Have you ever wondered how scientists find out how fast a chemical reaction happens or how long it takes for a substance to break down? That’s where integrated equations rate comes in. These equations help us understand how the concentration of reactants changes over time during a reaction. In this article, you’ll learn about the integrated rate equations for first order and second order reactions.

What is an Integrated Rate Equation?

The Integrated Rate Equation is a mathematical formula that shows how the concentration of a reactant changes over time during a chemical reaction. It helps us understand the relationship between time and concentration for different types of reactions. Instead of just telling us how fast a reaction is going at one moment (which is what the regular rate law does), the integrated form gives a complete picture over a period of time. This is especially useful when analysing data from experiments. 

Chemistry Notes Free PDFs

The equation varies depending on whether the reaction is zero, first order, or second order.

A first order reaction is one where the rate of reaction depends on the concentration of a single reactant. That means if you double the amount of the reactant, the reaction will happen twice as fast.

Integrated Rate Equation First Order

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

\ln [R]=\ln \left[R_0\right]-k t

Or, it can be written as:

[R]=\left[R_0\right] \cdot e^{-k t}

Here:

• R0 is the initial concentration
• R is the concentration at time t
• k is the rate constant 
• t is the time
• ln is the natural logarithm

This equation shows how the concentration of the reactant decreases exponentially over time.

Graph

Plotting in R against time t gives a straight line with a negative slope, confirming first order behaviour.

Unit of k:

The unit of the rate constant k is s⁻¹, meaning it depends only on time

Examples:

• Radioactive decay
• Decomposition of H2O2

Second Order Reactions

A second order reaction is one where the rate depends on the square of one reactant’s concentration or the product of two different reactant concentrations. For example:

Rate $\propto[A]^2$, or Rate $\propto[A][B]$

This means the reaction speed increases more sharply with concentration compared to a first order reaction.

Integrated Rate Law

For a simple second order reaction:

A→Products

The integrated rate equation is:

\frac{1}{[A]}=\frac{1}{\left[A_0\right]}+k t

Here:

• A0 is the initial concentration
• Ais the concentration at time t
• k is the rate constant 
• t is the time
• ln is the natural logarithm

This equation shows that as time increases, the inverse of concentration (1/A) increases linearly.

Graph

When you plot \frac{1}{[A]} \text { vs. } t, you get a straight line with a positive slope. This linear trend is used to identify second order reactions.

Units of rate Constant 

For second order reactions, the unit of k is:

\mathrm{mol}^{-1} \cdot \mathrm{~L} \cdot \mathrm{~s}^{-1}

Example

• Saponification of esters (like ethyl acetate with NaOH)
• Reaction between NO and O2 in the atmosphere

Quick comparison Table

Difference between First and Second Order Reactions

Importance of Integrated Rate Equation

It is crucial to understand the reaction rate because doing so enables us to modify the reaction's parameters for a more suitable rate and a more effective and efficient reaction.

If there are two competing reactions for a specific set of reactants, knowing the rate can assist us in carrying out the entire production of a specified single product.

Many industries and research fields depend on the ability to analyze data on reaction rates. The Haber process has one of the most significant uses in the industry. By observing the rates, Haber found that the reaction's speed greatly depended on how hard it was to break the triple bond in nitrogen.

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