First Order Integrated Rate Equation – Definition, Mathematical and Graphical Representation, Examples, Practice Problems and FAQ

Imagine that you and your four friends are invited to collaborate on a project at school. But a day before the final submission, your pals abandon you and urge you to finish the project even though all of their names will be included in it. Therefore, the project's completion timeline is entirely up to you. This is comparable to a first-order reaction, in which despite the presence of numerous reactants, only one reactant determines the rate of the reaction.

Without any further ado, let’s get to know more about the rate of reaction.

TABLE OF CONTENTS

Introduction
What is a First Order Reaction?
Differential and Integral Form of First Order Reaction
Half-life of First Order Reaction
Time of Completion of First Order Reaction
Examples of First Order Reaction
Practice Problems
Frequently Asked Questions – FAQ

Introduction

The rate law tells how the rate of the reaction depends on the concentration of reactant(s), but it does not tell anything about how the concentration changes with time.

Therefore, the integrated rate law method is used. This method, quantitatively, gives the concentration of reactant(s) as a function of time. The integrated rate law method is studied for zero, first, second and nth order reactions.

In this concept page, we will discuss the integrated rate equation for a first-order reaction in detail.

What is a First Order Reaction?

A first-order reaction is a chemical reaction in which the rate of the reaction depends upon the first power of concentration of reactants.

Consider a reaction, $a A \to P$

Mathematically, rate = - $\frac{d [A]}{d t}$ $\propto [A]^{1}$

Rate = - $\frac{d [A]}{d t}$ $= k [A]^{1} = k [A]$

Here, - d[A]dt refers to the rate of the reaction and k is the rate constant of the first-order reaction.

Differential and Integral Form of First Order Reaction

The differential form of a first-order reaction can be written as - $\frac{d [A]}{d t} = k [A]^{1} = k [A]$

On rearranging, $\frac{d [A]}{[A]} = - k d t$

Integrating on both the sides from time (t) = 0 to t

$\int_{{[A]_{0}}_{}}^{{[A]}_{}} \frac{d [A]}{[A]} = -$ $\int_{0}^{t} k . d t$

Here, [A]₀ and [A] are the concentrations of A at time (t) = 0 and t., respectively.

ln $\frac{{[A]}_{0}}{[A]} = k t$ is the required integrated form of the first-order reaction.

Or, $k = {\frac{2.303}{t}}_{} l o g$ $\frac{{[A]}_{0}}{[A]}$

On rearranging,

log[A] =log[A]₀- $\frac{k t}{2.303}$

On comparing the above equation with the equation of a straight line, y=mx+c

y=log[A]

y-intercept (c)=log[A]₀
Slope (m) = - $\frac{k t}{2.303}$

Or [A] =[A]₀ e^-kt

The unit of rate constant for a first-order reaction is s^-1 .

Half-life of First Order Reaction

It is the time at which the concentration of the reactant becomes half of its initial concentration.

At $t = t_{(} 1 / 2),$ , [A] = $\frac{{[A]}_{0}}{2}$

$t_{1 / 2}$ = $\frac{2.303}{k} l o g \frac{{[A]}_{0}}{{[A]}_{0} / 2}$

t1/2 = $\frac{0.693}{k}$

Time of Completion of First Order Reaction

Time of completion is the time at which the reactants are consumed completely, such that, [A]=0.

Substituting [A]=0 in the equation [A] =[A]₀ e^-kt

0 =[A]₀ e^-kt

0=e^-kt

$t = \infty$

Thus, the first-order reaction completes in infinite time.

Note: First-order reaction never completes.

Rate =k[A]

Examples of First Order Reaction

All radioactive disintegrations are first-order reactions.

Example: $_{88}^{226} R a \to_{86}^{222} R a + {_{2}^{4} H e}_{}^{}_{}^{}$

Decomposition of H₂O₂

$2 H_{2} O_{2} (a q) \to {2 H}_{2} O (l) + O_{2} (g)$

Practice Problems

Q1. The given plot represents the variation of the concentration of a reactant R with time. The order of the reaction is

A. 0
B. 2
C. 1
D. Can not be predicted

Answer: C

Solution: For a first-order reaction, R =[R]₀ e^-kt

Taking ln on both sides,

ln[R] =ln[R]₀ -kt

On comparing the above equation with the equation of a straight line, y=mx+c

y=ln[R]

y-intercept (c)=ln[R]₀
Slope (m) = -k

The slope is a straight line with a negative slope. Therefore the reaction is a first-order reaction.

So, option C is the correct answer.

Q2. The rate constant for a first-order reaction is $4.606 \times {10^{- 3}}^{} s^{- 1} .$ . The time required to reduce 2.0 g of the reactant to 0.2 g is

A. 500 s
B. 5000 s
C. 5 s
D. 50 s

Answer: A
Solution:

t= $\frac{2.303}{k}$ $l o g \frac{{[A]}_{0}}{[A]}$

t= $l o g \frac{2}{0.2}$

t= 500 s

So, option A is the correct answer.

Q3. The rate of a first-order reaction is 0.04 mol L^-1s^-1 at 10 s and 0.03 mol L^-1s^-1 at 20 s after the initiation of the reaction. The half-life period of the reaction is

A. 24.1 s
B. 34.1 s
C. 44.1 s
D. 54.1 s

Answer: A
Solution: For a first-order reaction, Rate (r) ∝ Concentration of the reactant (C)

$\frac{r_{1}}{r_{2}}$ = $\frac{C_{1}}{C_{2}}$ = $\frac{4}{3}$

∴ $k = {\frac{2.303}{t_{2} - t_{1}}}_{} l o g$ $\frac{C_{1}}{C_{2}}$

Substituting, t1/2 = $\frac{0.693}{k}$ in the above equation,

$\frac{0.693}{t_{1 / 2}}$ = ${\frac{2.303}{20 - 10}}_{} l o g$ $\frac{4}{3}$

t1/2 = 24.1 s

So, option A is the correct answer.

Q4. For a first-order reaction $A \to B$ , the reaction rate at the reactant concentration of 0.01 M is found to be $2.0 \times {10^{- 5}}^{} M s^{- 1}$ . The half-life period of the reaction is