Rate Equations | 速率方程

📚 Rate Equations | 速率方程

Rate equations describe how the speed of a chemical reaction depends on the concentrations of reactants. They are central to A-Level Chemistry because they link experimental measurements to reaction mechanisms.

速率方程描述化学反应速率如何随反应物浓度变化。它是 A-Level 化学的核心内容之一，因为它把实验测量与反应机理联系起来。

1. What is a rate equation? | 什么是速率方程？

For a reaction involving reactants A and B, the rate equation is usually written as:

对于涉及反应物 A 和 B 的反应，速率方程通常写作：

rate = k [A]ᵐ [B]ⁿ

Here k is the rate constant, [A] and [B] are the reactant concentrations, and m and n are the orders of reaction with respect to A and B.

其中 k 为速率常数，[A] 和 [B] 为反应物浓度，m 和 n 分别是对 A 和 B 的反应级数。

The orders m and n are usually 0, 1, or 2 at A-Level, although other values are possible in more advanced work.

在 A-Level 阶段，级数 m 和 n 通常为 0、1 或 2，尽管在更深入的内容中还可能出现其他数值。

2. Order of reaction | 反应级数

The order with respect to a reactant tells us how the rate responds to a change in that reactant’s concentration.

对某反应物的级数表示速率如何随该反应物浓度变化。

Zero order: doubling [A] has no effect on rate.
零级：[A] 加倍，速率不变。
First order: doubling [A] doubles the rate.
一级：[A] 加倍，速率加倍。
Second order: doubling [A] quadruples the rate.
二级：[A] 加倍，速率变为四倍。

The overall order is the sum of the individual orders: overall order = m + n.

总反应级数是各反应物级数之和：总级数 = m + n。

3. The rate constant k | 速率常数 k

The rate constant k is temperature-dependent but does not change when reactant concentrations change. Its units depend on the overall order of reaction.

速率常数 k 随温度变化，但不随反应物浓度变化。它的单位取决于总反应级数。

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

Always calculate units of k by rearranging the rate equation, not by memorising the table alone.

务必通过重排速率方程来计算 k 的单位，而不要只靠背诵表格。

4. Experimental method: initial rates | 实验方法：初始速率法

In the initial-rates method, the rate is measured at the very start of the reaction, where concentrations are known and reverse reaction is negligible.

在初始速率法中，速率在反应刚刚开始时测量，此时浓度已知，逆反应可以忽略。

A common technique is the iodine clock or a tangent drawn to a concentration-time curve at t = 0.

常用技术包括碘钟反应，或在浓度-时间曲线上于 t = 0 处作切线。

Experiment	[A] / mol dm⁻³	[B] / mol dm⁻³	Initial rate / mol dm⁻³ s⁻¹
1	0.10	0.10	2.0 × 10⁻⁴
2	0.20	0.10	4.0 × 10⁻⁴
3	0.10	0.20	8.0 × 10⁻⁴

Comparing experiments 1 and 2: doubling [A] doubles the rate, so A is first order. Comparing experiments 1 and 3: doubling [B] quadruples the rate, so B is second order.

比较实验 1 和 2：[A] 加倍，速率加倍，因此 A 为一级。比较实验 1 和 3：[B] 加倍，速率增加为四倍，因此 B 为二级。

rate = k [A] [B]²

5. Continuous monitoring and concentration-time graphs | 连续监测与浓度-时间图

Rather than only measuring initial rates, we can follow the concentration of a reactant over time. The rate at any point is the slope of the tangent to the curve.

除了只测量初始速率，我们还可以跟踪反应物浓度随时间的变化。曲线上任意点的速率等于该点切线的斜率。

Zero order: [A] decreases linearly with time.
零级：[A] 随时间线性下降。
First order: [A] decreases exponentially, and half-life is constant.
一级：[A] 呈指数下降，半衰期恒定。
Second order: [A] falls steeply at first and then more slowly than first order at low concentrations.
二级：[A] 先急剧下降，在低浓度时比一级反应下降得更慢。

6. Half-life and first-order reactions | 半衰期与一级反应

For a first-order reaction, the half-life is independent of the initial concentration. This is a key diagnostic test for first-order behaviour.

对于一级反应，半衰期与初始浓度无关。这是判断一级反应的重要特征。

t½ = ln 2 / k

Since ln 2 ≈ 0.693, the half-life is constant for a given temperature because k is constant.

由于 ln 2 ≈ 0.693，在一定温度下 k 为常数，因此半衰期也是常数。

For zero-order reactions, half-life decreases as concentration decreases. For second-order reactions, half-life increases as concentration decreases.

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增长。

7. Multi-step reactions and the rate-determining step | 多步反应与决速步骤

Many reactions occur in a sequence of elementary steps. The slowest step, called the rate-determining step, controls the overall rate.

许多反应由多个基元步骤组成。最慢的一步称为决速步骤，它决定总反应速率。

The species in the rate equation must be involved in or before the rate-determining step, but not necessarily in the overall stoichiometric equation.

速率方程中的物质必须参与决速步骤或在决速步骤之前生成，但不一定出现在总化学计量方程式中。

For example, if the rate equation is rate = k [RX], the rate-determining step might involve only RX breaking into an intermediate, followed by a fast attack by another reactant.

例如，若速率方程为 rate = k [RX]，决速步骤可能只涉及 RX 分解成中间体，随后另一反应物快速进攻。

8. Temperature and the Arrhenius equation | 温度与阿伦尼乌斯方程

Increasing temperature increases the rate constant k because more particles have energy greater than or equal to the activation energy Eₐ.

升高温度会增大速率常数 k，因为更多粒子具有大于或等于活化能 Eₐ 的能量。

The relationship is summarised by the Arrhenius equation:

其关系可用阿伦尼乌斯方程表示：

k = A e^(−Eₐ/RT)

Taking natural logarithms gives:

取自然对数得到：

ln k = ln A − Eₐ / RT

A is the pre-exponential factor, R is the gas constant 8.31 J K⁻¹ mol⁻¹, and T is the absolute temperature in kelvin.

A 为指前因子，R 为气体常数 8.31 J K⁻¹ mol⁻¹，T 为开尔文温标下的绝对温度。

9. Catalysts and rate equations | 催化剂与速率方程

A catalyst increases the rate of reaction by providing an alternative pathway with a lower activation energy. This increases k without being consumed in the overall reaction.

催化剂通过提供活化能较低的替代路径来提高反应速率。它增大 k，但在总反应中不被消耗。

At A-Level, a catalyst usually does not change the orders of reaction with respect to the reactants, but it may appear in the rate equation if it is involved in the rate-determining step of a homogeneous mechanism.

在 A-Level 阶段，催化剂通常不会改变各反应物的反应级数，但如果它参与均相机理的决速步骤，就可能出现在速率方程中。

10. Common exam skills and pitfalls | 常见考试技巧与易错点

Orders cannot be deduced from the balanced chemical equation; they must be found experimentally.
反应级数不能从配平的化学方程式直接推出，必须通过实验确定。
Use initial rates, not average rates, when determining orders.
确定级数时应使用初始速率，而不是平均速率。
The units of k must correspond to the overall order.
k 的单位必须与总反应级数对应。
Half-life is constant only for a first-order reaction.
只有一级反应的半衰期是恒定的。
Increasing temperature increases k; adding a catalyst also increases k by lowering Eₐ.
升高温度会增大 k；加入催化剂通过降低 Eₐ 也会增大 k。

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Rate Equations | 速率方程