📚 Year 12 CAIE Biology: Formula & Theorem Quick Reference | Year 12 CAIE 生物：公式定理速查手册
This handbook compiles the essential quantitative relationships and key theorems required throughout the Year 12 CAIE Biology (9700) course. From microscopy calculations to statistical tests, mastering these formulas is crucial for success in both theoretical papers and practical assessments. Each section presents the core equation or principle, explains its application, and highlights common pitfalls in exam settings.
本手册汇编了 Year 12 CAIE 生物 (9700) 课程中所有必需的定量关系和关键定理。从显微镜计算到统计检验，掌握这些公式对于理论考卷和实验评估的成功至关重要。每一节都展示了核心方程或原理，阐明了其应用，并指出了考试中的常见陷阱。
1. Magnification and Actual Size | 放大率与实际尺寸
In microscopy, the magnification (M) tells you how many times larger an image appears compared with the object's real size. The standard formula links image size (I) to actual size (A):
M = I / A
Rearrange to find actual size: A = I / M Both I and A must share the same unit; micrographs often give scale bars in micrometres (µm) or nanometres (nm). Remember that 1 mm = 1000 µm, and 1 µ check whether the image has been enlarged further from the eyepiece graticule calibration.
在显微镜检中，放大率 (M) 表示图像比实物真实尺寸大多少倍。标准公式为：M = I / A。重组后可得实际大小：A = I / M。I 与 A 必须使用相同单位；显微照片通常以微米 (µm) 或纳米 (nm) 标注比例尺。牢记 1 mm = 1000 µm，1 µm = 1000 nm。处理照片时，务必确认图像是否在目镜测微尺校准基础上又被额外放大了。
provides a scale bar and asks you to calculate magnification. First, measure the scale bar's length on the printed page (in mm), convert that length and the scale bar' express magnification to an appropriate number of significant figures and include the unit '×' (times).
常见考题提供比例尺并要求计算放大率。首先，测量打印页面上比例尺的长度 (以 mm 为单位)，将该长度与比例尺真实值转换为相同单位，然后使用 M = 图像长度 / 真实长度 计算。始终用适当有效数字表示放大倍数，并加上单位 '×' (倍)。
2. Haemocytometer Cell Counting | 血细胞计数板细胞计数
An improved Neubauer haemocytometer is used to estimate cell concentration, such as yeast or blood cells. The central counting grid has an area of 1 mm² and a depth of 0.1 mm, giving a volume of 0.1 mm³ (10⁻⁴ cm³ or 10⁻⁴ mL) over the entire ruled area. After loading the chamber, you count cells in a known number of squares and calculate the concentration:
ells mL⁻¹) = (Number of cells counted / Number of squares counted) × Dilution factor × 10⁴
uld be (120/5) × 10 × 10⁴ = 2.4 × 10⁶ cells mL⁻¹. Always include the dilution factor (e.g. 2 for 1 : 1 dilution) and count multiple squares to increase reliability.
改良的牛鲍尔血细胞计数板用于估算细胞浓度，如酵母或血细胞。中央计数网格面积为 1 mm²，深度为 0.1 mm，整个刻线区域的体积为 0.1 mm³ (10⁻⁴ cm³ 或 10⁻⁴ mL)。装载腔液后，对已知数量的方格内的细胞进行计数，并计算浓度：细胞浓度 (cells mL⁻¹) = (计数的细胞数 / 计数的方格数) × 稀释因子 × 10⁴。
系数 10⁴ 的出现是因为 0.1 mm³ 即 1/10⁴ mL。例如，若在 5 个小方格中计数到 120 个细胞，且稀释倍数为 1 : 10，则浓度为 (120/5) × 10 × 10⁴ = 2.4 × 10⁶ cells mL⁻¹。务必将稀释因子 (如 1 : 1 稀释即为 2) 纳入计算，并计数多个方格以提高可靠性。使用台盼蓝排除法可区分活细胞和死细胞。
3. Mitotic Index | 有丝分裂指数
The mitotic index quantifies the proportion of cells in a tissue that are undergoing mitosis. It is used to gauge how rapidly a cell population is dividing, for example in root tip meristems or tumours.
Mitotic index = (Number of cells in mitosis / Total number of cells counted) × 100
Cells in prophase, metaphase, anaphase and telophase are all counted as 'in mitosis'; interphase cells are not. Count a minimum of 200 cells to obtain a reliable value. A high mitotic index (e.g. >10%) may indicate a meristem or a fast‑growing tumour.
有丝分裂指数定量了组织中正在经历有丝分裂的细胞比例。它用于判断细胞群体分裂的速率，例如在根尖分生组织或肿瘤中。有丝分裂指数 = (处于有丝分裂的细胞数 / 计数的细胞总数) × 100。
处于前期、中期、后期和末期的细胞均被计为“正在有丝分裂”；间期细胞不计入。至少计数 200 个细胞以获得可靠数值。高有丝分裂指数 (如 >10%) 可能表明分生组织或快速生长的肿瘤。
4. Surface Area to Volume Ratio | 表面积与体积比
All cells rely on diffusion for transport of substances across their boundaries. The surface area to volume ratio (SA : V) is a critical factor limiting cell size. As an object increases in size, its volume grows faster than its surface area, reducing the SA : V ratio.
For a cube: SA = 6s², V = s³ ⇒ SA : V = 6 / s
For a sphere: SA = 4πr², V = 4/3 πr³ ⇒ SA : V = 3 / r
Because the ratio is inversely proportional to the linear dimension, large cells have a smaller SA : V ratio and therefore slower exchange of materials. Organisms have evolved specialised surfaces (alveoli, root hairs, villi) to increase the effective surface area without drastically increasing volume.
所有细胞依赖扩散进行跨膜物质运输。表面积与体积比 (SA : V) 是限制细胞大小的关键因素。随着物体增大，其体积比表面积增长得更快，从而降低了 SA : V 比率。对立方体：SA = 6s²，V = s³ ⇒ SA : V = 6 / s；对球体：SA = 4πr²，V = 4/3 πr³ ⇒ SA : V = 3 / r。
由于比率与线性尺寸成反比，大细胞具有较小的 SA : V 比，因此物质交换较慢。生物体演化出了特化的表面 (肺泡、根毛、绒毛)，在不显著增加体积的前提下增加了有效表面积。
5. Fick's Law of Diffusion | 菲克扩散定律
Fick's law describes the rate at which particles diffuse across a membrane or exchange surface. Although you do not need to plug values into a precise formula, you must understand the proportionality:
Rate of diffusion ∝ (Surface area × Concentration difference) / Thickness of exchange surface
To maximise diffusion rate, evolution favours large surface areas (e.g. folded membranes), a steep concentration gradient (maintained by ventilation and circulation), and a thin barrier (alveolar wall single‑cell thickness). Damage that thickens the barrier (fibrosis) or reduces area (emphysema) will lower the rate of gas exchange.
菲克定律描述了粒子跨膜或交换表面扩散的速率。虽然不需要将数值代入精确公式，但必须理解比例关系：扩散速率 ∝ (表面积 × 浓度差) / 交换表面厚度。
为使扩散速率最大化，演化倾向于大的表面积 (例如折叠膜)、陡峭的浓度梯度 (由通气和循环维持) 以及薄的屏障 (肺泡壁单细胞厚度)。使屏障增厚 (纤维化) 或减少面积 (肺气肿) 的损伤会降低气体交换速率。
6. Water Potential Equation | 水势方程
Water potential (Ψ) determines the direction of water movement across a partially permeable membrane. In plant physiology, it is a combination of solute potential (Ψₛ) and pressure potential (Ψₚ):
Ψ = Ψₛ + Ψₚ
Pure water at standard pressure has a water potential of 0 kPa. Dissolving solutes lowers the water potential (Ψₛ becomes negative), while turgor pressure exerted by the cell wall raises the pressure potential (Ψₚ positive). Water moves from a region of higher (less negative) water potential to lower (more negative) water potential.
水势 (Ψ) 决定了水分跨越部分透膜的移动方向。在植物生理学中，它是溶质势 (Ψₛ) 和压力势 (Ψₚ) 的组合：Ψ = Ψₛ + Ψₚ。
纯水在标准压力下水势为 0 kPa。溶解溶质会降低水势 (Ψₛ 为负)，而细胞壁施加的膨压则使压力势升高 (Ψₚ 为正)。水分从水势较高 (负值较小) 的区域流向水势较低 (负值较大) 的区域。
7. Serial Dilution Calculations | 稀释系列计算
Serial dilutions are used to produce a range of concentrations for investigations, such as measuring the effect of substrate concentration on enzyme activity. Starting from a stock solution of concentration C₁, a volumetric transfer followed by topping up with solvent yields a new concentration:
C₂ = C₁ × (V₁ / V₂)
ck to 9 parts solvent, giving DF = 10 and C₂ = C₁ / 10. Repeated 1 : 10 steps produce a logarithmic concentration series (10⁻¹, 10⁻², 10⁻³ …).
连续稀释在实验中用于生成一系列浓度，例如测定底物浓度对酶活性的影响。由浓度为 C₁ 的储备液出发，转移一定体积后用溶剂补足，可得新浓度：C₂ = C₁ × (V₁ / V₂)。
其中 V₁ 为取出的储备液体积，V₂ 为稀释后总终体积。稀释因子为 V₂ / V₁。进行 1 : 10 稀释时，取 1 份储备液加 9 份溶剂，DF = 10，C₂ = C₁ / 10。连续重复 1 : 10 步骤可产生对数浓度系列 (10⁻¹、10⁻²、10⁻³ …)。
8. Standard Deviation and Standard Error | 标准偏差与标准误
These statistics describe the spread and precision of data sets. For a sample of n measurements with mean x̄, the sample standard deviation (s) is:
s = √[ Σ(x − x̄)² / (n − 1) ]
Use (n − 1) for degrees of freedom in a sample. The standard error of the mean (SEM) estimates how far the sample mean is likely to be from the true population mean:
SEM = s / √n
A small SEM indicates high precision; overlapping SEM error bars on a bar chart suggest no significant difference, whereas non‑overlapping intervals often indicate a significant difference (though it is more rigorous to apply a t‑test).
这些统计量描述数据集的离散程度和精确度。对于一个有 n 个测量值且均值为 x̄ 的样本，样本标准偏差 (s) 为：s = √[ Σ(x − x̄)² / (n − 1) ]。使用 (n − 1) 作为样本的自由度。均值的标准误 (SEM) 估计样本均值与总体真实均值的差距：SEM = s / √n。
小的 SEM 表明高精确度；在条形图上，重叠的 SEM 误差棒暗示无显著差异，而非重叠的区间常表明存在显著差异 (尽管更严谨的做法是应用 t 检验)。
9. Student's t‑test | 学生 t 检验
The independent (unpaired) t‑test compares the means of two sets of continuous data to decide whether they are significantly different. The test statistic t is calculated as:
t = |x̄₁ − x̄₂| / √s₁²/n₁ + s₂²/n₂
− 2). If t > critical t, the difference is significant. The null hypothesis states there is no difference between means.
独立 (非配对) t 检验比较两组连续性数据均值，判断它们是否显著不同。检验统计量 t 计算如下：t = |x̄₁ − x̄₂| / √s₁²/n₁ + s₂²/n₂。
然后，将计算出的 t 值与选定概率水平 (通常 p = 0.05) 和自由度 (df = n₁ + n₂ − 2) 下的临界 t 值进行比较。若 t > 临界 t，差异显著。零假设声明两均值之间无差异。
10. Chi‑squared Test | 卡方检验
The chi‑squared test is used for categorical data to check if observed frequencies differ from expected frequencies. The formula is:
χ² = Σ (O − E)² / E
he number of categories to obtain the degrees of freedom. If χ² exceeds the critical value at p = 0.05, you reject the null hypothesis and conclude there is a significant difference.
卡方检验适用于分类数据，检查观察频数是否与期望频数相符。公式为：χ² = Σ (O − E)² / E。其中 O = 观察计数，E = 每个类别的期望计数。期望值根据理论或比例假设计算。将类别数减去 1 即得自由度。若 χ² 大于 p = 0.05 时的临界值，则拒绝零假设，并得出结论存在显著差异。
In Mendelian genetics, expected ratios (e.g. 3 : 1, 9 : 3 : 3 : 1) are used to compute E values. Always state the null hypothesis clearly and mention that chance alone could produce small deviations.
在孟德尔遗传学中，利用期望比值 (如 3 : 1、9 : 3 : 3 : 1) 计算 E 值。务必清楚地陈述零假设，并提及仅凭偶然也可能产生小的偏离。
11. Q10 Temperature Coefficient | Q10 温度系数
The Q10 value expresses how the rate of an enzyme‑catalysed reaction changes over a 10 °C temperature interval. It is a useful tool for comparing the temperature sensitivity of biological processes.
Q10 = Rate at (T + 10) °C / Rate at T °C
For a typical enzyme, Q10 is around 2 between 20 °C and 30 °C, meaning the rate doubles for every 10 °C rise. When Q10 falls below 1 as the temperature approaches the denaturation point, the rate drops sharply because of irreversible disruption of the tertiary structure. Always specify the temperature range because Q10 is not constant across all temperatures.
Q10 值表达了酶促反应速率在温度升高 10 °C 时的变化倍数。它是比较生物过程温度敏感性的有用工具。Q10 = 在 (T + 10) °C 时的速率 / 在 T °C 时的速率。
对于常见酶，在 20 °C 至 30 °C 范围 Q10 约为 2，即温度每升高 10 °C，速率加倍。当温度
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