
Derive an expression for the half-life of (a) a third order reaction;(b) a reaction whose order is

(a)
Interpretation:
The expression for half-life of third order reaction is to be stated.
Concept introduction:
The time required for the concentration of the reactant to reduce to half of its initial concentration gives the half-life of the reaction. The half-life of the reaction depends on the initial concentration of the reactant except for the first order reaction.
Answer to Problem 20.27E
The expression for half-life of third order reaction is
Explanation of Solution
The integrated rate law for the third-order kinetics is,
Where,
•
At half-life, that is, at time
Substitute
Thus, the half-life for third order kinetics is
The expression for half-life of third order reaction is

(b)
Interpretation:
The expression for half-life of a reaction whose order is
Concept introduction:
The time required for the concentration of the reactant to reduce to half of its initial concentration gives the half-life of the reaction. The half-life of the reaction depends on the initial concentration of the reactant except for the first order reaction.
Answer to Problem 20.27E
The expression for half-life for a reaction whose order is
Explanation of Solution
The rate law for a reaction whose order is
This law is rearranged as,
Where,
•
•
Integrate the given equation from
The constants are kept out of the integral and the equation is integrated.
Apply the limits in the above equation as shown below.
Thus, equation (1) represents the integrated rate law.
At half-life, that is, at time
Substitute
Thus, the half-life for a reaction whose order is
The expression for half-life for a reaction whose order is

(c)
Interpretation:
The expression for half-life of a reaction whose order is
Concept introduction:
The time required for the concentration of the reactant to reduce to half of its initial concentration gives the half-life of the reaction. The half-life of the reaction depends on the initial concentration of the reactant except for the first order reaction.
Answer to Problem 20.27E
The expression for half-life for a reaction whose order is
Explanation of Solution
The rate law for a reaction whose order is
This law is rearranged as,
Where,
•
•
Integrate the given equation from
The constants are kept out of the integral and the equation is integrated.
Apply the limits in the above equation as shown below.
Thus, equation (1) represents the integrated rate law.
At half-life, that is, at time
Substitute
Thus, the half-life for a reaction whose order is
The expression for half-life for a reaction whose order is
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Chapter 20 Solutions
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