The Ultimate Formula Cheat Sheet for Biology (IB & OCR) | IB 与 OCR 生物公式汇总手册

📚 The Ultimate Formula Cheat Sheet for Biology (IB & OCR) | IB 与 OCR 生物公式汇总手册

Quantitative skills are essential in both IB and OCR A Level Biology. From calculating cell size using a microscope to performing chi-squared tests on genetic data, a solid grasp of the key formulae can make the difference between a good grade and a great one. This handbook brings together all the essential equations you need to master, with clear explanations in both English and Chinese, matching the dual-language style of aleveler.com. Each formula is presented with its components defined, sample applications, and common pitfalls to avoid.

定量技能在 IB 和 OCR A Level 生物中都至关重要。从使用显微镜计算细胞大小到对遗传数据进行卡方检验，扎实掌握关键公式可能是优秀与卓越之间的分水岭。本手册汇集了所有必须掌握的核心方程式，并用中英双语进行清晰解释，与 aleveler.com 的双语风格保持一致。每个公式都配有定义、示例应用和常见错误的提示，帮助你精准记忆。

1. Magnification Equation | 放大倍率公式

The most frequently used formula in microscopy is the magnification equation. It allows you to determine either the magnification, the image size, or the actual size of a specimen. All quantities must be expressed in identical units before substitution.

显微镜学中最常用的公式是放大倍率公式。你可以用它计算放大倍率、图像尺寸或标本的实际尺寸。所有量在代入前必须转换为相同的单位。

Magnification = Image size ÷ Actual size

When rearranging, Actual size = Image size ÷ Magnification. Image size can be measured with a ruler on a photomicrograph, and magnification is usually given. A common mistake is forgetting to convert millimetres to micrometres (1 mm = 1000 μm).

变换公式可得 实际尺寸 = 图像尺寸 ÷ 放大倍率。图像尺寸可用直尺在显微照片上测量，放大倍率通常已知。一个常见错误是忘记将毫米转换为微米（1 mm = 1000 μm）。

2. Microscope Unit Conversions | 显微镜单位换算

Accurate unit conversion underpins every microscopic measurement. In IB and OCR exams, you are expected to move fluently between metres, millimetres, micrometres and nanometres. The standard prefixes must be memorised.

精确的单位换算是所有显微测量的基础。IB 和 OCR 考试要求你能够在米、毫米、微米和纳米之间自如转换。标准前缀必须牢记。

1 mm = 10⁻³ m , 1 μm = 10⁻⁶ m , 1 nm = 10⁻⁹ m

Or in a more practical form: 1 mm = 1000 μm ; 1 μm = 1000 nm. When a cell diagram has a scale bar labelled ’20 μm’, measure its length in mm, then convert that measured length into μm using the same ratio to find the actual size.

更实用的形式：1 mm = 1000 μm；1 μm = 1000 nm。当细胞图带有标有“20 μm”的比例尺时，先以毫米为单位测量比例尺长度，再将测量值按相同比例换算为微米，从而得出实际尺寸。

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

Biodiversity can be quantified using Simpson’s Diversity Index. The IB syllabus uses the formula D = 1 – Σ(n/N)², while OCR often uses the reciprocal form. Here we present the IB version, which yields a value between 0 (low diversity) and 1 (high diversity).

生物多样性可以用辛普森多样性指数进行量化。IB 大纲使用 D = 1 – Σ(n/N)²，而 OCR 常使用倒数形式。此处给出的是 IB 版本，其值介于 0（低多样性）与 1（高多样性）之间。

D = 1 – Σ (n / N)²

Here, n is the number of individuals of a particular species, and N is the total number of individuals of all species. The sum (Σ) is taken over all species present. A community with many equally abundant species gives a high D value, indicating a stable ecosystem.

式中，n 为某一特定物种的个体数，N 为所有物种的个体总数。求和符号 Σ 涵盖所有物种。由许多数量均匀的物种构成的群落会得到较高的 D 值，意味着生态系统较为稳定。

4. Lincoln Index (Capture-Mark-Recapture) | 林肯指数（标记–重捕法）

For estimating the population size of mobile animals, the Lincoln Index is a classic tool. It assumes that marked and unmarked individuals mix randomly and that no migration, births, or deaths occur between samples.

对于估计活动动物的种群大小，林肯指数是一种经典工具。它假设标记和未标记个体能随机混合，并且在两次取样之间没有迁移、出生或死亡。

N = (n₁ × n₂) ÷ m

Where n₁ is the number caught and marked in the first sample, n₂ is the total number caught in the second sample, and m is the number of marked individuals recaptured. Ethical considerations, such as ensuring tags do not harm the organism, are often examined alongside the calculation.

其中 n₁ 为第一次捕获并标记的数量，n₂ 为第二次捕获的总数，m 为第二次捕获中带标记的个体数。伦理考量（例如确保标记不会伤害生物）常常与计算题一起考查。

5. Net Primary Productivity and Ecological Efficiency | 净初级生产力与生态效率

Energy flows through ecosystems can be quantified using productivity equations. Net primary productivity (NPP) is the energy remaining after plant respiration, and ecological efficiency describes the percentage of energy transferred between trophic levels.

生态系统中的能量流动可以用生产力方程进行量化。净初级生产力（NPP）是植物呼吸作用后剩余的能量，而生态效率则描述营养级之间能量传递的百分比。

NPP = GPP – R

Ecological Efficiency (%) = (Energy transferred to next level ÷ Energy consumed) × 100

GPP is gross primary productivity, the total energy fixed by photosynthesis. R represents respiratory losses. Typical ecological efficiencies are around 10 %, explaining why food chains rarely exceed five trophic levels.

GPP 为总初级生产力，即光合作用固定的总能量。R 代表呼吸消耗。典型的生态效率约为 10 %，这也解释了为何食物链很少超过五个营养级。

6. Cardiac Output Equation | 心输出量公式

Cardiac output is a fundamental measure of heart performance. It links heart rate and stroke volume, and variations in these parameters are key to understanding exercise physiology and cardiovascular disease.

心输出量是衡量心脏功能的基本指标。它将心率和每搏输出量联系在一起，这些参数的变化是理解运动生理学和心血管疾病的关键。

Cardiac Output (CO) = Heart Rate (HR) × Stroke Volume (SV)

CO is typically expressed in litres per minute. During strenuous exercise, HR can triple and SV can double, causing CO to rise dramatically. Make sure to distinguish between resting and active values in exam questions.

心输出量通常以升每分钟表示。在剧烈运动期间，心率可增加两倍，每搏输出量可翻倍，使心输出量急剧上升。答题时务必区分静息值和运动值。

7. Lung Volume Calculations | 肺容积计算

Spirometer traces provide data to calculate tidal volume, vital capacity, and breathing rate. These basic respiratory equations are tested in both IB and OCR specifications, often alongside oxygen consumption.

肺活量描记图可提供数据用以计算潮气量、肺活量和呼吸频率。这些基础的呼吸公式在 IB 和 OCR 考纲中均有考查，常与氧耗量结合出题。

Tidal Volume (TV) = Volume per breath at rest

Vital Capacity (VC) = Maximum volume exhaled after maximum inhalation

Breathing Rate = Number of breaths per minute

To find minute ventilation, multiply tidal volume by breathing rate. Always check the y-axis scale on the spirometer graph; a single large square may not equal 1 dm³.

计算每分通气量时，将潮气量乘以呼吸频率即可。务必检查肺活量计图形的纵轴刻度；一个大方格不一定等于 1 dm³。

8. Hardy-Weinberg Equilibrium | 哈代–温伯格平衡

The Hardy-Weinberg principle is used to calculate allele and genotype frequencies in a non-evolving population. It is a core topic in IB Biology and also appears in OCR A Level genetics. The two fundamental equations must be memorised.

哈代–温伯格原理用于计算一个不发生进化的种群中的等位基因频率和基因型频率。它是 IB 生物的核心主题，也出现在 OCR A Level 遗传学中。两个基本方程必须熟记。

p + q = 1

p² + 2pq + q² = 1

Here, p is the frequency of the dominant allele, q is the frequency of the recessive allele. p² represents the frequency of homozygous dominant individuals, 2pq heterozygotes, and q² homozygous recessives. Always calculate q from the known recessive phenotype frequency first, then derive the rest.

式中 p 代表显性等位基因频率，q 为隐性等位基因频率。p² 表示显性纯合子频率，2pq 表示杂合子频率，q² 表示隐性纯合子频率。解题时一定要根据已知的隐性表型频率先求出 q，再推算出其余值。

9. Chi-Squared Test | 卡方检验

The chi-squared test is used to determine whether there is a significant difference between observed and expected frequencies. It is essential for analysing genetic crosses, ecological distributions, and any categorical data in both IB and OCR investigations.

卡方检验用于判断观测频率与预期频率之间是否存在显著差异。它是分析遗传杂交、生态分布以及任何分类数据（IB 与 OCR 均有涉及）的重要工具。

χ² = Σ (O – E)² ÷ E

O represents observed values and E expected values. The degrees of freedom (df) usually equal the number of categories minus one. Compare the calculated χ² to the critical value from a table at p = 0.05. If χ² calculated > χ² critical, reject the null hypothesis.

O 表示观测值，E 表示预期值。自由度（df）通常等于类别数减一。将计算得到的 χ² 与 p = 0.05 时的临界值表比较。如果 χ² 计算值大于临界值，则拒绝原假设。

10. Water Potential and Solute Potential | 水势与溶质势

Water movement in plants and animal cells is governed by water potential (ψ). In IB and OCR, the total water potential is the sum of solute potential and pressure potential. For solutions without a pressure component, solute potential alone determines osmosis.

植物和动物细胞中的水分运动由水势（ψ）决定。在 IB 和 OCR 中，总水势是溶质势与压力势之和。对于无压力组分的溶液，仅溶质势决定渗透方向。

ψ = ψₛ + ψₚ

ψₛ = –iCRT

In the van’t Hoff formula, i is the ionisation constant (e.g., 1 for sucrose, 2 for NaCl), C is the molar concentration, R is the pressure constant (0.0831 litre bar mol⁻¹ K⁻¹), and T is the absolute temperature in Kelvin. The negative sign indicates that solutes lower the water potential.

在范特霍夫公式中，i 是电离常数（如蔗糖为 1，NaCl 为 2），C 为摩尔浓度，R 为压力常数（0.0831 升·巴·摩尔⁻¹·开尔文⁻¹），T 是以开尔文为单位的绝对温度。负号表示溶质会降低水势。

11. pH and Absorbance | pH 值与吸光度

Enzyme activity is often monitored by measuring absorbance, and pH calculations help explain enzyme denaturation. The logarithmic relationship of pH and the Beer-Lambert law for absorbance appear in practical assessments.

酶活性常常通过测量吸光度来监测，而 pH 计算有助于解释酶变性。pH 的对数关系以及用于吸光度的比尔–朗伯定律常出现在实验评估中。

pH = –log₁₀ [H⁺]

Absorbance (A) = log₁₀ (I₀ / I) or A = –log₁₀ T

Where [H⁺] is hydrogen ion concentration in mol dm⁻³, I₀ is the incident light intensity, I is the transmitted light intensity, and T is transmittance. A higher absorbance corresponds to a darker solution or greater product formation in a colorimetric enzyme assay.

其中 [H⁺] 是以 mol dm⁻³ 为单位的氢离子浓度，I₀ 为入射光强，I 为透射光强，T 为透射率。吸光度越高，意味着溶液颜色越深或在比色酶实验中产物生成越多。

12. Serial Dilution Calculations | 连续稀释计算

Serial dilutions are used to create a range of concentrations for calibration curves or to reduce microbial colonies to a countable number. Mastering the dilution factor is critical for both IB internal assessments and OCR practical endorsements.

连续稀释用于制作一系列浓度以建立标准曲线，或将微生物菌落数量稀释至可计数范围。掌握稀释倍数对 IB 内部评估和 OCR 实践考核都至关重要。

C₁ × V₁ = C₂ × V₂

Dilution Factor = Volume of sample ÷ (Volume of sample + Volume of diluent)

When performing a serial dilution, the final concentration equals the initial concentration multiplied by the dilution factor raised to the power of the number of steps. Always record the units and check whether the question asks for the dilution factor or the final concentration.

执行连续稀释时，最终浓度等于初始浓度乘以稀释倍数的稀释步骤次方。务必记录单位并确认题目要求的是稀释倍数还是最终浓度。

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