Rate law and reaction order

Studying kinetics in general chemistry often involves understanding how the rate of a chemical reaction depends on the concentration of the reactants. This field is known as reaction kinetics, and it operates under specific principles called rate rules and reaction order.

Introduction to rate law

The rate law of a chemical reaction is an equation that relates the rate of the reaction to the concentrations of the reactants. To express this mathematically for a general reaction:

AA + BB → CC + DD

The rate law can be expressed as:

Rate = k[A] m [B] n

Where:

• rate is the speed of the reaction.
• k is the rate constant, a number that contains information about the reaction velocity.
• [A] and [B] are the molar concentrations of reactants A and B
• m and n are the reaction orders with respect to each reactant.

Understanding reaction order

Reaction order is an important concept in kinetics. It provides information about how the concentration of reactants affects the rate of a reaction. The overall order of a reaction is the sum of the exponents of the concentration terms in the rate law equation. Different orders can have markedly different effects on the reaction rate:

Zero-order reactions

A zero-order reaction means that the rate of the reaction is independent of the concentration of the reactants. The rate law for a zero-order reaction is:

Rate = K

Graphically, the concentration of the reactants decreases linearly with time on a concentration versus time plot. For example, consider the decomposition of ammonia on a platinum surface:

2NH 3 (g) → N 2 (g) + 3H 2 (g)

The velocity law for a zero-order reaction can be represented in a graph as follows:

First-order reactions

For a first-order reaction, the rate of the reaction is directly proportional to the concentration of one reactant:

Rate = k[A]

An example of a first-order reaction is the radioactive decay of an isotope or the decomposition of N ₂ O ₅ :

2N 2 O 5 → 4NO 2 + O 2

First-order reactions exhibit exponential decay behaviour which can be represented as follows:

Second order reactions

Second-order reactions depend on the square of the concentration of one reactant or on the concentration of two different reactants. The rate law for a second-order reaction is:

Rate = k[A] 2 or Rate = k[A][B]

A known second-order reaction involves the doubling of nitrogen dioxide:

2NO 2 → N 2 O 4

The graph of a second order reaction (with one reactant) looks like this:

Determination of reaction order and rate law

Determining the reaction order and rate law from experimental data is fundamental to kinetics. Two common methods used for this purpose are the initial rate method and the integrated rate law method.

Method of initial rates

This method uses the initial rates of the reaction obtained at different initial concentrations of reactants to determine the order with respect to each reactant. If you know the rate law is:

Rate = k[A] m [B] n

You can vary the concentration of A while keeping B constant to conclude m, then vary B while keeping A constant to conclude n. For example, let's assume these three experiments:

Experiment | [A] | [B] | Rate (mol/L*s)
1 | 0.1 | 0.1 | 0.005
2 | 0.2 | 0.1 | 0.01
3 | 0.1 | 0.2 | 0.01
    

From Experiments 1 and 2, doubling [A] also doubles the rate, indicating a first-order reaction with respect to A. From Experiments 1 and 3, doubling [B] also doubles the rate, indicating a first-order reaction with respect to B.

Thus, the rate law based on these data will be:

Rate = k[A][B]

Integrated rate law method

This method involves analyzing concentration versus time data to determine the reaction order. The integrated rate laws for zero, first, and second order reactions are as follows:

• Zero-order: [A] = [A] 0 - kt
• First-order: ln([A]/[A] 0 ) = -kt
• Second-order: 1/[A] = 1/[A] 0 + kt

In these equations, [A] 0 is the initial concentration of the reactant. By plotting the appropriate data transformation and checking for linearity, the order of the reaction can be determined.

Importance of the rate constant

The rate constant k is important in the rate law because it, combined with the concentration, gives the rate of the reaction. Each reaction has a unique rate constant that can change with varying temperatures. Arrhenius' law relates the rate constant to temperature, showing that:

k = a * e - ea/(rt)

Where:

• A is the frequency factor.
• _{E a} is the activation energy.
• R is the gas constant.
• T is the temperature in Kelvin.

The Arrhenius equation shows that the rate constant becomes larger with higher temperatures, generally resulting in a faster reaction.

Catalysis and reaction rate

Catalysts are substances that speed up the rate of a reaction without consuming it. Let's understand their role:

• Low activation energy: Catalysts provide an alternative pathway with lower energy barriers.
• Increased rate constant: As E a decreases, k becomes larger according to Arrhenius' law.

The important thing is that catalysts do not change the order of the reaction; they only increase the rate at which equilibrium is reached.

Conclusion

Mastering the concepts of rate laws and reaction orders is essential for predicting how chemical systems react to different conditions. Understanding these ideas enables chemists to make reactions safer, faster, or more energy-efficient, impacting industries from pharmaceuticals to environmental science.

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