Integrated Rate Equations (Zero, First and Second Order)

Chemical kinetics is a fascinating branch of chemistry that studies the speed or rate of chemical reactions. Understanding the rate of a chemical reaction helps us understand how quickly a reaction occurs, how the concentrations of reactants and products change over time, and how different conditions such as temperature and pressure affect the speed of reactions.

The core of chemical kinetics are rate laws and integrated rate equations. These equations allow us to relate reactant concentrations, rate constants, and time. In this lesson, we will explore integrated rate equations for zero-order, first-order, and second-order reactions. Each of these follows a unique mathematical relationship that helps predict how concentrations change over time.

Zero-order reactions

A zero-order reaction is one whose rate is independent of the concentration of the reactant(s). This means that the rate of the reaction is constant.

The rate law for a zero-order reaction can be expressed as:

Rate = K

Here, k is the rate constant, which remains constant with time.

Integrated rate equations for zero-order reactions

The integrated rate equation for a zero-order reaction is obtained by integrating the rate law over time:

[A] = [A]0 - kT

In this equation:

• [A] is the concentration of the reactant at time t.
• [A]0 is the initial concentration of the reactant.
• k is the rate constant.
• t is the elapsed time.

This equation shows that the concentration of the reactant decreases linearly with time.

Example

Consider a reaction where [A]0 = 0.5 M, and the rate constant k = 0.1 M/s. We can determine the concentration of A after 3 seconds:

[A] = 0.5 M – (0.1 M/s × 3 s) = 0.5 M – 0.3 M = 0.2 M

Visual example

First-order reactions

A first-order reaction is one in which the rate depends linearly on the concentration of a single reactant. These reactions are very common in nature and laboratory studies.

The rate law for a first order reaction is:

Rate = k[A]

Here, [A] is the concentration of reactant A, and k is the rate constant.

Integrated rate equations for first-order reactions

The integrated rate equation is obtained by integrating the rate law. The resulting expression is:

ln[A] = ln[A]0 - kt

Rearranging gives the concentration [A] at time t:

[A] = [A]0 e-kt

Example

If we have initial concentration [A]0 = 1.0 M and rate constant k = 0.2 s-1, what will be the concentration of A after 5 sec?

[A] = 1.0 M * e-0.2s-1 * 5s = 1.0 M * e-1 ≈ 1.0 M * 0.3679 ≈ 0.368 M

Visual example

Second-order reactions

In a second order reaction, the velocity is proportional to the square of the concentration of one reactant or the product of the concentrations of two different reactants.

The rate law for a second order reaction is:

Rate = k[A]2

Integrated rate equations for second-order reactions

For a single reactant, the integrated rate equation is:

1/[A] = 1/[A]0 + kt

Example

Let [A]0 = 0.2 M and k = 0.05 M-1s-1, find the concentration after 5 seconds:

1/[A] = 1/0.2 M + 0.05 M-1s-1 * 5 s

1/[A] = 5 + 0.25 = 5.25

[A] = 1 / 5.25 ≈ 0.190 M

Visual example

Summary

Integrated rate equations in chemical kinetics are important for predicting the concentrations of reactants over time. Understanding zero-order, first-order, and second-order reactions provides a foundation for studying more complex reactions. With these equations, chemists can not only better understand chemical processes but also control and optimize them in a variety of industrial and laboratory settings.

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