Rate Equations | 速率方程

📚 Rate Equations | 速率方程

In A-Level Chemistry, the rate equation is one of the most important tools for understanding how chemical reactions proceed. It links the rate of a reaction to the concentrations of the reactants, and it is determined experimentally, not from the balanced equation.

在A-Level化学中，速率方程是理解化学反应如何进行的最重要工具之一。它将反应速率与反应物浓度联系起来，并且必须通过实验确定，而不能从配平的化学方程式中直接推导。

1. What is a Rate Equation | 什么是速率方程

The rate equation expresses the rate of a reaction as a product of a rate constant and the concentrations of reactants, each raised to a power. For a general reaction aA + bB → products, the rate equation is written as:

速率方程将反应速率表示为速率常数与各反应物浓度（各带幂指数）的乘积。对于一般反应 aA + bB → 产物，速率方程写作：

Rate = k [A]ᵐ [B]ⁿ

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

其中，k 是速率常数，[A] 和 [B] 分别是反应物 A 和 B 的浓度（单位 mol dm⁻³），m 和 n 分别是反应对 A 和 B 的反应级数。反应的总级数为 m + n。

2. The Components: k, Orders, and Concentration | 三要素：速率常数、级数与浓度

The rate constant k is a proportionality constant that is specific to a particular reaction at a given temperature. Its value increases with temperature, and its units depend on the overall order of the reaction. The order with respect to a reactant tells us how the rate changes when the concentration of that reactant changes.

速率常数 k 是一个比例常数，在给定温度下对特定反应是唯一的。它的值随温度升高而增大，其单位取决于反应的总级数。对某一反应物的级数告诉我们，当该反应物浓度变化时，反应速率如何变化。

If m = 0, changing [A] has no effect on the rate. This is called zero order.
If m = 1, doubling [A] doubles the rate. This is called first order.
If m = 2, doubling [A] quadruples the rate. This is called second order.

如果 m = 0，改变 [A] 对速率没有影响，这称为零级反应。
如果 m = 1，[A] 加倍，速率也加倍，这称为一级反应。
如果 m = 2，[A] 加倍，速率变为原来的四倍，这称为二级反应。

3. Zero, First and Second Order | 零级、一级与二级反应

In a zero-order reaction, the rate is independent of the concentration of the reactant. This often occurs when the reactant is in large excess or when the reaction is catalysed by a surface that is saturated. The concentration-time graph is a straight line with a negative gradient.

在零级反应中，速率与反应物浓度无关。这通常发生在反应物大大过量，或反应由饱和的表面催化时。浓度-时间图是一条斜率為负的直线。

In a first-order reaction, the rate is directly proportional to the concentration of the reactant. The concentration-time graph shows a curved decline, and the rate-concentration graph is a straight line through the origin.

在一级反应中，速率与反应物浓度成正比。浓度-时间图呈弯曲下降的趋势，而速率-浓度图是一条通过原点的直线。

In a second-order reaction, the rate is proportional to the square of the concentration of the reactant. The concentration-time graph declines more steeply, and the rate-concentration graph is a curve that rises sharply.

在二级反应中，速率与反应物浓度的平方成正比。浓度-时间图下降更陡峭，速率-浓度图是一条急剧上升的曲线。

4. Using Graphs to Find Order | 用图像确定反应级数

The order of a reaction with respect to a reactant can be determined from a concentration-time graph by comparing successive half-lives. For a first-order reaction, the half-life is constant throughout the reaction. For a zero-order reaction, the half-life decreases as the concentration decreases. For a second-order reaction, the half-life increases as the concentration decreases.

反应对某一反应物的级数可以通过比较浓度-时间图中连续的半衰期来确定。对于一级反应，半衰期在整个反应过程中保持恒定。对于零级反应，半衰期随着浓度降低而减小。对于二级反应，半衰期随着浓度降低而增大。

Alternatively, a graph of rate against concentration can be plotted. If this graph is a horizontal line, the order is zero. If it is a straight line through the origin, the order is one. If it is a curve, the order is two.

另一种方法是绘制速率对浓度的图像。如果图像是水平线，则级数为零；如果是一条通过原点的直线，级数为一；如果是一条曲线，则级数为二。

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

The half-life, t½, is the time taken for the concentration of a reactant to fall to half of its initial value. For a first-order reaction, the half-life is independent of the initial concentration. This gives the useful relationship:

半衰期 t½ 是反应物浓度降到初始值一半所需的时间。对于一级反应，半衰期与初始浓度无关。由此得出一个有用的关系式：

t½ = ln 2 / k = 0.693 / k

This equation allows us to calculate the rate constant k if we measure the half-life from a concentration-time graph. It also means that the concentration halves in equal time intervals, which is a defining feature of first-order kinetics.

这个方程允许我们通过从浓度-时间图测得半衰期来计算速率常数 k。这也意味着浓度在相等的时间间隔内减半，这是一级动力学的一个典型特征。

6. The Initial Rates Method | 初始速率法

To determine the orders of a reaction, the initial rates method is often used. Several experiments are carried out in which the initial concentration of one reactant is changed while all other concentrations are kept constant. The initial rate is measured in each case, and the order with respect to that reactant is deduced from how the rate changes.

为了确定反应的级数，通常使用初始速率法。进行多组实验，在每组实验中只改变一种反应物的初始浓度，而保持其他所有浓度不变。分别测量每组实验的初始速率，通过速率变化的方式推断该反应物的级数。

For example, if doubling the concentration of reactant A quadruples the initial rate, then the reaction is second order with respect to A. If doubling it leaves the rate unchanged, then it is zero order with respect to A. If doubling it doubles the rate, then it is first order.

例如，如果反应物 A 的浓度加倍导致初始速率变为四倍，则反应对 A 为二级；如果浓度加倍而速率不变，则对 A 为零级；如果浓度加倍且速率也加倍，则对 A 为一级。

7. Clock Reactions | 时钟反应

Clock reactions provide a simple experimental method for measuring initial rate. In a clock reaction, a visible change such as a colour change or the appearance of a precipitate occurs after a known time. The time taken for this change is inversely proportional to the initial rate.

时钟反应提供了一种测量初始速率的简便实验方法。在时钟反应中，一个可见的变化（如颜色变化或沉淀生成）在已知时间后发生。该变化所花费的时间与初始速率成反比。

The initial rate is calculated as 1/t, where t is the time measured. By repeating the experiment with different initial concentrations of one reactant, the order with respect to that reactant can be found. The iodine clock reaction and the persulfate-iodide clock reaction are common examples in AQA practical work.

初始速率按 1/t 计算，其中 t 是测得的反应时间。通过用不同初始浓度的某一种反应物重复实验，可以得出该反应物的级数。碘时钟反应和过硫酸盐-碘化物时钟反应是 AQA 实验考察中的经典例子。

8. The Rate Constant k and Its Units | 速率常数 k 及其单位

The units of the rate constant depend on the overall order of the reaction. This is because the rate always has units of mol dm⁻³ s⁻¹, and the concentration terms in the rate equation have units of mol dm⁻³. The units of k must therefore compensate so that the overall expression has the correct units.

速率常数的单位取决于反应的总级数。这是因为速率的单位始终是 mol dm⁻³ s⁻¹，而速率方程中的浓度项具有 mol dm⁻³ 的单位。因此，k 的单位必须进行补偿，使得整个表达式的单位正确。

Overall Order	Units of k
Zero order	mol dm⁻³ s⁻¹
First order	s⁻¹
Second order	dm³ mol⁻¹ s⁻¹
Third order	dm⁶ mol⁻² s⁻¹

The pattern for the units of k is: (mol dm⁻³)¹⁻ⁿ s⁻¹, where n is the overall order of the reaction.

k 的单位规律为：(mol dm⁻³)¹⁻ⁿ s⁻¹，其中 n 是反应的总级数。

9. Temperature and the Arrhenius Equation | 温度与阿伦尼乌斯方程

The rate constant k changes with temperature. Increasing the temperature increases the proportion of molecules with energy greater than or equal to the activation energy, Eₐ. This means that more collisions are successful, and the rate increases. The Arrhenius equation describes this relationship mathematically:

速率常数 k 随温度变化。升高温度会增大具有大于或等于活化能 Eₐ 的能量分子所占的比例。这意味着更多碰撞是有效的，因此速率增加。阿伦尼乌斯方程从数学上描述了这一关系：

k = A e⁻ᵉᵃ/RT

In this equation, A is the Arrhenius constant, e is the base of natural logarithms, Eₐ is the activation energy in J mol⁻¹, R is the gas constant (8.31 J K⁻¹ mol⁻¹), and T is the temperature in kelvin. Taking natural logs of both sides gives the linear form:

在这个方程中，A 是阿伦尼乌斯常数，e 是自然对数的底数，Eₐ 是活化能（单位 J mol⁻¹），R 是气体常数（8.31 J K⁻¹ mol⁻¹），T 是热力学温度（单位 K）。对方程两边取自然对数，得到线性形式：

ln k = −Eₐ / RT + ln A

A plot of ln k against 1/T gives a straight line with gradient −Eₐ/R and intercept ln A. This is a common question in AQA exams and allows the activation energy to be determined from experimental data.

以 ln k 对 1/T 作图得到一条直线，斜率为 −Eₐ/R，截距为 ln A。这是 AQA 考试中常见的题型，可以从实验数据中确定活化能。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

One of the most common mistakes is trying to write the rate equation from the balanced chemical equation. The orders must always be determined experimentally. Another common error is confusing the order of reaction with the stoichiometric coefficients in the equation.

最常见的错误之一是试图从配平的化学方程式直接写出速率方程。级数必须始终通过实验确定。另一个常见错误是将反应级数与方程式中的化学计量系数混淆。

When calculating the units of k, be methodical: write out the rate equation, substitute the units, and rearrange. Also remember that the overall order is the sum of the individual orders, and that the half-life of a first-order reaction is constant regardless of concentration.

在计算 k 的单位时，要逐步进行：写出速率方程，代入单位，然后整理。还要记住总级数等于各分项级数之和，以及一级反应的半衰期与浓度无关，始终保持恒定。

11. Integrated Rate Equations and Concentration-Time Data | 积分速率方程与浓度-时间数据

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

对于一级反应，其积分速率方程为：

ln [A]₀ − ln [A]ₜ = kt

This can be rearranged to the linear form ln [A]ₜ = −kt + ln [A]₀. Plotting ln [A] against time yields a straight line with gradient −k, which is a powerful way to confirm first-order kinetics and calculate k from experimental concentration-time data.

该式可整理为线性形式 ln [A]ₜ = −kt + ln [A]₀。以 ln [A] 对时间作图得到一条斜率为 −k 的直线，这是确认一级动力学并从浓度-时间实验数据计算 k 的有力方法。

For a zero-order reaction, the integrated equation is [A]ₜ = [A]₀ − kt, and plotting [A] against time gives a straight line. For a second-order reaction, 1/[A] plotted against time gives a straight line with gradient k.

对于零级反应，积分方程为 [A]ₜ = [A]₀ − kt，以 [A] 对时间作图得直线。对于二级反应，以 1/[A] 对时间作图得到斜率为 k 的直线。

12. Summary | 本章小结

The rate equation is central to chemical kinetics. It shows how the rate depends on reactant concentrations, and it is determined experimentally. Orders, rate constants, half-life, clock reactions, initial rates, and the Arrhenius equation are all essential parts of this topic. Mastery of these concepts will allow you to analyse experimental data, calculate rate constants, and understand how temperature and concentration affect reaction rates.

速率方程是化学动力学的核心内容。它展示了速率如何依赖于反应物浓度，并且必须通过实验确定。反应级数、速率常数、半衰期、时钟反应、初始速率法和阿伦尼乌斯方程都是这一专题的重要组成部分。掌握这些概念将使你能够分析实验数据、计算速率常数，并理解温度和浓度如何影响反应速率。

Remember: in the exam, always link the theory to the data you are given. Read the question carefully, identify whether you are being asked for the order, the rate constant, or the units, and show every step of your working clearly.

请记住：考试中，始终将理论与所给数据联系起来。仔细阅读题目，明确题目问你的是级数、速率常数还是单位，并清晰地展示每一步计算过程。

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Rate Equations | 速率方程