IB Chemistry: Rate Equations and Determination of Reaction Order | IB化学：速率方程与反应级数求解

📚 IB Chemistry: Rate Equations and Determination of Reaction Order | IB化学：速率方程与反应级数求解

In IB Chemistry HL, chemical kinetics is not only about listing factors that affect reaction rate. You must also express rate as a mathematical relationship and use experimental data to find the powers in that relationship. These powers are called reaction orders, and together with the rate constant they form the rate equation.

在 IB 化学高阶（HL）中，化学动力学不仅要求列出影响反应速率的因素，还要用数学关系式表达反应速率，并通过实验数据确定关系式中的指数。这些指数称为反应级数，它们与速率常数一起构成速率方程。

1. What Is a Rate Equation? | 什么是速率方程？

A rate equation shows how the rate of a reaction depends on the concentrations of the species that affect it. For a general reaction A + B → products, the rate equation is usually written as:

速率方程表示反应速率如何受相关物质浓度的影响。对于一般反应 A + B → 生成物，速率方程通常写作：

rate = k[A]^m[B]ⁿ

Here, k is the rate constant, [A] and [B] are concentrations in mol dm⁻³, and m and n are the orders of reaction with respect to A and B. The overall order is m + n.

其中，k 是速率常数，[A] 和 [B] 是以 mol dm⁻³ 为单位的浓度，m 和 n 分别是相对于 A 和 B 的反应级数。总反应级数为 m + n。

The rate equation must be determined experimentally. It cannot be predicted reliably from the balanced chemical equation because the mechanism of the reaction often involves several elementary steps.

速率方程必须通过实验确定，不能仅由配平的化学方程式可靠预测，因为反应往往经由多个基元步骤组成的反应机理进行。

2. Reaction Orders: Zero, First and Second | 反应级数：零级、一级与二级

The order with respect to a reactant tells you how the rate changes when that reactant’s concentration changes, while other concentrations are kept constant.

某一反应物对应的级数表示在其他浓度保持不变时，该反应物浓度变化如何影响反应速率。

Zero order: rate is independent of [A]. Doubling [A] has no effect on rate.
First order: rate is proportional to [A]. Doubling [A] doubles the rate.
Second order: rate is proportional to [A]². Doubling [A] quadruples the rate.

零级：速率与 [A] 无关。将 [A] 加倍，速率不变。
一级：速率与 [A] 成正比。将 [A] 加倍，速率加倍。
二级：速率与 [A]² 成正比。将 [A] 加倍，速率变为原来的 4 倍。

These statements are only true when the concentration of every other reactant is fixed. In practice, you compare experiments in which only one concentration is changed at a time.

上述规律只在其他反应物浓度不变时才成立。实际操作中，应比较仅改变一种浓度的多组实验。

3. Initial Rates Method | 初始速率法

The initial rate is the instantaneous rate at the moment the reactants are mixed, t = 0. Measuring initial rates avoids complications from changing concentrations and reverse reactions.

初始速率是指反应物刚混合、t = 0 时的瞬时速率。利用初始速率可以避免浓度变化和逆反应带来的干扰。

Consider the following data for a reaction A + B → products.

以下为反应 A + B → 生成物的实验数据。

Experiment	[A] / mol dm⁻³	[B] / mol dm⁻³	Initial rate / mol dm⁻³ s⁻¹
1	0.10	0.10	2.0 × 10⁻³
2	0.20	0.10	8.0 × 10⁻³
3	0.20	0.20	8.0 × 10⁻³

From Experiment 1 to 2, [A] doubles while [B] is constant, and the rate quadruples. This means the order with respect to A is 2. From Experiment 2 to 3, [B] doubles while [A] is constant, and the rate does not change. Therefore the order with respect to B is 0.

从实验 1 到实验 2，[A] 加倍而 [B] 不变，速率变为原来的 4 倍，说明 A 的级数为 2。从实验 2 到实验 3，[B] 加倍而 [A] 不变，速率不变，说明 B 的级数为 0。

The rate equation is therefore rate = k[A]². To find k, substitute one experiment:

因此速率方程为 rate = k[A]²。代入任一组实验数据可求 k：

2.0 × 10⁻³ = k(0.10)² ➝ k = 0.20 mol⁻¹ dm³ s⁻¹

If more than one reactant appears in the rate equation, you can use the isolation method: keep all reactants except one in large excess. Their concentrations then stay almost constant, and the observed order is called a pseudo-order.

若速率方程涉及多个反应物，可采用“隔离法”：将除一种反应物外的所有反应物大量过量，使其浓度近似不变，此时测得的级数称为“假级数”。

4. Concentration–Time Graphs | 浓度–时间图像

Continuously monitoring a reactant’s concentration over time can reveal the order from the shape of the graph. However, straight-line graphs are more reliable than curve shapes.

连续监测某反应物浓度随时间变化可判断反应级数，但线性关系图比曲线形状更可靠。

For a zero-order reaction, a plot of [A] against t gives a straight line with slope −k.

对于零级反应，以 [A] 对 t 作图得到直线，斜率为 −k。

[A]_t = −kt + [A]₀

For a first-order reaction, the integrated rate law is:

对于一级反应，积分速率方程为：

ln[A]_t = −kt + ln[A]₀

Thus a plot of ln[A] against t is linear, with slope −k.

因此以 ln[A] 对 t 作图，得到斜率为 −k 的直线。

For a second-order reaction, the integrated rate law is:

对于二级反应，积分速率方程为：

1/[A]_t = kt + 1/[A]₀

This means a plot of 1/[A] against t is linear, with slope k.

也就是说，以 1/[A] 对 t 作图，得到斜率为 k 的直线。

In the exam, you may be given a set of concentration–time data and asked which plot produces a straight line. That plot identifies the order.

考试中可能会给出一组浓度–时间数据，要求判断哪张图呈直线。呈直线的图像即对应相应反应级数。

5. Half-Life and First-Order Reactions | 半衰期与一级反应

Half-life, t₁/₂, is the time required for the concentration of a reactant to fall to half its initial value. For a first-order reaction, the half-life is constant throughout the reaction.

半衰期 t₁/₂ 是反应物浓度降至初始值一半所需的时间。对于一级反应，半衰期在整个反应过程中保持不变。

t₁/₂ = ln 2 / k = 0.693 / k

Because t₁/₂ does not depend on [A]₀, you can check first-order behaviour by measuring successive half-lives. If each half-life is the same, the reaction is first order.

由于 t₁/₂ 与 [A]₀ 无关，可通过连续测量半衰期来判断一级反应。如果各段半衰期相等，则该反应为一级。

For zero-order reactions, the half-life becomes shorter as concentration decreases. For second-order reactions, the half-life becomes longer as concentration decreases. This difference helps distinguish the orders.

零级反应的半衰期随浓度降低而缩短；二级反应的半衰期随浓度降低而变长。这一差异可用于区分反应级数。

6. Units of the Rate Constant | 速率常数的单位

The units of k depend on the overall order of the reaction. They can be derived from the rate equation by rearranging k = rate / [A]^n, where n is the total order with respect to all species in the rate equation.

k 的单位取决于反应的总级数，可由速率方程变形 k = rate / [A]^n 推导，其中 n 为速率方程中涉及物种的总级数。

For an overall order n, the general units are:

对于总级数为 n 的反应，k 的一般单位为：

mol¹⁻ⁿ dm³ⁿ⁻³ s⁻¹

Overall order	Units of k
0	mol dm⁻³ s⁻¹
1	s⁻¹
2	mol⁻¹ dm³ s⁻¹
3	mol⁻² dm⁶ s⁻¹

Always write the units of k after calculation. Many IB questions award a mark for the correct unit.

计算后务必写出 k 的单位。许多 IB 题目会单独为正确单位给分。

7. Reaction Mechanisms and the Rate-Determining Step | 反应机理与限速步骤

A balanced equation gives only the overall stoichiometry. The rate equation actually reflects the slowest elementary step, called the rate-determining step.

配平方程式只给出总反应的化学计量关系。速率方程实际上反映的是最慢的基元步骤，即限速步骤。

For an elementary step, the power in the rate equation equals the stoichiometric coefficient of each reactant in that step. Reactants that appear only in fast steps before or after the slow step do not influence the rate.

对于基元步骤，速率方程中的指数等于该步骤中各反应物的化学计量系数。仅出现在慢步骤之前或之后快速步骤中的反应物不影响反应速率。

As an example, consider the overall reaction:

例如，考虑总反应：

NO₂ + CO → NO + CO₂

If the slow step is NO₂ + NO₂ → NO + NO₃, then a possible rate equation is rate = k[NO₂]², and the reaction is zero order with respect to CO, even though CO appears in the overall equation.

如果慢步骤是 NO₂ + NO₂ → NO + NO₃，则可能的速率方程为 rate = k[NO₂]²，该反应对 CO 为零级，尽管 CO 出现在总反应方程式中。

Therefore, when writing a rate equation, never copy coefficients from the overall balanced equation unless you are told the reaction is a single elementary step.

因此，在写速率方程时，除非题目明确说明反应是单步基元反应，否则

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IB Chemistry: Rate Equations and Determination of Reaction Order | IB化学：速率方程与反应级数求解