A-Level Chemistry: Rate Equations and Reaction Kinetics — Complete Guide / A-Level化学：速率方程与反应动力学 — 完整指南

Reaction kinetics is one of the most conceptually rich and mathematically demanding topics in A-Level Chemistry. Understanding how fast reactions proceed — and why — sits at the heart of physical chemistry, bridging thermodynamics (will a reaction happen?) with mechanism (how does it happen?). This article provides a rigorous, exam-focused treatment of rate equations, orders of reaction, the rate constant, the Arrhenius equation, and their application to reaction mechanisms.

反应动力学是A-Level化学中最具概念深度和数学挑战的主题之一。理解反应进行的速度及其原因，是物理化学的核心，它连接了热力学（反应会不会发生？）和机理（反应如何发生？）。本文对速率方程、反应级数、速率常数、阿伦尼乌斯方程及其在反应机理中的应用进行了严谨、面向考试的系统讲解。

1. The Rate of a Chemical Reaction / 化学反应速率

For a general reaction:

对于一般反应：

aA + bB → cC + dD

The rate of reaction can be expressed in terms of the change in concentration of any reactant or product per unit time:

反应速率可以用任一反应物或产物在单位时间内浓度的变化来表示：

rate = −(1/a) d[A]/dt = −(1/b) d[B]/dt = (1/c) d[C]/dt = (1/d) d[D]/dt

The negative sign for reactants reflects the fact that their concentrations decrease over time. The stoichiometric coefficients normalise the rates so that the numerical value of the rate is independent of which species is monitored.

反应物的负号反映了其浓度随时间减少的事实。化学计量系数对速率进行归一化，使得速率的数值与监测哪一种物质无关。

Experimentally, rate is measured as:

在实验中，速率通过以下方式测量：

rate = Δ[concentration] / Δt

Common experimental methods include monitoring gas volume evolved, colour change using a colorimeter, pH change, or mass loss (for reactions that produce a gas). The choice of method depends on the particular reaction under study.

常见的实验方法包括监测气体体积变化、使用比色计监测颜色变化、pH变化或质量损失（针对产生气体的反应）。方法的选择取决于所研究的特定反应。

2. The Rate Equation / 速率方程

The rate equation (also called the rate law) is the fundamental mathematical relationship linking the rate of reaction to the concentrations of the reactants:

速率方程（也称速率定律）是将反应速率与反应物浓度联系起来的基本数学关系：

rate = k [A]^m [B]ⁿ

Where:

• k = rate constant (速率常数) — temperature-dependent proportionality constant
• [A], [B] = concentrations of reactants A and B (反应物A和B的浓度)
• m = order of reaction with respect to A (对A的反应级数)
• n = order of reaction with respect to B (对B的反应级数)
• m + n = overall order of reaction (总反应级数)

3. Orders of Reaction / 反应级数

3.1 Zero Order (m = 0) / 零级反应

When the order with respect to a reactant is zero, the rate is independent of its concentration:

当对某反应物的级数为零时，反应速率与其浓度无关：

rate = k  (if zero order with respect to all reactants)

Explanation: The reactant is present in such excess that its concentration effectively remains constant, or it is not involved in the rate-determining step. On a concentration–time graph, zero order produces a straight line with constant negative gradient.

解释：该反应物的浓度远超其他物质，其浓度实际上保持恒定，或者它不参与速率决定步骤。在浓度-时间图中，零级反应产生具有恒定负斜率的直线。

Units of k for zero order: mol dm⁻³ s⁻¹

3.2 First Order (m = 1) / 一级反应

Rate is directly proportional to concentration:

速率与浓度成正比：

rate = k [A]

Key feature: The half-life (t_1/2) is constant. This is the diagnostic test for first-order kinetics — if the time taken for the concentration to halve is the same regardless of the starting concentration, the reaction is first order.

关键特征：半衰期（t_1/2）恒定。这是一级动力学的诊断性测试——如果浓度减半所需的时间与起始浓度无关，则该反应为一级反应。

t1/2 = ln 2 / k = 0.693 / k

Concentration–time graph: exponential decay. Rate–concentration graph: straight line through origin (gradient = k).

浓度-时间图：指数衰减。速率-浓度图：过原点的直线（斜率 = k）。

Units of k for first order: s⁻¹

3.3 Second Order (m = 2) / 二级反应

Rate is proportional to the square of the concentration:

速率与浓度的平方成正比：

rate = k [A]2

The half-life is not constant — each successive half-life doubles in length. On a rate–concentration graph, second order gives a parabola (quadratic curve through origin).

半衰期不恒定——每个连续的半衰期长度加倍。在速率-浓度图中，二级反应产生抛物线（过原点的二次曲线）。

Units of k for second order: dm³ mol⁻¹ s⁻¹

3.4 Summary Table / 汇总表

4. Determining Orders of Reaction / 确定反应级数

4.1 The Initial Rates Method / 初始速率法

The most common A-Level method. Multiple experiments are performed with different initial concentrations of reactants, and the initial rate is measured for each:

A-Level中最常用的方法。进行多次实验，每次使用不同的反应物初始浓度，并测量每次的初始速率：

The systematic approach involves comparing experiments where only one concentration changes, and determining the factor by which the rate changes:

• Rate unchanged → zero order (级数不变 → 零级)
• Rate ×2 when concentration ×2 → first order (浓度×2，速率×2 → 一级)
• Rate ×4 when concentration ×2 → second order (浓度×2，速率×4 → 二级)

4.2 Continuous Monitoring (Progress Curve) Method / 连续监测法

A single experiment is followed over time, producing a concentration–time curve. The gradient of the tangent at various points gives the rate at that concentration. Plotting rate against concentration then reveals the order — a linear relationship indicates first order; a curve indicates second order.

在单个实验中随时间跟踪反应，产生浓度-时间曲线。各点处的切线斜率给出该浓度下的速率。将速率对浓度作图即可揭示级数——线性关系表明一级反应；曲线表明二级反应。

4.3 Half-Life Method / 半衰期法

This is particularly useful for first-order reactions. Measure successive half-lives from a concentration–time graph:

这对一级反应特别有用。从浓度-时间图中测量连续的半衰期：

• Constant half-life → first order (半衰期恒定 → 一级)
• Half-life increases → zero order (半衰期增加 → 零级)
• Half-life doubles each time → second order (半衰期每次翻倍 → 二级)

5. The Rate Constant k / 速率常数k

The rate constant k is one of the most important quantities in chemical kinetics. It is:

• Temperature-dependent — k increases with temperature (k随温度升高而增大)
• Independent of concentration — k does not change when [reactant] changes (k不随反应物浓度变化而改变)
• Specific to a given reaction at a given temperature (在给定温度下对特定反应是唯一的)
• Units depend on the overall order of the reaction (k的单位取决于反应的总级数)

Calculating k from experimental data: Once the rate equation is determined, substitute any set of data to find k:

从实验数据计算k：一旦确定了速率方程，代入任一组数据即可求得k：

k = rate / ([A]m [B]n)

Using the example from Section 4.1:

k = (2.0 × 10−4) / (0.10 × 0.102) 
  = 2.0 × 10−4 / 1.0 × 10−3
  = 0.20 dm6 mol−2 s−1

6. Temperature Dependence: The Arrhenius Equation / 温度依赖性：阿伦尼乌斯方程

The rate constant’s dependence on temperature is described by the Arrhenius equation:

速率常数对温度的依赖性由阿伦尼乌斯方程描述：

k = A e^−E_a/RT

Where:

• A = pre-exponential factor / frequency factor (指前因子/频率因子) — related to collision frequency and orientation
• E_a = activation energy (活化能) in J mol⁻¹
• R = gas constant (气体常数) = 8.31 J K⁻¹ mol⁻¹
• T = absolute temperature (绝对温度) in Kelvin

6.1 Logarithmic Form / 对数形式

Taking natural logarithms of both sides yields the linear form, which is the basis for graphical determination of E_a:

对两边取自然对数得到线性形式，这是通过图形确定E_a的基础：

ln k = ln A − E_a/RT

Or, rearranged into the form y = mx + c:

或者，整理成 y = mx + c 的形式：

ln k = (−E_a/R)(1/T) + ln A

Graphical analysis / 图形分析：

• Plot ln k (y-axis) against 1/T (x-axis) / 以ln k为y轴，1/T为x轴作图
• Gradient = −E_a/R / 斜率 = −E_a/R
• y-intercept = ln A / y截距 = ln A
• E_a = −gradient × R

6.2 The Two-Point Form / 两点形式

If you know k at two different temperatures, you can calculate E_a without drawing a graph:

如果你知道两个不同温度下的k值，你可以在不画图的情况下计算E_a：

ln(k₂/k₁) = (E_a/R)(1/T₁ − 1/T₂)

7. Reaction Mechanisms and the Rate-Determining Step / 反应机理与速率决定步骤

A reaction mechanism is the step-by-step sequence of elementary reactions by which an overall chemical change occurs. The rate-determining step (RDS) is the slowest step in the mechanism — it acts as a bottleneck, controlling the overall rate.

反应机理是总体化学变化发生的逐步基元反应序列。速率决定步骤（RDS）是机理中最慢的步骤——它像一个瓶颈，控制着总反应速率。

7.1 The Connection Between Rate Equation and Mechanism / 速率方程与机理的联系

The rate equation is directly determined by the molecularity of the rate-determining step:

速率方程直接由速率决定步骤的分子数决定：

• If the RDS involves one molecule → the reaction is first order overall (如果RDS涉及一个分子 → 反应总体为一级)
• If the RDS involves two molecules colliding → the reaction is second order overall (如果RDS涉及两个分子碰撞 → 反应总体为二级)

7.2 Worked Example: Proposing a Mechanism / 例题：提出反应机理

Consider the reaction: 2NO(g) + 2H₂(g) → N₂(g) + 2H₂O(g)

The experimentally determined rate equation is: rate = k [NO]² [H₂]

Propose a two-step mechanism consistent with this rate equation.

Analysis / 分析：

• The rate equation has [NO]² in it, meaning two NO molecules are involved in the RDS
• [H₂] appears to the first power, meaning one H₂ is also in the RDS
• But three molecules colliding simultaneously (termolecular) is statistically improbable
• Therefore, the RDS likely involves NO + NO → intermediate, then the intermediate reacts with H₂

7.3 Key Principle / 关键原则

Any species that appears in the experimentally determined rate equation must be involved in the rate-determining step or in an equilibrium step immediately before it. Species that appear after the RDS (in fast subsequent steps) do not appear in the rate equation.

任何出现在实验确定的速率方程中的物种必须参与速率决定步骤或紧接其前的平衡步骤。在RDS之后出现的物种（在后续快速步骤中）不会出现在速率方程中。

8. Common Exam Questions / 常见考题

Question 1: Determining Rate Equation from Data

题目1：从数据确定速率方程

Solution / 解答：

Exp 1→2: [P] doubles, rate doubles → first order in P
Exp 1→3: [Q] doubles, rate unchanged → zero order in Q
Exp 1→4: [R] triples (×3), rate ×9 → second order in R
Rate equation: rate = k [P][R]²; Overall order: 3

Question 2: Arrhenius Calculation

题目2：阿伦尼乌斯计算

The rate constant for a reaction is 2.0 × 10⁻⁴ s⁻¹ at 300 K and 6.0 × 10⁻⁴ s⁻¹ at 320 K. Calculate the activation energy.

某反应在300 K时的速率常数为2.0 × 10⁻⁴ s⁻¹，在320 K时为6.0 × 10⁻⁴ s⁻¹。计算活化能。

ln(k₂/k₁) = (Eₐ/R)(1/T₁ − 1/T₂)
ln(6.0×10⁻⁴ / 2.0×10⁻⁴) = (Eₐ/8.31)(1/300 − 1/320)
ln(3.0) = (Eₐ/8.31)(0.003333 − 0.003125)
1.099 = (Eₐ/8.31)(2.083×10⁻⁴)
Eₐ = 1.099 × 8.31 / 2.083×10⁻⁴
Eₐ = 43,800 J mol⁻¹ = 43.8 kJ mol⁻¹

9. Summary: Key Points to Remember / 总结：需要记住的关键点

1. Orders are experimental, not stoichiometric — never assume m = a, n = b without data.
   级数是实验确定的，不是化学计量系数——没有数据，绝不要假设m=a、n=b。
2. Rate constant k is temperature-dependent only — changing concentration does not change k.
   速率常数k仅依赖于温度——改变浓度不会改变k。
3. Constant half-life = first order — the diagnostic test for first-order kinetics.
   恒定半衰期 = 一级反应——一级动力学的诊断性测试。
4. Rate-determining step determines the rate equation — the molecularity of the RDS dictates the orders.
   速率决定步骤决定速率方程——RDS的分子数决定级数。
5. Arrhenius: ln k vs 1/T is linear — gradient = −Eₐ/R, y-intercept = ln A.
   阿伦尼乌斯：ln k对1/T是线性的——斜率 = −Eₐ/R，y截距 = ln A。
6. Always check units of k — they reveal the overall order of reaction.
   始终检查k的单位——它们揭示了反应的总级数。

Mastering rate equations requires practice — work through past paper questions systematically, and always justify your reasoning. The mathematics is straightforward; the conceptual challenge lies in connecting the rate equation to the reaction mechanism.

掌握速率方程需要练习——系统地练习历年真题，并始终论证你的推理过程。数学本身很简单；概念上的挑战在于将速率方程与反应机理联系起来。