Pre-U CAIE Biology Equation & Theorem Quick Reference | Pre-U CAIE 生物：公式定理速查手册

📚 Pre-U CAIE Biology Equation & Theorem Quick Reference | Pre-U CAIE 生物：公式定理速查手册

This handbook collates essential equations, constants, and statistical tools encountered across the CAIE Pre-U Biology syllabus. Each entry states the formula, defines its components, and explains when it is applied—from microscopy and heart function to population genetics and biodiversity indices. Use it alongside practical questions and data-analysis exercises to reinforce your numerical fluency.

本手册汇集了 CAIE Pre-U 生物课程中涉及的核心公式、常数与统计工具。每个条目给出公式、定义各分量，并说明其应用场景——从显微镜操作和心脏功能，到群体遗传学与生物多样性指数。配合实验题和数据分析练习使用，可有效提升你的计算熟练度。

1. Magnification & Scale Conversion | 显微镜放大倍数与单位换算

Magnification is defined as the ratio of image size to actual object size. The formula requires both lengths to be in the same unit.

放大倍数定义为图像尺寸与实际物体尺寸之比。使用该公式时，两个长度需统一单位。

M = I / A

Where M = magnification, I = measured image length, A = actual object length. Commonly used with unit prefixes: 1 mm = 1000 µm, 1 µm = 1000 nm. Always convert to the same unit before division.

其中 M = 放大倍数，I = 实测图像长度，A = 实际物体长度。常用单位换算：1 mm = 1000 µm，1 µm = 1000 nm。计算前务必将单位统一。

For scale bar questions, measure the length of the scale bar in the image and divide by the length it represents in reality. This quotient is the magnification.

处理标尺题时，在图像上量取标尺长度，除以它所代表的实际长度。所得商即为放大倍数。

2. Osmolarity and Water Potential Relations | 渗透浓度与水势关系

Water potential (Ψ) determines the direction of water movement. It is the sum of solute potential (Ψₛ) and pressure potential (Ψₚ). In dilute solutions at atmospheric pressure, Ψₚ is often zero.

水势（Ψ）决定水分运动方向。它是溶质势（Ψₛ）与压力势（Ψₚ）之和。在常压下的稀溶液中，Ψₚ 通常为零。

Ψ = Ψₛ + Ψₚ

Solute potential can be estimated using the van ‘t Hoff relation: Ψₛ = −iCRT, where i is the ionization constant (1 for non-electrolytes like sucrose), C is molar concentration, R is the pressure constant (0.0831 L bar mol⁻¹ K⁻¹), and T is temperature in kelvin. For typical lab temperatures near 25 °C, Ψₛ ≈ −C × 2.48 bars.

溶质势可用范特霍夫关系估算：Ψₛ = −iCRT，其中 i 是电离常数（非电解质如蔗糖取 1），C 为摩尔浓度，R 为压力常数（0.0831 L bar mol⁻¹ K⁻¹），T 为开尔文温度。在 25 °C 附近的实验条件下，Ψₛ ≈ −C × 2.48 bars。

A practical conversion for dilute sucrose solutions at 20 °C: 1 mol m⁻³ ≈ −0.00276 MPa. Memorise the typical water potential values: pure water = 0 MPa; a plant cell fully turgid ≈ 0 MPa; flaccid cell ≈ −0.5 to −1.5 MPa.

20 °C 下稀蔗糖溶液的实用换算：1 mol m⁻³ ≈ −0.00276 MPa。记住典型水势值：纯水 = 0 MPa；充分膨胀的植物细胞 ≈ 0 MPa；萎蔫细胞 ≈ −0.5 至 −1.5 MPa。

3. Temperature Coefficient (Q₁₀) | 温度系数 Q₁₀

The Q₁₀ value indicates how much a reaction rate increases for a 10 °C rise in temperature. It is a dimensionless number.

Q₁₀ 值表示温度每升高 10 °C 时反应速率增加的倍数，是一个无量纲数值。

Q₁₀ = (R₂/R₁)^{10/(T₂−T₁)}

Here R₁ and R₂ are reaction rates at temperatures T₁ and T₂ (in °C). For typical enzyme-controlled processes, Q₁₀ ≈ 2, meaning the rate doubles with a 10 °C rise. Values near 1 suggest physical processes or limitations, while values significantly above 2 may indicate denaturation effects at the higher temperature.

式中 R₁ 和 R₂ 分别是温度 T₁ 和 T₂（单位为 °C）下的反应速率。对大多数酶控过程，Q₁₀ ≈ 2，即温度每升 10 °C 速率加倍。Q₁₀ 接近 1 往往表明物理过程或限制因素；显著大于 2 可能暗示高温下发生了变性。

If only a Q₁₀ value and one rate are known, the expected rate at a different temperature can be predicted by reversing the formula.

若已知 Q₁₀ 值和某一温度下的速率，可通过逆用公式预测另一温度下的预期速率。

4. Respiratory Quotient (RQ) | 呼吸商

The respiratory quotient compares the volume of CO₂ produced to the volume of O₂ consumed by an organism over a given time. It reveals the predominant respiratory substrate.

呼吸商比较了在一定时间内生物体释放的 CO₂ 体积与消耗的 O₂ 体积，可揭示正在利用的主要呼吸底物。

RQ = V(CO₂ produced) / V(O₂ consumed)

RQ = V(CO₂ 释放量) / V(O₂ 消耗量)

Typical RQ values: carbohydrate = 1.0; lipid ≈ 0.7; protein ≈ 0.9. Mixtures give intermediate values. In anaerobic respiration, RQ can exceed 1.0 because CO₂ is still produced without proportional O₂ uptake. In respirometers, measure gas volume changes with and without a CO₂ absorber to derive both volumes.

典型 RQ 值：糖类 = 1.0；脂质 ≈ 0.7；蛋白质 ≈ 0.9。混合底物给出中间值。无氧呼吸中，RQ 可超过 1.0，因为 CO₂ 仍被释放而无等比例的 O₂ 被摄入。使用呼吸计测量时，通过比较有无 CO₂ 吸收剂时的气体体积变化，即可分别求得两种气体体积。

5. Cardiac Output & Related Haemodynamics | 心输出量及相关血流动力学

Cardiac output (CO) is the volume of blood pumped by one ventricle per minute. It is the product of heart rate and stroke volume.

心输出量（CO）是每分钟一侧心室泵出的血量，等于心率与每搏输出量的乘积。

CO = HR × SV

CO = cardiac output (dm³ min⁻¹ or L min⁻¹), HR = heart rate (beats min⁻¹), SV = stroke volume (dm³ or L). Resting CO for an adult ≈ 5 L min⁻¹; during exercise it can rise 4-5 fold. Stroke volume itself depends on venous return and contractility (Starling’s law).

CO = 心输出量（dm³ min⁻¹ 或 L min⁻¹），HR = 心率（次/分钟），SV = 每搏输出量（dm³ 或 L）。成人静息时 CO ≈ 5 L min⁻¹；运动中可增至 4–5 倍。每搏输出量本身取决于静脉回流量和收缩力（斯塔林定律）。

Mean arterial pressure (MAP) can be estimated as MAP = DP + 1/3(SP−DP), where SP is systolic and DP diastolic pressure. This is less commonly tested but useful in physiological contexts.

平均动脉压（MAP）可估算为 MAP = 舒张压 + 1/3（收缩压－舒张压）。虽较少被直接考查，但在生理学情境中颇为实用。

6. Spirometry & Lung Volumes | 肺活量测定与肺容积

Spirometer traces allow calculation of tidal volume (TV), vital capacity (VC), inspiratory and expiratory reserve volumes, and respiratory frequency. The key relationship is:

肺量计记录可用于计算潮气量（TV）、肺活量（VC）、补吸气量、补呼气量和呼吸频率。关键关系式为：

VC = TV + IRV + ERV

肺活量 = 潮气量 + 补吸气量 + 补呼气量

Total lung capacity (TLC) = VC + residual volume (RV). Because RV cannot be exhaled, it must be estimated using a helium dilution method or given in exam data.

肺总量（TLC）= 肺活量 + 余气量（RV）。余气量无法被呼出，须用氦稀释法测定或由试题数据给出。

Minute ventilation = tidal volume × breathing rate.

每分通气量 = 潮气量 × 呼吸频率。

Minute ventilation = TV × f

每分通气量 = TV × f

Dead space (anatomical) typically ≈ 150 cm³ in adults; alveolar ventilation = (TV – dead space) × f. Emphasise that only alveolar ventilation participates in gas exchange.

解剖无效腔成人通常 ≈ 150 cm³；肺泡通气量 = (TV − 无效腔量) × 呼吸频率。需强调只有肺泡通气参与气体交换。

7. Mendelian Genetics & Probability Rules | 孟德尔遗传学与概率法则

Monohybrid crosses can be predicted using Punnett squares, but dihybrid crosses are streamlined with the product rule and sum rule.

单因子杂交可通过旁氏表预测，而双因子杂交可借助乘法法则和加法法则简化。

Product rule: probability of two independent events both occurring = P(A) × P(B).

乘法法则：两独立事件同时发生的概率 = P(A) × P(B)。

Sum rule: probability of either of two mutually exclusive events occurring = P(A) + P(B).

加法法则：两互斥事件任一发生的概率 = P(A) + P(B)。

For a dihybrid cross AaBb × AaBb, the expected phenotypic ratio is 9:3:3:1, derived as (3:1) × (3:1). The probability of a homozygous recessive phenotype (aabb) is ¼ × ¼ = 1/16.

对于双因子杂交 AaBb × AaBb，预期表型比为 9:3:3:1，由 (3:1) × (3:1) 得来。纯合隐性表型 (aabb) 的概率为 ¼ × ¼ = 1/16。

Chi-squared (χ²) tests are later used to compare observed ratios with expected Mendelian ratios. The degree of freedom for a monohybrid cross is 1; for a dihybrid cross (testing 9:3:3:1) it is 3.

后续可用卡方（χ²）检验比较观察值与预期孟德尔比例的差异。单因子杂交的自由度为 1；双因子杂交（检验 9:3:3:1）的自由度为 3。

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

Used to calculate allele and genotype frequencies in a non-evolving population.

用于计算非进化群体中的等位基因频率和基因型频率。

p + q = 1

p = frequency of dominant allele, q = frequency of recessive allele. The genotype frequencies for a two-allele system are:

p = 显性等位基因频率，q = 隐性等位基因频率。双等位基因体系的基因型频率为：

p² + 2pq + q² = 1

Where p² = frequency of homozygous dominant, 2pq = frequency of heterozygotes, q² = frequency of homozygous recessive. If the recessive phenotype frequency is known (q²), take the square root to find q, then p = 1−q. Exam questions often ask to calculate carrier (2pq) frequency for a recessive disorder.

其中 p² = 纯合显性频率，2pq = 杂合子频率，q² = 纯合隐性频率。若已知隐性表型频率 (q²)，开方求得 q，然后 p = 1−q。考试题常要求计算隐性遗传病的携带者频率 (2pq)。

Assumptions: large population, random mating, no mutation, no migration, no natural selection. Violations cause deviation from equilibrium.

假设条件：大群体、随机交配、无突变、无迁移、无自然选择。任何条件破坏都会导致偏离平衡。

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

Measures species biodiversity, considering both richness (number of species) and evenness (relative abundance).

衡量物种生物多样性，同时考虑丰富度（物种数）和均匀度（相对多度）。

D = 1 − ( Σ n(n−1) / N(N−1) )

n = number of individuals of a particular species; N = total number of all individuals of all species. D ranges from 0 (low diversity) to 1 (high diversity). An alternative formula sometimes used is D = Σ (n/N)², but Pre-U mainly uses the complement form above. High D suggests a stable ecosystem.

n = 某一特定物种的个体数；N = 所有物种的全部个体总数。D 值范围从 0（低多样性）到 1（高多样性）。有时也会用到 D = Σ (n/N)² 的形式，但 Pre-U 主要使用上述互补形式。高 D 值暗示稳定的生态系统。

When comparing two sites, compute D for each. Remember: D is sensitive to sample size; always use the same sampling effort.

比较两个样地时分别计算 D 值。注意：D 对样本量敏感；应始终采用相同的采样强度。

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

Estimates population size (N) for mobile organisms using proportion of marked individuals in a second sample.

通过第二次样本中标记个体所占比例来估算移动生物种群的大小（N）。

N = (M × C) / R

M = number marked and released initially; C = total number caught in second sample; R = number of marked individuals recaptured in second sample. Assumptions: closed population, marks are not lost, marked animals mix randomly, marking does not affect survival or catchability.

M = 初次标记并释放的数量；C = 第二次捕获的总数；R = 第二次捕获中带标记的个体数。假设条件：封闭种群、标记不丢失、标记个体随机混合、标记不影响存活率和可捕性。

If R is small, the estimate has large uncertainty. The 95% confidence interval can be approximated using 1.96 × √(M²C²/R³), but exams more commonly ask for the point estimate and critique of assumptions.

若 R 很小，估算值的不确定性很大。95% 置信区间可近似为 1.96 × √(M²C²/R³)，但考试更经常要求计算点估计值并评论假设条件。

11. Enzyme Kinetics & Rate Calculations | 酶动力学与速率计算

Initial rate of reaction (V₀) is often measured as product formed per unit time or substrate consumed per unit time. This is central to interpreting progress curves.

初始反应速率（V₀）通常以单位时间内产物生成量或底物消耗量来表示，这是解读反应进程曲线的核心。

V₀ = Δ[P] / Δt or −Δ[S] / Δt

V₀ = Δ[P] / Δt 或 −Δ[S] / Δt

Draw a tangent at time zero on a concentration–time graph; the slope is the initial rate. Units are typically mmol dm⁻³ s⁻¹, or arbitrary units if using absorbance.

在浓度–时间图上，于零时刻处作切线；切线斜率即为初始速率。单位通常为 mmol dm⁻³ s⁻¹，若使用吸光度则可为任意单位。

Michaelis-Menten kinetics describes hyperbolic relationship: V = (Vmax × [S]) / (Kₘ + [S]). The Michaelis constant (Kₘ) equals the substrate concentration at which V = Vmax/2. Vmax and Kₘ are obtained from a Lineweaver-Burk plot (1/V vs 1/[S]), but Pre-U papers usually focus on qualitative interpretation and calculation questions rather than complex transforms.

米氏动力学描述双曲线关系：V = (Vmax × [S]) / (Kₘ + [S])。米氏常数 Kₘ 等于反应速率达 Vmax/₂ 时的底物浓度。Vmax 和 Kₘ 可从 Lineweaver-Burk 图（1/V 对 1/[S]）获得，但 Pre-U 考卷更侧重于定性解释和计算，而非复杂的线性转换。

Inhibition effects: competitive inhibition increases apparent Kₘ, Vmax unchanged; non-competitive lowers Vmax, Kₘ unchanged.

抑制作用：竞争性抑制使表观 Kₘ 增大，Vmax 不变；非竞争性抑制降低 Vmax，Kₘ 不变。

12. Student’s t-test & Chi-squared (χ²) Test | 学生 t 检验与卡方检验

These are the two principal inferential statistics used in Pre-U biology to determine if results are significant. The t-test compares two means; the chi-squared test compares observed and expected frequencies.

这是 Pre-U 生物中两种主要的推断统计方法，用于判断结果是否显著。t 检验比较两个平均值；卡方检验比较观察频数与期望频数。

t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)

x̄ = sample mean, s² = variance, n = sample size. Use the unpaired two-sample t-test. Compare calculated |t| with critical value at n₁+n₂−2 degrees of freedom for a chosen significance level (usually p=0.05). If |t| > t_crit, reject null hypothesis.

x̄ = 样本均值，s² = 方差，n = 样本量。使用非配对双样本 t 检验。将计算得到的 |t| 值与自由度为 n₁+n₂−2、显著性水平（通常 p=0.05）下的临界值比较。若 |t| > t_crit，则拒绝原假设。

χ² = Σ (O−E)² / E

O = observed frequency, E = expected frequency. Calculate for each category, sum. Degrees of freedom = number of categories − 1 (for goodness-of-fit) or (rows−1) × (columns−1) for contingency tables. Compare with χ² critical value; if χ² > χ² crit, significant difference exists. Always state null hypothesis and conclusion.

O = 观察频数，E = 期望频数。按每个类别计算后求和。自由度 = 类别数 − 1（拟合优度检验），或 (行数−1) × (列数−1)（列联表检验）。与 χ² 临界值比较；若 χ² > χ² crit，则存在显著差异。务必陈述原假设并给出结论。

Quick reference for standard critical values at p=0.05: df=1 → χ²=3.84; df=2 → 5.99; df=3 → 7.81. For t-test df=10 → 2.23; df=20 → 2.09; df=∞ → 1.96.

标准临界值速查（p=0.05）：df=1 → χ²=3.84；df=2 → 5.99；df=3 → 7.81。t 检验中 df=10 → 2.23；df=20 → 2.09；df=∞ → 1.96。

Test	Formula	When to use
t-test	t = (x̄₁ − x̄₂) / SE_diff	Compare means of two samples (e.g. effect of a treatment on growth)
χ² test	χ² = Σ (O−E)² / E	Compare observed vs expected frequencies (e.g. genetic ratios, ecological distribution)

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Pre-U CAIE Biology Equation & Theorem Quick Reference | Pre-U CAIE 生物：公式定理速查手册