IB AQA 生物：公式汇总手册

。TITLE: IB AQA Biology: Formula Summary Handbook | IB AQA 生物：公式汇总手册

📚 IB AQA Biology: Formula Summary Handbook | IB AQA 生物：公式汇总手册

Mastering the core mathematical formulas in biology is essential for success in both IB and AQA examinations. This handbook compiles the most frequently tested equations — from magnification and genetics to ecology and membrane potentials — with clear explanations and paired bilingual paragraphs to support learners at every step. Use it as a quick reference or a revision companion to strengthen your data analysis and calculation skills.

掌握生物学中的核心数学公式对于 IB 和 AQA 考试的成功至关重要。本手册汇集了最常考的计算方程——从放大倍率、遗传学到生态学和膜电位——并配以清晰的双语段落解释，为学习者提供全面支持。将其用作快速参考或复习伴侣，以强化你的数据分析和计算能力。

1. Magnification Formula | 放大倍率公式

Magnification (M) in microscopy is calculated using the equation M = I / A, where I is the measured image size and A is the actual specimen size. This relationship is fundamental when working with scale bars and biological drawings.

显微镜下的放大倍率 (M) 使用公式 M = I / A 计算，其中 I 为测得的图像尺寸，A 为标本的实际尺寸。在处理比例尺和生物绘图时，这一关系至关重要。

M = I / A

Both I and A must be expressed in the same unit (e.g., mm, μm). Rearranging the formula allows you to calculate actual size from a known magnification: A = I / M.

I 和 A 必须使用相同的单位（如 mm、μm）。通过变形公式，可以根据已知放大倍率计算实际尺寸：A = I / M。

If a micrograph shows a cell with an image length of 45 mm and the magnification is ×400, the actual length is 45 / 400 = 0.1125 mm = 112.5 μm.

如果显微照片显示一个细胞的图像长度为 45 mm，放大倍率为 ×400，则实际长度为 45 / 400 = 0.1125 mm = 112.5 μm。

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2. Cell Counting (Haemocytometer) | 细胞计数（血球计数板）

When using an improved Neubauer haemocytometer, the number of cells per unit volume is determined by counting cells in a known grid area and depth. The formula is: Cell concentration = (total cells counted / number of squares) × dilution factor × 10⁴.

使用改进的 Neubauer 血球计数板时，通过计数已知网格面积和深度内的细胞来确定单位体积的细胞数。公式为：细胞浓度 = (计数细胞总数 / 方格数) × 稀释倍数 × 10⁴。

C = (N / n) × D × 10⁴ cells/mL

N is the total number of viable cells counted, n is the number of 0.1 mm³ squares used, D is the dilution factor, and 10⁴ converts per 0.1 mm³ to per mL. Always exclude cells touching two defined edges to avoid double counting.

N 为计数的活细胞总数，n 为使用的 0.1 mm³ 方格数，D 为稀释倍数，10⁴ 将每 0.1 mm³ 转换为每 mL。始终剔除接触两条指定边界的细胞，以避免重复计数。

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3. Hardy-Weinberg Principle | 哈代-温伯格原理

The Hardy-Weinberg equations describe allele and genotype frequencies in a non-evolving population. The two fundamental equations are p + q = 1 (allele frequencies) and p² + 2pq + q² = 1 (genotype frequencies).

哈代-温伯格方程描述了非进化种群中的等位基因和基因型频率。两个基本方程为：p + q = 1（等位基因频率）和 p² + 2pq + q² = 1（基因型频率）。

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 is the homozygous dominant frequency, 2pq heterozygotes, and q² homozygous recessive. Use these to predict carrier frequencies or test evolutionary change.

其中 p 为显性等位基因频率，q 为隐性等位基因频率，p² 为纯合显性频率，2pq 为杂合子频率，q² 为纯合隐性频率。利用这些方程可以预测携带者频率或检验进化变化。

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4. Chi-Squared Test | 卡方检验

The chi-squared (χ²) test evaluates whether observed results differ significantly from expected values. The formula sums the squared differences between observed (O) and expected (E) frequencies divided by the expected frequency for each category.

卡方 (χ²) 检验用于评估观测结果是否与预期值存在显著差异。该公式对每个类别的观测频率 (O) 与预期频率 (E) 之差的平方除以预期频率进行求和。

χ² = Σ (O − E)² / E

A high χ² value relative to the critical value (at a chosen probability, usually p=0.05, and degrees of freedom = number of categories − 1) indicates that the null hypothesis should be rejected. This test is widely used in genetics and ecology.

相对于临界值（通常在概率 p=0.05，自由度 = 类别数 − 1 时选取）较高的 χ² 值表明应该拒绝零假设。该检验广泛应用于遗传学和生态学。

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

Simpson’s Diversity Index (D) measures biodiversity by accounting for both species richness and evenness. The formula calculates the probability that two individuals randomly selected from a sample will belong to the same species and subtracts this from 1.

辛普森多样性指数 (D) 通过同时考虑物种丰富度和均匀度来衡量生物多样性。该公式计算从样本中随机抽取的两个个体属于同一物种的概率，并从 1 中减去该值。

D = 1 − ( Σ n(n − 1) / N(N − 1) )

n = total number of organisms of a particular species, N = total number of organisms of all species. D values range from 0 (low diversity) to 1 (high diversity). Higher D indicates greater biodiversity and ecosystem stability.

n = 某一特定物种的个体总数，N = 所有物种的个体总数。D 值范围从 0（低多样性）到 1（高多样性）。D 值越高，表明生物多样性越大，生态系统越稳定。

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6. Lincoln Index (Mark-Release-Recapture) | 林肯指数（标记重捕法）

For mobile animal populations, the Lincoln Index estimates population size (N) through a capture-mark-release-recapture method. The formula assumes a closed population and equal catchability.

对于移动性动物种群，林肯指数通过捕获-标记-释放-重捕的方法估算种群大小 (N)。该公式假设种群封闭且捕获概率均等。

N = (M × C) / R

M = number of individuals captured and marked in the first sample; C = total number captured in the second sample; R = number of marked individuals recaptured in the second sample. Ensure marking does not affect survival or behaviour.

M = 第一次捕获并标记的个体数；C = 第二次捕获的总个体数；R = 第二次捕获中带标记的个体数。确保标记不影响个体的存活或行为。

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7. Respiratory Quotient (RQ) | 呼吸商

The respiratory quotient is the ratio of carbon dioxide produced to oxygen consumed during respiration. It reveals which respiratory substrate is being metabolised and is particularly useful in respirometer experiments.

呼吸商是呼吸过程中产生的二氧化碳与消耗的氧气之比。它揭示了正在代谢的呼吸底物类型，在呼吸计实验中尤其有用。

RQ = CO₂ produced / O₂ consumed

Typical RQ values: carbohydrates ≈ 1.0, lipids ≈ 0.7, proteins ≈ 0.9. Values above 1.0 suggest anaerobic respiration. Temperature and substrate availability influence RQ.

典型 RQ 值：碳水化合物约为 1.0，脂质约为 0.7，蛋白质约为 0.9。RQ 值高于 1.0 提示存在无氧呼吸。温度和底物可用性影响 RQ。

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8. Rate of Photosynthesis | 光合作用速率

The rate of photosynthesis can be measured by oxygen production, carbon dioxide uptake, or changes in biomass. A common approach is to record the volume of O₂ evolved per unit time or the time required for a standardised indicator to change colour.

光合作用速率可通过氧气产生量、二氧化碳吸收量或生物量变化来测量。常用的方法是记录单位时间内释放的 O₂ 体积，或标准化指示剂变色所需的时间。

Rate = ΔO₂ volume / Δt

Using the reciprocal of ET₅₀ (time for 50% of leaf discs to float) provides a relative rate when investigating limiting factors such as light intensity, CO₂ concentration, or temperature.

在研究光强度、CO₂ 浓度或温度等限制因素时，使用 ET₅₀（叶圆片上浮 50% 所需时间）的倒数可获得相对速率。

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9. Osmolarity and Water Potential | 渗透压与水势

Water potential (Ψ) determines the direction of water movement in plants. It is the sum of solute potential (Ψₛ) and pressure potential (Ψₚ). The solute potential of a solution is calculated using the van’t Hoff relation.

水势 (Ψ) 决定了植物中水分的移动方向。它是溶质势 (Ψₛ) 与压力势 (Ψₚ) 之和。溶液的溶质势使用范特霍夫关系计算。

Ψ = Ψₛ + Ψₚ

Ψₛ = − i C R T

i = ionisation constant (e.g., 1 for sucrose, 2 for NaCl), C = molar concentration (mol L⁻¹), R = pressure constant (0.0831 L bar mol⁻¹ K⁻¹), T = temperature in Kelvin. The negative sign indicates that solutes lower water potential.

i = 解离常数（如蔗糖为 1，NaCl 为 2），C = 摩尔浓度 (mol L⁻¹)，R = 压力常数 (0.0831 L bar mol⁻¹ K⁻¹)，T = 开尔文温度。负号表明溶质会降低水势。

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10. Nernst Equation | 能斯特方程

The Nernst equation calculates the equilibrium potential for an ion across a membrane, based on the concentration gradient. It is fundamental for understanding resting membrane potentials and action potentials in neurones.

能斯特方程根据浓度梯度计算某种离子跨膜电位的平衡电势。它是理解静息膜电位和神经元动作电位的基础。

Eₓ = (RT / zF) ln([X]out / [X]in)

At 37°C, for a monovalent ion like K⁺, the equation simplifies to: Eₖ = 61.5 log₁₀([K⁺]out / [K⁺]in) mV. R = gas constant, T = temperature, z = ion charge, F = Faraday constant.

在 37°C 下，对于 K⁺ 等单价离子，方程可简化为：Eₖ = 61.5 log₁₀([K⁺]out / [K⁺]in) mV。R = 气体常数，T = 温度，z = 离子电荷，F = 法拉第常数。

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11. Enzyme Activity (Initial Rate) | 酶活性（初始速率）

Enzyme activity is often quantified by measuring the initial rate of reaction — the change in product concentration or substrate disappearance over the early linear phase. This avoids complications from product inhibition or substrate depletion.

酶活性通常通过测量反应的初始速率来量化——即反应早期线性阶段产物浓度或底物减少的变化。这避免了产物抑制或底物耗竭带来的复杂影响。

Initial rate = Δ[Product] / Δt

Alternatively, if using time for a fixed change (e.g., colour change), rate can be expressed as 1 / time. Enzyme kinetics are influenced by temperature, pH, and substrate concentration.

或者，如果使用发生固定变化（如颜色变化）所需的时间，速率可表示为 1 / 时间。酶动力学受温度、pH 和底物浓度的影响。

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12. Standard Deviation and Standard Error | 标准差与标准误

Standard deviation (s) measures the spread of data around the mean, while standard error (SE) indicates the precision of the sample mean estimate. Both are essential for error bars and t-tests in IA investigations.

标准差 (s) 衡量数据在均值周围的离散程度，而标准误 (SE) 则指示样本均值估计的精确度。两者对于 IA 调查中的误差线和 t 检验至关重要。

s = √( Σ (x − x̄)² / (n − 1) )

SE = s / √n

Here x represents each individual value, x̄ is the mean, and n is the sample size. SE decreases as sample size increases, reflecting greater confidence in the mean. Use (n − 1) for sample standard deviation.

其中 x 代表每个单独数值，x̄ 为均值，n 为样本容量。标准误随样本容量增大而减小，反映对均值的信心增强。计算样本标准差时使用 (n − 1)。

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