Methods of Determining Reaction Order

Introduction

The knowledge of the order of a reaction is among the major guidelines in the realm of chemistry. Such understanding is quite important as it explains how chemical reactions proceed with time. Understanding reaction rates serves not only in an academic setting, for instance, during examinations but also in many industrial applications, which range from drug design to environmental monitoring and other production processes. Now imagine that the processing of a drug from reactants to products requires an accurate prediction of that process for the development of a life-saving drug in these circumstances. If so, knowledge of the reaction order for process efficiency and quality of the product. The article will now discuss better methods for the determination of order in a chemical reaction: graphical and integrated rate law methods.

How to Determine Order of Reaction: Graphical Method

Here graphs are plotted between rate and concentration to find the order of the reaction.

Rate = k(concentration)ⁿ

Plots of Rate vs Concentration

Concept and Explanation

The graphical method bases itself on plotting experimental data to find trends that enable one to infer the reaction order. The plots of concentration versus time allow one to infer how the concentration of a reactant or product may change with time.

• Zero-Order Reactions: A plot of concentration versus time gives a straight line of negative gradient.
• First-order reactions: It is a straight line when one plots the natural logarithm of the concentration against time.
• Second-Order Reactions: The plot of the reciprocal of concentration vs. time is straight.

Integrated Rate Law Method

If the data for time(t) and [A] is given then this method is applicable. Thus follow the steps given below to find the order of reaction by using the integrated rate law method.

- Check for First Order:
1. Use the formula given below to find out the two values of k as $\mathrm{k}_1$ and $\mathrm{k}_2$.
$
\mathrm{k}=\frac{2.303}{\mathrm{t}} \log _{10}\left[\frac{\mathrm{A}_0}{\mathrm{~A}}\right]
$
2. If these two values $k_1$ and $k_2$ are the same, then this given reaction is of first order. But if $k_1 \neq k_2$, then check for zero-order.
- Check for Zero-Order:
1. Use the formula given below to find out the two values of k as $\mathrm{k}_1$ and $\mathrm{k}_2$.
$
\mathrm{k}=\frac{\mathrm{A}_0-\mathrm{A}}{\mathrm{t}}
$
2. Again, if these two values $\mathrm{k}_1$ and $\mathrm{k}_2$ are the same, then this given reaction is of zero order. But if $\mathrm{k}_1 \neq \mathrm{k}_2$, then check for second-order.
- Check for Third-Order:
1. Use the formula given below to find out the two values of $k$ as $k_1$ and $k_2$.
$
\mathrm{k}=\frac{1}{\mathrm{t}}\left[\frac{1}{\mathrm{~A}}-\frac{1}{\mathrm{~A}_0}\right]
$

Further, if these two values k₁ and k₂ are the same, then this given reaction is of second-order. But if k₁≠ k₂, then check for third-order and so on.

Half-Life Method

Concept and Explanation

The integrated rate law method is based on mathematical equations involved the concentration of reactants with time. The equations are associated with the identification of order of a reaction by the experimental data, which fits best with one of the mathematical models.

It is used when the rate law involves only one concentration term.
$
\begin{aligned}
& \mathrm{t}_{1 / 2} \propto(\mathrm{a})^{1-\mathrm{n}} \\
& \text { or } \\
& \mathrm{t}_{1 / 2} \propto 1 / \mathrm{a}^{\mathrm{n}-1}
\end{aligned}
$

For two different concentrations, we have:
$
\frac{\left(\mathrm{t}^{1 / 2}\right)_1}{\left(\mathrm{t}^{1 / 2}\right)_2}=\left(\frac{\mathrm{a}_2}{\mathrm{a}_1}\right)^{\mathrm{n}-1}
$

On taking logarithms on both sides, we get:
$
\log _{10} \frac{\left(\mathrm{t}_{1 / 2}\right)_1}{\left(\mathrm{t}_{1 / 2}\right)_2}=(\mathrm{n}-1) \log _{10}\left(\mathrm{a}_2 / \mathrm{a}_1\right)
$

Hence,
$
\mathrm{n}=1+\frac{\log \left(\mathrm{t}^{1 / 2}\right)_1-\log \left(\mathrm{t}^{1 / 2}\right)_2}{\log \mathrm{a}_2-\log \mathrm{a}_1}
$

Here, n is the order of the reaction.

Practical Applications

In practice, the order of reaction comes in by measuring the concentration of reactants as a function of time and fitting integrated rate laws. This is particularly important in research and industry that require highly accurate mathematical models of the behavior of reactions.

Importance and Applications

Academic Importance

Understanding reaction order is significant for all students of chemistry, researchers, and practicing experts. That provides the knowledge base for, among others, some of the advanced considerations in kinetics and proper design of experiments and their data interpretation.

Industrial Applications

Knowing the reaction order is, therefore, very important in industry, especially in optimizing their production processes. This is the case, especially in pharmaceuticals, in scaling up of reactions from the laboratory to meet industrial needs where knowledge of reaction order guides in effectiveness and cost-efficiency. It also guides environmental scientists in degrading pollutants and hygiene control strategies.

To determine the order of a reaction using the half-life method, we analyze how the half-life changes with initial concentrations.

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Let's go through the solutions for the given examples:

Example 1

Question:

A student has studied the decomposition of a gas at 25 degrees celcius. He obtained the following data.

The order of the reaction is

1)0 (zero) 2)0.5 3)1 4)2

Solution

$\begin{aligned} & \mathrm{t}^{1 / 2} \propto(\mathrm{Co})^{1-\mathrm{n}} \\ & =\frac{(\mathrm{t} 1 / 2)_1}{\left(\mathrm{t}^1 / 2\right)_2}=\left(\frac{\mathrm{P}_1}{\mathrm{P}_2}\right)^{1-\mathrm{n}} \\ & =\frac{4}{2}=\left(\frac{50}{100}\right)^{1-\mathrm{n}} \Rightarrow 2\left(\frac{1}{2}\right)^{1-\mathrm{n}} \\ & 2=(2)^{\mathrm{n}-1} \\ & \mathrm{n}=2\end{aligned}$

Example 2

Question:

Consider a reaction A B + C. If the initial concentration of A was reduced from 2M to 1M in 1 h and from 1 M to 0.25 M in 2 h, the order of the reaction is:

1) (correct)1

2)2

3)0

4)3

Solution:

Given reaction A B + C

If the initial concentration of A was reduced from 2M to 1M in 1 h and from 1 M to 0.25 M in 2 h

In case 1, the initial concentration becomes half of its initial value, taking 1 hr.

In case 2, the initial concentration becomes 1/4 of its initial value, taking 2 hours or 2 hours for 2 Half-Lives.

So, this relation is for a first-order reaction.

Hence, the answer is (1).

Summary

The determination of reaction order forms a fundamental but very important area in chemical kinetics, having broad applications both in academia and industry. The graphical method and the integrated rate law method are the two important ways of finding the order. The former makes use of plots of concentration data, while the latter does with mathematical equations relating concentration to time. Mastering these methodologies better will let the chemists know how reaction behavior goes and hence optimize industrial processes to the benefit of areas as diverse as pharmaceuticals and the environment.