# Chemical kinetics and half-life

Chemical kinetics differential rate low shows the dependence of the rate with the concentration of the reacting species.
But the integrated rate law of the chemical kinetics provides the concentration of these species at any time from the start of the reaction.

### Zero-order kinetics reactions

The rate of the zero-order chemical kinetics reaction does not depend on the concentration of the reactants. Chemical kinetics and half-life
Question
The rate constant of a chemical reaction is 5× 10⁻⁸  mol lit⁻¹sec⁻¹. What is the order of this reaction? How many seconds need to change concentration from 4 × 10⁻⁴ moles lit⁻¹ to 2 × 10⁻² moles lit⁻¹?
In chemical kinetics unit of the rate constant in nth order reaction
= (unit of concentration)1-n (unit of time)⁻¹
Given unit of the rate constant = mol lit⁻¹sec⁻¹
= (unit of concentration)(unit of time)⁻¹
Compare the above two equation
We have 1 -n = 1
or, n = 0
Thus the reaction is a zero-order reaction

And the integration rate equation at two times
(x₂ - x₁) = k (t₂ - t₁)

Here at the time t₂, x₂ = 2 × 10⁻² moles lit⁻¹ and at time t₁, x₁ = 4 × 10⁻⁴

Hence the time required to change the above concentration
(t₂ - t₁) = (x₂ - x₁)/t
= (2 × 10⁻² - 4 × 10⁻⁴)/5× 10⁻⁸ sec
= 3.92 × 10⁵ Sec
Question
The half-life of a zero-order reaction is x and the reaction is completed on t₁ time. What is the relation between x and t₁?
Form the zero-order kinetics half-life(t½) = [A]₀/2k
or, x = [A]₀/2k
or, [A]₀ = 2kx
Again for zero order chemical kinetics, [A]₀ - [A] = kt when the reaction completed concentration of [A] = 0.
Thus [A]₀ = kt₁
Compare the above two equation we have,
kt₁ = 2kx
or, t₁ = 2x
Question
When the rate of the reaction is equal to the rate constant, what is the order of the reaction?
Zero-order chemical kinetics, the rate of the reaction is proportional to the zero power of the reactant.
That means r ∝ [A]⁰
or, r = k
Thus the reaction is a zero-order reaction.
Question
For a reaction, N₂ + 3 H₂ → 2NH₃, if d[NH₃]/dt = 2 × 10⁻⁴ mol lit⁻¹sec⁻¹, what is the order and the value of - d[H₂]/dt of this reaction?
The unit of the rate constant, we can easily find out the order of this reaction. Here the unit of the rate constant is mol lit⁻¹sec⁻¹, this is the unit of zero-order reaction thus the reaction is zero-order chemical kinetics.
Rate of reaction of zero-order chemical kinetics is
- d[N₂]/dt = - ⅓ d[H₂]/dt
= ½ d[NH₃]/dt

Thus form the above equation, - ⅓ d[H₂]/dt = ½ d[NH₃]/dt
or, - d[H₂]/dt = (3/2) × d[NH₃]

Given d[NH₃]/dt = 2 × 10⁻⁴ mol lit⁻¹sec⁻¹

-d[H₂]/dt = (3 × 2 × 10⁻⁴ mol lit⁻¹sec⁻¹)/2
= 3 × 10⁻⁴ mol lit⁻¹sec⁻¹
Question
For a zero-order reaction N₂O₅ → 2NO₂ + ½ O₂, the rate of disappearance of N₂O₅ is 6.25 × 10⁻³  mol lit⁻¹sec⁻¹, what is the rate of formation of NO₂ and O₂ respectively?
Rate of reaction in zero-order chemical kinetics is
- d[N₂O₅]/dt = ½ d[NO₂]/dt
= 2d[NH₃]/dt

Rate of disappearance of N₂O₅ is,
6.25 × 10⁻³  mol lit⁻¹sec⁻¹ = - d[N₂O₅]/dt

Thus the rate of formation of NO₂
= (2 × 6.25 × 10⁻³  mol lit⁻¹sec⁻¹)
= 1.25 × 10⁻²  mol lit⁻¹sec⁻¹

Thus the rate of formation of O₂
= (6.25 × 10⁻³  mol lit⁻¹sec⁻¹)/2
= 3.125 × 10⁻²  mol lit⁻¹sec⁻¹
Question
For the reaction H₂ + Cl₂ → 2HCl on sunlight and taking place on the water. What is the order of the reaction?
This is a zero-order reaction in chemical kinetics.
Question
A reaction carried out within A and B. When the concentration of B doubled then the rate of the reaction is one-fourth of the initial rate. What is the order of B in this reaction?
Let the order of the reaction in terms of A is ɑ and in terms of B is β.
Rate of the reaction(r) = k [A]ɑ [B]β
where k is the rate constant of the reaction.

The initial concentration of A = [A]₀ and B = [B]₀

Initial rate of the reaction(r₀) = k [A]₀ɑ [B]₀β
When the concentration of B doubled then the rate of the reaction is one-fourth of the initial rate.
Thus, (r₀/4) = k [A]₀ɑ [2B]₀β

Compare these two equations we have, r₀/(r₀/4) = (k [A]₀ɑ [B]₀β)/(k [A]₀ɑ [2B]₀β)
or, 4 = 2
or, β = -2

#### Half-life in chemical kinetics

Question
In a chemical reaction, the rate constant of this reaction is 2.5 × 10⁻³  mol lit⁻¹sec⁻¹ and the initial concentration of the reactant is one.  Find out the half-life this reaction?
From the unit of the rate constant, we can easily find out the order of this reaction. Here the unit of the rate constant is mol lit⁻¹sec⁻¹, this is the unit of zero-order kinetics reaction thus the reaction is zero-order chemical kinetics.
Thus for the Zero-order kinetics half-life(t½)
= [A]₀/2k
or, t½ = 1/(2.5 × 10⁻³) sec
= 0.4 × 10³ sec

#### First-order chemical kinetics

Question
In a radioactive reaction, the rate constant of this reaction is 2.5 × 10⁻³ sec⁻¹. what is the order of this reaction?
Unit of the rate constant in nth order reaction
= (unit of concentration)1-n(unit of time)⁻¹

Given unit of the rate constant, = sec⁻¹
= (unit of concentration)⁰(unit of time)⁻¹

Compare the above two-equation,
we have 1 -n = 0
or, n = 1
The reaction is the first-order reaction.

Chemical kinetics, zero-order, first-order, second-order, and half-life in chemical kinetics with questions answers