The concept of half life is commonly used in A Levels H2 Chemistry syllabus.
Many of us know that the half life of 1st order reaction is constant, but why is it constant?
What is meant by constant?
And how about other orders?
What is half life (t 1/2)?
It is the time needed for the concentration of the reactant to reach half its original concentration.
Determining half lives.
For a ZERO order reaction with respect to reactant A in the following reaction,
αA → βB
Rate = k[A]0 = k
| Since the instantaneous rate of change of [A] = d[A] / dt |
(instantaneous gradient of a [A] vs time graph (which has a negative sign))
| – |
|
= k (since k > 0) |
Separating the variables, we obtain
-d[A] = kdt
Integrating [A] between the limits of [A]o to 1/2[A]o and t from 0 to t 1/2
Solving and simplifying,
½ [A]o – [A]o = -k t ½ ; (where [A]o = initial concentration of A)
t 1/2 = [A]o / (2k) ……….. ZERO order half life
For a FIRST order reaction with respect to reactant A in the following reaction,
αA → βB
Rate = k[A]1
| Since the instantaneous rate of change of [A] = d[A]/dt |
(instantaneous gradient of a [A] vs time graph (which has a negative sign))
| – |
|
= k[A] (since k > 0) |
Separating the variables, we obtain
| – |
|
= kdt |
Integrating 1/[A] between the limits of [A]o to 1/2[A]o and t from 0 to t 1/2
Solving and simplifying,
ln ( 1/2[A]o) – ln[A]o = -k t ½
ln (½ +1n[A]o) – ln[A]o = -k t ½
t 1/2 = ln/k ……….. FIRST order half life
By similar mathematical operations,
For SECOND order reactions
Rate = k[A]2
obtain
| – |
|
= kdt |
Integrating 1/ [A]2 between the limits of [A]o to 1/2[A]o and t from 0 to t 1/2
Solving and simplifying,
|
– |
|
= k t ½ |
t 1/2 = 1/(k[A]o) ……….. SECOND order half life
In summary
| Half Life | |||
| Zero order |
|
||
| First order |
|
||
| Second order |
|
Notice that the 1st order half life is the only expression of the three that does not have a “[A]o” term?
What is the meaning of [A]o in the calculation of half life?
It is the concentration of A from which half life is taken. Since this concentration decreases with time, thus,
ZERO order half life decreases with time and
SECOND order half life increases with time.
Thus, since the half life of a FIRST order reaction does not have the term “[A]o”, its half life is CONSTANT with time.
This applies whether we are looking at reactant or product graphs with time, since the order does not change when we track either reactant or product.
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