AS OCR Biology: Formula & Theorem Quick-Reference Handbook | AS OCR 生物：公式定理速查手册

📚 AS OCR Biology: Formula & Theorem Quick-Reference Handbook | AS OCR 生物：公式定理速查手册

This handbook brings together the essential formulae and key theorems that every AS OCR Biology student must master. From magnification calculations to Simpson’s Diversity Index, these mathematical relationships sit at the heart of exam questions and practical investigations. Each entry is presented with its standard notation, units, and an example of typical application, helping you revise efficiently and avoid common pitfalls.

本手册汇集了每位 AS OCR 生物学生必须掌握的核心公式与关键定理。从显微镜放大倍数计算到辛普森多样性指数，这些数学关系是考试题目和实验探究的核心。每一条公式都给出了标准符号、单位以及典型应用示例，帮助你高效复习并避开常见错误。

1. Magnification and Actual Size | 显微镜放大倍数与实际大小

Magnification (M) tells you how many times larger an image appears compared to the real object. The actual size (A) of a specimen can be calculated if you know the image size (I) and the magnification.

放大倍数 (M) 表示图像比实际物体大多少倍。如果已知图像大小 (I) 和放大倍数，就可以计算出标本的实际大小 (A)。

Magnification = Image size / Actual size     M = I / A

Rearranging the formula gives Actual size = Image size / Magnification. All measurements must be in the same unit. In light microscopy, image size is often given in millimetres (mm) while actual size is in micrometres (μm). Remember that 1 mm = 1000 μm, so converting units correctly is essential before substituting numbers into the formula.

公式变形可得 实际大小 = 图像大小 / 放大倍数。所有测量必须使用相同单位。在光学显微镜中，图像大小常以毫米 (mm) 给出，而实际大小则以微米 (μm) 表示。请牢记 1 mm = 1000 μm，因此在将数字代入公式前正确转换单位至关重要。

For example, if a cell image measures 22 mm under a magnification of ×4000, the actual size is 22 / 4000 = 0.0055 mm, which converts to 5.5 μm. Always show your working step by step in exam answers.

例如，如果一个细胞图像在放大 4000 倍下测得 22 mm，则实际大小为 22 / 4000 = 0.0055 mm，转换后为 5.5 μm。在考试作答时，一定要逐步展示计算过程。

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2. Cardiac Output | 心输出量

Cardiac output (CO) is the volume of blood pumped by one ventricle of the heart per minute. It depends on the stroke volume (SV) – the volume ejected per beat – and the heart rate (HR).

心输出量 (CO) 是指心脏每侧心室每分钟泵出的血液体积。它取决于每搏输出量 (SV)（每次心跳射出的血量）和心率 (HR)。

Cardiac output = Stroke volume × Heart rate     CO = SV × HR

Typical units are cm³ min⁻¹ or L min⁻¹ for CO, cm³ or mL for SV, and beats per minute (bpm) for HR. Because units must match, if SV is given in mL and you want CO in L min⁻¹, remember that 1000 mL = 1 L. Often data will show SV in mL and CO in L min⁻¹, so a division by 1000 is needed.

常见单位是：CO 用 cm³ min⁻¹ 或 L min⁻¹，SV 用 cm³ 或 mL，HR 用每分钟心跳次数 (bpm)。由于单位必须匹配，若 SV 以 mL 给出而你想得到以 L min⁻¹ 为单位的 CO，请记住 1000 mL = 1 L。题目中常出现 SV 用 mL，CO 用 L min⁻¹ 的情况，因此需要除以 1000。

For instance, an athlete with a resting heart rate of 50 bpm and a stroke volume of 90 mL has a cardiac output of 90 × 50 = 4500 mL min⁻¹, or 4.5 L min⁻¹. During exercise, both SV and HR increase, substantially raising CO to meet the demand for oxygen.

例如，一位运动员静息心率为 50 bpm，每搏输出量为 90 mL，则心输出量为 90 × 50 = 4500 mL min⁻¹，即 4.5 L min⁻¹。运动时，SV 和 HR 均会增加，从而显著提高 CO 以满足对氧气的需求。

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3. Respiratory Quotient (RQ) | 呼吸商

The respiratory quotient (RQ) is the ratio of the volume of carbon dioxide produced to the volume of oxygen consumed by an organism over a given time. RQ gives an indication of which respiratory substrate is being metabolised.

呼吸商 (RQ) 是指在特定时间内，有机体产生的二氧化碳体积与消耗的氧气体积之比。RQ 值能反映机体正在代谢哪种呼吸底物。

RQ = Volume of CO₂ produced / Volume of O₂ consumed

Pure carbohydrate respiration gives an RQ of approximately 1.0, lipid respiration yields about 0.7, and protein gives around 0.9. When measuring RQ using a respirometer, careful control of temperature and pressure is required. Any increase in CO₂ absorbed by soda lime must be accounted for in the calculation.

纯粹的碳水化合物呼吸产生的 RQ 值约为 1.0，脂类呼吸的 RQ 值约为 0.7，蛋白质则约为 0.9。使用呼吸计测量 RQ 时，需要严格控制温度和压力。苏打石灰吸收的 CO₂ 量在计算中必须予以考虑。

To calculate RQ, convert the distance moved by the manometer fluid into volumes using the known diameter of the capillary tube. The volume of O₂ consumed is found from the change with CO₂ absorber present, and the CO₂ produced is determined by comparing runs with and without the absorber.

计算 RQ 时，将测压管液柱移动的距离通过已知毛细管直径换算为体积。在装有 CO₂ 吸收剂的情况下，根据变化求得 O₂ 消耗体积；再与无吸收剂时的实验相比较，求出 CO₂ 产生体积。

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4. Fick’s Law of Diffusion | 菲克扩散定律

Fick’s Law describes the factors that determine the rate of diffusion across a membrane. It is fundamental to understanding gas exchange in the lungs and at tissues.

菲克定律描述了决定跨膜扩散速率的因素。它是理解肺部及组织气体交换的基础。

Rate of diffusion ∝ (Surface area × Concentration gradient) / Diffusion distance

The rate is proportional to the surface area available and the concentration difference across the membrane, and inversely proportional to the thickness of the diffusion path. Adaptations such as the numerous alveoli (large surface area), continuous ventilation and blood flow (maintaining steep concentration gradients), and the thin squamous epithelium (short diffusion distance) all maximise diffusion rate.

扩散速率与可用的表面积和跨膜浓度差成正比，与扩散距离的厚度成反比。众多肺泡（扩大表面积）、持续的通风与血流（维持陡峭的浓度梯度）以及薄的单层扁平上皮（缩短扩散距离）等适应特征，都能使扩散速率最大化。

Although a formal proportionality constant is not required at AS, you should be able to explain how changes in each factor influence the overall rate. When writing exam answers, always link structural features back to these three components of Fick’s Law.

尽管在 AS 阶段不要求使用正式的比例常数，但你应该能够解释每一因素的变化如何影响整体速率。在考试作答时，务必将结构特征与菲克定律的这三个要素联系起来。

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5. Surface Area to Volume Ratio | 表面积与体积比

As an organism or structure increases in size, its volume grows faster than its surface area. The surface area to volume ratio (SA:V) determines how efficiently substances can be exchanged with the environment.

随着生物体或结构的增大，其体积的增长速度快于表面积的增长。表面积与体积比 (SA:V) 决定了物质与环境交换的效率。

SA:V = Total surface area / Total volume

For a cube of side length L, the surface area is 6L² and the volume is L³, giving SA:V = 6/L. Small organisms, such as bacteria, have a very high SA:V and can rely on simple diffusion for their needs. Larger multicellular organisms have a low SA:V, making specialised exchange surfaces and transport systems essential.

对于边长为 L 的立方体，表面积为 6L²，体积为 L³，SA:V = 6/L。细菌等小型生物具有很高的 SA:V，可以依赖简单扩散维持生命活动。较大的多细胞生物 SA:V 低，因此专门的交换表面积和运输系统必不可少。

You may be asked to calculate the SA:V for simple geometric shapes, such as cubes or spheres, and to interpret the biological significance of the result. Remember that SA:V is a ratio, so the unit is typically expressed as “per unit length”, e.g. μm⁻¹.

考试中可能要求你计算简单几何形状（如立方体或球体）的 SA:V，并解释结果的生物学意义。请记住 SA:V 是一个比率，通常以“每单位长度”表示，例如 μm⁻¹。

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6. Simpson’s Index of Diversity | 辛普森多样性指数

Simpson’s Index (D) is used in ecology to quantify the biodiversity of a habitat. It accounts for both species richness and species evenness. The index ranges from 0 (low diversity) to 1 (very high diversity).

辛普森指数 (D) 在生态学中用于量化栖息地的生物多样性。它同时考虑了物种丰富度和物种均匀度。该指数的范围为 0（低多样性）至 1（极高多样性）。

D = 1 – ∑ (n/N)²

Here, n is the total number of organisms of a particular species, and N is the total number of organisms of all species. ∑ means ‘sum of’ – you calculate (n/N)² for each species and then add all these values together. In the OCR specification, the formula D = 1 – ∑(n/N)² is the version you are expected to use.

其中，n 为某一特定物种的全部个体数，N 为所有物种的总个体数。∑ 表示“求和”——你需要计算每个物种的 (n/N)²，然后将这些值全部相加。根据 OCR 大纲要求，你应该使用 D = 1 – ∑(n/N)² 这一公式。

A habitat with only one species gives D = 0, whereas a community with many species, each with similar abundance, gives D close to 1. When answering questions, avoid stating that a high index means the habitat is “stable”; instead, say it supports a wide variety of species with high species richness and evenness.

若栖息地仅有一个物种，则 D = 0；而一个拥有许多物种且每个物种个体数相近的群落，其 D 值接近 1。在作答时，不要声称高指数意味着栖息地“稳定”，而应表达为它支持了丰富的物种，且物种丰富度与均匀度都很高。

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7. Dilution and Molarity Calculations | 稀释与摩尔浓度计算

Producing a solution of required concentration is a core practical skill. The dilution equation links the concentration and volume of a stock solution to the desired working concentration and volume.

配制所需浓度的溶液是一项核心实验技能。稀释公式将储备液的浓度和体积与所需的工作浓度和体积联系起来。

C₁V₁ = C₂V₂

In this relationship, C₁ and V₁ are the concentration and volume of the starting solution, while C₂ and V₂ refer to the diluted solution. Units must be consistent: if concentrations are in mol dm⁻³, volumes should be in dm³ (or both in cm³). The amount of solute (n) can be found using n = C × V.

在此关系中，C₁ 和 V₁ 为初始溶液的浓度和体积，而 C₂ 和 V₂ 指稀释后的溶液。单位必须保持一致：若浓度用 mol dm⁻³，体积应用 dm³（或两者都用 cm³）。溶质的物质的量 (n) 可通过 n = C × V 得出。

For example, to prepare 250 cm³ of 0.2 mol dm⁻³ sucrose solution from a 1.0 mol dm⁻³ stock, the volume needed is V₁ = (C₂V₂)/C₁ = (0.2 × 0.25)/1.0 = 0.05 dm³ (50 cm³). You would then add distilled water to reach a final volume of 250 cm³. Always remember to mix thoroughly.

例如，欲用 1.0 mol dm⁻³ 的储备液配制 250 cm³ 的 0.2 mol dm⁻³ 蔗糖溶液，所需体积 V₁ = (C₂V₂)/C₁ = (0.2 × 0.25)/1.0 = 0.05 dm³（即 50 cm³）。然后加蒸馏水至最终体积 250 cm³。务必充分混匀。

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8. Initial Rate of Reaction | 初始反应速率

The initial rate of an enzyme-catalysed reaction is the fastest rate, occurring before substrate depletion or product inhibition affects the reaction. It is obtained from the tangent to the progress curve at time zero.

酶促反应的初始速率是最快速率，发生于底物耗尽或产物抑制产生影响之前。它由时间零点的进程曲线切线求得。

Initial rate = Δ[Product] / Δt    or    Δ[Substrate] / Δt

You can measure either the appearance of product or the disappearance of substrate per unit time. Data might be presented as a graph of concentration against time. Draw a straight line that follows the initial, steepest part of the curve and calculate its gradient. The units could be, for instance, mmol dm⁻³ min⁻¹.

你可以测量单位时间内产物的生成量或底物的消耗量。数据可能以浓度对时间的曲线图呈现。作一条与曲线最初、最陡部分相吻合的直线，并计算其斜率（梯度）。单位可以是 mmol dm⁻³ min⁻¹ 等。

At low substrate concentrations, the initial rate is roughly proportional to substrate concentration. At saturating concentrations, Vmax is approached. Understanding how to calculate the initial rate allows you to construct Michaelis–Menten and Lineweaver–Burk plots at a later stage, though at AS the emphasis is on practical evaluation and interpreting rate data.

在低底物浓度时，初始速率近似与底物浓度成正比。在饱和浓度下，速率趋近 Vmax。理解如何计算初始速率有助于后续构建米氏方程图和双倒数图，不过 AS 阶段的重点是实验评估和对速率数据的解读。

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9. Boyle’s Law and Ventilation | 波义耳定律与肺通气

Boyle’s Law describes the inverse relationship between the pressure and volume of a gas at constant temperature. It is crucial for explaining how air is drawn into and expelled from the lungs.

波义耳定律描述了恒温条件下气体压力与体积之间的反比关系。这对于解释空气如何被吸入和呼出肺部至关重要。

P₁V₁ = P₂V₂    (at constant temperature)

During inspiration, the diaphragm contracts and the ribcage moves upwards and outwards, increasing the volume of the thoracic cavity. According to Boyle’s Law, the increase in volume leads to a decrease in pressure inside the lungs below atmospheric pressure, so air rushes in. During expiration, the volume decreases, raising the pressure above atmospheric and forcing air out.

吸气时，膈肌收缩，胸廓向上向外移动，增大了胸腔体积。根据波义耳定律，体积增大导致肺内压力下降至低于大气压，于是空气涌入。呼气时，体积减小，压力上升至高于大气压，从而将空气挤出。

Although AS exams do not usually require numerical calculations with Boyle’s Law in breathing, you must be able to apply the pressure–volume principle to explain the mechanics of ventilation. Always refer to the movement of the diaphragm and intercostal muscles and the resulting pressure changes.

尽管 AS 考试通常不要求运用波义耳定律对呼吸进行数值计算，但你必须能够运用压力–体积原理来解释通气力学。作答时，务必要提及膈肌和肋间肌的运动以及由此产生的压力变化。

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10. Percentage Change and Ratio Calculations | 百分比变化与比率计算

Many OCR AS Biology questions ask you to process quantitative data, and percentage change is one of the most common calculations. It allows you to compare changes fairly when the starting values differ.

许多 OCR AS 生物题目要求你处理定量数据，而百分比变化是最常见的计算之一。当起始值不同时，它可以让你公平地比较变化。

Percentage change = [(Final value – Initial value) / Initial value] × 100%

If the final value is smaller than the initial value, the percentage change will be negative, indicating a decrease. Ratios, such as fresh mass : dry mass or surface area : volume, are expressed in their simplest form by dividing both numbers by the smaller value. Always state what the ratio represents, for example ‘1 : 0.6’.

如果最终值小于初始值，百分比变化将为负值，表示减少。比率，如鲜重 : 干重或表面积 : 体积，可用两数同时除以较小值来化为最简比。始终要说明比率所代表的意义，例如“1 : 0.6”。

When plotting data, you might need to convert raw numbers into percentage change to show relative trends, especially when investigating the effect of a factor on growth, enzyme activity or membrane permeability. Accuracy is essential – double-check that you have used the correct initial value as the denominator.

在绘制数据图表时，你可能需要将原始数值转换为百分比变化来展示相对趋势，尤其是在研究某一因素对生长、酶活性或膜通透性的影响时。准确性至关重要——务必再次确认分母使用了正确的初始值。

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