📚 Pre-U CCEA Biology Formula & Theorem Quick Reference | Pre-U CCEA 生物公式定理速查手册
This quick reference guide consolidates the essential formulae, theorems and calculations required for CCEA Pre-U Biology. From microscopy to ecological statistics, mastering these quantitative skills is vital for practical assessments and data-response questions. Each entry includes the relevant formula, key definitions and typical applications, so that you can revise the mathematics behind the biology quickly before a test, a practical or a written paper.
本速查手册汇总了 CCEA Pre-U 生物学所需的基本公式、定理和计算方法。从显微镜操作到生态统计，掌握这些定量技能对实验评估和数据分析题至关重要。每个条目都包含相关公式、关键定义和典型应用，方便你在测验、实验或笔试前快速复习生物学背后的数学。
1. Microscopy & Magnification | 显微镜与放大倍数
Magnification (M) is calculated by dividing the measured image size (I) by the actual size of the specimen (A). Always convert both measurements to the same unit before calculating, because mixing millimetres with micrometres is the most common source of lost marks in practical questions.
放大倍数 (M) 等于图像测量尺寸 (I) 除以实际标本尺寸 (A)。计算前务必将两者换算成相同单位，因为毫米与微米混用是实验题中最常见的失分原因。
M = I / A
ize (A = I / M) or the image size (I = M × A). Common unit conversions include 1 mm = 1000 µm and 1 µm = 1000 nm, so 1 mm = 1 × 10⁶ nm and 1 cm = 10 mm = 10⁴ µm.
公式可变形为 A = I / M（求实际尺寸）或 I = M × A（求图像尺寸）。常用单位换算：1 mm = 1000 µm，1 µm = 1000 nm，因此 1 mm = 1 × 10⁶ nm；1 cm = 10 mm = 10⁴ µm。
- 1 m = 1000 mm; 1 mm = 1000 µm; 1 µm = 1000 nm (1 m = 10⁹ nm)
- 1 cm = 10 mm = 10⁴ µm; 1 cm³ = 1 ml = 1000 mm³
length. If 100 e.u. correspond to 0.5 mm at a given objective, then 1 e.u. = 0.005 mm = 5 µm, and this value must be recalibrated whenever the objective lens is changed.
使用目镜测微尺时，必须用镜台测微尺校准，将目镜格值 (e.u.) 转换为实际长度。若某物镜下 100 个目镜格值对应 0.5 mm，则 1 e.u. = 0.005 mm = 5 µm；更换物镜后必须重新校准。
Magnification and resolution are not the same thing. Magnification enlarges the image, whereas resolution is the ability to distinguish two separate points and depends on the wavelength of the radiation used. A scale bar, usually 1-10 µm long on an electron micrograph, allows you to calculate magnification directly from the printed image.
放大倍数与分辨率并不相同。放大倍数只是把图像放大，而分辨率是区分两个相邻点的能力，取决于所用辐射的波长。电镜照片上的比例尺（通常 1-10 µm 长）可以让你直接从图像计算放大倍数。
2. Dilutions & Serial Dilutions | 稀释与连续稀释
The dilution equation is based on conservation of solute: the number of moles of solute before and after dilution is unchanged, so concentration multiplied by volume remains constant.
稀释公式基于溶质守恒：稀释前后溶质的物质的量不变，因此浓度与体积的乘积保持不变。
C₁V₁ = C₂V₂
work out the concentration of an unknown after dilution. For example, to make 250 cm³ of 0.2 mol dm⁻³ solution from a 2.0 mol dm⁻³ stock: V₁ = (0.2 × 250) / 2.0 = 25 cm³ of stock, made up to 250 cm³ with distilled water.
其中 C 为浓度，V 为体积。该公式可用于计算配制所需工作浓度时所需的母液体积，或推算稀释后未知溶液的浓度。例如用 2.0 mol dm⁻³ 母液配制 250 cm³ 0.2 mol dm⁻³ 溶液：V₁ = (0.2 × 250) / 2.0 = 25 cm³ 母液，加蒸馏水定容至 250 cm³。
The dilution factor (DF) is the ratio of final total volume to initial sample volume:
稀释倍数 (DF) 是最终总体积与初始取样体积之比：
DF = Vtotal / Vsample
n 10 dilution or 10⁻¹. Diluting the resulting solution tenfold again gives an overall DF of 100, or 10⁻².
将 1 cm³ 样品加入 9 cm³ 稀释液，DF = 10，通常写作 1:10 稀释或 10⁻¹。若再把所得溶液稀释十倍，总稀释倍数即为 100，也就是 10⁻²。
DF₁ × DF₂ × ... × DFₙ. After n successive tenfold dilutions the concentration is C₀ / 10ⁿ.
连续稀释中每一步的稀释倍数相乘，总稀释倍数为各步之积：DF_total = DF₁ × DF₂ × ... × DFₙ。经过 n 次十倍稀释后，浓度为 C₀ ÷ 10ⁿ。
/ volume plated in cm³. Only plates with roughly 30-300 colonies should be used, because fewer colonies produce large random errors while more colonies overlap and merge.
活菌计数时，将平板菌落数换算为每 cm³ 菌落形成单位：CFU cm⁻³ = (菌落数 × 稀释倍数) ÷ 接种体积 (cm³)。只应使用菌落数大致在 30-300 之间的平板，因为菌落过少随机误差大，过多则重叠融合难以计数。
3. Surface Area to Volume Ratio | 表面积与体积比
The surface area to volume ratio (SA:V) controls how quickly an organism or a cell can exchange substances with its environment. Surface area scales with the square of linear dimension while volume scales with the cube, so larger objects have proportionally smaller exchange surfaces.
表面积与体积比 (SA:V) 决定生物体或细胞与环境交换物质的快慢。表面积随线性尺寸的平方增长，而体积随立方增长，因此物体越大，其交换表面相对越小。
SA:V = surface area / volume
3)πr³, so SA:V = 3/r. For a cylinder of radius r and length h: SA = 2πr² + 2πrh, V = πr²h.
边长为 a 的立方体：SA = 6a²，V = a³，故 SA:V = 6/a。半径为 r 的球体：SA = 4πr²，V = (4/3)πr³，故 SA:V = 3/r。半径 r、长 h 的圆柱体：SA = 2πr² + 2πrh，V = πr²h。
- Cube of side 1 cm: SA = 6 cm², V = 1 cm³, SA:V = 6:1
- Cube of side 3 cm: SA = 54 cm², V = 27 cm³, SA:V = 2:1
Diffusion is described by Fick's law: rate of diffusion ∝ rence) / diffusion distance. This explains why alveoli, villi and root hair cells are numerous, thin-walled and highly folded, and why large animals need specialised transport systems.
扩散速率可用菲克定律描述：扩散速率 ∝ (表面积 × 浓度差) ÷ 扩散距离。这解释了为何肺泡、小肠绒毛和根毛细胞数量多、壁薄且高度折叠，也解释了大型动物为何需要专门的运输系统。
4. Enzyme Kinetics & Rate Calculations | 酶动力学与速率计算
The rate of an enzyme-controlled reaction is the change in amount of product or substrate divided by the time taken, usually expressed in mol dm⁻³ s⁻¹, cm³ min⁻¹ or absorbance units per minute.
酶促反应速率等于产物或底物量的变化量除以所用时间，常用单位为 mol dm⁻³ s⁻¹、cm³ min⁻¹ 或每分钟吸光度变化。
rate = Δ(amount) / Δ(time)
bstrate concentration is accurately known and product inhibition is minimal at that point.
初速率通过曲线在 t = 0 处最陡部分作切线求得，因为此时底物浓度准确已知且产物抑制最小。
The temperature coefficient Q₁₀ compares rates at temperatures 10 °C apart:
温度系数 Q₁₀ 比较相差 10 °C 的两个温度下的速率：
Q₁₀ = rate at (T + 10 °C) / rate at T
Between roughly 0 °C and the optimum temperature Q₁₀ is usually about 2, because kinetic energy and enzyme-substrate collisions increase. Above the optimum the enzyme denatures, the rate falls sharply and Q₁₀ becomes less than 1.
在约 0 °C 至最适温度之间，Q₁₀ 通常约为 2，因为分子动能增加、
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