📚 AS Cambridge Biology: Quick Reference Handbook of Formulas and Theorems | AS 剑桥生物：公式定理速查手册
This quick reference handbook brings together all the essential quantitative relationships, equations and principles that appear throughout the AS Cambridge Biology syllabus. Whether you are calculating cell dimensions, analysing diffusion rates or determining water potential, a firm grasp of these formulas will boost your confidence in both theory and practical assessments. Each entry is explained with clear notation, worked-through logic and a note on when it is used.
这本速查手册汇集了AS剑桥生物课程中涉及的所有重要定量关系、方程和原理。无论是计算细胞尺寸、分析扩散速率还是确定水势，牢固掌握这些公式都能提升你在理论与实验评估中的信心。每个条目都配有清晰的符号、推导思路以及适用场景的说明。
1. Magnification and Actual Size | 放大倍数与实际尺寸
Magnification (M) is the number of times an image is larger than the actual object. The formula is central to any microscopic drawing or photograph.
放大倍数(M)是指图像比实物扩大的倍数。该公式是所有显微镜绘图或照片计算的核心。
Magnification = Image size / Actual size M = I / A
Both image size and actual size must be expressed in identical units before the calculation. Magnification has no units.
计算前，图像尺寸与实际尺寸必须使用相同单位。放大倍数不带单位。
to find the magnification of the micrograph. To find actual size, rearrange to A = I / M.
如果提供了比例尺，先测量图像中比例尺的长度，再除以它所代表的真实长度，即可获得显微照片的放大倍数。若要求实际尺寸，可将公式变形为 A = I / M。
2. Unit Conversions in Microscopy | 显微镜中的单位换算
Working with cells demands fluency in metric conversions. The most common units are millimetre (mm), micrometre (µm) and nanometre (nm).
研究细胞需要熟练进行公制单位换算。最常用的单位是毫米(mm)、微米(µm)和纳米(nm)。
| Conversion | Equivalent |
| 1 mm | 1000 µm |
| 1 µm | 1000 nm |
| 1 mm | 1 000 000 nm |
Always convert all measurements to the same unit before substituting into the magnification formula. A common mistake is leaving image size in mm and actual size in µm.
在代入放大倍数公式前，务必将所有测量值转换为同一单位。常见错误是把图像尺寸留在毫米而实际尺寸用微米。
3. Surface Area to Volume Ratio | 表面积与体积比
The surface-area-to-volume ratio (SA:V) limits cell size because exchange of materials depends on surface area while metabolic demand depends on volume.
表面积与体积之比(SA:V)限制了细胞的大小，因为物质交换取决于表面积，而代谢需求取决于体积。
length L: Surface area = 6L², Volume = L³, so SA:V = 6/L. As L increases, SA:V decreases.
对于边长为 L 的立方体：表面积 = 6L²，体积 = L³，因此 SA:V = 6/L。随着 L 增大，SA:V 减小。
Large organisms overcome this by having specialised exchange surfaces (flattened shape, villi, extensive capillary networks). In adaptation questions, calculate the ratio to show an increase in surface area relative to volume.
大型生物通过特化的交换表面（扁平体型、绒毛、庞大的毛细血管网）来克服这一限制。在适应性考题中，可计算比值以显示表面积相对于体积的增加。
4. Fick's Law of Diffusion | 菲克扩散定律
Fick's Law expresses the factors that affect the rate of diffusion across a membrane or an exchange surface. It is not usually given on the data sheet, so memorising the proportional relationship is valuable.
菲克定律表达了影响跨膜或交换表面扩散速率的因素。它通常不会出现在公式表上，因此记住比例关系很有价值。
Rate of diffusion ∝ (Surface area × Concentration gradient) / Diffusion distance
Surface area: larger area allows more particles to cross per second. Concentration gradient: a steeper difference drives faster net movement. Thickness of membrane: shorter distance means a quicker journey for molecules.
表面积：面积越大，每秒通过的粒子越多。浓度梯度：梯度越陡，净移动越快。膜的厚度：距离越短，分子穿越越快。
When explaining adaptations such as the alveoli, apply Fick's Law: large surface area, short diffusion distance and maintained steep gradient through ventilation and blood flow.
在解释肺泡等适应性结构时，可运用菲克定律：表面积大、扩散距离短，并通过通气和血流维持陡峭的浓度梯度。
5. Dilution Calculations | 稀释计算
Dilutions are routine when preparing solutions for enzyme assays or serial dilutions of a stock solution. The relationship between initial and final concentrations and volumes is expressed as:
在制备酶活性测定溶液或对储备液进行系列稀释时，稀释是常规操作。初始与最终浓度及体积的关系可表示为：
C₁V₁ = C₂V₂
Where C₁ is the initial concentration, V₁ the initial volume taken, C₂ the final concentration and V₂ the final total volume. This formula assumes the amount of solute remains constant.
其中 C₁ 为初始浓度，V₁ 为取出的初始体积，C₂ 为最终浓度，V₂ 为最终总体积。该公式基于溶质的含量保持不变。
For a 1 in 10 dilution, one part of stock is added to nine parts of diluent, giving a final concentration that is one-tenth of the original. In serial dilutions, the concentration halves or decreases by a factor of 10 at each step, which is useful for generating a standard curve.
进行 1:10 稀释时，取 1 份储备液加入 9 份稀释液，得到原浓度十分之一的最终溶液。在系列稀释中，每步浓度减半或降低至十分之一，常用以绘制标准曲线。
6. Enzyme Reaction Rate | 酶促反应速率
Reaction rate in enzyme experiments can be calculated as the change in product concentration per unit time or the disappearance of substrate per unit time. The initial rate is the most reliable part of the progress curve.
酶实验中的反应速率可表示为单位时间内产物浓度的变化或底物浓度的减少。初始速率（反应进程曲线最前段）是最可靠的部分。
Rate = Product formed / Time taken or Rate = Substrate used / Time taken
If a graph of product concentration against time is given, draw a tangent at time zero for the true initial rate. Avoid using the later plateau where substrate becomes limiting.
当给出产物浓度-时间图时，应在时间为零处作切线以获得真实的初速率。避免使用后期底物成为限制因素时的平台段。
Units are typically concentration min⁻¹, e.g. µmol min⁻¹. Remember to convert volumes and concentrations appropriately when calculating molar quantities.
单位通常为浓度·min⁻¹，例如 µmol min⁻¹。计算摩尔数时注意体积与浓度的适当换算。
7. Cell Counting with Haemocytometer | 血球计数板细胞计数
When estimating cell density using a haemocytometer, a standard formula converts the average count per square into cells per millilitre. Each improved Neubauer counting chamber has a fixed volume for a large square.
使用血球计数板评估细胞密度时，一个标准公式可将每格平均计数转换为每毫升细胞数。改良牛鲍计数板的每个大方格具有固定的体积。
l⁻¹ = (Average count per large square × Dilution factor) / Volume of one large square (ml)
e square is 0.1 mm³, which equals 1 × 10⁻⁴ ml. Therefore, the formula often simplifies to:
一个大方格的体积为 0.1 mm³，即 1 × 10⁻⁴ ml。因此公式常简化为：
Cells ml⁻¹ = Average count per large square × Dilution factor × 10⁴
Count both live and dead cells if using a vital stain. Several squares are counted to obtain a reliable average. This method is frequently assessed in practical examinations (paper 3).
若使用活体染色剂，需分别计数活细胞和死细胞。通常会计数多个方格以得到可靠的平均值。这个方法在实验考试（Paper 3）中常被考查。
8. Water Potential Equation | 水势方程
Water potential (ψ) is the measure of free energy of water per unit volume and determines the direction of water movement. In plant cells it is the sum of two components.
水势(ψ)是单位体积水的自由能的量度，决定水运动的方向。在植物细胞中，它是两个组分的总和。
ψ = ψₛ + ψₚ
Here ψₛ is the solute potential (also called osmotic potential), always negative because solutes reduce the tendency of water to move. ψₚ is the pressure potential, which is usually positive when a cell pushes against the cell wall.
其中 ψₛ 是溶质势（也称渗透势），始终为负值，因为溶质降低了水移动的趋势。ψₚ 是压力势，当细胞推挤细胞壁时通常为正值。
Water moves from regions of higher (less negative) water potential to regions of lower (more negative) water potential. Plasmolysis occurs when a plant cell is placed in a hypertonic solution, causing ψₚ to drop to zero.
水从水势较高（负值较小）的区域移向水势较低（负值更大）的区域。当植物细胞置于高渗溶液中时，会失去水分发生质壁分离，此时 ψₚ 降至零。
9. Temperature Coefficient (Q10) | 温度系数 (Q10)
Q10 quantifies how the rate of an enzyme-controlled reaction changes with a 10 °C rise in temperature. It is widely used to assess the sensitivity of biological processes to temperature.
Q10 可量化当温度上升 10 °C 时酶控反应速率的变化。它被广泛用来评估生物过程对温度的敏感性。
Q10 = Rate at (T + 10)°C / Rate at T°C
For most enzyme reactions between 10 °C and 40 °C, Q10 is around 2, meaning the rate roughly doubles for every 10 °C increase. Above the optimum, denaturation causes a sharp drop and Q10 becomes meaningless.
在 10 °C 到 40 °C 之间，大多数酶反应的 Q10 约为 2，即温度每升高 10 °C，速率大约加倍。超过最适温度后，变性导致速率急剧下降，此时计算 Q10 没有意义。
When presented with rate data at different temperatures, apply the formula to two values exactly 10 °C apart. Variations may ask for Q10 over a smaller interval, but the same principle holds.
当给出不同温度下的速率数据时，选取恰好相差 10 °C 的两个值代入公式。题目有时会要求计算更小温度区间的 Q10，但原理一致。
10. Calculating Percentage Change and Ratios | 计算百分比变化与比率
Percentage change is a basic statistic that appears when analysing mass change in osmosis experiments or growth data. It standardises differences, making comparisons between different-sized organs or organisms meaningful.
百分比变化是一种基本统计量，常用于分析渗透实验中的质量变化或生长数据。它将差异标准化，使不同大小的器官或生物之间的比较更有意义。
Percentage change = (Final value - Initial value) / Initial value × 100%
A positive value indicates an increase; a negative value indicates a decrease. In a potato osmoscope, the percentage change in mass is plotted against solute concentration to estimate the water potential.
正值表示增加，负值表示减少。在土豆渗透实验中，将质量变化百分比对溶质浓度作图，可估算土豆的水势。
Ratios, such as the ratio of surface area to volume, are used to quantify relationships. Express them in the simplest whole-number form and be precise about what is being compared.
像表面积与体积比这样的比率，可用于定量描述关系。应表示为最简整数比，并明确比较的对象。
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