always count a large number of cells in several fields of view to improve reliability.
务必在多个视野内计数大量细胞以提高可靠性。
3. Surface Area to Volume Ratio | 表面积与体积比
As an object increases in size, its volume grows faster than its surface area. For a cube of side l, surface area = 6l² and volume = l³, so SA:V = 6/l. For a sphere of radius r, surface area = 4πr², volume = (4/3)πr³, giving SA:V = 3/r.
当物体增大时，其体积的增长快于表面积。对边长为 l 的立方体，表面积 = 6l²，体积 = l³，故 SA:V = 6/l。对半径为 r 的球体，表面积 = 4πr²，体积 = (4/3)πr³，SA:V = 3/r。
SA:V ratio = Surface area ÷ Volume
A high SA:V ratio favours efficient diffusion of gases, nutrients and waste across the cell membrane. This explains why most cells are microscopically small and why specialised exchange surfaces (alveoli, root hairs, villi) are folded or flattened.
高表面积体积比有利于气体、营养物质和废物在细胞膜上进行高效扩散。这就解释了为什么大多数细胞都极其微小，也揭示了为什么肺泡、根毛、小肠绒毛等特化交换面都呈折叠或扁平状。
4. Transport & Fick’s Law | 运输与菲克定律
Fick’s Law describes the factors affecting the rate of diffusion across an exchange surface:
Rate of diffusion ∝ (Surface area × Concentration difference) ÷ Thickness of diffusion pathway
菲克定律描述了影响跨交换面扩散速率的因素：
扩散速率 ∝ (表面积 × 浓度差) ÷ 扩散距离（膜的厚度）
To maximise diffusion, organisms increase surface area (e.g. gill filaments, capillary networks), maintain steep concentration gradients (ventilation and blood flow) and minimise the thickness of exchange surfaces (squamous epithelium in alveoli). The proportional relationship is a guiding principle for all analysis of gas exchange and nutrient uptake.
为使扩散速率最大，生物体往往增大表面积（如鳃丝、毛细血管网）、维持陡峭的浓度梯度（通过通气与血流）并减小交换面的厚度（如肺泡的扁平上皮）。这一正比关系是分析所有气体交换和营养吸收问题的指导性原则。
5. Water Potential & Osmosis | 水势与渗透
Water potential (ψ, psi) determines the direction of water movement. The total water potential of a plant cell is the sum of its solute potential (ψₛ) and pressure potential (ψₚ):
ψ = ψₛ + ψₚ
水势 (ψ) 决定了水分移动的方向。植物细胞的总水势是其溶质势 (ψₛ) 与压力势 (ψₚ) 之和：
ψ = ψₛ + ψₚ
Pure water at atmospheric pressure has ψ = 0 kPa. Solute potential (ψₛ) is always negative or zero; adding solutes lowers water potential. Pressure potential (ψₚ) is usually positive inside a turgid cell. Water always moves from a region of higher (less negative) water potential to a region of lower (more negative) water potential across a partially permeable membrane.
标准大气压下的纯水 ψ = 0 kPa。溶质势 (ψₛ) 永为负值或零；加入溶质会降低水势。压力势 (ψₚ) 在膨压细胞中通常为正值。水分总是跨过半透膜从较高（负值较小）水势区域向较低（负值较大）水势区域移动。
In questions, you may need to compare ψ values to predict whether a cell will lose or gain water, or determine if a solution is hypotonic, isotonic or hypertonic relative to the cell.
考题中你可能需要比较 ψ 值，以预测细胞会失水还是吸水，或判断溶液相对于细胞是低渗、等渗还是高渗。
6. Cardiopulmonary Calculations | 心肺系统计算
Cardiac output (CO) is the volume of blood pumped by one ventricle per minute. It is the product of heart rate (HR, beats per minute) and stroke volume (SV, volume per beat):
Cardiac Output = Heart rate × Stroke volume (CO = HR × SV)
心输出量 (CO) 是指一侧心室每分钟泵出的血液体积。它是心率 (HR，次/分钟) 与每搏输出量 (SV，每次搏出体积) 的乘积：
心输出量 = 心率 × 每搏输出量
Lung volumes are often measured using a spirometer. Vital capacity (VC) – the maximum volume that can be exhaled after a maximum inhalation – is the sum of inspiratory reserve volume (IRV), tidal volume (TV) and expiratory reserve volume (ERV).
Vital Capacity = IRV + TV + ERV
Total lung capacity (TLC) adds the residual volume (RV) that cannot be exhaled:
Total Lung Capacity = VC + RV
肺容积常用肺活量计测定。肺活量 (VC)——最大吸气后再全力呼出的最大气量——等于补吸气量 (IRV)、潮气量 (TV) 与补呼气量 (ERV) 之和。
肺活量 = IRV + TV + ERV
总肺容量 (TLC) 还需加上无法呼出的残气量 (RV)：
总肺容量 = 肺活量 + 残气量
Be prepared to read spirometer traces and calculate these volumes and rates such as ventilation rate.
要准备好读懂肺活量图线，并计算这些气量和诸如通气速率等参数。
7. Enzyme Kinetics & Dilution | 酶动力学与稀释
The initial rate of an enzyme-catalysed reaction is often measured as the change in product concentration (or substrate disappearance) per unit time. From a progress curve, draw a tangent at time zero to find the gradient.
Initial rate = Δ[Product] ÷ Δtime
酶促反应的初始速率通常用单位时间内产物浓度的变化（或底物消耗）来衡量。在反应进程曲线上，在零点绘制切线，其斜率即为初始速率。
初始速率 = Δ[产物] ÷ Δ时间
In experiments involving serial dilutions of an enzyme or substrate, the simple dilution equation is essential:
C₁ V₁ = C₂ V₂
where C₁ and V₁ are the concentration and volume of the stock solution, and C₂ and V₂ are the desired concentration and final volume.
在涉及酶或底物梯度稀释的实验中，这个简单的稀释方程至关重要：
C₁ V₁ = C₂ V₂
其中 C₁ 和 V₁ 为母液浓度与体积，C₂ 和 V₂ 为目标浓度与最终体积。
Always ensure the volumes are in the same units and remember that a ‘1 in 10’ dilution means 1 part stock plus 9 parts diluent, giving a final concentration of 1/10 of the original.
务必确保体积单位相同，并记住“1比10稀释”意味着1份母液加9份稀释液，最终浓度为原浓度的1/10。
8. Population Estimation: Mark-Release-Recapture | 种群估计：标记重捕法
For motile organisms, the Lincoln index estimates population size (N) after a capture–mark–release–recapture procedure:
N = (M × C) ÷ R
where M = number initially captured, marked and released; C = total number captured in the second sample; R = number of marked individuals recaptured in the second sample.
对于活动范围较大的动物，林肯指数通过捕获–标记–放回–重捕程序来估算种群数量 (N)：
N = (M × C) ÷ R
其中 M = 首次捕获、标记并放回的个体数；C = 第二次样本总捕获数；R = 第二次样本中带标记的个体数。
This method assumes that marked and unmarked individuals mix randomly, no significant births, deaths or migration occur between samples, and marks are not lost or overlooked. Any violation of these assumptions leads to under- or overestimates.
此方法假设标记与未标记个体随机混合，两次取样间无显著出生、死亡或迁入迁出，且标记不脱落、不被忽视。任何假设的违背都会导致低估或高估。
9. Species Diversity: Simpson’s Index | 物种多样性：辛普森指数
Simpson’s index of diversity (D) measures biodiversity by accounting for both species richness and evenness. The formula most frequently used in CAIE is:
D = 1 – Σ (n ÷ N)²
where n = total number of individuals of a particular species, N = total number of individuals of all species, and Σ means ‘sum of’ for all species. The value of D ranges from 0 (low diversity, one species dominates) to nearly 1 (high diversity, even distribution).
辛普森多样性指数 (D) 通过兼顾物种丰富度和均匀度来衡量生物多样性。CAIE 最常用的公式是：
D = 1 – Σ (n / N)²
其中 n 为某一特定物种的个体数，N 为所有物种的个体总数，Σ 表示对各物种求和。D 值介于 0（低多样性，单一物种
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